# 5 Multivariate Linear Models

McElreath’s listed reasons for multivariable regression include

- statistical control for confounds,
- multiple causation, and
- interactions.

We’ll approach the first two in this chapter. Interactions are reserved for Chapter 7.

#### 5.0.0.1 Rethinking: Causal inference.

“Despite its central importance, there is no unified approach to causal inference yet in the sciences or in statistics” (McElreath, 2015, p. 120). McElreath didn’t cover this much in this edition of the text. However, he peppered causal inference throughout his second edition (McElreath, 2020b). To dip into the topic, you might check out the recent blog post by Finn Lattimore and David Rohde, *Causal inference with Bayes rule*.

## 5.1 Spurious associations

Load the Waffle House data.

Unload rethinking and load brms and, while we’re at it, the tidyverse.

I’m not going to show the output, but you might go ahead and investigate the data with the typical functions. E.g.,

Now we have our data, we can reproduce Figure 5.1. One convenient way to get the handful of sate labels into the plot was with the `geom_text_repel()`

function from the ggrepel package (Slowikowski, 2020). But first, we spent the last few chapters warming up with ggplot2. Going forward, each chapter will have its own plot theme. In this chapter, we’ll characterize the plots with `theme_bw() + theme(panel.grid = element_rect())`

and coloring based off of `"firebrick"`

.

```
# install.packages("ggrepel", dependencies = T)
library(ggrepel)
d %>%
ggplot(aes(x = WaffleHouses/Population, y = Divorce)) +
stat_smooth(method = "lm", fullrange = T, size = 1/2,
color = "firebrick4", fill = "firebrick", alpha = 1/5) +
geom_point(size = 1.5, color = "firebrick4", alpha = 1/2) +
geom_text_repel(data = d %>% filter(Loc %in% c("ME", "OK", "AR", "AL", "GA", "SC", "NJ")),
aes(label = Loc),
size = 3, seed = 1042) + # this makes it reproducible
scale_x_continuous("Waffle Houses per million", limits = c(0, 55)) +
ylab("Divorce rate") +
coord_cartesian(xlim = c(0, 50), ylim = c(5, 15)) +
theme_bw() +
theme(panel.grid = element_blank())
```

Since these are geographically-based data, we might plot our three major variables in a map format. The urbnmapr package (Urban Institute, 2020) provides latitude and longitude data for the 50 states and the `geom_sf()`

function for plotting them. We’ll use the `left_join()`

function to combine those data with our primary data `d`

.

```
# devtools::install_github("UrbanInstitute/urbnmapr")
library(urbnmapr)
left_join(
# get the map data
get_urbn_map(map = "states", sf = TRUE),
# add the primary data
d %>%
# tandardize the three variables to put them all on the same scale
mutate(Divorce_z = (Divorce - mean(Divorce)) / sd(Divorce),
MedianAgeMarriage_z = (MedianAgeMarriage - mean(MedianAgeMarriage)) / sd(MedianAgeMarriage),
Marriage_z = (Marriage - mean(Marriage)) / sd(Marriage),
# convert the state names to characters to match the map data
state_name = Location %>% as.character()) %>%
select(Divorce_z:Marriage_z:state_name),
by = "state_name"
) %>%
# convert to the long format for faceting
pivot_longer(ends_with("_z")) %>%
# plot!
ggplot() +
geom_sf(aes(fill = value, geometry = geometry),
size = 0) +
scale_fill_gradient(low = "#f8eaea", high = "firebrick4") +
theme_void() +
theme(legend.position = "none",
strip.text = element_text(margin = margin(0, 0, .5, 0))) +
facet_wrap(~name)
```

One of the advantages of this visualization method is it just became clear that Nevada is missing from the `WaffleDivorce`

data. Execute `d %>% distinct(Location)`

to see for yourself and click here to find out why it’s missing. Those missing data should motivate the skills we’ll cover in Chapter 14. But let’s get back on track.

