linearhypothesis r

Linear Hypothesis Tests

Most regression output will include the results of frequentist hypothesis tests comparing each coefficient to 0. However, in many cases, you may be interested in whether a linear sum of the coefficients is 0. For example, in the regression

You may be interested to see if $GoodThing$ and $BadThing$ (both binary variables) cancel each other out. So you would want to do a test of $\beta_1 - \beta_2 = 0$.

Alternately, you may want to do a joint significance test of multiple linear hypotheses. For example, you may be interested in whether $\beta_1$ or $\beta_2$ are nonzero and so would want to jointly test the hypotheses $\beta_1 = 0$ and $\beta_2=0$ rather than doing them one at a time. Note the and here, since if either one or the other is rejected, we reject the null.

Keep in Mind

Be sure to carefully interpret the result. If you are doing a joint test, rejection means that at least one of your hypotheses can be rejected, not each of them. And you don’t necessarily know which ones can be rejected!
Generally, linear hypothesis tests are performed using F-statistics. However, there are alternate approaches such as likelihood tests or chi-squared tests. Be sure you know which on you’re getting.
Conceptually, what is going on with linear hypothesis tests is that they compare the model you’ve estimated against a more restrictive one that requires your restrictions (hypotheses) to be true. If the test you have in mind is too complex for the software to figure out on its own, you might be able to do it on your own by taking the sum of squared residuals in your original unrestricted model ($SSR_{UR}$), estimate the alternate model with the restriction in place ($SSR_R$) and then calculate the F-statistic for the joint test using $F_{q,n-k-1} = ((SSR_R - SSR_{UR})/q)/(SSR_{UR}/(n-k-1))$.

Also Consider

The process for testing a nonlinear combination of your coefficients, for example testing if $\beta_1\times\beta_2 = 1$ or $\sqrt{\beta_1} = .5$, is generally different. See Nonlinear hypothesis tests .

Implementations

Linear hypothesis test in R can be performed for most regression models using the linearHypothesis() function in the car package. See this guide for more information.

Tests of coefficients in Stata can generally be performed using the built-in test command.

Test Linear Hypothesis

Description.

Generic function for testing a linear hypothesis, and methods for linear models, generalized linear models, and other models that have methods for coef and vcov .

Computes either a finite sample F statistic or asymptotic Chi-squared statistic for carrying out a Wald-test-based comparison between a model and a linearly restricted model. The default method will work with any model object for which the coefficient vector can be retrieved by coef and the coefficient-covariance matrix by vcov (otherwise the argument vcov. has to be set explicitely). For computing the F statistic (but not the Chi-squared statistic) a df.residual method needs to be available. If a formula method exists, it is used for pretty printing.

The method for "lm" objects calls the default method, but it changes the default test to "F" , supports the convenience argument white.adjust (for backwards compatibility), and enhances the output by residual sums of squares. For "glm" objects just the default method is called (bypassing the "lm" method).

The function lht also dispatches to linear.hypothesis .

The hypothesis matrix can be supplied as a numeric matrix (or vector), the rows of which specify linear combinations of the model coefficients, which are tested equal to the corresponding entries in the righ-hand-side vector, which defaults to a vector of zeroes.

Alternatively, the hypothesis can be specified symbolically as a character vector with one or more elements, each of which gives either a linear combination of coefficients, or a linear equation in the coefficients (i.e., with both a left and right side separated by an equals sign). Components of a linear expression or linear equation can consist of numeric constants, or numeric constants multiplying coefficient names (in which case the number precedes the coefficient, and may be separated from it by spaces or an asterisk); constants of 1 or -1 may be omitted. Spaces are always optional. Components are separated by positive or negative signs. See the examples below.

An object of class "anova" which contains the residual degrees of freedom in the model, the difference in degrees of freedom, Wald statistic (either "F" or "Chisq" ) and corresponding p value.

Achim Zeleis and John Fox [email protected]

Fox, J. (1997) Applied Regression, Linear Models, and Related Methods. Sage.

anova , Anova , waldtest , hccm , vcovHC , vcovHAC , coef , vcov

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How to Use the linearHypothesis() Function in R

In statistics, understanding how variables relate to each other is crucial. This helps in making smart decisions. When we build regression models, we need to check if certain combinations of variables are statistically significant. In R Programming Language a tool called linear hypothesis () in the “car” package for this purpose. Also, this article gives a simple guide on using linear hypothesis () in R for such analyses.

What is a linear hypothesis?

The linear hypothesis () function is a tool in R’s “car” package used to test linear hypotheses in regression models. It helps us to determine if certain combinations of variables have a significant impact on our model’s outcome.

H 0 : Cβ = 0

H 0 denotes the null hypothesis.
C is a matrix representing the coefficients of the linear combination being tested.
β represents the vector of coefficients in the regression model.
0 signifies a vector of zeros.

linearHypothesis(model, hypothesis.matrix)

model is the fitted regression model for which hypotheses are to be tested.
hypothesis.matrix is a matrix specifying the linear hypotheses to be tested.

Implemention of linearHypothesis() Function in R

The hypothesis being tested is whether the coefficients of X1 and X2 are equal (i.e., X1 – X2 = 0).

The output provides the results of the linear hypothesis test, including the Residual Degrees of Freedom (Res.Df), Residual Sum of Squares (RSS), the change in the RSS between the restricted and full models, the F-statistic (F), and the corresponding p-value (Pr(>F)).
In this case, the p-value is 0.5459, which indicates that we fail to reject the null hypothesis at conventional significance levels, suggesting that there is no significant difference between the coefficients of X1 and X2.

Perform linearHypothesis() Function on mtcars dataset

The hypothesis being tested is whether the coefficients of ‘cyl’ and ‘disp’ sum up to zero (i.e., cyl + disp = 0).

In this case, the p-value is 0.3321, which indicates that we fail to reject the null hypothesis at conventional significance levels. Therefore, we don’t have sufficient evidence to conclude that the sum of the coefficients of ‘cyl’ and ‘disp’ is significantly different from zero.

The `linearHypothesis()` function in R’s “car” package provides a straightforward method for testing linear hypotheses in regression models. The significance of specific variable combinations, analysts can make informed decisions about the model’s predictive power.

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How to Use the linearHypothesis() Function in R

The linearHypothesis() function is a valuable statistical tool in R programming. It’s provided in the car package and is used to perform hypothesis testing for a linear model’s coefficients.

To fully grasp the utility of linearHypothesis() , we must understand the basic principles of linear regression and hypothesis testing in the context of model fitting.

Understanding Hypothesis Testing in Regression Analysis

In regression analysis, it’s common to perform hypothesis tests on the model’s coefficients to determine whether the predictors are statistically significant. The null hypothesis asserts that the predictor has no effect on the outcome variable, i.e., its coefficient equals zero. Rejecting the null hypothesis (based on a small p-value, usually less than 0.05) suggests that there’s a statistically significant relationship between the predictor and the outcome variable.

The linearHypothesis( ) Function

linearHypothesis() is a function in R that tests the general linear hypothesis for a model object for which a formula method exists, using a specified test statistic. It allows the user to define a broader set of null hypotheses than just assuming individual coefficients equal to zero.

The linearHypothesis() function can be especially useful for comparing nested models or testing whether a group of variables significantly contributes to the model.

Here’s the basic usage of linearHypothesis() :

In this function:

model is the model object for which the linear hypothesis is to be tested.
hypothesis.matrix specifies the null hypotheses.
rhs is the right-hand side of the linear hypotheses; typically set to 0.
... are additional arguments, such as the test argument to specify the type of test statistic to be used (“F” for F-test, “Chisq” for chi-squared test, etc.).

Installing and Loading the Required Package

linearHypothesis() is part of the car package. If you haven’t installed this package yet, you can do so using the following command:

Once installed, load it into your R environment with the library() function:

Using linearHypothesis( ) in Practice

Let’s demonstrate the use of linearHypothesis() with a practical example. We’ll use the mtcars dataset that’s built into R. This dataset comprises various car attributes, and we’ll model miles per gallon (mpg) based on horsepower (hp), weight (wt), and the number of cylinders (cyl).

We first fit a linear model using the lm() function:

Let’s say we want to test the hypothesis that the coefficients for hp and wt are equal to zero. We can set up this hypothesis test using linearHypothesis() :

This command will output the Residual Sum of Squares (RSS) for the model under the null hypothesis, the RSS for the full model, the test statistic, and the p-value for the test. A low p-value suggests that we should reject the null hypothesis.

Using linearHypothesis( ) for Testing Nested Models

linearHypothesis() can also be useful for testing nested models, i.e., comparing a simpler model to a more complex one where the simpler model is a special case of the complex one.

For instance, suppose we want to test if both hp and wt can be dropped from our model without a significant loss of fit. We can formulate this as the null hypothesis that the coefficients for hp and wt are simultaneously zero:

This gives a p-value for the F-test of the hypothesis that these coefficients are zero. If the p-value is small, we reject the null hypothesis and conclude that dropping these predictors from the model would significantly degrade the model fit.

Limitations and Considerations

The linearHypothesis() function is a powerful tool for hypothesis testing in the context of model fitting. However, it’s important to consider the limitations and assumptions of this function. The linearHypothesis() function assumes that the errors of the model are normally distributed and have equal variance. Violations of these assumptions can lead to incorrect results.

As with any statistical function, it’s crucial to have a good understanding of your data and the theory behind the statistical methods you’re using.

The linearHypothesis() function in R is a powerful tool for testing linear hypotheses about a model’s coefficients. This function is very flexible and can be used in various scenarios, including testing the significance of individual predictors and comparing nested models.

Understanding and properly using linearHypothesis() can enhance your data analysis capabilities and help you extract meaningful insights from your data.

The Complete Guide: Hypothesis Testing in R

A hypothesis test is a formal statistical test we use to reject or fail to reject some statistical hypothesis.

This tutorial explains how to perform the following hypothesis tests in R:

One sample t-test
Two sample t-test
Paired samples t-test

We can use the t.test() function in R to perform each type of test:

x, y: The two samples of data.
alternative: The alternative hypothesis of the test.
mu: The true value of the mean.
paired: Whether to perform a paired t-test or not.
var.equal: Whether to assume the variances are equal between the samples.
conf.level: The confidence level to use.

The following examples show how to use this function in practice.

Example 1: One Sample t-test in R

A one sample t-test is used to test whether or not the mean of a population is equal to some value.

For example, suppose we want to know whether or not the mean weight of a certain species of some turtle is equal to 310 pounds. We go out and collect a simple random sample of turtles with the following weights:

Weights : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

The following code shows how to perform this one sample t-test in R:

From the output we can see:

t-test statistic: -1.5848
degrees of freedom: 12
p-value: 0.139
95% confidence interval for true mean: [303.4236, 311.0379]
mean of turtle weights: 307.230

Since the p-value of the test (0.139) is not less than .05, we fail to reject the null hypothesis.

This means we do not have sufficient evidence to say that the mean weight of this species of turtle is different from 310 pounds.

Example 2: Two Sample t-test in R

A two sample t-test is used to test whether or not the means of two populations are equal.

For example, suppose we want to know whether or not the mean weight between two different species of turtles is equal. To test this, we collect a simple random sample of turtles from each species with the following weights:

Sample 1 : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

Sample 2 : 335, 329, 322, 321, 324, 319, 304, 308, 305, 311, 307, 300, 305

The following code shows how to perform this two sample t-test in R:

t-test statistic: -2.1009
degrees of freedom: 19.112
p-value: 0.04914
95% confidence interval for true mean difference: [-14.74, -0.03]
mean of sample 1 weights: 307.2308
mean of sample 2 weights: 314.6154

Since the p-value of the test (0.04914) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean weight between the two species is not equal.

Example 3: Paired Samples t-test in R

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

For example, suppose we want to know whether or not a certain training program is able to increase the max vertical jump (in inches) of basketball players.

To test this, we may recruit a simple random sample of 12 college basketball players and measure each of their max vertical jumps. Then, we may have each player use the training program for one month and then measure their max vertical jump again at the end of the month.

The following data shows the max jump height (in inches) before and after using the training program for each player:

Before : 22, 24, 20, 19, 19, 20, 22, 25, 24, 23, 22, 21

After : 23, 25, 20, 24, 18, 22, 23, 28, 24, 25, 24, 20

The following code shows how to perform this paired samples t-test in R:

t-test statistic: -2.5289
degrees of freedom: 11
p-value: 0.02803
95% confidence interval for true mean difference: [-2.34, -0.16]
mean difference between before and after: -1.25

Since the p-value of the test (0.02803) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean jump height before and after using the training program is not equal.

Additional Resources

Use the following online calculators to automatically perform various t-tests:

One Sample t-test Calculator Two Sample t-test Calculator Paired Samples t-test Calculator

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How to Use the linearHypothesis() Function in R

You can use the linearHypothesis() function from the car package in R to test linear hypotheses in a specific regression model.

This function uses the following basic syntax:

This particular example tests if the regression coefficients var1 and var2 in the model called fit are jointly equal to zero.

The following example shows how to use this function in practice.

Example: How to Use linearHypothesis() Function in R

Suppose we have the following data frame in R that shows the number of hours spent studying, number of practice exams taken, and final exam score for 10 students in some class:

Now suppose we would like to fit the following multiple linear regression model in R:

Exam score = β 0 + β 1 (hours) + β 2 (practice exams)

We can use the lm() function to fit this model:

Now suppose we would like to test if the coefficient for hours and prac_exams are both equal to zero.

We can use the linearHypothesis() function to do so:

The hypothesis test returns the following values:

F test statistic : 14.035
p-value : .003553

This particular hypothesis test uses the following null and alternative hypotheses:

H 0 : Both regression coefficients are equal to zero.
H A : At least one regression coefficient is not equal to zero.

Since the p-value of the test (.003553) is less than .05, we reject the null hypothesis.

In other words, we don’t have sufficient evidence to say that the regression coefficients for hours and prac_exams are both equal to zero.

