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.
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 heterogenous data.
library(mgcv)
#> Loading required package: nlme
#> This is mgcv 1.8-31. For overview type 'help("mgcv-package")'.
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.
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
.
Next, we plot the separate estimates together with the meta-analytic fit. We clearly 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.
The plotting function for the meta-analysis object is a ggplot2-object, and can thus be altered using standard ggplot syntax. To learn more about customization of ggplot2-objects, please see the ggplot2 documentation.
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.
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.
Cochran, William G. 1954. “The Combination of Estimates from Different Experiments.” Biometrics 10 (1): 101.