Author: Michael Friendly
Date: 2023-01-29

# Continuous predictors

#### Michael Friendly

#### 2023-01-29

When continuous predictors are available—and potentially important—in explaining a categorical outcome, models for that outcome include: logistic regression (binary response), the proportional odds model (ordered polytomous response), multinomial (generalized) logistic regression. Many of these are special cases of the generalized linear model using the `"poisson"` or `"binomial"` family and their relatives.

## Spine and conditional density plots

I don’t go into fitting such models here, but I would be remiss not to illustrate some visualizations in `vcd` that are helpful here. The first of these is the spine plot or spinogram (Hummel, 1996), produced with `spine()`. These are special cases of mosaic plots with specific spacing and shading to show how a categorical response varies with a continuous or categorical predictor.

They are also a generalization of stacked bar plots where not the heights but the widths of the bars corresponds to the relative frequencies of `x`. The heights of the bars then correspond to the conditional relative frequencies of `y` in every `x` group.

Example: For the `Arthritis` data, we can see how `Improved` varies with `Age` as follows. `spine()` takes a formula of the form `y ~ x` with a single dependent factor and a single explanatory variable `x` (a numeric variable or a factor). The range of a numeric variable`x` is divided into intervals based on the `breaks` argument, and stacked bars are drawn to show the distribution of `y` as `x` varies. As shown below, the discrete table that is visualized is returned by the function.

    (spine(Improved ~ Age, data = Arthritis, breaks = 3))
    ##          Improved
    ## Age       None Some Marked
    ##   [20,40]   10    3      2
    ##   (40,60]   21    3     17
    ##   (60,80]   11    8      9
    (spine(Improved ~ Age, data = Arthritis, breaks = "Scott"))
    ##          Improved
    ## Age       None Some Marked
    ##   [20,30]    6    1      0
    ##   (30,40]    4    2      2
    ##   (40,50]    9    0      3
    ##   (50,60]   12    3     14
    ##   (60,70]   11    8      8
    ##   (70,80]    0    0      1

Spine plots for the `Arthritis` data

The conditional density plot (Hofmann & Theus, 2005) is a further generalization. This visualization technique is similar to spinograms, but uses a smoothing approach rather than discretizing the explanatory variable. As well, it uses the original `x` axis and not a distorted one.

    cdplot(Improved ~ Age, data = Arthritis)

Conditional density plot for the `Arthritis` data showing the variation of Improved with Age.

In such plots, it is useful to also see the distribution of the observations across the horizontal axis, e.g., with a `rug()` plot. @ref{fig:cd-plot} uses `cdplot()` from the `graphics` package rather than `cd_plot()` from `vcd`, and is produced with

    cdplot(Improved ~ Age, data = Arthritis)
    with(Arthritis, rug(jitter(Age), col="white", quiet=TRUE))

From this figure it can be easily seen that the proportion of patients reporting Some or Marked improvement increases with Age, but there are some peculiar bumps in the distribution. These may be real or artifactual, but they would be hard to see with most other visualization methods. When we switch from non-parametric data exploration to parametric statistical models, such effects are easily missed.

## Model-based plots: effect plots and `ggplot2 plots`

The nonparametric conditional density plot uses smoothing methods to convey the distributions of the response variable, but displays that are simpler to interpret can often be obtained by plotting the predicted response from a parametric model.

For complex `glm()` models with interaction effects, the `effects` package provides the most useful displays, plotting the predicted values for a given term, averaging over other predictors not included in that term. I don’t illustrate this here, but see Fox & Weisberg (2018),Fox (2003) and `help(package="effects")`.

Here I just briefly illustrate the capabilities of the `ggplot2` package for model-smoothed plots of categorical responses in `glm()` models.

Example: The `Donner` data frame in `vcdExtra` gives details on the survival of 90 members of the Donner party, a group of people who attempted to migrate to California in 1846. They were trapped by an early blizzard on the eastern side of the Sierra Nevada mountains, and before they could be rescued, nearly half of the party had died. What factors affected who lived and who died?

    data(Donner, package="vcdExtra")
    str(Donner)
    ## 'data.frame':    90 obs. of  5 variables:
    ##  $ family  : Factor w/ 10 levels "Breen","Donner",..: 9 1 1 1 1 1 1 1 1 1 ...
    ##  $ age     : int  23 13 1 5 14 40 51 9 3 8 ...
    ##  $ sex     : Factor w/ 2 levels "Female","Male": 2 2 1 2 2 1 2 2 2 2 ...
    ##  $ survived: int  0 1 1 1 1 1 1 1 1 1 ...
    ##  $ death   : POSIXct, format: "1846-12-29" NA ...

A potential model of interest is the logistic regression model for \(Pr(survived)\), allowing separate fits for males and females as a function of `age`. The key to this is the `stat_smooth()` function, using `method = "glm", method.args = list(family = binomial)`. The `formula = y ~ x` specifies a linear fit on the logit scale (@ref{fig:donner3}, left)

    # separate linear fits on age for M/F
    ggplot(Donner, aes(age, survived, color = sex)) +
      geom_point(position = position_jitter(height = 0.02, width = 0)) +
      stat_smooth(method = "glm",
                  method.args = list(family = binomial),
                  formula = y ~ x,
                  alpha = 0.2, size=2, aes(fill = sex))

Alternatively, we can allow a quadratic relation with `age` by specifying `formula = y ~ poly(x,2)` (, right).

    # separate quadratics
    ggplot(Donner, aes(age, survived, color = sex)) +
      geom_point(position = position_jitter(height = 0.02, width = 0)) +
      stat_smooth(method = "glm",
                  method.args = list(family = binomial),
                  formula = y ~ poly(x,2),
                  alpha = 0.2, size=2, aes(fill = sex))

Logistic regression plots for the `Donner` data showing survival vs. age, by sex. Left: linear logistic model; right: quadratic model {#fig:donner3}

These plots very nicely show (a) the fitted \(Pr(survived)\) for males and females; (b) confidence bands around the smoothed model fits and (c) the individual observations by jittered points at 0 and 1 for those who died and survived, respectively.

# References

Fox, J. (2003). Effect displays in R for generalised linear models. Journal of Statistical Software, 8(15), 1–27. http://doi.org/10.18637/jss.v008.i15
Fox, J., & Weisberg, S. (2018). Visualizing fit and lack of fit in complex regression models with predictor effect plots and partial residuals. Journal of Statistical Software, 87(9), 1–27. http://doi.org/10.18637/jss.v087.i09
Hofmann, H., & Theus, M. (2005). Interactive graphics for visualizing conditional distributions.
Hummel, J. (1996). Linked bar charts: Analysing categorical data graphically. Computational Statistics, 11, 23–33.