Additional Resources

The following tutorials provide additional information about linear regression in R:

How to Interpret Regression Output in R How to Perform Simple Linear Regression in R How to Perform Multiple Linear Regression in R How to Perform Logistic Regression in R

R: Count Number of NA Values in Each Column

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car Companion to Applied Regression

Using car functions inside user functions
Anova: Anova Tables for Various Statistical Models
avPlots: Added-Variable Plots
bcPower: Box-Cox, Box-Cox with Negatives Allowed, Yeo-Johnson and...
Boot: Bootstrapping for regression models
boxCox: Graph the profile log-likelihood for Box-Cox transformations...
boxCoxVariable: Constructed Variable for Box-Cox Transformation
Boxplot: Boxplots With Point Identification
boxTidwell: Box-Tidwell Transformations
brief: Print Abbreviated Ouput
car-defunct: Defunct Functions in the car Package
car-deprecated: Deprecated Functions in the car Package
carHexsticker: View the Official Hex Sticker for the car Package
car-internal: Internal Objects for the 'car' package
carPalette: Set or Retrieve 'car' Package Color Palette
carWeb: Access to the R Companion to Applied Regression Website
ceresPlots: Ceres Plots
compareCoefs: Print estimated coefficients and their standard errors in a...
Contrasts: Functions to Construct Contrasts
crPlots: Component+Residual (Partial Residual) Plots
deltaMethod: Estimate and Standard Error of a Nonlinear Function of...
densityPlot: Nonparametric Density Estimates
dfbetaPlots: dfbeta and dfbetas Index Plots
durbinWatsonTest: Durbin-Watson Test for Autocorrelated Errors
Ellipses: Ellipses, Data Ellipses, and Confidence Ellipses
Export: Export a data frame to disk in one of many formats
hccm: Heteroscedasticity-Corrected Covariance Matrices
hist.boot: Methods Functions to Support 'boot' Objects
Import: Import data from many file formats
infIndexPlot: Influence Index Plot
influence-mixed-models: Influence Diagnostics for Mixed-Effects Models
influencePlot: Regression Influence Plot
invResPlot: Inverse Response Plots to Transform the Response
invTranPlot: Choose a Predictor Transformation Visually or Numerically
leveneTest: Levene's Test
leveragePlots: Regression Leverage Plots
linearHypothesis: Test Linear Hypothesis
logit: Logit Transformation
marginalModelPlot: Marginal Model Plotting
mcPlots: Draw Linear Model Marginal and Conditional Plots in Parallel...
ncvTest: Score Test for Non-Constant Error Variance
outlierTest: Bonferroni Outlier Test
panel.car: Panel Function for Coplots
pointLabel: Label placement for points to avoid overlaps
poTest: Test for Proportional Odds in the Proportional-Odds...
powerTransform: Finding Univariate or Multivariate Power Transformations
Predict: Model Predictions
qqPlot: Quantile-Comparison Plot
recode: Recode a Variable
regLine: Plot Regression Line
residualPlots: Residual Plots for Linear and Generalized Linear Models
Browse all...

linearHypothesis : Test Linear Hypothesis In car: Companion to Applied Regression

Test linear hypothesis, description.

Generic function for testing a linear hypothesis, and methods for linear models, generalized linear models, multivariate linear models, linear and generalized linear mixed-effects models, generalized linear models fit with svyglm in the survey package, robust linear models fit with rlm in the MASS package, and other models that have methods for coef and vcov . For mixed-effects models, the tests are Wald chi-square tests for the fixed effects.

linearHypothesis computes either a finite-sample F statistic or asymptotic Chi-squared statistic for carrying out a Wald-test-based comparison between a model and a linearly restricted model. The default method will work with any model object for which the coefficient vector can be retrieved by coef and the coefficient-covariance matrix by vcov (otherwise the argument vcov. has to be set explicitly). For computing the F statistic (but not the Chi-squared statistic) a df.residual method needs to be available. If a formula method exists, it is used for pretty printing.

The method for "lm" objects calls the default method, but it changes the default test to "F" , supports the convenience argument white.adjust (for backwards compatibility), and enhances the output by the residual sums of squares. For "glm" objects just the default method is called (bypassing the "lm" method). The "svyglm" method also calls the default method.

Multinomial logit models fit by the multinom function in the nnet package invoke the default method, and the coefficient names are composed from the response-level names and conventional coefficient names, separated by a period ( "." ): see one of the examples below.

The function lht also dispatches to linearHypothesis .

The hypothesis matrix can be supplied as a numeric matrix (or vector), the rows of which specify linear combinations of the model coefficients, which are tested equal to the corresponding entries in the right-hand-side vector, which defaults to a vector of zeroes.

If the user sets the arguments coef. and vcov. , then the computations are done without reference to the model argument. This is like assuming that coef. is normally distibuted with estimated variance vcov. and the linearHypothesis will compute tests on the mean vector for coef. , without actually using the model argument.

A linear hypothesis for a multivariate linear model (i.e., an object of class "mlm" ) can optionally include an intra-subject transformation matrix for a repeated-measures design. If the intra-subject transformation is absent (the default), the multivariate test concerns all of the corresponding coefficients for the response variables. There are two ways to specify the transformation matrix for the repeated measures:

The transformation matrix can be specified directly via the P argument.

A data frame can be provided defining the repeated-measures factor or factors via idata , with default contrasts given by the icontrasts argument. An intra-subject model-matrix is generated from the one-sided formula specified by the idesign argument; columns of the model matrix corresponding to different terms in the intra-subject model must be orthogonal (as is insured by the default contrasts). Note that the contrasts given in icontrasts can be overridden by assigning specific contrasts to the factors in idata . The repeated-measures transformation matrix consists of the columns of the intra-subject model matrix corresponding to the term or terms in iterms . In most instances, this will be the simpler approach, and indeed, most tests of interests can be generated automatically via the Anova function.

matchCoefs is a convenience function that can sometimes help in formulating hypotheses; for example matchCoefs(mod, ":") will return the names of all interaction coefficients in the model mod .

For a univariate model, an object of class "anova" which contains the residual degrees of freedom in the model, the difference in degrees of freedom, Wald statistic (either "F" or "Chisq" ), and corresponding p value. The value of the linear hypothesis and its covariance matrix are returned respectively as "value" and "vcov" attributes of the object (but not printed).

For a multivariate linear model, an object of class "linearHypothesis.mlm" , which contains sums-of-squares-and-product matrices for the hypothesis and for error, degrees of freedom for the hypothesis and error, and some other information.

The returned object normally would be printed.

Achim Zeileis and John Fox [email protected]

Fox, J. (2016) Applied Regression Analysis and Generalized Linear Models , Third Edition. Sage.

Fox, J. and Weisberg, S. (2019) An R Companion to Applied Regression , Third Edition, Sage.

Hand, D. J., and Taylor, C. C. (1987) Multivariate Analysis of Variance and Repeated Measures: A Practical Approach for Behavioural Scientists. Chapman and Hall.

O'Brien, R. G., and Kaiser, M. K. (1985) MANOVA method for analyzing repeated measures designs: An extensive primer. Psychological Bulletin 97 , 316–333.

anova , Anova , waldtest , hccm , vcovHC , vcovHAC , coef , vcov

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Principles of Econometrics with $R$

Chapter 6 further inference in multiple regression.

New package: broom (Robinson 2016 ) .

6.1 Joint Hypotheses and the F-statistic

In Equation \ref{eq:FstatFormula6} the subscript $U$ stands for “unrestricted,” that is, the initial regression equation; the “restricted” equation is a new equation, obtained from the initial one, with the relationships in the null hypothesis assumed to hold. For example, if the initial equation is Equation \ref{eq:initeqF6} and the null hypothesis is Equation \ref{eq:nullHforF6}, then the restricted equation is Equation \ref{eq:restreqforF6}.

The symbol $J$ in the $F$ formula (Equation \ref{eq:FstatFormula6}) is the first ( numerator ) degrees of freedom of the $F$ statistic and is equal to the number of simultaneous restrictions in the null hypothesis (Equation \ref{eq:nullHforF6}); the second (the denominator ) degrees of freedom of the $F$ -statistic is $N-K$ , which is the usual degrees of freedom of the unrestricted regression model (Equation \ref{eq:initeqF6}). The practical procedure to test a joint hypothesis like the one in Equation \ref{eq:nullHforF6} is to estimate the two regressions (unrestricted and restricted) and to calculate the $F$ -statistic.

6.2 Testing Simultaneous Hypotheses

Let’s look, again, at the quadratic form of the andy equation (Equation \ref{eq:quadraticandyagain6}).

Equation \ref{eq:quadraticandyagain6} has two terms that involve the regressor $advert$ , of which at least one needs to be significant for a relationship between $advert$ and $sales$ to be established. To test if such a relationship exists, we can formulate the following test:

I have already mentioned that $R$ can do an $F$ test quite easily (remember the function linearHypothesis ?), but for learning purposes let us calculate the $F$ -statistic in steps. The next code sequence uses information in the anova -type object, which, remember, can be visualized simply by typing the name of the object in the RStudio’s Console window.

The calculated $F$ -statistic is $fval=8.441$ and the critical value corresponding to a significance level $\alpha =0.05$ is $3.126$ , which rejects the null hypothesis that both $\beta_{3}$ and $\beta_{4}$ are zero. The $p$ -value of the test is $p=0.0005$ .

Using the linearHypothesis() function should produce the same result:

The table generated by the linearHypothesis() function shows the same values of the $F$ -statistic and $p$ -value that we have calculated before, as well as the residual sum of squares for the restricted and unrestricted models. Please note how I formulate the joint hypothesis as a vector of character values in which the names of the variables perfectly match those in the unrestricted model.

Testing the overall significance of a model amounts to testing the joint hypothesis that all the slope coefficients are zero. $R$ does automatically this test and the resulting $F$ -statistic and $p$ -value are reported in the regression output.

The $F$ -statistic can be retrieved from summary(mod1) or by using the function glance(modelname) in package broom , as shown in the following code lines. The function tidy , also from package broom organizes regression output (mainly the coefficients and their statistics) in a neat table. Both glance and tidy create output in the form of data.frame , which makes it suitable for use by other functions such as kable and ggplot2 . Please also note that tidy(mod1) and tidy(summary(mod1)) produce the same result, as shown in Tables 6.1 and 6.2 . As always, we can use the function names to obtain a list of the quantities available in the output of the glance function.

Table 6.3 shows a summary of the quadratic andy model ( mod1 ), where I have changed the names of various items so that the table fits the width of the page. When retrieving these variables, make sure you use the original names as indicated by the names(glance(mod1)) command.

When testing a two-tail single (not joint) null hypothesis, the $t$ and $F$ tests are equivalent. However, one-tail tests, single or joint cannot be easily performed by an $F$ test.

Let us solve one more exercise involving a joint hypothesis with linear combinations of regression coefficients. Suppose we want to test the simultaneous hypotheses that the monthly advertising expenditure $advert_{0}=\$1900$ in the quadratic andy model (Equation \ref{eq:quadraticandyagain6}) satisfies the profit-maximizing condition $\beta_{3}+2\beta_{4}advert_{0}=1$ , and that, when $price=\$6$ and $advert=\$1900$ sales revenue is $\$80\,000$ .

Table 6.4 includes the $F$ -statistic of the test, $F=5.741229$ and its $p$ -value $p=0.004885$ . Please be aware that the function tidy changes the names in the output of linearHypothesis . A useful exercise is to compare the raw output of linearHypothesis with the output generated by tidy(linearHypothesis(mod1)) . There are several other possibilities to compare two regression models, such as a restricted and unrestricted ones in $R$ , such as the anova() function or Wald tests. These are going to be mentioned in later chapters.

6.3 Omitted Variable Bias

Suppose we are only interested in estimating $\beta_{2}$ , but there is no data available for $x_{3}$ , or for other reasons $x_{3}$ is omitted from the model in Equation \ref{eq:omittedgeneral}. What is the error in the estimate of $\beta_{2}$ introduced by omitting $x_{3}$ ? Equation \ref{eq:omittedequation6} shows what is left of the true model after omitting $x_{3}$ .

Let $b_{2}^*$ be the estimate of $\beta_{2}$ when $x_{3}$ is omitted. Equation \ref{eq:omittedbiasformula6} gives the bias in this simple, two-regressor case. The formula shows that bias depends on the direct relationship between the omitted regressor and response through $\beta_{3}$ , as well as the correlation between the omitted and the included regressors. When $\beta_{3}$ and $cov(x_{2},\,x_{3})$ are both positive or both negative the bias is positive (the incorrect model overestimates the true $\beta_{2}$ ), and when they are of opposite signs the bias is negative ( $\beta_{2}$ is underestimated)

The example in this section uses the dataset $edu\_inc$ , and two models: one where the response variable family income ( $faminc$ ) is explained by the regressors husband’s education ( $he$ ) and wife’s education ( $we$ ), and another model, where the $we$ regressor is omitted. The purpose is to compare the estimates coming from the two models and see if there is a significant difference between them.

The marginal effect of husband’s education is much lower in the incorrect model. Let us apply the logic of Equation \ref{eq:omittedbiasformula6} to the $edu\_inc$ model. The direct effect of the omitted regressor ( $we$ ) on response ( $faminc$ ) is likely to be positive in theory (higher education generates higher income); the correlation between husband’s and wife’s education is also likely to be positive if we believe that people generally marry persons within their entourage. Thus, we should expect that omitting the regressor $we$ should produce an overestimated marginal effect of $he$ . Our data happen to confirm this supposition, though there is some chance that they might not.

Understanding the problem of omitted variable is very important because it can justify the choice of a particular model. If one is not interested in the effect of variable $x_{3}$ on $y$ and can convince that $x_{3}$ is uncorrelated with $x_{2}$ , one can argue with criticism about omitting the important regressor $x_{3}$ .

6.4 Irrelevant Variables

We have seen the effect of omitting a relevant regressor (the effect is biased estimates and lower variances of the included regressors). But what happens if irrelevant variables are incorrectly included? Not surprisingly, this increases the variances (lowers the precision) of the other variables in the model. The next example uses the same ( $edu\_inc$ ) dataset as above, but includes two artificially generated variables, $xtra\_x5$ and $xtra\_x6$ that are correlated with $he$ and $we$ but, obviously, have no role in determining $y$ . Let us compare two models, of which one includes these irrelevant variables.

A comparison of the two models shown in Tables 6.7 and 6.8 indicates that the inclusion of the two irrelevant variables has increased the marginal effects, standard errors, and the $p$ -values of $he$ and $we$ . Thus, including irrelevant variables may incorrectly diminish the significance of the “true” regressors.

6.5 Model Selection Criteria

The main tools of building a model should be economic theory, sound reasoning based on economic principles, and making sure that the model satisfies the Gauss-Markov assumptions. One should also consider the possibility of omitted variable bias and the exclusion of irrelevant variables that may increase variablility in the estimates. After all these aspects have been considered and a model established, there are a few quantities that help comparing different models. These are $R^2$ , adjusted $R^2$ ( $\bar R ^2$ ), the Akaike information criterion ( $AIC$ ), and the Schwarz (or Bayesian information) criterion ( $SC$ or $BIC$ ).

Among several models, the best fit is the one that maximizes $R^2$ or $\bar R^2$ . On the contrary, the best model must minimize $AIC$ or $BIC$ . Some computer packages, $R$ included, calculate $AIC$ and $BIC$ differentlly than Equations \ref{eq:aicformula6} and \ref{eq:scformula6} indicate. However, the ranking of the various models is the same.

The following code sequence needs some explanation. Function as.numeric extracts only the numbers from an object such as glance(mod1) , which also contains row and column names. The purpose is to put together a table with information comming from several models. Function rbind gathers several rows in a matrix, which then is made into a data.frame and given the name tab . The part of code [,c(1,2,8,9)] at the end of rbind instructs $R$ to pick all rows, but only columns 1, 2, 8, and 9 from the glance table. Function row.names assigns or changes the row names in a data frame; finally, kable , which we have already encountered several times, prints the table, assigns column names, and gives a caption to the table. While there are many ways to create a table in $R$ , I use kable from package knitr because it allows me to cross-reference tables within this book. kable only works with data frames.

Tabla 6.9 shows the four model selection criteria for four different models based on the $edu\_inc$ dataset, with the first column showing the variables included in each model. It is noticeable that three of the criteria indicate the third model as the best fit, while one, namely $R^2$ prefers the model that includes the irrelevant variables $xtra\_x5$ and $xtra\_x6$ .

As a side note, a quick way of extracting the information criteria from an lm() object is illustrated in the following code fragment.

