Prussian Horse Kicks II: Mixed Effects Models

r
stats
glms
regression
mixed-effects
Published

November 1, 2025

Last time we used a quirky dataset to get acquainted with Generalised Linear Models. If you recall, the dataset is a count of deaths by horse kick, where every year all the corps in the army had some non-negative integer number of deaths and we were predicting the mean of the distribution for each year. Being count data we had to adapt the linear model to use a Poisson distribution, which itself entailed using a log link function. All hail the GLM.

Here’s a dimension we didn’t consider though: are all corps equally death-prone? Let’s explore the data in R.

First some setup.

library(tidyverse)
library(knitr)
library(Horsekicks) # Thanks to Antony Unwin and Bill Venables

prussian.colours <- c("#333333", "#F9BE11", "#BE0007") # Prussia is/was Germany!

hk.tidy <- hkdeaths |>
  pivot_longer(
    c(kick, drown, fall),
    names_to = "death.type",
    values_to = "death.count"
  )

Let’s plot the deaths over time for each corp. We’re looking to see if they’re very similar, or if there is noticeable difference between corps. Since Unwin and Venables provided three types of accidental death (or cunningly disguised murder… we will never know) we can inspect all three.

Code 
ggplot(hk.tidy) +
  aes(x = year, y = death.count, colour = death.type) +
  facet_grid(
    rows = vars(death.type),
    cols = vars(corps),
    scale = "free_y",
    switch = "y"
  ) +
  scale_colour_manual(values = prussian.colours, name = "Death type") +
  geom_line(show.legend = FALSE) +
  labs(x = "Year", y = "Number of deaths") +
  stat_smooth(
    method = "glm",
    se = F,
    formula = y ~ x,
    method.args = list(family = poisson),
    show.legend = F
  ) +
  theme_minimal() +
  theme(
    axis.text.x = element_blank(),
    axis.text.y = element_blank(),
    panel.grid = element_blank(),
    panel.border = element_rect(colour = "grey", fill = "transparent")
  )
Figure 1: Deaths per year by corp and accident type.

This fits a GLM to each, so we can see the trend for each corps more easily. And lo, there is clear difference. For example, corps XIV stands alone in having increased deaths by drowning over time, but showed the biggest improvement in deaths by horse kick. That said, these are visibly not the best fits - the amount of data for each corp is quite low.

This might be explainable by having changing numbers of soldiers (or horses) with deaths per capita remaining the same. Sadly the dataset doesn’t include the number of soldiers (or horses) but it does show that the number of regiments was the same every year for each corps. Luckily we’re not too concerned with the actual causes here, this is just a statistical plaything. The Prussian army no longer has a problem with horse-related deaths anyway.

Mixed Effects Models

So, how can we improve our model to accommodate these corps-level differences? Introducing… Mixed Effects Models!

Mixed Effects Models, also known as Multilevel Models, Hierarchical Models, Random Effects Models, or simply Mixed Models (pick a lane, statisticans!) describe the response variable as the result of fixed effects and cluster level random effects. This means as well as a global trend across all observations, there are also trends for each group (for us that’s each corps).

The fixed effects are the same as in a linear model or GLM, in this case the coefficient for the year and an intercept. However we also try to fit coefficients for each corps, taking into account the fixed effects.

Let’s consider it as a linear model first, rather than a GLM. Here’s what that looks like:

\[ E[Y]_{corps} = \beta_0 + \beta_1 X + ( \beta'_0 + \beta'_1 X )_{corps} + \epsilon \]

Here the response variable Y is the number of deaths, and the predictor X is the year. We have two pairs of coefficients. The coefficents are the fixed effects, while the ’ coefficients are the random effects for each corps. These are assumed to be normally distributed, with a mean of 0. The variance of the random effects is something we’ll learn. is the (normal) error. All this is to say that for each corps, the expectation of the number of deaths is a linear combination of the fixed effects and the corps-specific random effects, plus error.

Extending this to a generalised linear mixed model (GLMM) is conceptually easy - pick a distribution and a link function. As per the previous post, for this dataset that would be the Poisson distribution and the log link function.

Here’s how we fit it in R, using the lme4 package. This model fits deaths by drowning.

library(lme4)

midyear <- quantile(hkdeaths$year, 0.5)

fit <- glmer(
  drown ~ I(year - midyear) + (1 + I(year - midyear) | corps),
  family = poisson,
  data = hkdeaths
)
summary(fit)
Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: poisson  ( log )
Formula: drown ~ I(year - midyear) + (1 + I(year - midyear) | corps)
   Data: hkdeaths

      AIC       BIC    logLik -2*log(L)  df.resid 
   2076.0    2096.7   -1033.0    2066.0       457 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.2885 -0.8548 -0.0857  0.6398  4.7076 

Random effects:
 Groups Name              Variance  Std.Dev. Corr
 corps  (Intercept)       6.600e-02 0.256900     
        I(year - midyear) 9.499e-05 0.009746 0.12
Number of obs: 462, groups:  corps, 14

Fixed effects:
                   Estimate Std. Error z value Pr(>|z|)    
(Intercept)        1.364285   0.072815   18.74  < 2e-16 ***
I(year - midyear) -0.023013   0.003624   -6.35 2.15e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
            (Intr)
I(yer-mdyr) 0.130

Note that this time the year predictor is redefined as a distance from the median year, which avoids convergence issues and makes the intercepts easier to interpret. The bit of the formula with (...| corps) defines the corps-specific intercept and slope.

We can inspect the random effects for each corps. Remember this is in addition to the fixed effects, so a positive slope here means less decline than average, not necessarily an increase in deaths.

ranef(fit)$corps |> kable(digits = 3)
Table 1: Random effects for each corps.
(Intercept) I(year - midyear)
G -0.246 0.003
I 0.548 0.014
II 0.151 -0.007
III 0.230 -0.007
IV -0.168 -0.003
V 0.000 0.001
VI 0.062 -0.001
VII -0.019 0.005
VIII 0.057 0.000
IX 0.009 -0.007
X -0.151 -0.005
XI -0.188 -0.010
XIV -0.473 0.013
XV 0.257 0.006

This model actually considers corps I to have a slightly more positive slope than corps XIV. That’s a surprise, although somewhat understandable if you eyeball the earlier chart.

So there we have it, a simple example of using a Mixed Effects Model to model global fixed effects and cluster-level random effects. For more on Mixed Effects Models, I strongly recommend Michael Clark’s online book.

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