Modelling with ALE and GAM
With GAM, only numeric variables can be smoothed, not binary or
categorical ones. However, smoothing does not always help improve the
model since some variables are not related to the outcome and some that
are related actually do have a simple linear relationship. To keep this
demonstration simple, we have done some earlier analysis (not shown
here) that determines where smoothing is worthwhile on the modified
var_cars dataset, so only some of the numeric variables are
smoothed. Our goal here is not to demonstrate the best modelling
procedure but rather to demonstrate the flexibility of the
ale package.
cm <- mgcv::gam(mpg ~ cyl + disp + hp + drat + wt + s(qsec) +
vs + am + gear + carb + country,
data = var_cars)
summary(cm)
#>
#> Family: gaussian
#> Link function: identity
#>
#> Formula:
#> mpg ~ cyl + disp + hp + drat + wt + s(qsec) + vs + am + gear +
#> carb + country
#>
#> Parametric coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -7.84775 12.47080 -0.629 0.54628
#> cyl 1.66078 1.09449 1.517 0.16671
#> disp 0.06627 0.01861 3.561 0.00710 **
#> hp -0.01241 0.02502 -0.496 0.63305
#> drat 4.54975 1.48971 3.054 0.01526 *
#> wt -5.03736 1.53979 -3.271 0.01095 *
#> vsTRUE 12.45630 3.62342 3.438 0.00852 **
#> amTRUE 8.77813 2.67611 3.280 0.01080 *
#> gear.L 0.53111 3.03337 0.175 0.86525
#> gear.Q 0.57129 1.18201 0.483 0.64150
#> carb -0.34479 0.78600 -0.439 0.67223
#> countryItaly -0.08633 2.22316 -0.039 0.96995
#> countryJapan -3.31948 2.22723 -1.490 0.17353
#> countrySweden -3.83437 2.74934 -1.395 0.19973
#> countryUK -7.24222 3.81985 -1.896 0.09365 .
#> countryUSA -7.69317 2.37998 -3.232 0.01162 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Approximate significance of smooth terms:
#> edf Ref.df F p-value
#> s(qsec) 7.797 8.641 5.975 0.0101 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> R-sq.(adj) = 0.955 Deviance explained = 98.8%
#> GCV = 6.4263 Scale est. = 1.6474 n = 32Now we generate ALE data from the var_cars GAM model and
plot it.
cars_ale <- ale(var_cars, cm)
#>
Calculating ALE ■ 0% | ETA: ?
Calculating ALE ■■■■■■ 18% | ETA: 1s
Calculating
#> ALE ■■■■■■■■■■■■■■■ 45% | ETA: 0s
Calculating ALE ■■■■■■■■■■■■■■■■■■■■■■■ 73% |
#> ETA: 0s
# Print all plots
gridExtra::grid.arrange(grobs = cars_ale$plots, ncol = 2)
We can see that ale has no trouble modelling any of the
datatypes in our sample (logical, factor, ordered, integer, or double).
It plots line charts for the numeric predictors and column charts for
everything else.
The numeric predictors have rug plots that indicate in which ranges
of the x (predictor) and y (mpg) values data actually
exists in the dataset. This helps us to not over-interpret regions where
data is sparse. Since column charts are on a discrete scale, they cannot
rug plots. Instead, the percentage of data represented by each column is
displayed.
We can also generate and plot the ALE data for all two-way interactions.
cars_ale_ixn <- ale_ixn(var_cars, cm)
#>
Calculating ALE interactions ■ 0% | ETA: ?
Calculating ALE interactions ■■■■■
#> 14% | ETA: 2s
Calculating ALE interactions ■■■■■■■■■■ 29% | ETA: 2s
Calculating
#> ALE interactions ■■■■■■■■■■■■■■ 43% | ETA: 1s
Calculating ALE interactions
#> ■■■■■■■■■■■■■■■■■■ 57% | ETA: 1s
# Print plots
cars_ale_ixn$plots |>
purrr::walk(\(.x1) { # extract list of x1 ALE outputs
gridExtra::grid.arrange(grobs = .x1, ncol = 2) # plot all x1 plots
})
There are no interactions in this dataset. (To see what ALE interaction plots look like in the presence of interactions, see the ALEPlot comparison vignette, which explains the interaction plots in more detail.)
Finally, as explained in the vignette on modelling with small datasets, a more appropriate modelling workflow would require bootstrapping the entire model, not just the ALE data. So, let’s do that now.
mb <- model_bootstrap(
var_cars,
'mgcv::gam(mpg ~ cyl + disp + hp + drat + wt + s(qsec) +
vs + am + gear + carb + country)',
boot_it = 10, # 100 by default but reduced here for a faster demonstration
silent = TRUE # progress bars disabled for the vignette
)
gridExtra::grid.arrange(grobs = mb$ale$plots, ncol = 2)
(By default, model_bootstrap creates 100 bootstrap
samples but, so that this illustration runs faster, we demonstrate it
here with only 10 iterations.)
With such a small dataset, the bootstrap confidence interval always overlap with the middle band, indicating that this dataset cannot support any claims that any of its variables has a meaningful effect on fuel efficiency (mpg). Considering that the average bootstrapped ALE values suggest various intriguing patterns, the problem is no doubt that the dataset is too small–if more data were collected and analyzed, some of the patterns would probably be confirmed.