Another potentially useful tool for building an appropriate model is the Ramsey specification test, RESET. This method automatically adds higher-order polynomial terms to your model and tests the joint hypothesis that their coefficients are all zero. Thus, the null hypothesis of the test is $H_{0}$ : “No higher-order polynomial terms are necessary”; if we reject the null hypothesis we need to consider including such terms.

The $R$ function that performs a RESET test is resettest , which requires the following argumets: formula , the formula of the model to be tested or the name of an already calculated lm object; power , a set of integers indicating the powers of the polynomial terms to be included; type , which could be one of “fitted”, “regressor”, or “princomp”, indicating whether the aditional terms should be powers of the regressors, fitted values, or the first principal component of the regressor matrix; and, finally, data , which specifies the dataset to be used if a formula has been provided and not a model object. The following code applies the test to the complete $faminc$ model, first using only quadratic terms of the fitted values, then using both quadratic and cubic terms.

The number labeled as RESET in the output is the $F$ -statistic of the test under the null hypothesis followed by the two types of degrees of freedom of the $F$ distribution and the $p$ -value. In our case both $p$ -values are slightly lower than $0.05$ , indicating that the model marginally fails the specification test and some higher order terms may be necessary.

6.6 Collinearity

There is collinearity among regressors when two or more regressors move closely with each other or display little variability. A consequence of collinearity is large variance in the estimated parameters, which increases the chances of not finding them significantly different from zero. The estimates are, however, unbiased since (imperfect) collinearity does not technically violate the Gauss-Markov assumptions. Collinearity tends to show insignificant coefficients even when measures of goodness-of-fit such as $R^2$ or overall significance (the $F$ -statistic) may be quite large.

Let us consider the example of the dataset cars , where mpg is miles per gallon, cyl is number of cylinders, eng is engine displacement in cubic inches, and wgt is the weight of the vehicle in pounds.

This naive model suggests a strong effect of the number of cylinders on fuel economy, but if we introduce more terms in the equation this result changes substantially.

In the model summarized in Table 6.11 the number of cylinders becomes insignificant alltogether, a sharp change with respect to the previous specification. This high sensitivity of the estimates when other variables are introduced is also a sign of collinearity. Indeed, it is reasonable to believe that the characteristics of the vehicles vary together: heavier vehicles have more cylinders and bigger engines.

The results in Table 6.12 show that the regressors cyl and eng fail the collinearity test, having $VIF$ s greater than 10.

6.7 Prediction and Forecasting

We have previously discussed the semantic difference between prediction and forecasting, with prediction meaning the estimation of an expected value of the response and forecasting meaning an estimate of a particular value of the response. We mentioned that, for the same vector of regressors, prediction has a narrower confidence interval than forecasting because forecasting includes, besides the uncertainty of the expected value of the response, the variablility of a particular observation about its mean. I essentially repeat the same procedure here for the quadratic andy model, which regresses $sales$ on $price$ , $advert$ , and $advert^2$ . The key $R$ function to calculate both predictions and forecasts is the function predict , with the following arguments: model , which is the name of a model object; newdata , which contains the new data points where prediction is desired; if newdata is missing, predictions are calculated for all observations in the dataset; interval , which can be “none”, “confidence”, or “prediction”, and tells $R$ whether we want only a point estimate of the response, a prediction with its confidence interval, or a forecast with its confidence interval; level , which is the confidence level we want; if missing, level is $95\%$ ; other arguments (see ?predict() for more details).

Table 6.13 displays the point estimate and forecast interval estimate for the data point $price=\$6$ and $advert=1.9$ , which, remember, stands for advertising expenditure of $\$1900$ .

Robinson, David. 2016. Broom: Convert Statistical Analysis Objects into Tidy Data Frames . https://CRAN.R-project.org/package=broom .

Test Linear Hypothesis

Description.

The method for "lm" objects calls the default method, but it changes the default test to "F" , supports the convenience argument white.adjust (for backwards compatibility), and enhances the output by the residual sums of squares. For "glm" objects just the default method is called (bypassing the "lm" method). The "svyglm" method also calls the default method.

The function lht also dispatches to linearHypothesis .

The hypothesis matrix can be supplied as a numeric matrix (or vector), the rows of which specify linear combinations of the model coefficients, which are tested equal to the corresponding entries in the right-hand-side vector, which defaults to a vector of zeroes.

The transformation matrix can be specified directly via the P argument.

matchCoefs is a convenience function that can sometimes help in formulating hypotheses; for example matchCoefs(mod, ":") will return the names of all interaction coefficients in the model mod .

The returned object normally would be printed.

Achim Zeileis and John Fox [email protected]

Fox, J. (2016) Applied Regression Analysis and Generalized Linear Models , Third Edition. Sage.

Fox, J. and Weisberg, S. (2019) An R Companion to Applied Regression , Third Edition, Sage.

Hand, D. J., and Taylor, C. C. (1987) Multivariate Analysis of Variance and Repeated Measures: A Practical Approach for Behavioural Scientists. Chapman and Hall.

O'Brien, R. G., and Kaiser, M. K. (1985) MANOVA method for analyzing repeated measures designs: An extensive primer. Psychological Bulletin 97 , 316–333.

anova , Anova , waldtest , hccm , vcovHC , vcovHAC , coef , vcov

linearHypothesis.systemfit: Test Linear Hypothesis

Description.

Testing linear hypothesis on the coefficients of a system of equations by an F-test or Wald-test.

An object of class anova , which contains the residual degrees of freedom in the model, the difference in degrees of freedom, the test statistic (either F or Wald/Chisq) and the corresponding p value. See documentation of linearHypothesis

in package "car".

a fitted object of type systemfit .

matrix (or vector) giving linear combinations of coefficients by rows, or a character vector giving the hypothesis in symbolic form (see documentation of linearHypothesis in package "car" for details).

optional right-hand-side vector for hypothesis, with as many entries as rows in the hypothesis matrix; if omitted, it defaults to a vector of zeroes.

character string, " FT ", " F ", or " Chisq ", specifying whether to compute Theil's finite-sample F test (with approximate F distribution), the finite-sample Wald test (with approximate F distribution), or the large-sample Wald test (with asymptotic Chi-squared distribution).

a function for estimating the covariance matrix of the regression coefficients or an estimated covariance matrix (function vcov is used by default).

further arguments passed to linearHypothesis.default (package "car").

Arne Henningsen [email protected]

Theil's $F$ statistic for sytems of equations is $$F = \frac{ ( R \hat{b} - q )' ( R ( X' ( \Sigma \otimes I )^{-1} X )^{-1} R' )^{-1} ( R \hat{b} - q ) / j }{ \hat{e}' ( \Sigma \otimes I )^{-1} \hat{e} / ( M \cdot T - K ) } $$ where $j$ is the number of restrictions, $M$ is the number of equations, $T$ is the number of observations per equation, $K$ is the total number of estimated coefficients, and $\Sigma$ is the estimated residual covariance matrix. Under the null hypothesis, $F$ has an approximate $F$ distribution with $j$ and $M \cdot T - K$ degrees of freedom (Theil, 1971, p. 314).

The $F$ statistic for a Wald test is $$ F = \frac{ ( R \hat{b} - q )' ( R \, \widehat{Cov} [ \hat{b} ] R' )^{-1} ( R \hat{b} - q ) }{ j } $$ Under the null hypothesis, $F$ has an approximate $F$ distribution with $j$ and $M \cdot T - K$ degrees of freedom (Greene, 2003, p. 346).

The $\chi^2$ statistic for a Wald test is $$ W = ( R \hat{b} - q )' ( R \widehat{Cov} [ \hat{b} ] R' )^{-1} ( R \hat{b} - q ) $$ Asymptotically, $W$ has a $\chi^2$ distribution with $j$ degrees of freedom under the null hypothesis (Greene, 2003, p. 347).

Greene, W. H. (2003) Econometric Analysis, Fifth Edition , Prentice Hall.

Theil, Henri (1971) Principles of Econometrics , John Wiley & Sons, New York.

systemfit , linearHypothesis (package "car"), lrtest.systemfit

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Published: 26 May 2024

A double machine learning model for measuring the impact of the Made in China 2025 strategy on green economic growth

Jie Yuan 1 &
Shucheng Liu 2

Scientific Reports volume 14 , Article number: 12026 ( 2024 ) Cite this article

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Environmental economics
Environmental impact
Sustainability

The transformation and upgrading of China’s manufacturing industry is supported by smart and green manufacturing, which have great potential to empower the nation’s green development. This study examines the impact of the Made in China 2025 industrial policy on urban green economic growth. This study applies the super-slacks-based measure model to measure cities’ green economic growth, using the double machine learning model, which overcomes the limitations of the linear setting of traditional causal inference models and maintains estimation accuracy under high-dimensional control variables, to conduct an empirical analysis based on panel data of 281 Chinese cities from 2006 to 2021. The results reveal that the Made in China 2025 strategy significantly drives urban green economic growth, and this finding holds after a series of robustness tests. A mechanism analysis indicates that the Made in China 2025 strategy promotes green economic growth through green technology progress, optimizing energy consumption structure, upgrading industrial structure, and strengthening environmental supervision. In addition, the policy has a stronger driving effect for cities with high manufacturing concentration, industrial intelligence, and digital finance development. This study provides valuable theoretical insights and policy implications for government planning to promote high-quality development through industrial policy.

The influence of AI on the economic growth of different regions in China

The trilemma of sustainable industrial growth: evidence from a piloting OECD’s Green city

Green technology advancement, energy input share and carbon emission trend studies

Introduction.

Since China’s reform and opening up, the nation’s economy has experienced rapid growth for more than 40 years. According to the National Bureau of Statistics, China’s per capita GDP has grown from 385 yuan in 1978 to 85,698 yuan in 2022, with an average annual growth rate of 13.2%. However, obtaining this growth miracle has come at considerable social and environmental costs 1 . Current pollution prevention and control systems have not yet fundamentally alleviated the structural and root causes, impairing China’s economic progress toward high-quality development 2 . The report of the 20th National Congress of the Communist Party of China proposed that the future will be focused on promoting the formation of green modes of production and lifestyles and advancing the harmonious coexistence of human beings and nature. This indicates that transforming the mode of economic development is now the focus of the government’s attention, calling for advancing the practices of green growth aimed at energy conservation, emissions reduction, and sustainability while continuously increasing economic output 3 . As a result, identifying approaches to balance economic growth and green environmental protection in the development process and realize green economic growth has become an arduous challenge and a crucially significant concern for China’s high-quality economic development.

An intrinsic driver of urban economic growth, manufacturing is also the most energy-intensive and pollution-emitting industry, and greatly constrains urban green development 4 . China’s manufacturing industry urgently needs to advance the formation of a resource-saving and environmentally friendly industrial structure and manufacturing system through transformation and upgrading to support for green economic growth 5 . As an incentive-based industrial policy that emphasizes an innovation-driven and eco-civilized development path through the development and implementation of an intelligent and green manufacturing system, Made in China 2025 is a significant initiative for promoting the manufacturing industry’s transformation and upgrading, providing solid economic support for green economic growth 6 . To promote the effective implementation of this industrial policy, fully mobilize localities to explore new modes and paths of manufacturing development, and strengthen the urban manufacturing industry’s influential demonstration role in advancing the green transition, the Ministry of Industry and Information Technology of China successively launched 30 Made in China 2025 pilot cities (city clusters) in 2016 and 2017. The Pilot Demonstration Work Program for “Made in China 2025” Cities specified that significant results should be achieved within three to 5 years. After several years of implementation, has the Made in China 2025 pilot policy promoted green economic growth? What are the policy’s mechanisms of action? Are there differences in green economic growth effects in pilot cities based on various urban development characteristics? This study’s theoretical interpretation and empirical examination of the above questions can add to the growing body of related research and provide valuable insights for cities to comprehensively promote the transformation and upgrading of manufacturing industry to advance China’s high-quality development.

This study constructs an analytical framework at the theoretical level to analyze the impact of the Made in China 2025 strategy on urban green economic growth, and uses the double machine learning (ML) model to test its green economic growth effect. The contributions of this study are as follows. First, focusing on the field of urban green development, the study incorporates variables representing the potential economic and environmental effects of the Made in China 2025 policy into a unified framework to systematically examine the impact of the Made in China 2025 pilot policy on the urban green economic growth, providing a novel perspective for assessing the effects of industrial policies. Second, we investigate potential transmission mechanisms of the Made in China 2025 strategy affecting green economic growth from the perspectives of green technology advancement, energy consumption structure optimization, industrial structure upgrading, and environmental supervision strengthening, establishing a useful supplement for related research. Third, leveraging the advantage of ML algorithms in high-dimensional and nonparametric prediction, we apply a double ML model assess the policy effects of the Made in China 2025 strategy to avoid the “curse of dimensionality” and the inherent biases of traditional econometric models, and improve the credibility of our research conclusions.

The remainder of this paper is structured as follows. Section “ Literature review ” presents a literature review. Section “ Policy background and theoretical analysis ” details our theoretical analysis and research hypotheses. Section “ Empirical strategy ” introduces the model setting and variables selection for the study. Section “ Empirical result ” describes the findings of empirical testing and analyzes the results. Section “ Conclusion and policy recommendation ” summarizes our conclusions and associated policy implications.

Literature review

Measurement and influencing factors of green economic growth.

The Green Economy Report, which was published by the United Nations Environment Program in 2011, defined green economy development as facilitating more efficient use of natural resources and sustainable growth than traditional economic models, with a more active role in promoting combined economic development and environmental protection. The Organization for Economic Co-operation and Development defined green economic growth as promoting economic growth while ensuring that natural assets continue to provide environmental resources and services; a concept that is shared by a large number of institutions and scholars 7 , 8 , 9 . A considerable amount of research has assessed green economic growth, primarily using three approaches. First, single-factor indicators, such as sulfur dioxide emissions, carbon dioxide emissions intensity, and other quantified forms; however, this approach neglects the substitution of input factors such as capital and labor for the energy factor, which has certain limitations 5 , 10 . Second, studies have been based on neoclassical economic growth theory, incorporating factors of capital, technology, energy, and the environment, and constructing a green Solow model to measure green total factor productivity (GTFP) 11 , 12 . Third, based on neoclassical economic growth theory, some studies have simultaneously considered desirable and undesirable output, applying Shepard’s distance function, the directional distance function, and data envelopment analysis to measure GTFP 13 , 14 , 15 .

Economic growth is an extremely complex process, and green economic growth is also subject to a combination of multiple complex factors. Scholars have explored the influence mechanisms of green economic growth from perspectives of resource endowment 16 , technological innovation 17 , industrial structure 18 , human capital 19 , financial support 20 , government regulation 21 , and globalization 22 . In the field of policy effect assessment, previous studies have confirmed the green development effects of pilot policies such as innovative cities 23 , Broadband China 24 , smart cities 25 , and low-carbon cities 26 . However, few studies have focused on the impact of Made in China 2025 strategy on urban green economic growth and identified its underlying mechanisms.

