Using R to make sense of the generalised linear model

Statistical models

Modern science rests on several foundational pillars. Among these is the ability to construct sufficiently solid theoretical abstractions that are able to explain concrete observable aspects of the world. Once the applicability range of a theoretical model has been defined, it is usually compared to another model that is similar in its goal (i.e., another model that aims to explain similar aspects of the world). However, we rarely directly compare theoretical models. Instead, we are brought to compare statistical models that aim to represent theories.

According to Rouder, Morey, & Wagenmakers (2016), “Models are devices that connect theories to data. A model is an instantiation of a theory as a set of probabilistic statements”. One common and convenient example of such instantiation is the linear model, which –in its general form– allows to predict parameter(s) of a distribution, which is supposed to reflect the distribution from which the observed data is issued (the data generation process).

But what does it mean for a model to predict something? In the current post, I focus on four R functions (the predict, fitted, residuals and simulate functions), exploring the similarities and differences between these functions, to illustrate the mechanisms and assumptions of the generalised linear model.

The usual linear model is of the following form.

\[ \begin{aligned} y_{i} &\sim \mathrm{Normal}(\mu_{i}, \sigma) \\ \mu_{i} &= \alpha + \beta \cdot \text{x}_{i} \\ \end{aligned} \]

Where, in Bayesian terms, the first line of the model corresponds to the likelihood of the model, which is the assumption made about the data generation process. We make the assumption that the outcomes \(y_{i}\) are normally distributed around a mean \(\mu_{i}\) with some error \(\sigma\). This is equivalent to say that the errors are normally distributed around \(0\).

Of course, the distributional assumption is not restricted to be Gaussian, and can be adapted to whatever distribution that makes sense in consideration of the data at hand. The linear aspect of the linear model actually refers to the second line of the above description, in which one tries to predict parameters of the distribution (e.g., \(\mu_{i}\) or \(\sigma\)) by a linear combination of some predictor variable(s). Generalising to other distributions, the generalised linear model can be rewritten as:

\[ \begin{aligned} y_{i} &\sim \mathrm{D}(f(\eta_{i}), \theta) \\ \eta &= \mathbf{X} \beta \\ \end{aligned} \]

Where the response \(y_{i}\) is predicted through the linear combination \(\eta\) of predictors transformed by the inverse link function \(f\), assuming a certain distribution \(D\) for \(y\) (also called the family), and family-specific parameters \(\theta\) (Bürkner, 2017).

Below, we illustrate a simple Gaussian linear model using the Howell1 dataset from the rethinking package (McElreath, 2016), which contains data about 544 individuals, including height (centimetres), weight (kilograms), age (years) and gender (0 indicating female and 1 indicating male).

library(rethinking)
library(tidyverse)
library(ggExtra)

data(Howell1)
d <- Howell1 %>% filter(age >= 18)

p <- d %>%
  ggplot(aes(x = weight, y = height) ) +
  geom_point(pch = 21, color = "white", fill = "black", size = 2, alpha = 0.8) +
  geom_smooth(method = "lm", colour = "black") +
  theme_bw(base_size = 12)

ggMarginal(p, type = "histogram")

A quick visual exploration of the dataset reveals a positive relationship between height and weight. The above plotted regression line corresponds to the following model, where we assume a normal likelihood:

\[ \begin{aligned} \text{height}_{i} &\sim \mathrm{Normal}(\mu_{i}, \sigma) \\ \mu_{i} &= \alpha + \beta \cdot \text{weight}_{i} \\ \end{aligned} \]

This model can be fitted easily in R with the following syntax.

(mod1 <- lm(height ~ weight, data = d) )
## 
## Call:
## lm(formula = height ~ weight, data = d)
## 
## Coefficients:
## (Intercept)       weight  
##     113.879        0.905

The intercept (113.879) represents the predicted height when weight is at 0 (which makes no much sense in our case), whereas the slope (0.905) represents the change in height when weight increases by one unit (i.e., one kilogram).