Here we’ll officially standardize the predictor, `MedianAgeMarriage`

.

```
d <-
d %>%
mutate(MedianAgeMarriage_s = (MedianAgeMarriage - mean(MedianAgeMarriage)) /
sd(MedianAgeMarriage))
```

Now we’re ready to fit the first univariable model.

```
b5.1 <-
brm(data = d,
family = gaussian,
Divorce ~ 1 + MedianAgeMarriage_s,
prior = c(prior(normal(10, 10), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.01")
```

Check the summary.

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: Divorce ~ 1 + MedianAgeMarriage_s
## Data: d (Number of observations: 50)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 9.69 0.22 9.26 10.11 1.00 4789 4092
## MedianAgeMarriage_s -1.04 0.21 -1.45 -0.62 1.00 5772 4345
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.52 0.16 1.24 1.88 1.00 4928 4064
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

We’ll employ `fitted()`

to make Figure 5.2.b. In preparation for `fitted()`

we’ll make a new tibble, `nd`

, composed of a handful of densely-packed values for our predictor, `MedianAgeMarriage_s`

. With the `newdata`

argument, we’ll use those values to return model-implied expected values for `Divorce`

.

```
# define the range of `MedianAgeMarriage_s` values we'd like to feed into `fitted()`
nd <- tibble(MedianAgeMarriage_s = seq(from = -3, to = 3.5, length.out = 30))
# now use `fitted()` to get the model-implied trajectories
f <-
fitted(b5.1, newdata = nd) %>%
as_tibble() %>%
# tack the `nd` data onto the `fitted()` results
bind_cols(nd)
# plot
ggplot(data = f,
aes(x = MedianAgeMarriage_s, y = Estimate)) +
geom_smooth(aes(ymin = Q2.5, ymax = Q97.5),
stat = "identity",
fill = "firebrick", color = "firebrick4", alpha = 1/5, size = 1/4) +
geom_point(data = d,
aes(y = Divorce),
size = 2, color = "firebrick4") +
ylab("Divorce") +
coord_cartesian(xlim = range(d$MedianAgeMarriage_s),
ylim = range(d$Divorce)) +
theme_bw() +
theme(panel.grid = element_blank())
```

Before fitting the next model, we’ll standardize `Marriage`

.

We’re ready to fit our second univariable model, this time with `Marriage_s`

as the predictor.

```
b5.2 <-
brm(data = d,
family = gaussian,
Divorce ~ 1 + Marriage_s,
prior = c(prior(normal(10, 10), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.02")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: Divorce ~ 1 + Marriage_s
## Data: d (Number of observations: 50)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 9.68 0.25 9.20 10.18 1.00 5481 4184
## Marriage_s 0.64 0.24 0.15 1.11 1.00 5931 4592
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.76 0.18 1.44 2.16 1.00 4304 3983
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

Now we’ll wangle and plot our version of Figure 5.2.a.

```
nd <- tibble(Marriage_s = seq(from = -2.5, to = 3.5, length.out = 30))
f <-
fitted(b5.2, newdata = nd) %>%
as_tibble() %>%
bind_cols(nd)
ggplot(data = f,
aes(x = Marriage_s, y = Estimate)) +
geom_smooth(aes(ymin = Q2.5, ymax = Q97.5),
stat = "identity",
fill = "firebrick", color = "firebrick4", alpha = 1/5, size = 1/4) +
geom_point(data = d,
aes(y = Divorce),
size = 2, color = "firebrick4") +
ylab("Divorce") +
coord_cartesian(xlim = range(d$Marriage_s),
ylim = range(d$Divorce)) +
theme_bw() +
theme(panel.grid = element_blank())
```

But merely comparing parameter means between different bivariate regressions is no way to decide which predictor is better Both of these predictors could provide independent value, or they could be redundant, or one could eliminate the value of the other. So we’ll build a multivariate model with the goal of measuring the partial value of each predictor. The question we want answered is:

What is the predictive value of a variable, once I already know all of the other predictor variables?(p. 123,emphasisin the original)

### 5.1.1 Multivariate notation.

Now we’ll get both predictors in there with our very first multivariable model. We can write the statistical model as

\[\begin{align*} \text{Divorce}_i & \sim \operatorname{Normal}(\mu_i, \sigma) \\ \mu_i & = \alpha + \beta_1 \text{Marriage_s}_i + \beta_2 \text{MedianAgeMarriage_s}_i \\ \alpha & \sim \operatorname{Normal}(10, 10) \\ \beta_1 & \sim \operatorname{Normal}(0, 1) \\ \beta_2 & \sim \operatorname{Normal}(0, 1) \\ \sigma & \sim \operatorname{Uniform}(0, 10). \end{align*}\]

It might help to read the \(+\) symbols as “or” and then say:

A State’s divorce rate can be a function of its marriage rate. The “or” indicates independent associations, which may be purely statistical or rather causal. (p. 124,orits median age at marriageemphasisin the original)

### 5.1.2 Fitting the model.

Much like we used the `+`

operator to add single predictors to the intercept, we just use more `+`

operators in the `formula`

argument to add more predictors. Also notice we’re using the same prior `prior(normal(0, 1), class = b)`

for both predictors. Within the brms framework, they are both of `class = b`

. But if we wanted their priors to differ, we’d make two `prior()`

statements and differentiate them with the `coef`

argument. You’ll see examples of that later on.