The impact of Made in China 2025 strategy

Since the industrial policy of Made in China 2025 was proposed, scholars have predominantly focused on exploring its economic effects on technological innovation 27 , digital transformation 28 , and total factor productivity (TFP) 29 , while the potential environmental effects have been neglected. Chen et al. (2024) 30 found that Made in China 2025 promotes firm innovation through tax incentives, public subsidies, convenient financing, academic collaboration and talent incentives. Xu (2022) 31 point out that Made in China 2025 policy has the potential to substantially improve the green innovation of manufacturing enterprises, which can boost the green transformation and upgrading of China’s manufacturing industry. Li et al. (2024) 32 empirically investigates the positive effect of Made in China 2025 strategy on digital transformation and exploratory innovation in advanced manufacturing firms. Moreover, Liu and Liu (2023) 33 take “Made in China 2025” as an exogenous shock and find that the pilot policy has a positive impact on the high-quality development of enterprises and capital markets. Unfortunately, scholars have only discussed the impact of Made in China 2025 strategy on green development and environmental protection from a theoretical perspective and lack empirical analysis. Li (2018) 27 has compared Germany’s “Industry 4.0” and China’s “Made in China 2025”, and point out that “Made in China 2025” has clear goals, measures and sector focus. Its guiding principles are to enhance industrial capability through innovation-driven manufacturing, optimize the structure of Chinese industry, emphasize quality over quantity, train and attract talent, and achieve green manufacturing and environment. Therefore, it is necessary to systematically explore the impact and mechanism of Made in China 2025 strategy on urban green economic growth from both theoretical and empirical perspectives.

Causal inference based on double ML

The majority of previous studies have used traditional causal inference models to assess policy effects; however, some limitations are inherent to the application of these models. For example, the parallel trend test of the difference-in-differences model has stringent requirements on appropriate sample data; the synthetic control method can construct a virtual control group that conforms to the parallel trend, but it requires that the treatment group does not have the extreme value characteristics, and it is only applicable to “one-to-many” circumstances; and the propensity score matching (PSM) method involves a considerable amount of subjectivity in selecting matching variables. To compensate for the shortcomings of traditional models, scholars have started to explore the application of ML in the field of causal inference 34 , 35 , 36 , and double ML is a typical representative.

Double ML was formalized in 2018 34 , and the relevant research falls into two main categories. The first strand of literature applies double ML to assess causality concerning economic phenomena. Yang et al. (2020) 37 applied double ML using a gradient boosting algorithm to explore the average treatment effect of top-ranked audit firms, verifying its robustness compared with the PSM method. Zhang et al. (2022) 38 used double ML to quantify the impact of nighttime subway services on the nighttime economy, house prices, traffic accidents, and crime following the introduction of nighttime subway services in London in 2016. Farbmacher et al. (2022) 39 combined double ML with mediating effects analysis to assess the causal relationship between health insurance coverage and youth wellness and examine the indirect mechanisms of regular medical checkups, based on a national longitudinal health survey of youth conducted by the US Bureau of Labor Statistics. The second strand of literature has innovated methodological theory based on double ML. Chiang et al. (2022) 40 proposed an improved multidirectional cross-fitting double ML method, obtaining regression results for high-dimensional parameters while estimating robust standard errors for dual clustering, which can effectively adapt to multidirectional clustered sampled data and improve the validity of estimation results. Bodory et al. (2022) 41 combined dynamic analysis with double ML to measure the causal effects of multiple treatment variables over time, using weighted estimation to assess the dynamic treatment effects of specific subsamples, which enriched the dynamic quantitative extension of double ML.

In summary, previous research has conducted some useful investigations regarding the impact of socioeconomic policies on green development, but limited studies have explored the relationship between the Made in China 2025 strategy and green economic growth. This study takes 281 Chinese cities as the research object, and applies the super-slacks-based measure (SBM) model to quantify Chinese cities’ green economic growth from 2006 to 2021. Based on a quasi-natural experiment of Made in China 2025 pilot policy implementation, we use the double ML model to test the impact and transmission mechanisms of the policy on urban green economic growth. We also conduct a heterogeneity analysis of cities based on different levels of manufacturing agglomeration, industrial intelligence, and digital finance. This study applies a novel approach and provides practical insights for research in the field of industrial policy assessment.

Policy background and theoretical analysis

Policy background.

The Made in China 2025 strategy aims to encourage and support local exploration of new paths and models for the transformation and upgrading of the manufacturing industry, and to drive the improvement of manufacturing quality and efficiency in other regions through demonstration effects. According to the Notice of Creating “Made in China 2025” National Demonstration Zones issued by the State Council, municipalities directly under the central government, sub-provincial cities, and prefecture-level cities can apply for the creation of demonstration zones. Cities with proximity and high industrial correlation can jointly apply for urban agglomeration demonstration zones. The Notice clarifies the goals and requirements for creating demonstration zones in areas such as green manufacturing, clean production, and environmental protection. In 2016, Ningbo became the first Made in China 2025 pilot city, and a total of 12 cities and 4 city clusters were included in the list of Made in China 2025 national demonstration zones. In 2018, the State Council issued the Evaluation Guidelines for “Made in China 2025” National Demonstration Zone, which further clarified the evaluation process and indicator system of the demonstration zone. Seven primary indicators and 29 secondary indicators were formulated, including innovation driven, quality first, green development, structural optimization, talent oriented, organizational implementation, and coordinated development of urban agglomerations. This indicator system can evaluate the creation process and overall effectiveness of pilot cities (city clusters), which is beneficial for the promotion of successful experiences and models in demonstration areas.

Advancing green urban development is a complex systematic project that requires structural adjustment and technological and institutional changes in the socioeconomic system 42 . The Made in China 2025 strategy emphasizes the development and application of smart and green manufacturing systems, which can unblock technological bottlenecks in the manufacturing sector in terms of industrial production, energy consumption, and waste emissions, and empower cities to operate in a green manner. In addition, the Made in China 2025 policy established requirements for promoting technological innovation to advance energy saving and environmental protection, improving the rate of green energy use, transforming traditional industries, and strengthening environmental supervision. For pilot cities, green economy development requires the support of a full range of positive factors. Therefore, this study analyzes the mechanisms by which the Made in China 2025 strategy affects urban green economic growth from the four paths of green technology advancement, energy consumption structure optimization, industrial structure upgrading, and environmental supervision strengthening.

Theoretical analysis and research hypotheses

As noted, the Made in China 2025 strategy emphasizes strengthening the development and application of energy-saving and environmental protection technologies to advance cleaner production. Pilot cities are expected to prioritize the driving role of green innovation, promote clustering carriers and innovation platforms for high-tech enterprises, and guide the progress of enterprises’ implementation of green technology. Specifically, pilot cities are encouraged to optimize the innovation environment by increasing scientific and technological investment and financial subsidies in key areas such as smart manufacturing and high-end equipment and strengthening intellectual property protection to incentivize enterprises to conduct green research and development (R&D) activities. These activities subsequently promote the development of green innovation technologies and industrial transformation 43 . Furthermore, since quality human resources are a core aspect of science and technology innovation 44 , pilot cities prioritize the cultivation and attraction of talent to establish a stable human capital guarantee for enterprises’ ongoing green technology innovation, transform and upgrade the manufacturing industry, and advance green urban development. Green technology advances also contribute to urban green economic growth. First, green technology facilitates enterprises’ adoption of improved production equipment and innovation in green production technology, accelerating the change of production mode and driving the transformation from traditional crude production to a green and intensive approach 45 , promoting green urban development. Second, green technology advancement accelerates green innovations such as clean processes, pollution control technologies, and green equipment, and facilitates the effective supply of green products, taking full advantage of the benefits of green innovations 46 and forming a green economic development model to achieve urban green economic growth.

The Made in China 2025 pilot policy endeavors to continuously increase the rate of green and low-carbon energy use and reduce energy consumption. Under target constraints of energy saving and carbon control, pilot cities will accelerate the cultivation of high-tech industries in green environmental protection and high-end equipment manufacturing with advantages of sustainability and low resource inputs 47 to improve the energy consumption structure. Pilot cities also advance new energy sector development by promoting clean energy projects, subsidizing new energy consumption, and supporting green infrastructure construction and other policy measures 48 to optimize the energy consumption structure. Energy consumption structure optimization can have a profound impact on green economy development. Optimization means that available energy tends to be cleaner, which can reduce the manufacturing industry’s dependence on traditional fossil energy and raise the proportion of clean energy 49 , ultimately promoting green urban development. Pilot cities also provide financial subsidies for new energy technology R&D, which promotes the innovation and application of new technologies, energy-saving equipment, efficient resource use, and energy-saving diagnostics, which allow enterprises to save energy and reduce consumption and improve energy use efficiency and TFP 50 , advancing the growth of urban green economy.

At its core, the Made in China 2025 strategy promotes the transformation and upgrading of the manufacturing sector. Pilot cities guide and develop technology-intensive high-tech industries, adjust the proportion of traditional heavy industry, and improve the urban industrial structure. Pilot cities also implement the closure, merger, and transformation of pollution-intensive industries; guide the fission of professional advantages of manufacturing enterprises 51 ; and expand the establishment and development of service-oriented manufacturing and productive service industries to promote the evolution of the industrial structure toward rationalization and high-quality development 52 . Upgrading the industrial structure can also contribute to urban green economic growth. First, industrial structure upgrading promotes the transition from labor- and capital-intensive industries to knowledge- and technology-intensive industries, which optimizes the industrial distribution patterns of energy consumption and pollutant emissions and promotes the transformation of economic growth dynamics and pollutant emissions control, providing a new impetus for cities’ sustainable development 53 . Second, changes in industrial structure and scale can have a profound impact on the type and quantity of pollutant emissions. By introducing high-tech industries, service-oriented manufacturing, and production-oriented service industries, pilot cities can promote the transformation of pollution-intensive industries, promoting the adjustment and optimization of industrial structure and scale 54 to achieve the purpose of driving green urban development.

The Made in China 2025 strategy proposes strengthening green supervision and conducting green evaluations, establishing green development goals for the manufacturing sector in terms of emissions and consumption reduction and water conservation. This requires pilot cities to implement stringent environmental regulatory policies, such as higher energy efficiency and emissions reduction targets and sewage taxes and charges, strict penalties for excess emissions, and project review criteria 55 , which consolidates the effectiveness of green development. Under the framework of environmental authoritarianism, strengthening environmental supervision is a key measure for achieving pollution control and improving environmental quality 56 . Therefore, environmental regulatory enhancement can help cities achieve green development goals. First, according to the Porter hypothesis 57 , strong environmental regulatory policies encourage firms to internalize the external costs of environmental supervision, stimulate technological innovation, and accelerate R&D and application of green technologies. This response helps enterprises improve input–output efficiency, achieve synergy between increasing production and emissions reduction, partially or completely offset the “environmental compliance cost” from environmental supervision, and realize the innovation compensation effect 58 . Second, strict environmental regulations can effectively mitigate the complicity of local governments and enterprises in focusing on economic growth while neglecting environmental protection 59 , urging local governments to constrain enterprises’ emissions, which compels enterprises to conduct technological innovation and pursue low-carbon transformation, promoting urban green economic growth.

Based on the above analysis, we propose the mechanisms that promote green economic growth through Made in China 2025 strategy, as shown in Fig. 1 . The proposed research hypotheses are as follows:

Mechanism analysis of Made in China 2025 strategy and green economic growth.

Hypothesis 1

The Made in China 2025 strategy promotes urban green economic growth.

Hypothesis 2

The Made in China 2025 strategy drives urban green economic growth through four channels: promoting green technology advancement, optimizing energy consumption structure, upgrading industrial structure, and strengthening environmental supervision.

Empirical strategy

Double ml model.

Compared with traditional causal inference models, double ML has unique advantages in variable selection and model estimation, and is also more applicable to the research problem of this study. Green economic growth is a comprehensive indicator of transformative urban growth that is influenced by many socioeconomic factors. To ensure the accuracy of our policy effects estimation, the interference of other factors on urban green economic growth must be controlled as much as possible; however, when introducing high-dimensional control variables, traditional regression models may face the “curse of dimensionality” and multicollinearity, rendering the accuracy of the estimates questionable. Double ML uses ML and regularization algorithms to automatically filter the preselected set of high-dimensional control variables to obtain an effective set of control variables with higher prediction accuracy. This approach avoids the “curse of dimensionality” caused by redundant control variables and mitigates the estimation bias caused by the limited number of primary control variables 39 . Furthermore, nonlinear relationships between variables are the norm in the evolution of economic transition, and ordinary linear regression may suffer from model-setting bias producing estimates that lack robustness. Double ML effectively overcomes the problem of model misspecification by virtue of the advantages of ML algorithms in handling nonlinear data 37 . In addition, based on the idea of instrumental variable functions, two-stage predictive residual regression, and sample split fitting, double ML mitigates the “regularity bias” in ML estimation and ensures unbiased estimates of the treatment coefficients in small samples 60 .

Based on the analysis above, this study uses the double ML model to assess the policy effects of the Made in China 2025 strategy. The partial linear double ML model is constructed as follows:

where i denotes the city, t denotes the year, and Y it represents green economic growth. Policy it represents the policy variable of Made in China 2025, which is set as 1 if the pilot is implemented and 0 otherwise. θ 0 is the treatment coefficient that is the focus of this study. X it denotes the set of high-dimensional control variables, and the ML algorithm is used to estimate the specific functional form $\hat{g}(X_{it} )$ . U it denotes the error term with a conditional mean of zero.

Direct estimation of Eqs. ( 1 ) and ( 2 ) yields the following estimate of the treatment coefficient:

where n denotes the sample size.

Notably, the double ML model uses a regularization algorithm to estimate the specific functional form $\hat{g}(X_{it} )$ , which prevents the variance of the estimate from being too large, but inevitably introduces a “regularity bias,” resulting in a biased estimate. To speed up the convergence of the $\hat{g}(X_{it} )$ directions so that the estimates of the treatment coefficients satisfy unbiasedness with small samples, the following auxiliary regression is constructed:

where $m(X_{it} )$ is the regression function of the treatment variable on the high-dimensional control variable, using ML algorithms to estimate the specific functional form $\hat{m}(X_{it} )$ . V it is the error term with a conditional mean of zero.

The specific operation process follows three stages. First, we use the ML algorithm to estimate the auxiliary regression $\hat{m}(X_{it} )$ and take its residuals $\hat{V}_{it} = Policy_{it} - \hat{m}(X_{it} )$ . Second, we use the ML algorithm to estimate $\hat{g}(X_{it} )$ and change the form of the main regression $Y_{it} - \hat{g}(X_{it} ) = \theta_{0} Policy_{it} + U_{it}$ . Finally, we regress $\hat{V}_{it}$ as an instrumental variable for Policy it , obtaining unbiased estimates of the treatment coefficients as follows:

Variable selection

Green economic growth

We apply the super-SBM model to measure urban green economic growth. The super-SBM model is compatible with radial and nonradial characteristics, which avoids inflated results due to ignoring slack variables and deflated results due to ignoring the linear relationships between elements, and can truly reflect relative efficiency 61 . The SBM model reflects the nature of green economic growth more accurately compared with other models, and has been widely adopted by scholars 62 . The expression of the super-SBM model considering undesirable output is as follows:

where x is the input variable; y and z are the desirable and undesirable output variables, respectively; m denotes the number of input indicators; s 1 and s 2 represent the respective number of indicators for desirable and undesirable outputs; k denotes the period of production; i , r , and t are the decision units for the inputs, desirable outputs, and undesirable outputs, respectively; $s^{ - }$ , $s^{ + }$ , and $s^{z - }$ are the respective slack variables for the inputs, desirable outputs, and undesirable outputs; and γ is a vector of weights. A larger $\rho_{SE}$ value indicates greater efficiency. If $\rho_{SE}$ = 1, the decision unit is effective; if $\rho_{SE}$ < 1, the decision unit is relatively ineffective, indicating a loss of efficiency.

