The metagam package offers a way to visualize the heterogeneity of the estimated smooth functions over the range of explanatory variables. This will be illustrated here.

`library("metagam")`

We start by simulating 5 datasets using the `gamSim()`

function from mgcv. We use the response \(y\) and the explanatory variable \(x_{2}\), but add an additional shift \(\beta x_{2}^{2}\) where \(\beta_{2}\) differs between datasets,
yielding heterogeneous data.

```
library("mgcv")
#> Loading required package: nlme
#> This is mgcv 1.8-42. For overview type 'help("mgcv-package")'.
set.seed(1233)
<- c(0, .5, 1, 0, -1)
shifts <- lapply(shifts, function(x) {
datasets ## Simulate data
<- gamSim(scale = .1, verbose = FALSE)
dat ## Add a shift
$y <- dat$y + x * dat$x2^2
dat## Return data
dat })
```

Next, we analyze all datasets, and strip individual participant data.

```
<- lapply(datasets, function(dat){
models <- gam(y ~ s(x2, bs = "cr"), data = dat)
b strip_rawdata(b)
})
```

Next, we meta-analyze the models. Since we only have a single smooth
term, we use `type = "response"`

to get the response
function. This is equivalent to using `type = "iterms"`

and
`intercept = TRUE`

.

`<- metagam(models, type = "response") meta_analysis `

Next, we plot the separate estimates together with the meta-analytic fit. We see that dataset 3, which had a positive shift \(\beta=1 x_{2}^2\), lies above the others for \(x_{2}\) close to 1, and opposite for dataset 5.

`plot(meta_analysis, legend = TRUE)`

We can investigate this further using a heterogeneity plot, which visualizes Cochran’s Q-test (Cochran (1954)) as a function of \(x_{2}\). By default, the test statistic (Q), with 95 % confidence bands, is plotted. We can see that the confidence band for Q is above 0 for \(x_{2}\) larger than about 0.7.

`plot_heterogeneity(meta_analysis)`

We can also plot the \(p\)-value of Cochran’s Q-test. The dashed line shows the value \(0.05\). The \(p\)-value plot is in full agreement with the Q-statistic plot above: There is evidence that the underlying functions from each dataset are different for values from about 0.7 and above.

`plot_heterogeneity(meta_analysis, type = "p")`

Cochran, William G. 1954. “The Combination of Estimates from
Different Experiments.” *Biometrics* 10 (1): 101.