Meta-Analysis: Meta-Regression

#math #datascience #statistics #tutorial

Adapted from an appendix of my MS thesis.

Meta-Regression

Usually, regression models are based on data comprising individual persons or specimens, for which both the independent $x$ and dependent $y$ values are measured. In meta-regression, this concept is applied to entire studies. The variable $x$ represents characteristics of studies, for example, the year in which it was conducted. Based on this information, a meta-regression model tries to predict $y$ , the study’s effect size [1].

In a conventional regression, we want to estimate the value $y_ i$ of a person or specimen $i$ using a predictor (or covariate) $x_ i$ with a regression coefficient $\beta$ [1].

y_ i = \beta_ 0 + \beta_ 1 x_ i.

In meta-regression, the variable $y$ we want to predict is the observed effect size $\hat{\theta}_ k$ of study $k$ [1].

\hat{\theta}_ k = \theta + \beta x_ k + \epsilon_ k + \zeta_ k.

This formula contains two extra terms, $\epsilon_ k$ and $\zeta_ k$ . The same terms found in the random-effects model from the equation. The first one, $\epsilon_ k$ , is the sampling error through which the effect size of a study deviates from its true effect. The second error, $\zeta_ k$ , denotes that even the true effect size of the study is only sampled from an overarching distribution of effect sizes. Since the meta-regression formula includes a fixed effect (the $\beta$ coefficient) as well as a random effect ( $\zeta_ k$ ), the model used in meta-regression is often called a mixed-effects model [1].

Meta-Regression with a Categorical Predictor

Subgroup analysis (from the companion Subgroup Analysis post) is nothing else than a meta-regression with a categorical predictor, where such categorical variables can be included through dummy-coding, such as the following [1].

D_ g = \begin{cases} 0: &\text{Subgroup A} \\ 1: &\text{Subgroup B} \\ \end{cases}

To specify a subgroup analysis in the form of a meta-regression, we simply have to replace the covariate $x_ k$ with $D_ g$ [1].

\hat{\theta}_ k = \theta + \beta D_ g + \epsilon_ k + \zeta_ k.

Meta-Regression with a Continuous Predictor

When people speak of a meta-regression, they usually think of models in which a continuous variable is used as the predictor. The aim of the meta-regression model is to find values of $\theta$ and $\beta$ which minimize the difference between the predicted effect size and the true effect size of studies. Taking into account both the sampling error $\epsilon_ k$ and between-study heterogeneity $\zeta_ k$ , meta-regression tried to find a model that generalizes well, not only to the observed effect sizes but to the universe of all possible studies of interest [1].

An important detail about meta-regression models is that they can be seen as an extension of the random-effects model. The random-effects model is nothing but a meta-regression without a slope term. Since it contains no slope, the random-effects model simply predicts the same value for all studies. That is, the estimate of the pooled effect size $\mu$ , which is equivalent to the intercept [1].

In the first step, the calculation of a meta-regression therefore closely resembles the one of a random-effects meta-analysis, in that the between-study heterogeneity $\tau^ 2$ is estimated. In the next step, the fixed weights $\theta$ and $\beta$ are estimated. In meta-regression, weighted least squares (WLS) is used to find the regression line that fits the data best. WLS makes sure that studies with a smaller standard error are given a higher weight [1].

If the meta-regression model fits the data well, the true effect sizes should deviate less from the regression line compared to the pooled effect $\hat{\mu}$ . If this is the case, the predictor $x$ explains some of the heterogeneity variance in our meta-analysis. The fit of the meta-regression model can therefore be assessed by checking how much of the heterogeneity variance it explains [1].

The predictors included in the mixed-effects model should minimize the amount of residual heterogeneity variance denoted by $\hat{\tau}_ \text{MEM}^ 2$ . The $R_ \ast^ 2$ index is used to quantify the percentage of variation explained by the meta-regression model. The asterisk is added to differentiate from the $R^ 2$ index used in conventional regressions, since meta-regression deals with true effect sizes instead of observed data points [1].

$R_ \ast^ 2$ uses the amount of residual heterogeneity variance that even the meta-regression slope cannot explain. It is put in relation to the total heterogeneity that we initially found in our random-effects model $\hat{\tau}_ \text{REM}^ 2$ . Subtracting this fraction from 1 leaves us with the percentage of between-study heterogeneity explained by the predictor, and is expressed in the equation. We can also says that it is how much the mixed-effects model has reduced the heterogeneity variance compared to the initial random-effects pooling model, in percent [1].

R_ \ast^ 2 = 1 - \frac{\hat{\tau}_ \text{MEM}^ 2}{\hat{\tau}_ \text{REM}^ 2} = \frac{\hat{\tau}_ \text{REM}^ 2-\hat{\tau}_ \text{MEM}^ 2}{\hat{\tau}_ \text{REM}^ 2}.

Usually, we are not only in the amount of heterogeneity explained by the regression model but also if the regression weight of our predictor $x$ is significant. Both in conventional and meta-regression, the significance of a regression weight is commonly assessed through a Wald test. This involves calculating the test statistic $z$ , by dividing the estimate of $\beta$ through its standard error [1].

z = \frac{\beta}{SE_ {\hat{\beta}}}.

Under the null hypothesis that $\beta=0$ , this $z\text{-statistic}$ follows a standard normal distribution. This allows us to calculate a corresponding $p\text{-value}$ . Like in normal meta-analysis models, we can also use the Knapp-Hartung adjustment, which results in a test statistic based on the $t\text{-distribution}$ [1].