Referencing Sarkodie et al. (2023) 63 , the evaluation index system of green economic growth is constructed as shown in Table 1 .

Made in China 2025 pilot policy

The list of Made in China 2025 pilot cities (city clusters) published by the Ministry of Industry and Information Technology of China in 2016 and 2017 is matched with the city-level data to obtain 30 treatment group cities and 251 control group cities. The policy dummy variable of Made in China 2025 is constructed by combining the implementation time of the pilot policies.

Mediating variables

This study also examines the transmission mechanism of the Made in China 2025 strategy affecting urban green economic growth from four perspectives, including green technology advancement, energy consumption structure optimization, industrial structure upgrading, and strengthening of environmental supervision. (1) The number of green patent applications is adopted to reflect green technology advancement. (2) Energy consumption structure is quantified using the share of urban domestic electricity consumption in total energy consumption. (3) The industrial structure upgrading index is calculated using the formula $\sum\nolimits_{i = 1}^{3} {i \times (GDP_{i} /GDP)}$ , where GDP i denotes the added value of primary, secondary, or tertiary industries. (4) The frequency of words related to the environment in government work reports is the proxy for measuring the intensity of environmental supervision 64 .

Control variables

Double ML can effectively accommodate the case of high-dimensional control variables using regularization algorithms. To control for the effect of other urban characteristics on green economic growth, this study introduces the following 10 control variables. We measure education investment by the ratio of education expenditure to GDP. Technology investment is the ratio of technology expenditure to GDP. The study measures urbanization using the share of urban built-up land in the urban area. Internet penetration is the number of internet users as a share of the total population at the end of the year. We measure resident consumption by the total retail sales of consumer goods per capita. The unemployment rate is the ratio of the number of registered unemployed in urban areas at the end of the year to the total population at the end of the year. Financial scale is the ratio of the balance of deposits and loans of financial institutions at the end of the year to the GDP. Human capital is the natural logarithm of the number of students enrolled in elementary school, general secondary schools, and general tertiary institutions per 10,000 persons. Transportation infrastructure is the natural logarithm of road and rail freight traffic. Finally, openness to the outside world is reflected by the ratio of actual foreign investment to GDP. Quadratic terms for the control variables are also included in the regression analysis to improve the accuracy of the model’s fit. We introduce city and time fixed effects as individual and year dummy variables to avoid missing information on city and time dimensions.

Data sources

This study uses 281 Chinese cities spanning from 2006 to 2021 as the research sample. Data sources include the China City Statistical Yearbook, the China Economic and Social Development Statistics Database, and the EPS Global Statistics Database. We used the average annual growth rate method to fill the gaps for the minimal missing data. To remove the effects of price changes, all data measured in monetary units are deflated using the consumer price index for each province for the 2005 base period. The descriptive statistics of the data are presented in Table 2 .

Empirical result

Baseline results.

The sample split ratio of the double ML model is set to 1:4, and we use the Lasso algorithm to predict and solve the main and auxiliary regressions, presenting the results in Table 3 . Column (1) does not control for fixed effects or control variables, column (2) introduces city and time fixed effects, and columns (3) and (4) add control variables to columns (1) and (2), respectively. The regressions in columns (1) and (2) are highly significant, regardless of whether city and time fixed effects are controlled. Column (4) controls for city fixed effects, time fixed effects, and the primary term of the control variable over the full sample interval, revealing that the regression coefficient of the Made in China 2025 pilot policy on green economic growth is positive and significant at the 1% level, confirming that the Made in China 2025 strategy significantly promotes urban green economic growth. Column (5) further incorporates the quadratic terms of the control variables and the regression coefficients remain significantly positive with little change in values. Therefore, Hypothesis 1 is verified.

Parallel trend test

The prerequisite for the establishment of policy evaluation is that the development status of cities before the pilot policy is introduced is similar. Referring to Liu et al. (2022) 29 , we adopt a parallel trend test to verify the effectiveness of Made in China 2025 pilot policy. Figure 2 shows the result of parallel trend test. None of the coefficient estimates before the Made in China 2025 pilot policy are significant, indicating no significant difference between the level of green economic growth in pilot and nonpilot cities before implementing the policy, which passes the parallel trend test. The coefficient estimates for all periods after the policy implementation are significantly positive, indicating that the Made in China 2025 pilot policy can promote urban green economic growth.

Parallel trend test.

Robustness tests

Replace explained variable.

Referencing Oh and Heshmati (2010) 65 and Tone and Tsutsui (2010) 66 , we use the Malmquist–Luenberger index under global production technology conditions (GML) and an epsilon-based measure (EBM) model to recalculate urban green economic growth. The estimation results in columns (1) and (2) of Table 4 show that the estimated coefficients of the Made in China 2025 pilot policy remain significantly positive after replacing the explanatory variables, validating the robustness of the baseline findings.

Adjusting the research sample

Considering the large gaps in the manufacturing development base between different regions in China, using all cities in the regression analysis may lead to biased estimation 67 . Therefore, we exclude cities in seven provinces with a poor manufacturing development base (Gansu, Qinghai, Ningxia, Xinjiang, Tibet, Yunnan, and Guizhou) and four municipalities with a better development base (Beijing, Tianjin, Shanghai, and Chongqing). The other city samples are retained to rerun the regression analysis, and the results are presented in column (3) of Table 4 . The first batch of pilot cities of the Made in China 2025 strategy was released in 2016, and the second batch of pilot cities was released in 2017. To exclude the effect of point-in-time samples that are far from the time of policy promulgation, the regression is also rerun by restricting the study interval to the three years before and after the promulgation of the policy (2013–2020), and the results are presented in column (4) of Table 4 . The coefficients of the Made in China 2025 pilot policy effect on urban green economic growth decrease after adjusting for the city sample and the time interval, but remain significantly positive at the 1% level. This, once again, verifies the robustness of the benchmark regression results.

Eliminating the impact of potential policies

During the same period of the Made in China 2025 strategy implementation, urban green economy growth may be affected by other relevant policies. To ensure the accuracy of the policy effect estimates, four representative policy categories overlapping with the sample period, including smart cities, low-carbon cities, Broadband China, and innovative cities, were collected and organized. Referencing Zhang and Fan (2023) 25 , dummy variables for these policies are included in the benchmark regression model and the results are presented in Table 5 . The estimated coefficient of the Made in China 2025 pilot policy decreases after controlling for the effects of related policies, but remains significantly positive at the 1% level. This suggests that the positive impact of the Made in China 2025 strategy on urban green economic growth, although overestimated, does not affect the validity of the study’s findings.

Reset double ML model

To avoid the impact of the double ML model imparting bias on the conclusions, we conduct robustness tests by varying the sample splitting ratio, the ML algorithm, and the model estimation form. First, we change the sample split ratio of the double ML model from 1:4 to 3:7 and 1:3. Second, we replace the Lasso ML algorithm with random forest (RF), gradient boosting (GBT), and BP neural network (BNN). Third, we replace the partial linear model based on the dual ML with a more generalized interactive model, using the following main and auxiliary regressions for the analysis:

among them, the meanings of each variable are the same as Eqs. ( 1 ) and ( 2 ).

The estimated coefficients for the treatment effects are obtained from the interactive model as follows:

Table 6 presents the regression results after resetting the double ML model, revealing that the sample split ratio, ML algorithm, and the model estimation form in double ML model did not affect the conclusion that the Made in China 2025 strategy promotes urban green economic growth, and only alters the magnitude of the policy effect, once again validating the robustness of our conclusions.

Difference-in-differences model

To further verify the robustness of the estimation results, we use traditional econometric models for regression. Based on the difference-in-differences (DID) model, a synthetic difference-in-differences (SDID) model is constructed by combining the synthetic control method 68 . It constructs a composite control group with a similar pre-trend to the treatment group by linearly combining several individuals in the control group, and compares it with the treatment group 69 . Table 7 presents the regression results of traditional DID model and SDID model. The estimated coefficient of the Made in China 2025 policy remains significantly positive at the 1% level, which once again verifies the robustness of the study’s findings.

Mechanism verification

This section conducts mechanism verification from four perspectives of green technology advancement, energy consumption structure, industrial structure, and environmental supervision. The positive impacts of the Made in China 2025 strategy on green technology advancement, energy consumption structure optimization, industrial structure upgrading, and strengthening environmental supervision are empirically examined using a dual ML model (see Table A.1 in the Online Appendix for details). Referencing Farbmacher et al. (2022) 39 for causal mediating effect analysis of double ML (see the Appendix for details), we test the transmission mechanism of the Made in China 2025 strategy on green economic growth based on the Lasso algorithm, presenting the results in Table 8 . The findings show that the total effects under different mediating paths are all significantly positive at the 1% level, verifying that the Made in China 2025 strategy positively promotes urban green economic growth.

Mechanism of green technology advancement

The indirect effect of green technological innovation is significantly positive for both the treatment and control groups. After stripping out the path of green technology advancement, the direct effects of the treatment and control groups remain significantly positive, indicating that the increase in the level of green technological innovation brought about by the Made in China 2025 strategy significantly promotes urban green economic growth. The Made in China 2025 strategy proposes to strengthen financial and tax policy support, intellectual property protection, and talent training systems. Through the implementation of policy incentives, pilot cities have fostered the concentration of high-technology enterprises and scientific and technological talent cultivation, exerting a knowledge spillover effect that further promotes green technology advancement. At the same time, policy preferences have stimulated the demand for innovation in energy conservation and emissions reduction, which raises enterprises’ motivation to engage in green innovation activities. Green technology advancement helps cities achieve an intensive development model, bringing multiple dividends such as lower resource consumption, reduced pollution emissions, and improved production efficiency, which subsequently promotes green economic growth.

Mechanism of energy consumption structure

The indirect effect of energy consumption structure is significantly positive for the treatment and control groups, while the direct effect of the Made in China 2025 pilot policy on green economic growth remains significantly positive, indicating that the policy promotes urban green economic growth through energy consumption structure optimization. The policy encourages the introduction of clean energy into production processes, reducing pressure on enterprise performance and the cost of clean energy use, which helps enterprises to reduce traditional energy consumption that is dominated by coal and optimize the energy structure to promote green urban development.

Mechanism of industrial structure

The indirect effects of industrial structure on the treatment and control groups are significantly positive. After stripping out the path of industrial structure upgrading, the direct effects remain significantly positive for both groups, indicating that the Made in China 2025 strategy promotes urban green economic growth through industrial structure optimization. Deepening the restructuring of the manufacturing industry is a strategic task specified in Made in China 2025. Pilot cities focus on transforming and guiding the traditional manufacturing industry toward high-end, intelligent equipment upgrades and digital transformation, driving the regional industrial structure toward rationalization and advancement to achieve rational allocation of resources. Upgrading industrial structure is a prerequisite for cities to advance intensive growth and sustainable development. By assuming the roles of “resource converter” and “pollutant controller,” industrial upgrading can continue to release the dividends of industrial structure, optimize resource allocation, and improve production efficiency, establishing strong support for green economic growth.

Mechanism of environmental supervision

The treatment and control groups of environmental supervision has a positive indirect effect in the process of the Made in China 2025 pilot policy affecting green economic growth that is significant at the 1% level, affirming the transmission path of environmental supervision. The Made in China 2025 strategy states that energy consumption, material consumption, and pollutant emissions per unit of industrial added value in key industries should reach the world’s advanced level by 2025. This requires pilot cities to consolidate and propagate the effectiveness of green development by strengthening environmental supervision while promoting the manufacturing sector’s green development. Strengthening environmental supervision promotes enterprises’ energy saving and emissions reduction through innovative compensation effects, while restraining enterprises’ emissions behaviors by tightening environmental protection policies, promoting environmental legislation, and increasing penalties to advance green urban development. Based on the above analysis, Hypothesis 2 is validated.

Heterogeneity analysis

Heterogeneity of manufacturing agglomeration.

To reduce production and transaction costs and realize economies of scale and scope, the manufacturing industry tends to accelerate its growth through agglomeration, exerting an “oasis effect” 70 . Cities with a high degree of manufacturing agglomeration are prone to scale and knowledge spillover effects, which amplify the agglomeration functions of talent, capital, and technology, strengthening the effectiveness of pilot policies. Based on this, we use the locational entropy of manufacturing employees to measure the degree of urban manufacturing agglomeration in the year (2015) before policy implementation, using the median to divide the full sample of cities into high and low agglomeration groups. Columns (1) and (2) in Table 9 reveal that the Made in China 2025 pilot policy has a stronger effect in promoting green economic growth in cities with high manufacturing concentration compared to those with low concentration. The rationale for this outcome may be that cities with a high concentration of manufacturing industries has large population and developed economy, which is conducive to leveraging agglomeration economies and knowledge spillover effects. Meanwhile, they are able to offer greater policy concessions by virtue of economic scale, public services, infrastructure, and other advantages. These benefits can attract the clustering of productive services and the influx of innovative elements such as R&D talent, accelerating the transformation and upgrading of the manufacturing industry and the integration and advancement of green technologies, empowering the green urban development.

Heterogeneity of industrial intelligence

As a landmark technology for the integration of the new scientific and technological revolution with manufacturing, industrial intelligence is a new approach for advancing the green transformation of manufacturing production methods. Based on this, we use the density of industrial robot installations to measure the level of industrial intelligence in cities in the year (2015) prior to policy implementation 71 , using the median to classify the full sample of cities into high and low level groups. Columns (3) and (4) in Table 9 reveals that the Made in China 2025 pilot policy has a stronger driving effect on the green economic growth of highly industrial intelligent cities. The rationale for this outcome may be that with the accumulation of smart factories, technologies, and equipment, a high degree of industrial intelligence is more likely to leverage the green development effects of pilot policies. For cities where the development of industrial intelligence is in its infancy or has not yet begun, the cost of information and knowledge required for enterprises to undertake technological R&D is higher, reducing the motivation and incentive to conduct innovative activities, diminishing the pilot policy’s contribution to green economic growth.

Heterogeneity of digital finance

As a fusion of traditional finance and information technology, digital finance has a positive impact on the development of the manufacturing industry by virtue of its advantages of low financing thresholds, fast mobile payments, and wide range of services 72 . Cities with a high degree of digital finance development have abundant financial resources and well-developed financial infrastructure that provide enterprises with more complete financial services, with subsequent influence on the effects of pilot policies. We use the Peking University Digital Inclusive Finance Index to measure the level of digital financial development in cities in the year (2015) prior to policy implementation, using the median to divide the full sample of cities into high and low level groups. Columns (5) and (6) in Table 9 reveal that the Made in China 2025 pilot policy has a stronger driving effect on the green economic growth of cities with highly developed digital finance. The rationale for this outcome may be that cities with a high degree of digital finance development can fully leverage the universality of financial resources, provide financial supply for environmentally friendly and technology-intensive enterprises, effectively alleviate the mismatch of financial capital supply, and provide financial security for enterprises to conduct green technology R&D. Digital finance also makes enterprises’ information more transparent through a rich array of data access channels, which strengthens government pollution regulation and public environmental supervision and compels enterprises to engage in green technological innovation to promote green economic growth.

