mgcViz: visual tools for Generalized Additive Models

This R package offers visual tools for Generalized Additive Models (GAMs). Most of the tools provided by mgcViz fall in one of the following categories:

1. Layered smooth effect plots;

2. Traditional and layered model checks;

The layering system has been implemented by wrapping several ggplot2 layers and integrating them with computations specific to GAM models. All methods are meant to work with large datasets ($n \approx 10^7$), by adopting discretization and/or sub-sampling.

1 Layered smooth effect plots

Here we start by describing smooth-specific plotting methods and then we move to the new plot.gam function, which wraps several plots together.

1.1 Smooth-specific plots

Let's start with a simple example with two smooth effects:

library(mgcViz)
n  <- 1e3
x1 <- rnorm(n)
x2 <- rnorm(n)
dat <- data.frame("x1" = x1, "x2" = x2,
                  "y" = sin(x1) + 0.5 * x2^2 + pmax(x2, 0.2) * rnorm(n))
b <- gam(y ~ s(x1)+s(x2), data = dat, method = "REML")

Now we convert the fitted object to the gamViz class. Doing this allows to save quite a lot of time when producing multiple plots using the same fitted GAM model.

b <- getViz(b)

We extract the first smooth component using the sm function and we plot it. The resulting o object contains, among other things, a ggplot object. This allows us to add several layers.

o <- plot( sm(b, 1) )
( o <- o + l_fitLine(colour = "red") + l_rug(mapping = aes(x=x, y=y), alpha = 0.8) )

We added only the fitted smooth effect and rugs on the x and y axes. Now we add confidence lines at 1 and 5 standard deviations, partial residual points and we change the theme to ggplot2::theme_classic.

o + l_ciLine(mul = 1) + 
    l_ciLine(mul = 5, colour = "blue", linetype = 2) + 
    l_points(shape = 19, size = 1, alpha = 0.1) + 
    theme_classic()

Functions such as l_fitLine or l_rug provide smooth-specific layers. To see all the layers available for each smooth effect plot we can do:

listLayers(o)

Similar methods exist for 2D smooth effect plots, for instance if we fit:

b <- gam(y ~ s(x1, x2), data = dat, method = "REML")
b <- getViz(b)

we can do

plot(sm(b, 1)) + l_fitRaster() + l_fitContour() + l_points()

This can be converted to an interactive plotly plot as follows:

library(plotly)
ggplotly( plot(sm(b, 1)) + l_fitRaster() + l_points() + l_fitContour()  )

If needed, we can convert a gamViz object back to its original form by doing:

b <- getGam(b)
class(b)

1.2 The new plot.gam method

The new plot.gam function masks mgcv::plot.gam when mgcViz is loaded. This function wraps together the plotting methods related to each specific smooth effect, which can save time when doing GAM modelling. Consider this model:

dat <- gamSim(1,n=1e3,dist="normal",scale=2)
b <- gam(y~s(x0)+s(x1, x2)+s(x3), data=dat)

To plot all the effects we do:

b <- getViz(b)
plot(b)         # Calls print.plotGam()

Here getViz is not strictly necessary, but converting to a gamViz object first saves time when we need to call plot.gam several times. To see all three plots on one page we can do:

print(plot(b) + labs(title = NULL), pages = 1)

where we have also removed the titles. Notice that plot.gam returns an object of class plotGam, which is initially empty. The layers in the previous plots (e.g. the rug and the confidence interval lines) have been added by print.plotGam, which adds some default layers to empty plotGam objects. This can be avoided by setting addLay = FALSE in the call to print.plotGam. A plotGam object in considered not empty if we added an object of class gamLayer to it, for instance:

pl <- plot(b) + l_points() + l_fitLine(linetype = 3) + l_fitContour() + 
       l_ciLine(colour = 2) + theme_get() + labs(title = NULL)
print(pl, pages = 1)
pl$empty # FALSE: because we added gamLayers

here all the functions starting with l_ return gamLayer objects. Notice that some layers are not relevant to all smooths. For instance, l_fitContour is added only to the second smooth. The +.plotGam method automatically adds each layer only to compatible smooth effect plots.

