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07-metaregression.Rmd
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# Meta-Regression
Conceptually, **Meta-Regression** does not differ much from a **subgroup analysis**. In fact, subgroup analyses with more than two groups are nothing more than a meta-regression with categorial covariates. However, meta-regression does also allow us to use **continuous data** as covariates and check weather values of this variable are associated with effect size.
<<<<<<< HEAD
**The idea behind meta-regression**
You may have already performed regressions in regular data where participants or patients are the **unit of analysis**. In typical Meta-Analyses, we do not have the individual data for each participant available, but only the **aggregated effects**, which is why we have to perform meta-regressions with covariates at the **study level**. This also means that while we conduct analyses on participant samples much larger than usual in single studies, it is still very likely that **we don't have enough studies for a meta-regression to be sensible**. In [Chapter 7](#subgroup), we told you that subgroup analyses make no sense when *k*<10. For **meta-regression**, Borenstein and colleages [@borenstein2011] recommend that **each covariate should at least contain ten studies**, although this should not be seen as clear rule. In a conventional regression, we want to estimate a parameter $y$ using a covariate $x_i$ with $n$ regression coefficients $\beta$. A standard regression equation therefore looks like this:
=======
```{block,type='rmdinfo'}
**The idea behind meta-regression**
You may have already performed regressions in regular data where participants or patients are the **unit of analysis**. In typical Meta-Analyses, we do not have the individual data for each participant available, but only the **aggregated effects**, which is why we have to perform meta-regressions with covariates at the **study level**. This also means that while we conduct analyses on participant samples much larger than usual in single studies, it is still very likely that **we don't have enough studies for a meta-regression to be sensible**. In [Chapter 7](#subgroup), we told you that subgroup analyses make no sense when *k*<10. For **meta-regression**, Borenstein and colleages [@borenstein2011] recommend that **each covariate should at least contain ten studies**, although this should not be seen as clear rule.
In a conventional regression, we want to estimate a parameter $y$ using a covariate $x_i$ with $n$ regression coefficients $\beta$. A standard regression equation therefore looks like this:
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$$y=\beta_0 + \beta_1x_1 + ...+\beta_nx_n$$
In a meta-regression, we want to estimate the **effect size** $\theta$ for different values of the covariate(s), so our regression looks like this:
$$\hat \theta_k = \theta + \beta_1x_{1k} + ... + \beta_nx_{nk} + \epsilon_k + \zeta_k$$
<<<<<<< HEAD
You might have seen that when estimating the effect size $\theta_k$ of a study $k$ in our regression model, there are two **extra terms in the equation**, $\epsilon_k$ and $\zeta_k$. The same terms can also be found in the equation for the random-effects-model in [Chapter 4.2](#random). The two terms signify two types of **independent errors** which cause our regression prediction to be **imperfect**. The first one, $\epsilon_k$, is the sampling error through which the effect size of the study deviates from its "true" effect. The second one, $\zeta_k$, denotes that even the true effect size of the study is only sampled from **an overarching distribution of effect sizes** (see the Chapter on the [Random-Effects-Model](#random) for more details). In a **fixed-effect-model**, we assume that all studies actually share the **same true effect size** and that the **between-study heterogeneity** $\tau^2 = 0$. In this case, we do not consider $\zeta_k$ in our equation, but only $\epsilon_k$. As the equation above has includes **fixed effects** (the $\beta$ coefficients) as well as **random effects** ($\zeta_k$), the model used in meta-regression is often called **a mixed-effects-model**. Mathematically, this model is identical to the **mixed-effects-model** we described in [Chapter 7](#subgroup) where we explained how **subgroup analyses** work.Indeed **subgroup analyses with more than two subgroups** are nothing else than a **meta-regression** with a **categorical predictor**. For meta-regression, these subgroups are then **dummy-coded**, e.g.