Conclusion and policy recommendation

Conclusions.

This study examines the impact of the Made in China 2025 strategy on urban green economic growth using the double ML model based on panel data for 281 Chinese cities from 2006 to 2021. The relevant research results are threefold. First, the Made in China 2025 strategy significantly promotes urban green economic growth; a conclusion that is supported by a series of robustness tests. Second, regarding mechanisms, the Made in China 2025 strategy promotes urban green economic growth through green technology advancement, energy consumption structure optimization, industrial structure upgrading, and strengthening of environmental supervision. Third, the heterogeneity analysis reveals that the Made in China 2025 strategy has a stronger driving effect on green economic growth for cities with a high concentration of manufacturing and high degrees of industrial intelligence and digital finance.

policy recommendations

We next propose specific policy recommendations based on our findings. First, policymakers should summarize the experience of building pilot cities and create a strategic model to advance the transformation and upgrading of the manufacturing industry to drive green urban development. The Made in China 2025 pilot policy effectively promotes green economic growth and highlights the significance of the transformation and upgrading of the manufacturing industry to empower sustainable urban development. The government should strengthen the model and publicize summaries of successful cases of manufacturing development in pilot cities to promote the experience of manufacturing transformation and upgrading by producing typical samples to guide the transformation of the manufacturing industry to intelligence and greening. Policies should endeavor to optimize the industrial structure and production system of the manufacturing industry to create a solid real economy support for high-quality urban development.

Second, policymakers should explore the multidimensional driving paths of urban green economic growth and actively stimulate the green development dividend of pilot policies by increasing support for enterprise-specific technologies, subsidizing R&D in areas of energy conservation and emissions reduction, consumption reduction and efficiency, recycling and pollution prevention, and promoting the progress of green technologies. The elimination of outdated production capacity must be accelerated and the low-carbon transformation of traditional industries must be targeted, while guiding the clustering of high-tech industries, optimizing cities’ industrial structure, and driving industrial structure upgrading. Policymakers can regulate enterprises’ production practices and enhance the effectiveness of environmental supervision by improving the system of environmental information disclosure and mechanisms of rewards and penalties for pollution discharge. In addition, strategies should consider cities’ own resource endowment, promote large-scale production of new energy, encourage enterprises to increase the proportion of clean energy use, and optimize the structure of energy consumption.

Third, policymakers should engage a combination of urban development characteristics and strategic policy implementation to empower green urban development, actively promoting optimization of manufacturing industry structure, and accelerating the development of high-technology industries under the guidance of policies and the market to promote high-quality development and agglomeration of the manufacturing industry. At the same time, the government should strive to popularize the industrial internet, promote the construction of smart factories and the application of smart equipment, increase investment in R&D to advance industrial intelligence, and actively cultivate new modes and forms of industrial intelligence. In addition, new infrastructure construction must be accelerated, the application of information technology must be strengthened, and digital financial services must be deepened to ease the financing constraints for enterprises conducting R&D on green technologies and to help cities develop in a high-quality manner.

Data availability

The datasets used or analysed during the current study are available from the corresponding author on reasonable request.

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This work was supported by the Major Program of National Fund of Philosophy and Social Science of China (20&ZD133).

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Yuan, J., Liu, S. A double machine learning model for measuring the impact of the Made in China 2025 strategy on green economic growth. Sci Rep 14 , 12026 (2024). https://doi.org/10.1038/s41598-024-62916-0

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Int J Neuropsychopharmacol
v.24(10); 2021 Oct

Pharmacodynamic Interactions Between Ketamine and Psychiatric Medications Used in the Treatment of Depression: A Systematic Review

Jolien k e veraart.

1 University of Groningen, University Medical Center Groningen, Department of Psychiatry, Groningen, the Netherlands

1a PsyQ Haaglanden, Parnassia Psychiatric Institute, The Hague, the Netherlands

Sanne Y Smith-Apeldoorn

Iris m bakker, berber a e visser, jeanine kamphuis, robert a schoevers.

3a University of Groningen, Research School of Behavioural and Cognitive Neurosciences (BCN), Groningen, the Netherlands

Daan J Touw

4 Department of Clinical Pharmacy and Pharmacology, University of Groningen, University Medical Center Groningen, Groningen, The Netherlands

Associated Data

The use of ketamine for depression has increased rapidly in the past decades. Ketamine is often prescribed as an add-on to other drugs used in psychiatric patients, but clear information on drug-drug interactions is lacking. With this review, we aim to provide an overview of the pharmacodynamic interactions between ketamine and mood stabilizers, benzodiazepines, monoamine oxidase-inhibitors, antipsychotics, and psychostimulants.

MEDLINE and Web of Science were searched.

Twenty-four studies were included. For lithium, no significant interactions with ketamine were reported. Two out of 5 studies on lamotrigine indicated that the effects of ketamine were attenuated. Benzodiazepines were repeatedly shown to reduce the duration of ketamine’s antidepressant effect. For the monoamine oxidase-inhibitor tranylcypromine, case reports showed no relevant changes in vital signs during concurrent S-ketamine use. One paper indicated an interaction between ketamine and haloperidol, 2 other studies did not. Four papers investigated risperidone, including 3 neuroimaging studies showing an attenuating effect of risperidone on ketamine-induced brain perfusion changes. Clozapine significantly blunted ketamine-induced positive symptoms in patients with schizophrenia but not in healthy participants. One paper reported no effect of olanzapine on ketamine’s acute psychotomimetic effects.

Current literature shows that benzodiazepines and probably lamotrigine reduce ketamine’s treatment outcome, which should be taken into account when considering ketamine treatment. There is evidence for an interaction between ketamine and clozapine, haloperidol, and risperidone. Due to small sample sizes, different subject groups and various outcome parameters, the evidence is of low quality. More studies are needed to provide insight into pharmacodynamic interactions with ketamine.

Introduction

Since 2000 the rapid and robust antidepressant effects of ketamine have been reported repeatedly ( Han et al., 2016 ; Kishimoto et al., 2016 ). Ketamine is increasingly being used off label for the treatment of depression in the United States ( Wilkinson and Sanacora, 2017 ) and at a slightly slower pace in Europe ( López-Díaz et al., 2019 ), often as an add-on to other psychiatric medication. Furthermore, the US Food and Drug Administration (FDA) and European Medicines Agency (EMA) recently approved intranasal S-ketamine for treatment-resistant depression (TRD) in conjunction with an oral antidepressant and for depression with imminent risk of suicide ( EMA, 2019 ; FDA, 2019 ). However, to date, there are no clinical practice guidelines recommending the use of S-ketamine in depression ( López-Díaz et al., 2019 ; Malhi et al., 2021 ). The use of ketamine in clinical practice entails different challenges, one of them being the acute and bothersome psychotomimetic side effects such as anxiety, perceptual changes, and dissociation ( Krystal et al., 1994 ). In case of severe agitation or anxiety, clinicians may resort to benzodiazepines or antipsychotics as rescue medication ( Kasper et al., 2020 ). Furthermore, strategies to maintain the antidepressant effects of ketamine are considered a major unmet need ( Papakostas, 2020 ). Ketamine is often combined with other psychiatric drugs and continuation of this psychiatric pharmacotherapy has been proposed to prevent relapse, a strategy that is already proven effective after successful electroconvulsive therapy ( Sackeim et al., 2001 ). Since most patients for whom ketamine treatment is considered are already being prescribed psychiatric drugs, knowledge on pharmacodynamic interactions is crucial.

Ketamine is a noncompetitive N-methyl-d-aspartate (NMDA) glutamate receptor antagonist. In addition, it shows affinity for multiple other receptors ( Mathew et al., 2016 ). Action at the µ, k, and σ receptors may contribute to its analgesic effects. Furthermore, there is inhibition at muscarinic and nicotinic receptors and activity as a cholinesterase inhibitor. Modulation of monoaminergic systems occurs via inhibition of reuptake transporters (dopamine, serotonin, and noradrenaline) and via direct activity at the monoamine receptors (dopamine and serotonin). Inhibition of the NMDA receptor also leads to downstream enhancement of monoaminergic activity ( Mathew et al., 2016 ). The main mechanism of the antidepressive action is believed to stem from antagonism of the NMDA receptor on γ-aminobutyric acid (GABA) releasing interneurons. After a reduction in GABA inhibition, the pyramidal cells in the prefrontal cortex (PFC) release glutamate. This results in enhanced stimulation of α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors, activating a signaling cascade that raises neurotrophic factors and induces synaptic plasticity ( Krystal et al, 1994 ; Mathew et al., 2016 ). Modulation of brain areas induced by ketamine are most notably found in the subgenual anterior cingulate cortex, posterior cingulate cortex, PFC, and hippocampus ( Ionescu et al., 2018 ).

Many trials have studied the efficacy and safety of ketamine treatment as add-on to regular antidepressant agents such as selective serotonin reuptake inhibitors, serotonin-norepinephrine reuptake inhibitors, tricyclic antidepressants, bupropion, mirtazapine, etc. ( Kishimoto et al, 2016 ; Kryst et al., 2020 ; Memon et al., 2020 ). However, other psychiatric drugs are also often used in combination with ketamine. The pharmacodynamic actions of ketamine may provide leads on how to optimize treatment. For instance, hypotheses on synergistic effects of ketamine and lithium through inhibition of glycogen synthase kinase 3 have been proposed ( Ghasemi et al., 2010 ). On the other hand, it can be hypothesized that lamotrigine might influence the effects of ketamine by reducing glutamate transmission as a result of sodium channel blockage. Anecdotal evidence and personal experience suggest that in some patients, lamotrigine might counteract the antidepressant effects of ketamine. A case report by Ford et al. (2015) suggests that higher doses of benzodiazepines attenuate the antidepressant effect of ketamine. While previous data show that ketamine addition to regular antidepressants is safe and accelerates the antidepressant response ( Hu et al., 2016 ; Arabzadeh et al., 2018 ), there are studies presenting enhanced monoamine levels in the rat brain effectuated by ketamine ( Nishimura and Sato, 1999 ; Tso et al., 2004 ). Consequently, monoamine oxidase inhibitors (MAOIs) in combination with ketamine might increase heart rate and blood pressure or cause a serotonergic syndrome. Given the affinity of ketamine for monoamine receptors, interactions may also be expected for ketamine and antipsychotics or psychostimulants. The pharmacological interactions between ketamine and other psychiatric medications commonly used in patients with TRD have not been systematically reviewed.

The aim of this review is to systematically summarize the current knowledge from human studies on combined use of ketamine with psychiatric drugs commonly prescribed to patients with depression.

The electronic databases MEDLINE and Web of Science were searched on July 9, 2020, for studies that examined a pharmacodynamic interaction between ketamine and a psychiatric drug consisting of antidepressants, mood stabilizers, antipsychotics, benzodiazepine(-agonist)s, or psychostimulants. Filters for English, Dutch, and human studies were applied. The search strategy was built as follows: (‘generic name’ OR ‘brand name’) AND (‘ketamine’) including multiple generic and brand names for each drug. The search strategies can be found in the supplementary Material (Appendix 1 ).

We included human studies with a randomized controlled design. In case limited information from controlled trials was available (only 1 or no randomized controlled trials [RCTs]), we included studies with lower levels of evidence (post-hoc analyses from controlled trials and open label studies). Furthermore, case reports were only included if they provided supporting evidence regarding an interaction between ketamine and psychiatric medication. Grey literature (e.g., conference abstracts) was included in the review. Studies that investigated the antidepressant effect of ketamine added to a regular antidepressant (except MAOIs) were excluded because the safety and additive antidepressant effect of ketamine has already been demonstrated irrefutably. Furthermore, studies investigating anesthetic doses of ketamine for induction or sedation were excluded. The references were imported in the reference manager Endnote. After removal of duplicates ( Bramer et al., 2016 ), 2 reviewers (I.B. and B.V.) independently screened the titles and abstracts and full texts. Discrepancies were solved by discussion, and if necessary, a third or fourth reviewer (J.V. or D.T.) assisted to come to an agreement. Systematic reviews were screened for studies that possibly met our criteria and were not found with the literature search. We selected information on the study design, sample size, population characteristics, details on the intervention, clinical outcomes, and hypothesis on the mechanisms of action. The process is shown in Figure 1 . Joanna Briggs Institute’s critical appraisal tools for RCTs, quasi-experimental studies, and case reports were used for quality assessment of the included studies ( JBI, 2020 ). This systematic review was written according to the Preferred Reporting Items for Systematic Reviews and Meta-Analysis statement.

An external file that holds a picture, illustration, etc.
Object name is pyab039f0001.jpg

PRISMA flowchart

Studies Retrieved

In total, 4887 studies were identified through database searching. Screening of the reference lists of systematic reviews did not result in any additional articles. Five additional papers were proposed during the review process of this manuscript. Deduplication resulted in 3845 articles. By screening titles and abstracts, 3723 studies were excluded and 1228 articles remained for full-text assessment. Subsequently, 98 studies were excluded for the following reasons: anesthetic dosages (n = 20), other outcome measures (n = 22), other drugs than ketamine examined (n = 5), animal models (n = 12), a study design that does not allow conclusions on an interaction (n = 25), or for a combination of these reasons (n = 14). Finally, 24 studies met our eligibility criteria and were included in this systematic review ( Figure 1 ). The results of the quality assessment can be found in the supplementary Material (Appendix 2 ).

We included 2 RCTs on lithium and 7 studies on lamotrigine. Furthermore, we retrieved 1 case report, 3 post-hoc analyses, and a placebo-controlled study on the interaction between ketamine and benzodiazepines. We found 2 case reports on concomitant use of ketamine and tranylcypromine. Three double-blind, placebo-controlled studies investigated an interaction with haloperidol. An interaction between risperidone and ketamine was studied in 4 articles (3 were already included for lamotrigine, and these report on different outcome results from the same study). One RCT reported on an interaction between olanzapine and ketamine, and 3 on clozapine and ketamine. No studies were found investigating the concomitant use of ketamine with psychostimulants.

Based on the mechanism of action of lithium and ketamine, one would expect synergistic effects in patients with depression. The study by Costi et al. (2019) included 42 patients with major depressive disorder (MDD) who showed an antidepressant response to ketamine. Patients received 600–1200 mg of lithium or placebo in combination with 0.5 mg/kg intravenous (IV) ketamine. No significant difference was found on the Montgomery Ǻsberg Depression Rating Scale (MADRS) scores between the 2 groups. The study by Xu et al. (2015) investigated 0.5 mg/kg IV ketamine in 36 patients with bipolar TRD that were maintained on a therapeutic dose of lithium (plasma target level of 0.6–1.2 mEq/L) or valproate. Depressive symptoms improved significantly after ketamine in both the lithium and valproate group, and there were no significant differences between the 2 mood stabilizers. Taken together, lithium did not seem to potentiate ketamine’s antidepressant effect in patients with depression ( Table 1 ).

Abbreviations: AD, antidepressant; IV, intravenous; MADRS, Montgomery-Asberg Depression Rating Scale; MDD, major depressive disorder; QIDS-SR, Quick Inventory of Depressive Symptomatology - Self Report; TRD, treatment-resistant depression.

a Time interval uncertain.

b Dosage adjusted to a target blood level in the range of 0.6–0.9 mEq/L.

c Route of administration not reported.

