Heterogeneity Plots

In metagam: Meta-Analysis of Generalized Additive Models

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.

Simulation

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
})

Fit GAMs

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)  
})

Meta-Analysis

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")

References

metagam documentation built on May 31, 2023, 6:43 p.m.