```
b5.3 <-
brm(data = d,
family = gaussian,
Divorce ~ 1 + Marriage_s + MedianAgeMarriage_s,
prior = c(prior(normal(10, 10), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.03")
```

Our multivariable summary will have multiple rows below the ‘Intercept’ row.

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: Divorce ~ 1 + Marriage_s + MedianAgeMarriage_s
## Data: d (Number of observations: 50)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 9.69 0.22 9.25 10.12 1.00 5725 4039
## Marriage_s -0.13 0.29 -0.70 0.46 1.00 3893 4305
## MedianAgeMarriage_s -1.12 0.29 -1.69 -0.55 1.00 4219 4563
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.52 0.16 1.24 1.88 1.00 5236 4082
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

The `mcmc_plot()`

function is an easy way to get a default coefficient plot. You just put the brmsfit object into the function.

There are numerous ways to make a coefficient plot. Another is with the `mcmc_intervals()`

function from the bayesplot package. A nice feature of the bayesplot package is its convenient way to alter the color scheme with the `color_scheme_set()`

function. Here, for example, we’ll make the theme `red`

. But note how the `mcmc_intervals()`

function requires you to work with the `posterior_samples()`

instead of the brmsfit object.

```
# install.packages("bayesplot", dependencies = T)
library(bayesplot)
post <- posterior_samples(b5.3)
color_scheme_set("red")
mcmc_intervals(post[, 1:4],
prob = .5,
point_est = "median") +
ggtitle("My fancy bayesplot-based coefficient plot") +
theme_bw() +
theme(axis.text.y = element_text(hjust = 0),
axis.ticks.y = element_blank(),
panel.grid = element_blank())
```

Because bayesplot produces a ggplot2 object, the plot was adjustable with familiar ggplot2 syntax. For more ideas, check out Gabry’s (2020b) vignette, *Plotting MCMC draws using the bayesplot package*.

The `tidybaes::stat_pointinterval()`

function offers a third way, this time with a more ground-up ggplot2 workflow.

```
library(tidybayes)
post %>%
select(b_Intercept:sigma) %>%
gather() %>%
ggplot(aes(x = value, y = reorder(key, value))) + # note how we used `reorder()` to arrange the coefficients
geom_vline(xintercept = 0, color = "firebrick4", alpha = 1/10) +
stat_pointinterval(point_interval = mode_hdi, .width = .95,
size = 3/4, color = "firebrick4") +
labs(title = "My tidybayes-based coefficient plot",
x = NULL, y = NULL) +
theme_bw() +
theme(axis.text.y = element_text(hjust = 0),
axis.ticks.y = element_blank(),
panel.grid = element_blank(),
panel.grid.major.y = element_line(color = alpha("firebrick4", 1/4), linetype = 3))
```

The substantive interpretation of all those coefficient plots is: “*Once we know median age at marriage for a State, there is little or no additive predictive power in also knowing the rate of marriage in that State*” (p. 126, *emphasis* in the original).

### 5.1.3 Plotting multivariate posteriors.

McElreath’s prose is delightfully deflationary. “There is a huge literature detailing a variety of plotting techniques that all attempt to help one understand multiple linear regression. None of these techniques is suitable for all jobs, and most do not generalize beyond linear regression” (p. 126). Now you’re inspired, let’s learn three:

- Predictor residual plots,
- Counterfactual plots, and
- Posterior prediction plots.

#### 5.1.3.1 Predictor residual plots.

To get ready to make our residual plots, we’ll predict `Marriage_s`

with `MedianAgeMarriage_s`

.