Lamotrigine

Since lamotrigine reduces glutamatergic activity, an impeding interaction with ketamine can be hypothesized. Seven studies reported on the effects of 300 mg lamotrigine prior to IV ketamine administration in dosages ranging from 0.12 to 0.5 mg/kg. Except from the study of Mathew et al. (2010) , which included patients with MDD, these studies investigated the effects in healthy individuals ( Doyle, 1990 ; Anand et al., 2000 ; Deakin et al., 2008 ; Joules et al., 2015 ; Shcherbinin et al., 2015 ; Abdallah et al., 2017 ). Lamotrigine did not block the ketamine-induced psychotomimetic effects measured by the Brief Psychiatric Rating Scale (BPRS) and Clinician-Administered Dissociative States Scale (CADSS) in the study by Abdallah et al. (2017) . Anand et al. (2000) reported that pretreatment with lamotrigine in healthy individuals showed a significant reduction in ketamine-induced Hopkins Verbal Learning Test impairment and a significant decrease in ketamine-induced symptoms measured by the CADSS and BPRS scores. They found an increasing effect on mood elevation after ketamine measured by the Young Mania Rating Scale with lamotrigine. Mathew et al. (2010) observed no attenuated psychotomimetic effects in patients with MDD randomized to lamotrigine prior to ketamine. Deakin et al. (2008) recorded a significant reduction in ketamine-induced total CADSS and BPRS scores when pretreated with lamotrigine. Furthermore, areas showing blood oxygenation level-dependent (BOLD) responses to ketamine assessed by pharmacomagnetic resonance imaging (phMRI) revealed greater responses after placebo infusion than after lamotrigine infusion. In addition, a trial by Doyle et al. (2013) reported that lamotrigine administration resulted in a relatively consistent attenuation of the BOLD response in frontal and thalamic regions effectuated by IV ketamine. Joules et al. (2015) and Shcherbinin et al. (2015) (reporting other outcomes from the same study as Doyle et al.) found no interaction between lamotrigine and ketamine on the degree-centrality pattern and resting state brain perfusion. To summarize, the studies on lamotrigine seem to confirm an antagonistic interaction with ketamine, but the relevance for clinical outcomes is still unclear ( Table 2 ).

Abbreviations: AD, antidepressant; BOLD, blood oxygenation level- dependent; BPRS, Brief Psychiatric Rating Scale; CADSS, Clinician-Administered Dissociative States Scale; GBCr, global brain connectivity with global signal regression; HVLT, Hopkins Verbal Learning Test; IDS-C 30 , Inventory of Depressive Symptomatology - Clinician Rated; IV, intravenous; MADRS, Montgomery-Asberg Depression Rating Scale; MDD, major depressive disorder; NMDA, N-methyl-D-aspartate; TRD, therapy resistant depression; vPFC, ventral prefrontal cortex; YMRS, Young Mania Rating Scale.

b Total duration of infusion not reported.

c Dosage adjusted to a target plasma level of 75 ng/mL in accordance with the subject’s height and weight.

Benzodiazepines

Ford et al. (2015) suggested that benzodiazepines could attenuate the action of ketamine in their case of a patient with bipolar disorder who experienced a severe episode of depression with no response to several antidepressants and antipsychotics. He was treated with lorazepam and responded to repeated doses of ketamine (0.5 mg/kg IV). The antidepressant response was prolonged from 2–3 days to 10–14 days after lorazepam (3.5 mg/d) was withdrawn.

In addition to this case report, we found 4 papers reporting on an interaction between ketamine and benzodiazepines. In a post-hoc analysis of an open label study with patients with TRD, Albott et al. (2015) found no significant difference between 4 benzodiazepine users (mean daily dose 2.75 mg lorazepam equivalents) and 9 non-users in depression response rate (MADRS score ≤ 50%), remission (MADRS < 10) or depression relapse (MADRS ≥ 50% of baseline) after ketamine infusions (0.5 mg/kg/40 min). However, benzodiazepine users showed a significantly longer time to antidepressant response ( P = .029), longer time to depression remission ( P = .042), and a shorter time to depression relapse ( P = .020). In a post-hoc analysis of Frye et al. (2015) , benzodiazepines gave no significant difference in response rate in 10 patients with TRD. Nevertheless, they found that the mean benzodiazepine dose (0.75 mg lorazepam equivalents) was significantly lower ( P = .026) in the responder group (response ≥50% reduction in MADRS scores) than in the non-responder group (3 mg). Andrashko et al. (2019) presented an analysis of 2 randomized controlled cross-over trials with 47 patients with MDD. This study found a significant difference in benzodiazepine dosage between responders (response ≥50% reduction in MADRS scores) and non-responders. Logistic regression analysis showed that benzodiazepine medication (10 mg and more in diazepam equivalent) after ketamine infusion of 0.54 mg/kg predicted nonresponse anytime during 1-week follow-up (odds ratio = 1.5, P = .0445). In a study of Krystal et al. (1998) , lorazepam 2 mg reduced emotional distress, and it tended to reduce distorted sensory perceptions associated with ketamine infusion (0.26 mg/kg bolus, followed by 0.65 mg/kg/h) in 30 healthy individuals. It also reduced the inability to produce a proverb interpretation, perhaps an indication of reduced thought blocking. However, it failed significantly to block psychotogenic, perceptual, cognitive, neuroendocrine, and physiological ketamine responses. In conclusion, these studies indicate that higher doses of benzodiazepines can delay the time to response and remission and shorten the antidepressant effects of ketamine. Furthermore, higher doses of benzodiazepines predicted nonresponse ( Table 3 ).

Abbreviations: AMPA = α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid; ATH = Antidepressant Treatment History; BD II= bipolar disorder type 2; BPRS = Brief Psychiatric Rating Scale; CADSS = Clinician-Administered Dissociative States Scale; GABA(ergic) = γ-aminobutyric acid–ergic; IV = Intravenous; MADRS = Montgomery-Asberg Depression Rating Scale; MDD = major depressive disorder; NMDA(R) = N-methyl-D-aspartate (receptor); SD = standard deviation; TRD = Treatment resistant depression; VAS = Visual Analog mood Scale; WCST = The Wisconsin Card Sorting Test.

Tranylcypromine

No reports were found that confirmed the hypothesized risk of hypertensive crisis or serotonergic syndrome caused by the combination of MAOIs and ketamine. We retrieved 2 case reports about patients with depression receiving tranylcypromine in dosages ranging from 10 to 80 mg daily in combination with S-ketamine ( Bartova et al., 2015 ; Dunner et al., 2019 ). Two patients were administered IV S-ketamine (dosages 12.5–75 mg) while the other patient received intranasal (IN) S-ketamine (dosages 28–56 mg). No relevant cardiovascular or hemodynamic changes were observed ( Table 4 ).

Abbreviations: IV, intravenous; MDD, major depressive disorder; TRD, treatment resistant depression.

a Time interval uncertain

b Total duration of infusion not reported

c Route of administration not reported

Haloperidol

Even though the clinical impact in humans remains unknown, ketamine has shown dopamine-enhancing effects in vitro. The dopamine D 2 receptor antagonist haloperidol might therefore interact with ketamine’s effects. Krystal et al. (1999) reported that in 35 healthy individuals, impairments in executive cognitive functions (assessed with the Wisconsin card sorting test and the Gorham’s Proverb’s test) produced by ketamine infusion, bolus of 0.26 mg/kg followed by 0.65 mg/kg/h, were reduced by haloperidol pretreatment (5 mg). In addition, haloperidol reduced the anxiogenic effects and increased the sedative and prolactin responses to ketamine.

The study of Oranje et al. (2009) found an improving effect of 2 mg of oral haloperidol on ketamine-induced (0.3 mg/kg IV) reduction of processing negativity ( P < .05) in 18 healthy male individuals. Processing negativity is a negative deflection in the event-related potential, representing the ability to focus on 1 source of information, when multiple sources of information are present.

Lahti et al. (1995) found that a fixed high dose of haloperidol (0.3 mg/kg/d) did not blunt ketamine-induced psychosis in 9 patients with schizophrenia receiving ketamine (injections of 0.1, 0.3, and 0.5 mg/kg) while on and off haloperidol. In fact, the patients experienced greater increases in psychosis ratings after ketamine administration during haloperidol treatment compared with a drug-free period. Interestingly, ketamine did not produce a significant worsening in the latter group. In summary, some acute effects of ketamine (impairments in executive cognitive functions and anxiogenic effects) were reduced by haloperidol pretreatment, whereas no blunting of ketamine-induced psychosis was achieved with haloperidol in patients with schizophrenia ( Table 5 ).

Abbreviations: BPRS, Brief Psychiatric Rating Scale; D1/2/5, dopamine receptor 1/2/5;-aspartate receptor; ERP, event-related potential; IV, Intravenous; NMDAR, N-methyl-D; VAS, Visual Analog mood Scale; WCST, The Wisconsin Card Sorting Test.

Risperidone

The strong antiserotonergic and antidopaminergic effects of risperidone might counteract ketamine’s monoaminergic enhancement. For the combined use of risperidone and ketamine, we found 3 phMRI studies and 1 study investigating oculomotor performance. Schmechtig et al. (2013) reported no effect of 2 mg of risperidone on ketamine-induced (IV targeted to a concentration of 100 ng/mL) eye movement changes. In the study of Doyle et al. (2013) in healthy individuals analysis of different regions of interest (ROI) revealed significant positive and negative BOLD responses to ketamine infusion (targeted to a plasma level of 75 ng/mL). Risperidone 2 mg attenuated the ketamine effect across most ROIs, including the medial prefrontal and cingulate regions and the thalamus. For the negatively responding regions in the subgenual cingulate and ventromedial prefrontal cortex, risperidone strongly attenuated the negative BOLD response. In the same sample, Joules et al. (2015) found that ketamine increased the degree centrality, defined as the number of links incident on a node in phMRI. Risperidone pre-treatment significantly modulated the ketamine-induced centrality changes in 20 healthy males. Shcherbinin et al. (2015) performed an analysis of resting brain perfusion in the same individuals. Ketamine showed positive weights (post-ketamine > pre-ketamine) in prefrontal and cingulate regions, thalamus and lateral parietal cortex in comparison with pre-ketamine in phMRI, with strongest negative contributions in the occipital lobes. Pretreatment with risperidone significantly increased the ketamine-induced perfusion changes ( P < .02). To summarize, an interaction between ketamine and risperidone was found based on brain perfusion changes in imaging studies, but no clinical studies have yet investigated whether this would also result in an attenuation of the antidepressant effects ( Table 6 ).

Abbreviations: 5-HT2AR, 5-hydroxytryptamine 2A receptor; AS, antisaccades; BOLD, blood oxygenation level- dependent; D2, dopamine receptor 2; DC, degree centrality; IV, Intravenous; NMDAR, N-methyl-D-aspartate receptor; ORGP, Ordinal regression using Gaussian processes; phMRI, pharmacological magnetic resonance imaging; PS, prosaccades; ROI, Regions of interest; SPEM, smooth pursuit eye movement.

Clozapine has a wide spectrum of pharmacological actions, including blockade of the dopamine D 1 - and serotonergic 5-hydroxytryptamine 2 receptors. Hypotheses on possible interactions are not straightforward. Lipschitz et al. (1997) showed that clozapine 50 mg pretreatment before 0.5 mg IV ketamine did not reduce the BPRS 5 key positive or 3 key negative scores in 7 healthy individuals, but there was a trend for a reduction in perceptual alteration as measured by the CADSS ( P = .09). Clozapine at a mean dose of 430 mg/d for several weeks significantly blunted the positive symptoms induced by ketamine in 10 patients with schizophrenia, but not the negative symptoms as reported by Malhotra et al. (1997) . An imaging study ( Vollenweider et al., 2012 ) with 20 healthy individuals shows that a low dose of clozapine (30 mg oral) reduced some of the S-ketamine (0.006 mg/kg/min) responses. The results of these 3 studies are inconsistent on interactions between ketamine and clozapine ( Table 7 ).

Abbreviations: 5-HT2AR, 5-hydroxytryptamine 2A receptor; APD free, Antipsychotic drug free; BPRS, Brief Psychiatric Rating Scale; CADSS, Clinician-Administered Dissociative States Scale; DSM, Diagnostic and Statistical Manual; H215O-PET, H215O-positron emission tomography; IV, Intravenous; NMDAR, N-methyl-D-aspartate receptor.

The antagonistic effects of olanzapine on serotonin and dopamine receptors might cause an interaction with ketamine. One paper studied an interaction between olanzapine and ketamine. Lahti et al. (1999) investigated ketamine administration (0.3 mg/kg) in 5 healthy individuals treated with a low dose of 5 mg olanzapine and an unknown number of patients with schizophrenia treated with 10 mg. They found no difference in blocking ketamine-induced psychotic symptoms between olanzapine and placebo ( Table 8 ).

Abbreviations: IV, intravenous.

This review summarizes currently available information on interactions between ketamine and other psychiatric drugs, derived from human studies investigating clinical, neuroimaging, and electrophysiological outcomes. Knowledge on potentiating and antagonistic interactions are of utmost importance for the optimization of ketamine treatment for depression.

Mood Stabilizers

We found no evidence for interactions between lithium and ketamine, nor for the hypothesis that lithium would strengthen the antidepressant effect through inhibition of glycogen synthase kinase 3. Three studies ( Anand et al., 2000 ; Deakin et al., 2008 ; Doyle et al., 2013 ) reported attenuating effects of lamotrigine pretreatment on ketamine-induced effects on the BPRS, dissociative symptoms, and BOLD responses. As lamotrigine reduces glutamate transmission by sodium channel blockage, these ketamine responses are thought to result from an increase in glutamatergic activity. Four other studies ( Mathew et al., 2010 ; Joules et al., 2015 ; Shcherbinin et al., 2015 ; Abdallah et al., 2017 ) did not confirm the interaction. This might be explained by differences in dose or infusion speed of ketamine or differences in drug metabolism and functional brain characteristics between TRD patients and healthy controls could contribute to the contrasting results. Interestingly, Anand et al. (2000) described an increasing effect on mood elevation in healthy individuals. These participants may have benefitted from the decrease of dissociative, psychiatric, and learning impairment symptoms, which could have resulted in increased awareness of mood-elevating effects. This, however, could reflect a different mechanism than antidepressant effects in patients with depression.

All papers reporting on benzodiazepines in combination with ketamine indicate an interaction. This could be the result of GABA-receptor agonism by benzodiazepines. By increasing the inhibitory tone of GABAergic interneurons, benzodiazepines might decrease excitatory glutamatergic signal transduction and attenuate the antidepressant effects of ketamine. Moreover, an animal study ( Eintrei et al., 1999 ) has shown a ketamine-induced enhanced metabolism in the limbic system, and this action is selectively blocked by administration of diazepam. Ketamine-induced dopamine release was similarly blocked by diazepam as reported by another animal study ( Irifune et al., 1998 ).

Both included case reports on S-ketamine in combination with tranylcypromine correspond to previous literature stating no relevant cardiovascular or hemodynamic changes ( Doyle, 1990 ; Bartova et al., 2015 ; Dunner et al., 2019 ). Also, no serotonergic syndrome has been reported in the combination of MAOIs and ketamine, but clinical experience is limited. An increase in monoamines is probably not the most relevant pharmacodynamic effect of ketamine.