Prediction is key: predict and fitted

The main advantage of the previous model is that it allows to make predictions for any value of \(\text{weight}\). In R, this is done using the aptly named predict function. For instance, we can ask our model what is the expected height for an individual of weight 43, which is equal to \(\alpha + \beta \cdot 43\).

wght <- 43

d %>%
  ggplot(aes(x = weight, y = height) ) +
  geom_line(aes(y = predict(mod1) ), size = 1) +
  geom_point(size = 2, alpha = 0.2) +
  geom_segment(
    x = wght, xend = wght,
    y = 0, yend = predict(mod1, newdata = data.frame(weight = wght) ),
    linetype = 2, lwd = 0.5
    ) +
  geom_segment(
    x = 0, xend = wght,
    y = predict(mod1, newdata = data.frame(weight = wght) ),
    yend = predict(mod1, newdata = data.frame(weight = wght) ),
    linetype = 2, lwd = 0.5
    ) +
  theme_bw(base_size = 12)

Implementing the function predict by hand is quite easy and will allow us to better understand how it works. This function is actually simply retrieving parameters of the fitted model (in our case, the intercept and the slope) to make predictions about the outcome variable, given some values of the predictor(s). In other words, it corresponds to the second line of our model.

d <- d %>%
  mutate(
    pred_mod1 = predict(mod1),
    pred_mod1_2 = coef(mod1)[1] + coef(mod1)[2] * weight
    )

head(d)
##    height   weight age male pred_mod1 pred_mod1_2
## 1 151.765 47.82561  63    1  157.1630    157.1630
## 2 139.700 36.48581  63    0  146.9001    146.9001
## 3 136.525 31.86484  65    0  142.7180    142.7180
## 4 156.845 53.04191  41    1  161.8839    161.8839
## 5 145.415 41.27687  51    0  151.2362    151.2362
## 6 163.830 62.99259  35    1  170.8895    170.8895

We could also be interested in predicting the height of individuals with other weights that the weights we observed (e.g., weights between 80 and 100 kgs). Below we simulate new data from our model (i.e., we simulate heights) and predictions for this new set of data (i.e., the \(\mu_{i}\)).

# generating weights from 80 to 100 kgs
data.frame(weight = 80:100) %>%
  # retrieving mod1 predictions
  mutate(pred = predict(mod1, newdata = .) ) %>%
  # simulating data from our model, taking into account sigma
  mutate(sim =  pred + rnorm(length(weight), 0, sd(residuals(mod1) ) ) ) %>%
  # or using sigma(mod1) instead of sd(residuals(mod1) ), as suggested
  # by Tristan Mahr (https://www.tjmahr.com)
  mutate(sim =  pred + rnorm(length(weight), 0, sigma(mod1) ) ) %>%
  # plotting these predictions
  ggplot(aes(x = weight, y = pred) ) +
  geom_line(size = 1) +
  geom_point(aes(x = weight, y = sim), size = 2) +
  geom_segment(
      aes(xend = weight, yend = sim),
      size = 0.5, alpha = 0.5, lineend = "round"
      ) +
  theme_bw(base_size = 12)

Where the vertical lines represent deviations from the predicted values. OK, so we’ve seen the the predict function simply uses the linear model to make predictions about the \(\mu_{i}\).

You might know that there exists a similar function, the fitted function, which allows to extract fitted values of a model.

d <- d %>% mutate(fitted_mod1 = fitted(mod1) )
head(d)
##    height   weight age male pred_mod1 pred_mod1_2 fitted_mod1
## 1 151.765 47.82561  63    1  157.1630    157.1630    157.1630
## 2 139.700 36.48581  63    0  146.9001    146.9001    146.9001
## 3 136.525 31.86484  65    0  142.7180    142.7180    142.7180
## 4 156.845 53.04191  41    1  161.8839    161.8839    161.8839
## 5 145.415 41.27687  51    0  151.2362    151.2362    151.2362
## 6 163.830 62.99259  35    1  170.8895    170.8895    170.8895

Surprisingly, the predict and fitted functions seem to do the exact same thing (at least their results are the same)… but do they? To answer this question, let’s ask another one.

Errors and residuals: the residuals function

A very common fallacy about the assumptions of the linear (Gaussian) model is that the outcome variable should be normally distributed. Instead, this assumption concerns the distribution of the outcome variable around its predicted value (i.e., the distribution of the errors).