```
b5.4 <-
brm(data = d,
family = gaussian,
Marriage_s ~ 1 + MedianAgeMarriage_s,
prior = c(prior(normal(0, 10), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.04")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: Marriage_s ~ 1 + MedianAgeMarriage_s
## Data: d (Number of observations: 50)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.00 0.10 -0.20 0.20 1.00 5597 4332
## MedianAgeMarriage_s -0.71 0.10 -0.91 -0.51 1.00 5311 4407
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.72 0.08 0.59 0.88 1.00 4927 4109
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

With `fitted()`

, we compute the expected values for each state (with the exception of Nevada). Since the `MedianAgeMarriage_s`

values for each state are in the date we used to fit the model, we’ll omit the `newdata`

argument.

```
## # A tibble: 6 x 19
## Estimate Est.Error Q2.5 Q97.5 Location Loc Population MedianAgeMarria… Marriage Marriage.SE
## <dbl> <dbl> <dbl> <dbl> <fct> <fct> <dbl> <dbl> <dbl> <dbl>
## 1 0.430 0.121 0.189 0.674 Alabama AL 4.78 25.3 20.2 1.27
## 2 0.487 0.126 0.240 0.740 Alaska AK 0.71 25.2 26 2.93
## 3 0.142 0.106 -0.0657 0.355 Arizona AZ 6.33 25.8 20.3 0.98
## 4 1.00 0.178 0.650 1.35 Arkansas AR 2.92 24.3 26.4 1.7
## 5 -0.432 0.120 -0.667 -0.198 Califor… CA 37.2 26.8 19.1 0.39
## 6 0.200 0.108 -0.0140 0.418 Colorado CO 5.03 25.7 23.5 1.24
## # … with 9 more variables: Divorce <dbl>, Divorce.SE <dbl>, WaffleHouses <int>, South <int>,
## # Slaves1860 <int>, Population1860 <int>, PropSlaves1860 <dbl>, MedianAgeMarriage_s <dbl>,
## # Marriage_s <dbl>
```

After a little data processing, we can make Figure 5.3.

```
f %>%
ggplot(aes(x = MedianAgeMarriage_s, y = Marriage_s)) +
geom_point(size = 2, shape = 1, color = "firebrick4") +
geom_segment(aes(xend = MedianAgeMarriage_s, yend = Estimate),
size = 1/4) +
geom_line(aes(y = Estimate),
color = "firebrick4") +
coord_cartesian(ylim = range(d$Marriage_s)) +
theme_bw() +
theme(panel.grid = element_blank())
```

We get the residuals with the well-named `residuals()`

function. Much like with `brms::fitted()`

, `brms::residuals()`

returns a four-vector matrix with the number of rows equal to the number of observations in the original data (by default, anyway). The vectors have the familiar names: `Estimate`

, `Est.Error`

, `Q2.5`

, and `Q97.5`

. See the brms reference manual (Bürkner, 2020b) for details.

With our residuals in hand, we just need a little more data processing to make Figure 5.4.a.

```
r <-
residuals(b5.4) %>%
# to use this in ggplot2, we need to make it a tibble or data frame
as_tibble() %>%
bind_cols(d)
# for the annotation at the top
text <-
tibble(Estimate = c(- 0.5, 0.5),
Divorce = 14.1,
label = c("slower", "faster"))
# plot
r %>%
ggplot(aes(x = Estimate, y = Divorce)) +
stat_smooth(method = "lm", fullrange = T,
color = "firebrick4", fill = "firebrick4",
alpha = 1/5, size = 1/2) +
geom_vline(xintercept = 0, linetype = 2, color = "grey50") +
geom_point(size = 2, color = "firebrick4", alpha = 2/3) +
geom_text(data = text,
aes(label = label)) +
scale_x_continuous("Marriage rate residuals", limits = c(-2, 2)) +
coord_cartesian(xlim = range(r$Estimate),
ylim = c(6, 14.1)) +
theme_bw() +
theme(panel.grid = element_blank())
```