Antipsychotics

An interaction with the dopamine D 2 receptor antagonist haloperidol can be expected because ketamine shows modest activity at the dopamine transporter at subanesthetic doses. Two studies in healthy individuals reported a reduction in ketamine-induced effects (impairments in executive cognitive functions, anxiogenic effects, and processing negativity) with haloperidol pretreatment. This implies that these effects of ketamine are caused by its (direct or indirect) agonistic effect on dopaminergic D 2 receptor activity. The only study investigating patients with schizophrenia on and off a high dose of haloperidol did not find blunting effects on ketamine-induced psychosis. These results suggest that the mechanism by which ketamine causes psychotic symptoms is not affected by D 2 blockade.

We found no studies investigating the effects of an interaction between risperidone and ketamine on the clinical antidepressant effect or side effects. Risperidone had no effect on ketamine-induced changes in eye movements, which are biomarkers in drug development and the evaluation of treatment effects ( Sweeney et al., 1998 ; Levy et al., 2010 ). However, 3 imaging studies showed attenuating effects of risperidone on ketamine’s brain perfusion changes in healthy individuals ( Doyle et al., 2013 ; Joules et al., 2015 ; Shcherbinin et al., 2015 ). The opposing effects of ketamine and risperidone at the D 2 receptor, similar to the interaction with haloperidol, may play a role here. Furthermore, 5-hydroxytryptamine 2A receptor antagonism by risperidone might also mitigate the effects of ketamine. Although no clear conclusions can be drawn from these case reports, a few examples of concomitant ketamine and risperidone use in patients with depression suggest that use of risperidone does not attenuate ketamine’s antidepressant effects in doses of 1–4 mg/d ( Zigman and Blier, 2013 ; López-Díaz et al., 2017 ; Zheng et al., 2018 ).

The results of 3 studies investigating clozapine and ketamine were inconsistent. No significant interaction was reported in the study investigating a low dose of clozapine (50 mg) on ketamine’s effects on the BPRS and CADSS in healthy individuals. On the other hand, an even lower dose of 30 mg clozapine did reduce some of the S-ketamine responses on neuroimaging outcomes. Furthermore, a higher dose of clozapine (mean 430 mg) did alter ketamine’s positive symptoms in patients with schizophrenia. The differences in participant characteristics, dosages, and ketamine formulation (racemic vs S-ketamine) are reason to compare and interpret the results with caution. Dose-response studies with clozapine show that doses of at least 300–600 mg/d are required to achieve a therapeutic response in patients ( VanderZwaag et al., 1996 ; Simpson et al., 1999 ). Clozapine is a relatively weak antagonist at striatal dopamine D 2 -receptors and produces a more potent blockade of central dopamine D 1 , cholinergic, serotonergic 5HT 2 , histamine H 1 , and adrenergic α 1 and α 2 receptors ( Fitton and Heel, 1990 ).

One paper found no blocking of ketamine induced psychotic symptoms with olanzapine, which corresponds to the results with haloperidol but is in contrast with higher doses of clozapine. Some evidence from uncontrolled trials suggests that concomitant use of olanzapine may attenuate the antidepressant effects of ketamine or S-ketamine. One patient with TRD receiving olanzapine 10 mg (in addition to lithium, mirtazapine, and zopiclone 11.25 mg) did not achieve response or remission (Hamilton Depression Rating Scale scores decreased from 19 to 11) after 6 infusions 0.25 mg/kg S-ketamine ( Segmiller et al., 2013 ). The open label study of Zheng et al. (2018) reports olanzapine use (2.5–20 mg/d) in 22.7% of responders and 25.8% of non-responders to 6 ketamine infusions of 0.5 mg/kg in patients with depression. It should be noted that other causes of reduced antidepressant effect cannot be excluded in these reports.

Limitations

The present study should be interpreted in light of some limitations. First, our review contains multiple conference abstracts (n = 4) that have not been published in peer-reviewed full text. Second, the studies have small sample sizes, ranging from 1 to 72 participants. As the different outcomes in patients and healthy individuals of the clozapine studies show, the underlying mechanism of the interaction may be different in patients than in healthy individuals. In addition, there may be a concentration-dependent effect regarding interactions. Many studies included in this review used ketamine as a model for psychosis. In these studies, the focus is not necessarily on exploring an interaction with respect to the antidepressant effect of ketamine. Furthermore, results were based on various intervention methods (racemic vs S-ketamine and different dosages, intermittent/bolus administration vs continuous infusion) and outcome measures, which complicates comparison of the trials. Finally, the case reports gave new insights but have to be interpreted with great caution due to limited generalizability.

In conclusion, this systematic review provides new insights for the clinical practitioner. The pharmacodynamic interactions between ketamine and generally prescribed psychiatric drugs were analyzed based on published evidence. The literature provided no reports of an interaction between lithium and ketamine. Some but not all studies on lamotrigine provided evidence for attenuation of ketamine-induced effects. Based on the available literature, it is very likely that benzodiazepines shorten the duration of ketamine’s antidepressant effects. This should be taken into account when considering ketamine treatment. In addition, several cases suggested that no relevant changes in vital signs occurred when ketamine was combined with the MAOI tranylcypromine. There are indications for an interaction between the antipsychotic drugs haloperidol, risperidone, and clozapine but not evidently for olanzapine in combination with ketamine. No information was found yet about ketamine’s possible interactions with psychostimulants.

Clinical Implications and Future Directions

Clinical practitioners should be aware of possible drug-to-drug interactions when prescribing ketamine. To optimize treatment effects, we would recommend minimizing benzodiazepine (and Z-drug) use in patients receiving ketamine for depression. Furthermore, a possible interaction with lamotrogine should be considered when patients show a lack of response to ketamine during concomitant lamotrigine use. We believe prescribing ketamine as add-on to lower doses of MAOI is possible but only with careful monitoring of vital signs.

For future studies, we recommend trials focusing on the antidepressant efficacy and side effects of the combination of ketamine and psychiatric drugs in patients with depression. It would, for instance, be worthwhile to investigate whether lower doses of benzodiazepines, or temporary benzodiazepine withdrawal before ketamine administration, could prevent an efficacy attenuating interaction. Additional research on the safety of combining ketamine and MAOIs is warranted. Furthermore, we suggest that researchers and clinicians systematically record and report the use of co-medication in ketamine trials to detect possible interactions.

Supplementary Material

Pyab039_suppl_supplementary_material_1, pyab039_suppl_supplementary_material_2, pyab039_suppl_supplementary_material_3, author contributions.

JKEV: provided the conception and design of the study, acquisition of data, analysis and interpretation of data, drafted the article, revised it critically for important intellectual content, final approval of the version to be submitted, and agreement to be accountable for all aspects of the work. SYSA: interpretation of data, critical revision for important intellectual content, final approval of the version to be submitted and agreement to be accountable for all aspects of the work. IMB: acquisition of data, analysis and interpretation of data, drafted the article, final approval of the version to be submitted and agreement to be accountable for all aspects of the work. BAEV: acquisition of data, analysis and interpretation of data, drafted the article, final approval of the version to be submitted and agreement to be accountable for all aspects of the work. JK: interpretation of data, critical revision for important intellectual content, final approval of the version to be submitted and agreement to be accountable for all aspects of the work. RAS: interpretation of data, critical revision for important intellectual content, final approval of the version to be submitted and agreement to be accountable for all aspects of the work. DJT: provided the conception and design of the study, analysis and interpretation of data, critical revision for important intellectual content, final approval of the version to be submitted and agreement to be accountable for all aspects of the work.

Statement of Interest

J.K.E. Veraart received a speakers fee from Janssen Pharmaceuticals, outside the submitted work. R.A. Schoevers received research funding for 2 randomized clinical trials with generic oral esketamine from the Netherlands Organisation for Health Research & Development and the National Health Care Institute, a speakers fee from Janssen Pharmaceuticals, and consultancy fee from Clexio biosciences, both outside the submitted work. D.J. Touw received research grants from ZONMW and Chiesi Pharmaceuticals outside this work. He is a member of the advisory boards from Sanquin and PureIMS outside this work. S.Y. Smith-Apeldoorn, I.M. Bakker, B.A.E. Visser, and J. Kamphuis report no competing interests.

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Calculate Non Linear Relationships such as Exponential, Logarithmic, Quadratic and Cubic using Python

By: Hristo Hristov | Updated: 2023-05-05 | Comments | Related: > Python

Considering two variables, we want to determine to what extent they are correlated. There are two types of correlation analysis depending on how the two variables relate: linear and non-linear. How do we check if two variables are non-linearly correlated? How do we measure and express their degree of correlation?

Two variables are non-linearly correlated when their dependence cannot be mapped to a linear function. In a non-linear correlation, one variable increases or decreases with a variable ratio relative to the other. This behavior contrasts with linear dependence, where the correlation between the variables maps to a linear function and is constant. Therefore, in a linear dependence, you see a straight line on the scatter plot, while with a non-linear dependence, the line is curved upwards or downwards or represents another complex shape. Examples of non-linear relationships are exponential, logarithmic, quadratic, or cubic.

Like the last experiment with the linear correlation, we will use a dataset containing machining data but focus on different variables. You can download it here . Here is a preview of the first five out of the total ten thousand rows:

One of the easiest ways to examine visually multivariate data is to generate a pair plot. This type of plot creates a scatter plot for each pair of numeric variables. This is how we can identify a pair that looks non-linearly correlated. A pair plot is easy to generate with the seaborn package. Let's select only the numerical features of interest:

pariplot generated with the seaborn package

Now let's focus on the relation between Rotational speed in revolutions per minute and torque in Newton meters. The shape of the dot cloud is not a straight line, i.e., the dependency between these two vectors is non-linear for the most part.

Non-linear Correlation

There are many types of non-linear relationships: quadratic, cubic, exponential, or logarithmic. In our example, we can say that as the rotational speed increases, the torque decreases and vice-versa. According to the scatter plot, this relationship appears to be non-linear, i.e., maps to a non-linear function. To calculate the correlation coefficient, we should experiment with methods for calculating non-linear relationships. However, note that our focus here is to express the strength of the relationship rather than to find the closest function that describes the relations. Therefore, doing a regression analysis, which tries to approximate a function close to reality, is out of the scope of this tip.

First, let's define our variables:

Next, let's examine various methods for calculating the non-linear correlation coefficient.

Distance Correlation

This metric can be applied to both linear and non-linear data. Its advantages are that it does not assume the normality of the input vectors, and the presence of outliers has a reduced influence on it. The results range from 0 to 2, where 0 means perfect correlation and 2 means perfect negative correlation.

The result is 1.85, so as expected, there is a strong reverse dependence between torque and rotational speed:

Mutual Information

Mutual information (MI) between two random variables is a non-negative value ranging from 0 to +∞. Results around 0 mean the variables are independent. On the other hand, the higher the value, the more interdependent the two variables are. MI is called this because it quantifies the information the two variables share. However, results can be difficult to interpret because there is no higher bound to the max MI possible. For example:

Let's break it down:

To calculate MI, we need the scikit learn package, so we import it on line 1.
On line 3, we define a function accepting two vectors and returning a list of floats, the MI coefficient for each pair (here just one pair)
On lines 4 and 5, we convert the pd.Series objects to numpy arrays.
Finally, on line 6, we calculate MI.

The result is:

Again, we get a strong correlation, considering the higher bound is non-existent.

Kendall's Tau

Kendall rank correlation coefficient, or Kendall's τ coefficient, is a statistic used to measure the dependence between two ordinal values. Kendall's tau is a useful measure of dependence in cases where the data is not normally distributed or where outliers may be present. While our variables are not ordinal, we can still use Kendall's coefficient like we used Spearman's to measure linear dependence. The coefficient ranges from -1 to 1.

The result is -0.75, meaning a strong negative correlation. The probability value ( p_value ) of 0 indicates we can reject the null hypothesis of an absence of association (where tau would be 0).

Maximal Information Coefficient

Finally, we can also calculate the maximal information coefficient (MIC). As we have shown previously, this robust correlation measure applies equally well to both linearly and non-linearly correlated data. The coefficient ranges between 0 and 1. Therefore, it is not helpful in showing the direction of the dependence.

This article examined four methods for calculating the correlation coefficient between non-linearly correlated vectors: distance correlation, mutual information, Kendall's tau, and Mutual Information Coefficient. Each has different bounds and captures the relationship between the variables differently. Therefore, using more than one to corroborate or reject a certain theory is standard practice.

Distance correlation
Mutual info regression
Kendall's tau
Calculating MIC

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linearHypothesis function
vcov. a function for estimating the covariance matrix of the regression coefficients, e.g., hccm, or an estimated covariance matrix for model. See also white.adjust. For the "lmList" and "nlsList" methods, vcov. must be a function (defaulting to vcov) to be applied to each model in the list. coef.
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Learn how to perform hypothesis testing for a linear model's coefficients using the linearHypothesis() function in R. See examples of testing individual predictors, comparing nested models, and checking assumptions.
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This is a beginner's guide to applied econometrics using the free statistics software R. PoE with R. 1 Introduction. 1.1 The RStudio Screen. 1.1.1 The Script, or data view window; 1.1.2 The console, or output window; ... First, the linearHypothesis() function creates an $R$ object that contains several items, one of which is the \(F ...
linear.hypothesis function
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Function for testing a linear hypothesis. Along with metalm (or others depending on the class, e.g. "lm" ), it can be used to carry out meta-regression, meta-comparisons, analyses of variance and covariance, and etcetera. RDocumentation. Learn R. Search all packages and functions ...
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The table generated by the linearHypothesis() function shows the same values of the $F$-statistic and $p$-value that we have calculated before, as well as the residual sum of squares for the restricted and unrestricted models.Please note how I formulate the joint hypothesis as a vector of character values in which the names of the variables perfectly match those in the unrestricted model.
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a = rnorm(100) b = rnorm(100) c = rnorm(100) y = 4 + 1*a + 3*b + 0.5*c + 2*a*b + rnorm(100) mod = lm(y ~ a:b + a + b + c) >car::linearHypothesis(mod, c("a","a:b")) Linear hypothesis test Hypothesis: a = 0 a:b = 0 Model 1: restricted model Model 2: y ~ a:b + a + b + c Res.Df RSS Df Sum of Sq F Pr(>F) 1 97 657.20 2 95 116.31 2 540.88 220.89 < 2.2 ...
linearHypothesis function in R and interaction terms
ZR8. 1. 1. An asterisk can be used to indicate both the main effects and the corresponding interaction term in an R formula, but it can't be used for car 's linearHypothesis. Use a colon instead: "index1 + index1:gender = 0". - Daeyoung. Mar 14, 2022 at 19:28.
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a fitted object of type systemfit. matrix (or vector) giving linear combinations of coefficients by rows, or a character vector giving the hypothesis in symbolic form (see documentation of linearHypothesis in package "car" for details). optional right-hand-side vector for hypothesis, with as many entries as rows in the hypothesis matrix; if ...
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Now let's focus on the relation between Rotational speed in revolutions per minute and torque in Newton meters. The shape of the dot cloud is not a straight line, i.e., the dependency between these two vectors is non-linear for the most part.

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