We can plot individuals effects by using the select arguments. For instance:

plot(b, select = 1)

where only the default layers are added. Obviously we can have our custom layers instead:

plot(b, select = 1) + l_dens(type = "cond") + l_fitLine() + l_ciLine()

where the l_dens layer represents the conditional density of the partial residuals.

1.3 Interactive rgl smooth effect plots

mgcViz provides tools for generating interactive plots of multidimensional smooths via the rgl R package. Here is an example where we are plotting a 2D slice of a 3D smooth effect with confidence intervals:

library(mgcViz)
n <- 500
x <- rnorm(n); y <- rnorm(n); z <- rnorm(n)
ob <- (x-z)^2 + (y-z)^2 + rnorm(n)
b <- gam(ob ~ s(x, y, z))
b <- getViz(b)

plotRGL(sm(b, 1), fix = c("z" = 0), residuals = TRUE)

The fix argument is used to determine the slice along the z-axis. The plot also shows some residuals (colour-coded depending on sign) that fall close (in term of Euclidean distance) to the selected slice.
Notice that plotRGL is not layered at the moment, and most options need to be specified in the initial function call. But the interactive plot can still be manipulated once the rgl window is open, for instance here we change the aspect ratio:

aspect3d(1, 2, 1)

We then close the window using:

rgl.close()

2 Model checking

2.1 New version of traditional model checks

Most of the model checks provided by mgcv are contained in qq.gam and gam.check. mgcViz provides a new version of qq.gam (which masks the one provided by mgcv) and substitutes gam.check with the check.gam method.

2.1.1 The new qq.gam function

Consider the following model with binomial responses:

set.seed(0)
n.samp <- 400
dat <- gamSim(1,n = n.samp, dist = "binary", scale = .33)
p <- binomial()$linkinv(dat$f) ## binomial p
n <- sample(c(1, 3), n.samp, replace = TRUE) ## binomial n
dat$y <- rbinom(n, n, p)
dat$n <- n
lr.fit <- gam(y/n ~ s(x0) + s(x1) + s(x2) + s(x3)
              , family = binomial, data = dat,
              weights = n, method = "REML")

We can get a QQ-plot of the residuals as follows:

qq.gam(lr.fit, method = "simul1", 
       a.qqpoi = list("shape" = 1), 
       a.ablin = list("linetype" = 2))

Here method determines the method used to compute the QQ-plot, while the arguments starting with a. are lists that will be passed directly to the corresponding ggplot2 layer (geom_point and geom_abline here). We can remove the confidence intervals and show all simulated (model-based) QQ-curves as follows:

qq.gam(lr.fit, rep = 20, show.reps = T, CI = "none",
       a.qqpoi = list("shape" = 19),
       a.replin = list("alpha" = 0.2))

Importantly, mgcViz::qq.gam can handle large datasets by discretizing the QQ-plot before plotting. For instance, let's increase n.samp in the previous example:

set.seed(0)
n.samp <- 50000
dat <- gamSim(1,n=n.samp,dist="binary",scale=.33)
p <- binomial()$linkinv(dat$f) ## binomial p
n <- sample(c(1,3),n.samp,replace=TRUE) ## binomial n
dat$y <- rbinom(n,n,p)
dat$n <- n
lr.fit <- bam(y/n ~ s(x0) + s(x1) + s(x2) + s(x3)
              , family = binomial, data = dat,
              weights = n, method = "fREML", discrete = TRUE)

Here the discrete argument determines whether the QQ-plot is discretized or not. Notice that we can compute the QQ-plot, store it in o and then plot it (via print.qqGam).

o <- qq.gam(lr.fit, rep = 10, method = "simul1", CI = "normal", show.reps = TRUE, 
            a.replin = list(alpha = 0.1), discrete = TRUE)
o 