To get the `MedianAgeMarriage_s`

residuals, we have to fit the corresponding model first.

```
b5.4b <-
brm(data = d,
family = gaussian,
MedianAgeMarriage_s ~ 1 + Marriage_s,
prior = c(prior(normal(0, 10), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.04b")
```

And now we’ll get the new batch of residuals, do a little data processing, and make a plot corresponding to Figure 5.4.b.

```
# redefine the annotation data
text <-
tibble(Estimate = c(- 0.7, 0.5),
Divorce = 14.1,
label = c("younger", "older"))
# extract the residuals
residuals(b5.4b) %>%
as_tibble() %>%
bind_cols(d) %>%
# plot
ggplot(aes(x = Estimate, y = Divorce)) +
stat_smooth(method = "lm", fullrange = T,
color = "firebrick4", fill = "firebrick4",
alpha = 1/5, size = 1/2) +
geom_vline(xintercept = 0, linetype = 2, color = "grey50") +
geom_point(size = 2, color = "firebrick4", alpha = 2/3) +
geom_text(data = text,
aes(label = label)) +
scale_x_continuous("Age of marriage residuals", limits = c(-2, 3)) +
coord_cartesian(xlim = range(r$Estimate),
ylim = c(6, 14.1)) +
theme_bw() +
theme(panel.grid = element_blank())
```

So what’s the point of all this? There’s direct value in seeing the model-based predictions displayed against the outcome, after subtracting out the influence of other predictors. The plots in Figure 5.4 do this. But this procedure also brings home the message that regression models answer with the remaining association of each predictor with the outcome, after already knowing the other predictors. In computing the predictor residual plots, you had to perform those calculations yourself. In the unified [multivariable] model, it all happens automatically. (p. 129)

#### 5.1.3.2 Counterfactual plots.

A second sort of inferential plot displays the implied predictions of the model. I call these plots

counterfactual, because they can be produced for any values of the predictor variable you like, even unobserved or impossible combinations like very high median age of marriage and very high marriage rate. There are no States with this combination, but in a counterfactual plot, you can ask the model for a prediction for such a State. (p. 129,emphasisin the original)

Making Figure 5.5.a requires a little more data wrangling than before.

```
# we need new `nd` data
nd <-
tibble(Marriage_s = seq(from = -3, to = 3, length.out = 30),
MedianAgeMarriage_s = mean(d$MedianAgeMarriage_s))
fitted(b5.3, newdata = nd) %>%
as_tibble() %>%
# since `fitted()` and `predict()` name their intervals the same way,
# we'll need to `rename()` them to keep them straight
rename(f_ll = Q2.5,
f_ul = Q97.5) %>%
# note how we're just nesting the `predict()` code right inside `bind_cols()`
bind_cols(
predict(b5.3, newdata = nd) %>%
as_tibble() %>%
# since we only need the intervals, we'll use `transmute()` rather than `mutate()`
transmute(p_ll = Q2.5,
p_ul = Q97.5),
# now tack on the `nd` data
nd) %>%
# we're finally ready to plot
ggplot(aes(x = Marriage_s, y = Estimate)) +
geom_ribbon(aes(ymin = p_ll, ymax = p_ul),
fill = "firebrick", alpha = 1/5) +
geom_smooth(aes(ymin = f_ll, ymax = f_ul),
stat = "identity",
fill = "firebrick", color = "firebrick4", alpha = 1/5, size = 1/4) +
labs(subtitle = "Counterfactual plot for which\nMedianAgeMarriage_s = 0",
y = "Divorce") +
coord_cartesian(xlim = range(d$Marriage_s),
ylim = c(6, 14)) +
theme_bw() +
theme(panel.grid = element_blank())
```