=======
You might have seen that when estimating the effect size $\theta_k$ of a study $k$ in our regression model, there are two **extra terms in the equation**, $\epsilon_k$ and $\zeta_k$. The same terms can also be found in the equation for the random-effects-model in [Chapter 4.2](#random). The two terms signify two types of **independent errors** which cause our regression prediction to be **imperfect**. The first one, $\epsilon_k$, is the sampling error through which the effect size of the study deviates from its "true" effect. The second one, $\zeta_k$, denotes that even the true effect size of the study is only sampled from **an overarching distribution of effect sizes** (see the Chapter on the [Random-Effects-Model](#random) for more details). In a **fixed-effect-model**, we assume that all studies actually share the **same true effect size** and that the **between-study heterogeneity** $\tau^2 = 0$. In this case, we do not consider $\zeta_k$ in our equation, but only $\epsilon_k$.
As the equation above has includes **fixed effects** (the $\beta$ coefficients) as well as **random effects** ($\zeta_k$), the model used in meta-regression is often called **a mixed-effects-model**. Mathematically, this model is identical to the **mixed-effects-model** we described in [Chapter 7](#subgroup) where we explained how **subgroup analyses** work.
Indeed **subgroup analyses with more than two subgroups** are nothing else than a **meta-regression** with a **categorical predictor**. For meta-regression, these subgroups are then **dummy-coded**, e.g.
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$$ D_k = \{\begin{array}{c}0:ACT \\1:CBT \end{array}$$
$$\hat \theta_k = \theta + \beta x_{k} + D_k \gamma + \epsilon_k + \zeta_k$$
In this case, we assume the same **regression line**, which is simply "shifted" **up or down for the different subgroups** $D_k$.
<<<<<<< HEAD
=======
```
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```{r, echo=FALSE, fig.width=6,fig.cap="Visualisation of a Meta-Regression with dummy-coded categorial predictors",fig.align='center'}
library(png)
library(grid)
img <- readPNG("dummy.PNG")
grid.raster(img)
```
<<<<<<< HEAD
$~$
=======
<br><br>
```{block,type='rmdinfo'}
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**Assessing the fit of a regression model**
To evaluate the **statistical significance of a predictor**, we a **t-test** of its $\beta$-weight is performed.
$$ t=\frac{\beta}{SE_{\beta}}$$
<<<<<<< HEAD
Which provides a $p$-value telling us if a variable significantly predicts effect size differences in our regression model. If we fit a regression model, our aim is to find a model **which explains as much as possible of the current variability in effect sizes** we find in our data. In conventional regression, $R^2$ is commonly used to quantify the **goodness of fit** of our model in percent (0-100%). As this measure is commonly used, and many researchers know how to to interpret it, we can also calculate a $R^2$ analog for meta-regression using this formula:
=======
Which provides a $p$-value telling us if a variable significantly predicts effect size differences in our regression model.
If we fit a regression model, our aim is to find a model **which explains as much as possible of the current variability in effect sizes** we find in our data.
In conventional regression, $R^2$ is commonly used to quantify the **goodness of fit** of our model in percent (0-100%). As this measure is commonly used, and many researchers know how to to interpret it, we can also calculate a $R^2$ analog for meta-regression using this formula:
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$$R_2=\frac{\hat\tau^2_{REM}-\hat\tau^2_{MEM}}{\hat\tau^2_{REM}}$$
Where $\hat\tau^2_{REM}$ is the estimated total heterogenetiy based on the random-effects-model and $\hat\tau^2_{REM}$ the total heterogeneity of our mixed-effects regression model.
<<<<<<< HEAD
=======
```
<br><br>
---
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## Calculating meta-regressions in R
Meta-regressions can be conducted in R using the `metareg` function in `meta`. To show the similarity between `subgroup` analysis and `meta-regression` with categorical predictors, i'll first conduct a meta-regression with my variable "Control" as predictor again.
