knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )

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") set.seed(1233) shifts <- c(0, .5, 1, 0, -1) datasets <- lapply(shifts, function(x) { ## Simulate data dat <- gamSim(scale = .1, verbose = FALSE) ## Add a shift dat$y <- dat$y + x * dat$x2^2 ## Return data dat })

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

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

.

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

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 (@Cochran1954) 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")

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