We follow the same process for Figure 5.5.b.

```
# new data
nd <-
tibble(MedianAgeMarriage_s = seq(from = -3, to = 3.5, length.out = 30),
Marriage_s = mean(d$Marriage_s))
# `fitted()` + `predict()`
fitted(b5.3, newdata = nd) %>%
as_tibble() %>%
rename(f_ll = Q2.5,
f_ul = Q97.5) %>%
bind_cols(
predict(b5.3, newdata = nd) %>%
as_tibble() %>%
transmute(p_ll = Q2.5,
p_ul = Q97.5),
nd
) %>%
# plot
ggplot(aes(x = MedianAgeMarriage_s, y = Estimate)) +
geom_ribbon(aes(ymin = p_ll, ymax = p_ul),
fill = "firebrick", alpha = 1/5) +
geom_smooth(aes(ymin = f_ll, ymax = f_ul),
stat = "identity",
fill = "firebrick", color = "firebrick4", alpha = 1/5, size = 1/4) +
labs(subtitle = "Counterfactual plot for which\nMarriage_s = 0",
y = "Divorce") +
coord_cartesian(xlim = range(d$MedianAgeMarriage_s),
ylim = c(6, 14)) +
theme_bw() +
theme(panel.grid = element_blank())
```

A tension with such plots, however, lies in their counterfactual nature. In the small world of the model, it is possible to change median age of marriage without also changing the marriage rate. But is this also possible in the large world of reality? Probably not….

…If our goal is to intervene in the world, there may not be any realistic way to manipulate each predictor without also manipulating the others. This is a serious obstacle to applied science, whether you are an ecologist, an economist, or an epidemiologist [or a psychologist] (p. 131)

#### 5.1.3.3 Posterior prediction plots.

“In addition to understanding the estimates, it’s important to check the model fit against the observed data” (p. 131). For more on the topic, check out Gabry and colleagues’ (2019) *Visualization in Bayesian workflow* or Simpson’s related blog post, *Touch me, I want to feel your data*.

In this version of Figure 5.6.a, the thin lines are the 95% intervals and the thicker lines are \(\pm\) the posterior \(SD\), both of which are returned when you use `fitted()`

.

```
p1 <-
fitted(b5.3) %>%
as_tibble() %>%
bind_cols(d) %>%
ggplot(aes(x = Divorce, y = Estimate)) +
geom_abline(linetype = 2, color = "grey50", size = .5) +
geom_point(size = 1.5, color = "firebrick4", alpha = 3/4) +
geom_linerange(aes(ymin = Q2.5, ymax = Q97.5),
size = 1/4, color = "firebrick4") +
geom_linerange(aes(ymin = Estimate - Est.Error,
ymax = Estimate + Est.Error),
size = 1/2, color = "firebrick4") +
# Note our use of the dot placeholder, here: https://magrittr.tidyverse.org/reference/pipe.html
geom_text(data = . %>% filter(Loc %in% c("ID", "UT")),
aes(label = Loc),
hjust = 0, nudge_x = - 0.65) +
labs(x = "Observed divorce",
y = "Predicted divorce") +
theme_bw() +
theme(panel.grid = element_blank())
p1
```

In order to make Figure 5.6.b, we need to clarify the relationships among `fitted()`

, `predict()`

, and `residuals()`

. Here’s my attempt in a table.

```
tibble(`brms function` = c("fitted", "predict", "residual"),
mean = c("same as the data", "same as the data", "in a deviance-score metric"),
scale = c("excludes sigma", "includes sigma", "excludes sigma")) %>%
knitr::kable()
```

brms function | mean | scale |
---|---|---|

fitted | same as the data | excludes sigma |

predict | same as the data | includes sigma |

residual | in a deviance-score metric | excludes sigma |

Hopefully that clarified that if we want to incorporate the prediction interval in a deviance metric, we’ll need to first use `predict()`