<<<<<<< HEAD
$~$
=======
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```{r}
metareg(m.hksj,Control)
```
<<<<<<< HEAD
$~$
We see in the output that the `metareg` function uses the values of "Control" (i.e, the three different types of control groups) as a **moderator**. It takes **"information only"** as a dummy-coded *reference group*, and **"no intervention"** and **"WLC"** as dummy-coded **predictors**. Under `Test of Moderators`, we can see that control groups are not significantly associated with effect size differences $F_{2,15}=0.947$, $p=0.41$. Our regression model does not explain any of the variability in our effect size data ($R^2=0\%$). Below `Model Results`, we can also see the $\beta$-values (`estimate`) of both predictors, and their significance level `pval`. As we can see, both predictors were not significant.
$~$
**Continuous variables**
Let's assume i want to check if the **publication year** is associated with effect size. I have stored the variable `pub_year`, containing the publication year of every study in my dataset, and conducted the meta-analysis with it. I stored my meta-analysis output in the `m.pubyear` output. Now, i can use this predictor in a meta-regression.
$~$
=======
We see in the output that the `metareg` function uses the values of "Control" (i.e, the three different types of control groups) as a **moderator**. It takes **"information only"** as a dummy-coded *reference group*, and **"no intervention"** and **"WLC"** as dummy-coded **predictors**. Under `Test of Moderators`, we can see that control groups are not significantly associated with effect size differences $F_{2,15}=0.947$, $p=0.41$. Our regression model does not explain any of the variability in our effect size data ($R^2=0\%$).
Below `Model Results`, we can also see the $\beta$-values (`estimate`) of both predictors, and their significance level `pval`. As we can see, both predictors were not significant.
<br><br>
**Continuous variables**
Let's assume i want to check if the **publication year** is associated with effect size. I have stored the variable `pub_year`, containing the publication year of every study in my dataset, and conducted the meta-analysis with it. I stored my meta-analysis output in the `m.pubyear` output.
**Now, i can use this predictor in a meta-regression.**
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```{r,echo=FALSE}
load("Meta_Analysis_Data.RData")
madata<-Meta_Analysis_Data
madata$pub_year<-c(2001,2002,2011,2013,2013,2014,1999,2018,2001,2002,2011,2013,2013,2014,1999,2018,2003,2005)
madata$pub_year<-as.numeric(madata$pub_year)
m.pubyear<-metagen(TE,seTE,studlab = paste(Author),comb.fixed = FALSE,data=madata)
```
```{r}
output.metareg<-metareg(m.pubyear,pub_year)
output.metareg
```
<<<<<<< HEAD
$~$
As you can see from the output, `pub_year` was now included as a **predictor**, but it is not significantly associated with the effect size ($p=0.9412$).
=======
As you can see from the output, `pub_year` was now included as a **predictor**, but it is not significantly associated with the effect size ($p=0.9412$).
<br><br>
---
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## Plotting regressions
```{r, echo=FALSE, fig.width=4,fig.cap="A finished bubble plot",fig.align='center'}
library(png)
library(grid)
img <- readPNG("metareg.PNG")
grid.raster(img)
```
To plot our meta-regression output, we can use the `bubble` function in `meta`. Here a few parameters we can specify for this function.
```{r,echo=FALSE}
i<-c("xlim","ylim","xlab","ylab","col","lwd","col.line","studlab")
ii<-c("The x-axis limit of the plot. Must be specified as, e.g., 'xlim=c(0,1)'","The y-axis limit of the plot. Must be specified as, e.g., 'ylim=c(0,1)'","The label for the x axis","The label for the y axis","The color of the individual studies","The line width of the regression line","The color of the regression line","If the labels for each study should be printed within the plot (TRUE/FALSE)")
ms<-data.frame(i,ii)
names<-c("Parameter", "Function")
colnames(ms)<-names
kable(ms)
```
<<<<<<< HEAD
$~$
=======
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```{r,eval=FALSE}
bubble(output.metareg,
xlab = "Publication Year",
col.line = "blue",
studlab = TRUE)
```
<<<<<<< HEAD
=======
<br><br>
---
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