What we actually said above is that the errors \(\epsilon_{i}\) should be normally distributed around the predicted value. But the errors are the the non-observed (and non-observable) differences between the theoretical predicted value \(\mu\) and the observed outcomes. Consequently, we do not have access to it. Instead, what we can work with are the residuals \(e_{i}\), which can be seen as an estimate (from the sample) of the errors \(\epsilon_{i}\), in a similar way as \(b\) is an estimate of \(\beta\). To sum up, the residuals are the \(e_{i} = y_{i} - \bar{X}\) whereas the errors are the \(\epsilon_{i} = y_{i} - \mu_{i}\).

In other words, errors pertain to the data generation process, whereas residuals are the difference between the model’s estimation and the observed outcomes. Basically, the residuals are the difference between the observed value and the predicted value. We can obtain them easily using the residuals function (which is useful for more complex models), or by subtracting to each observed outcome \(y_{i}\) the predicted \(\mu_{i}\).

d <- d %>%
  mutate(
    res1 = residuals(mod1),
    res2 = height - pred_mod1
    )

d %>%
  select(height, weight, male, pred_mod1, res1, res2) %>%
  head
##    height   weight male pred_mod1      res1      res2
## 1 151.765 47.82561    1  157.1630 -5.397960 -5.397960
## 2 139.700 36.48581    0  146.9001 -7.200111 -7.200111
## 3 136.525 31.86484    0  142.7180 -6.193000 -6.193000
## 4 156.845 53.04191    1  161.8839 -5.038870 -5.038870
## 5 145.415 41.27687    0  151.2362 -5.821164 -5.821164
## 6 163.830 62.99259    1  170.8895 -7.059520 -7.059520

Below we plot these residuals, and make the alpha (i.e., the transparency) and the size of the points dependent on the distance to the predicted value (so that larger residuals appear as bigger and less transparent). This distance is also represented by the length of the vertical lines.

d %>%
  sample_frac(0.5) %>% # selecting a (50%) subsample of the data
  ggplot(aes(x = weight, y = height) ) +
  geom_line(aes(y = pred_mod1), size = 1) +
  geom_point(aes(alpha = abs(res1), size = abs(res1) ) ) +
  guides(alpha = FALSE, size = FALSE) +
  geom_segment(aes(xend = weight, yend = pred_mod1, alpha = abs(res1) ) ) +
  theme_bw(base_size = 12)

If we take all the above verticals bars (i.e., the residuals) and plot their distribution, we can compare them to a normal distribution with mean 0, and standard deviation \(\sigma\) equal to the standard deviation of the residuals, to check our assumption about the distribution of the residuals (here a normality assumption).

d %>%
  ggplot(aes(x = res1) ) +
  geom_histogram(aes(y = ..density..), bins = 20, alpha = 0.6) +
  geom_line(aes(y = dnorm(res1, mean = 0, sd = sd(res1) ) ), size = 1) +
  guides(fill = FALSE) +
  theme_bw(base_size = 12) +
  labs(x = "Residuals", y = "Density")

To sum up, the distributional assumption of the linear model (Gaussian or whatever) concerns the distribution of the errors, that we do not know. Instead, we can evaluate the distribution of the residuals, which are the observed differences between the outcome variable and the predictions of the model.

Prediction uncertainty: the simulate function

Recall that our model is:

\[ \begin{aligned} \text{height}_{i} &\sim \mathrm{Normal}(\mu_{i}, \sigma) \\ \mu_{i} &= \alpha + \beta \cdot \text{weight}_{i} \\ \end{aligned} \]

So far, we used the linear part of the linear model (i.e., the second line) to make predictions about the \(\mu_{i}\), from \(\alpha\) and \(\beta\). In other words, we made predictions about mean values of the Gaussian distribution for a specific value of weight.

Now, if we want to generate actual data (i.e., the \(y_{i}\)), from our model, we still have to include \(\sigma\). This can be done “by-hand”, by adding error (i.e., some random variation) to our predictions.

d <- d %>%
  mutate(
    sim1 = coef(mod1)[1] + coef(mod1)[2] * d$weight + rnorm(1, 0, sigma(mod1) )
    )

d %>%
  select(height, weight, male, pred_mod1, sim1) %>%
  head
##    height   weight male pred_mod1     sim1
## 1 151.765 47.82561    1  157.1630 162.2150
## 2 139.700 36.48581    0  146.9001 151.9521
## 3 136.525 31.86484    0  142.7180 147.7700
## 4 156.845 53.04191    1  161.8839 166.9359
## 5 145.415 41.27687    0  151.2362 156.2882
## 6 163.830 62.99259    1  170.8895 175.9415