and then subtract the intervals from their corresponding `Divorce`

values in the data.

```
p2 <-
residuals(b5.3) %>%
as_tibble() %>%
rename(f_ll = Q2.5,
f_ul = Q97.5) %>%
bind_cols(
predict(b5.3) %>%
as_tibble() %>%
transmute(p_ll = Q2.5,
p_ul = Q97.5),
d
) %>%
# here we put our `predict()` intervals into a deviance metric
mutate(p_ll = Divorce - p_ll,
p_ul = Divorce - p_ul) %>%
# now plot!
ggplot(aes(x = reorder(Loc, Estimate), y = Estimate)) +
geom_hline(yintercept = 0, size = 1/2,
color = "firebrick4", alpha = 1/10) +
geom_pointrange(aes(ymin = f_ll, ymax = f_ul),
size = 2/5, shape = 20, color = "firebrick4") +
geom_segment(aes(y = Estimate - Est.Error,
yend = Estimate + Est.Error,
x = Loc,
xend = Loc),
size = 1, color = "firebrick4") +
geom_segment(aes(y = p_ll, yend = p_ul,
x = Loc, xend = Loc),
size = 3, color = "firebrick4", alpha = 1/10) +
labs(x = NULL, y = NULL) +
coord_flip(ylim = c(-6, 5)) +
theme_bw() +
theme(axis.text.y = element_text(hjust = 0),
axis.ticks.y = element_blank(),
panel.grid = element_blank())
p2
```

Compared to the last couple plots, Figure 5.6.c is pretty simple.

```
p3 <-
residuals(b5.3) %>%
as_tibble() %>%
bind_cols(d) %>%
mutate(wpc = WaffleHouses / Population) %>%
ggplot(aes(x = wpc, y = Estimate)) +
geom_point(size = 1.5, color = "firebrick4", alpha = 1/2) +
stat_smooth(method = "lm", fullrange = T,
color = "firebrick4", size = 1/2,
fill = "firebrick", alpha = 1/5) +
geom_text_repel(data = . %>% filter(Loc %in% c("ME", "AR", "MS", "AL", "GA", "SC", "ID")),
aes(label = Loc),
seed = 5.6) +
scale_x_continuous("Waffles per capita", limits = c(0, 45)) +
ylab("Divorce error") +
coord_cartesian(xlim = range(0, 40)) +
theme_bw() +
theme(panel.grid = element_blank())
p3
```

For the sake of good practice, let’s use patchwork syntax to combine those three subplots like they appear in the text.

```
library(patchwork)
((p1 / p3) | p2) + plot_annotation(tag_levels = "a",
tag_prefix = "(",
tag_suffix = ")")
```

More McElreath inspiration: “No matter how many predictors you’ve already included in a regression, it’s still possible to find spurious correlations with the remaining variation” (p. 134).

##### 5.1.3.3.1 Rethinking: Stats, huh, yeah what is it good for?

To keep our deflation train going, it’s worthwhile repeating this message:

Often people want statistical modeling to do things that statistical modeling cannot do. For example, we’d like to know whether an effect is real or rather spurious. Unfortunately, modeling merely quantifies uncertainty in the precise way that the model understands the problem. Usually answers to large world questions about truth and causation depend upon information not included in the model. For example, any observed correlation between an outcome and predictor could be eliminated or reversed once another predictor is added to the model. But if we cannot think of another predictor, we might never notice this. Therefore all statistical models are vulnerable to and demand critique, regardless of the precision of their estimates and apparent accuracy of their predictions. (p. 134)

##### 5.1.3.3.2 Overthinking: Simulating spurious association.

Simulate the spurious predictor data.

```
n <- 100 # number of cases
set.seed(5) # setting the seed makes the results reproducible
d <-
tibble(x_real = rnorm(n), # x_real as Gaussian with mean 0 and SD 1 (i.e., the defaults)
x_spur = rnorm(n, x_real), # x_spur as Gaussian with mean = x_real
y = rnorm(n, x_real)) # y as Gaussian with mean = x_real
```

Here are the quick `pairs()`

plots.