The coarseness of the discretization grid is determined by the ngr argument:

o <- qq.gam(lr.fit, rep = 10, method = "simul1", CI = "normal", show.reps = TRUE,
            ngr = 1e2, a.replin = list(alpha = 0.1), a.qqpoi = list(shape = 19))
o 

2.1.2 The check.gam method

The check.gam method is similar to mgcv::gam.check, with the difference that it produces a sequence of ggplot objects and that it sub-samples the residuals to avoid over-plotting (or stalling entirely) when dealing with large data sets. Here is an example:

set.seed(0)
dat <- gamSim(1, n = 200)
b <- gam(y ~ s(x0) + s(x1) + s(x2) + s(x3), data = dat)

check(b,
      a.qq = list(method = "tnorm", 
                  a.cipoly = list(fill = "light blue")), 
      a.respoi = list(size = 0.5), 
      a.hist = list(bins = 10))

The a.qq argument is a list that gets passed directly to mgcViz::qq.gam. Similarly, a.repoi is passed to ggplot2::geom_points and a.hist to ggplot2::geom_hist.

2.2 New layered model checks

The qq.gam and check.gam functions are not layered, and in fact require using lists of arguments to be passed to the underlying ggplot2 layers. Instead, the methods described in this section are fully layered, hence easy to extend and customize.

2.2.1 One dimensional checks using check1D

This function allows to verify how the residuals vary along one covariate. Consider the following model:

set.seed(4124)
n <- 1e4
x <- rnorm(n); y <- rnorm(n);

ob <- (x)^2 + (y)^2 + (0.2*abs(x) + 1)  * rnorm(n)
b <- bam(ob ~ s(x,k=30) + s(y, k=30), discrete = TRUE)

Here the responses variance varies a lot along $x$. Assume that we didn't know this, but that we wanted to find out whether the residuals are heteroscedastic. We can start by doing the following:

ck <- check1D(b, "x")
ck

This produces a view along $x$, but as you can see that plot is initially empty. We might want to add a layer showing the conditional distribution of the residuals along $x$ and another containing a rug:

ck + l_dens(type = "cond", alpha = 0.8) + l_rug(alpha = 0.2)

This suggests that the variance of the residuals might be lower in the middle ($x=0$), but it is not entirely clear. The l_densCheck layer gives a more clear answer in this case:

ck + l_densCheck()

This layers adds an heatmap proportional to ${p(r|x)^{1/2} - p_m(r|x)^{1/2}}^{1/3}$, where $r$ are the residuals, while $p$ and $p_m$ are their empirical and theoretical (model based) density. In particular, $p$ is estimated using the the fast k.d.e. method of Wand (1994) (implemented by the kernSmooth package) and $p_m$ is a standard normal density here. This plot makes clear that the residuals are over-dispersed when $x$ is far from zero.

The l_gridCheck1D provides another way of finding residuals patterns. For instance:

b <- getViz(b, nsim = 50)
check1D(b, "x") + l_gridCheck1D(gridFun = sd, show.reps = TRUE)

Before calling check1D we convert b using getViz. This is because l_gridCheck1D need some simulations to compute the confidence intervals. The simulations are done by getViz and then stored inside b. l_gridCheck1D simply bins the residuals according to their $x$ values, and evaluates a user-defined function (sd here) over the observed and simulated residuals.