The simulate function allows to automate the previous step and to run it nsims times.

nsims <- 1e4
sims <- simulate(mod1, nsim = nsims) %>% data.frame

sims[1:6, 1:6]
##      sim_1    sim_2    sim_3    sim_4    sim_5    sim_6
## 1 155.9598 155.3906 156.7651 152.7330 153.6077 156.5931
## 2 146.8243 141.4573 135.7357 153.7677 142.5962 150.1732
## 3 137.6233 141.3779 143.2266 149.4296 149.9607 144.1135
## 4 173.9980 162.2460 170.4402 157.2107 158.7464 159.1776
## 5 150.3885 156.8790 152.9237 140.2885 156.0903 144.1566
## 6 178.6628 166.6504 162.7225 167.7171 173.9149 179.0282
lower_ci_sim <- apply(sims, 1, function(x) quantile(x, probs = 0.025) )
upper_ci_sim <- apply(sims, 1, function(x) quantile(x, probs = 0.975) )

sims_summary <- data.frame(
  lower = lower_ci_sim,
  upper = upper_ci_sim
  )

d %>%
  ggplot(aes(x = weight, y = height) ) +
  geom_smooth(method = "lm", color = "black") +
  geom_point(size = 2, alpha = 0.3) +
  geom_ribbon(
    data = sims_summary, inherit.aes = FALSE,
    aes(x = d$weight, ymin = lower, ymax = upper), alpha = 0.1
    ) +
  theme_bw(base_size = 12)

The first shaded region around the regression line represents the 95% confidence interval around the regression line, whereas the second broader shaded area represents the 95% central quantiles of the Gaussian distribution at each value of weight. These are computed by simulating nsims height values at each weight value, and by taking the 95% mean values of this distribution.

Conclusions

In this post, we tried to shed light on some concepts of the generalised linear model by examining the similarities and differences between four common R functions. It should be acknowledged that these functions (predict, fitted, residuals, and simulate) work with many different types of models, including models fitted with lme4 or brms. For instance, here is an overview of the available methods in my current session.

methods(predict)
##  [1] predict.ar*                predict.Arima*            
##  [3] predict.arima0*            predict.brmsfit*          
##  [5] predict.bs*                predict.bSpline*          
##  [7] predict.glm                predict.glmmPQL*          
##  [9] predict.gls*               predict.gnls*             
## [11] predict.HoltWinters*       predict.lda*              
## [13] predict.lm                 predict.lme*              
## [15] predict.lmList*            predict.lmList4*          
## [17] predict.loess*             predict.lqs*              
## [19] predict.mca*               predict.merMod*           
## [21] predict.mlm*               predict.nbSpline*         
## [23] predict.nlme*              predict.nls*              
## [25] predict.npolySpline*       predict.ns*               
## [27] predict.pbSpline*          predict.polr*             
## [29] predict.poly*              predict.polySpline*       
## [31] predict.ppolySpline*       predict.ppr*              
## [33] predict.prcomp*            predict.princomp*         
## [35] predict.qda*               predict.rlm*              
## [37] predict.smooth.spline*     predict.smooth.spline.fit*
## [39] predict.StructTS*         
## see '?methods' for accessing help and source code

References

Click to expand

Bürkner, P.-C. (2017). brms: An R package for bayesian multilevel models using Stan. Journal of Statistical Software, 80 (1), 1–28. doi:10.18637/jss.v080.i01

McElreath, R. (2016). Statistical Rethinking (p. 469). Chapman; Hall/CRC.

Rouder, J. N., Morey, R. D., & Wagenmakers, E.-J. (2016). The Interplay between Subjectivity, Statistical Practice, and Psychological Science. Collabra, 2(1), 1–12. doi:10.1525/collabra.28

Notes


  1. For more details on the logit model, please check this previous post.↩︎

Ladislas Nalborczyk
Ladislas Nalborczyk
Postdoctoral researcher

My main research interests span cognitive science at large and more specifically include motor cognition, motor imagery, overt and covert speech production, computational and statistical modelling, machine learning, and deep learning.

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