2.2.2 Two dimensional checks using check2D

check2D is quite similar to check1D, but looks at the residuals along two covariates. Here is an example where the mean effect follows the Rosenbrock function:

set.seed(566)
n <- 1e4
X <- data.frame("x1"=rnorm(n, 0.5, 0.5), "x2"=rnorm(n, 1.5, 1))
X$y <- (1-X$x1)^2 + 100*(X$x2 - X$x1^2)^2 + rnorm(n, 0, 2)
b <- bam(y ~ te(x1, x2, k = 5), data = X, discrete = TRUE)
b <- getViz(b, nsim = 50)

We start by generating a 2D view:

ck <- check2D(b, x1 = "x1", x2 = "x2")

Then we add the l_gridCheck2D layer:

ck + l_gridCheck2D(gridFun = mean)

l_gridCheck2D bins the observed and simulated residuals, summarizes them using a scalar-valued function (mean here), and adds an heatmap proportional to the observed summary in each cell, normalized using the nsim summaries obtained using the simulations. Here the pattern in the residual means is not very well visible, due to outliers on the far right. The pattern is made more visible by zooming on the center of the distribution and by changing the size of the bins:

ck + l_gridCheck2D(bw = c(0.05, 0.1)) + xlim(-1, 1) + ylim(0, 3)

As for smooth effect plots, we can list the available layers by doing:

listLayers( ck ) 

The most sophisticated layer is probably l_glyphs2D which we illustrate here using an heteroscedastic model:

set.seed(4124)
n <- 1e4
dat <- data.frame("x1" = rnorm(n), "x2" = rnorm(n))
dat$y <- (dat$x1)^2 + (dat$x2)^2 + (1*abs(dat$x1) + 1)  * rnorm(n)
b <- bam(y ~ s(x1,k=30) + s(x2, k=30), data = dat, discrete = TRUE)

ck <- check2D(b, x1 = "x1", x2 = "x2", type = "tnormal")

Similarly to l_gridCheck2D, l_glyphs2D bins the residuals according to two covariates, but the user-defined function used to summarize the residuals in each bin has to return a data.frame rather than a scalar. Here is an example:

glyFun <- function(.d){
  .r <- .d$z
  .qq <- as.data.frame( density(.r)[c("x", "y")], n = 100 )
  .qq$colour <- rep(ifelse(length(.r)>50, "black", "red"), nrow(.qq))
  return( .qq )
}

ck + l_glyphs2D(glyFun = glyFun, ggLay = "geom_path", n = c(8, 8),
                 mapping = aes(x=gx, y=gy, group = gid, colour = I(colour)), 
                 height=1.5, width = 1)

Each glyph represend a kernel density of the residuals, with colours indicating whether we have more (black) or less (red) that 50 observations in that bin. It is clear that the residuals are much less variable for $x \approx 0$ than elsewhere. We can do the same using binned worm-plots:

glyFun <- function(.d){
  n <- nrow(.d)
  px <- qnorm( (1:n - 0.5)/(n) )
  py <- sort( .d$z )
  clr <- if(n > 50) { "black" } else { "red" }
  clr <- rep(clr, n)
  return( data.frame("x" = px, "y" = py - px, "colour" = clr))
}

ck + l_glyphs2D(glyFun = glyFun, ggLay = "geom_point", n = c(10, 10),
                mapping = aes(x=gx, y=gy, group = gid, colour = I(colour)),
                height=2, width = 1, size = 0.2) 

Notice that worm-plots (Buuren and Fredriks, 2001) are simply rotated QQ-plots. An horizontal plot indicates well specified residual model. An increasing (decreasing) worm indicates over (under) dispersion.

References

• Buuren, S. v. and Fredriks, M. (2001) Worm plot: a simple diagnostic device for modelling growth reference curves, Statistics in medicine, 20, 1259–1277.

• Murdoch, D. (2001) Rgl: An r interface to opengl, in Proceedings of DSC, p. 2.

• Wand, M. P. (1994) Fast computation of multivariate kernel estimators, Journal of Computational and Graphical Statistics, 3, 433–445

• Wickham, H. (2009) ggplot2: Elegant Graphics for Data Analysis, Springer-Verlag New York.

• Wickham, H. (2010) A layered grammar of graphics, Journal of Computational and Graphical Statistics, 19, 3–28.

• Wood, S.N. (2017) Generalized Additive Models: An Introduction with R (2nd edition). Chapman and Hall/CRC.