We may as well fit a model.

```
b5.0_spur <-
brm(data = d,
family = gaussian,
y ~ 1 + x_real + x_spur,
prior = c(prior(normal(0, 10), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.00_spur")
```

```
## Estimate Est.Error Q2.5 Q97.5
## Intercept 0.00 0.10 -0.19 0.19
## x_real 0.98 0.15 0.68 1.28
## x_spur 0.06 0.10 -0.13 0.25
```

## 5.2 Masked relationship

A second reason to use more than one predictor variable is to measure the direct influences of multiple factors on an outcome, when none of those influences is apparent from bivariate relationships. This kind of problem tends to arise when there are two predictor variables that are correlated with one another. However, one of these is positively correlated with the outcome and the other is negatively correlated with it. (p. 134)

Let’s load the Hinde & Milligan (2011) milk data.

Unload rethinking and load brms.

You might inspect the data like this.

By just looking at that mess, do you think you could describe the associations of `mass`

and `neocortex.perc`

with the criterion, `kcal.per.g`

? I couldn’t. It’s a good thing we have math.

McElreath has us start of with a simple univariable `milk`

model.

```
b5.5 <-
brm(data = d,
family = gaussian,
kcal.per.g ~ 1 + neocortex.perc,
prior = c(prior(normal(0, 100), class = Intercept),
prior(normal(0, 1), class = b),
prior(cauchy(0, 1), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.05")
```

The uniform prior was difficult on Stan. After playing around a bit, I just switched to a unit-scale half Cauchy. Similar to the rethinking example in the text, brms warned that “Rows containing NAs were excluded from the model.” This isn’t necessarily a problem; the model fit just fine. But we should be ashamed of ourselves and look eagerly forward to Chapter 14 where we’ll learn how to do better.

To compliment how McElreath removed cases with missing values on our variables of interest with base R `complete.cases()`

, here we’ll do so with `tidyr::drop_na()`

and a little help with `ends_with()`

.

But anyway, let’s inspect the parameter summary.

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: kcal.per.g ~ 1 + neocortex.perc
## Data: d (Number of observations: 17)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 0.355 0.555 -0.741 1.432 1.000 5715 4132
## neocortex.perc 0.004 0.008 -0.012 0.021 1.000 5697 3989
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.193 0.040 0.134 0.286 1.000 3012 3091
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

Did you notice now we set `digits = 3`

within `print()`

much the way McElreath set `digits=3`

within `precis()`

?

To get the brms answer to what McElreath did with `coef()`

, we can use the `fixef()`

function.

`## [1] 0.09388076`

Yes, indeed, “that’s less than 0.1 kilocalories” (p. 137).

Just for kicks, we’ll superimpose 50% intervals atop 95% intervals for the next few plots. Here’s Figure 5.7, top left.

```
nd <- tibble(neocortex.perc = 54:80)
fitted(b5.5,
newdata = nd,
probs = c(.025, .975, .25, .75)) %>%
as_tibble() %>%
bind_cols(nd) %>%
ggplot(aes(x = neocortex.perc, y = Estimate)) +
geom_ribbon(aes(ymin = Q2.5, ymax = Q97.5),
fill = "firebrick", alpha = 1/5) +
geom_smooth(aes(ymin = Q25, ymax = Q75),
stat = "identity",
fill = "firebrick4", color = "firebrick4", alpha = 1/5, size = 1/2) +
geom_point(data = dcc,
aes(y = kcal.per.g),
size = 2, color = "firebrick4") +
ylab("kcal.per.g") +
coord_cartesian(xlim = range(dcc$neocortex.perc),
ylim = range(dcc$kcal.per.g)) +
theme_bw() +
theme(panel.grid = element_blank())
```

Do note the `probs`

argument in the `fitted()`

code, above. Let’s make the `log_mass`

variable.

Now we use `log_mass`

as the new sole predictor.

```
b5.6 <-
brm(data = dcc,
family = gaussian,
kcal.per.g ~ 1 + log_mass,
prior = c(prior(normal(0, 100), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 1), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.06")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: kcal.per.g ~ 1 + log_mass
## Data: dcc (Number of observations: 17)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 0.704 0.058 0.590 0.817 1.000 4923 3220
## log_mass -0.032 0.024 -0.080 0.015 1.001 4201 3299
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.182 0.037 0.126 0.270 1.000 3767 3668
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

Make Figure 5.7, top right.

```
nd <- tibble(log_mass = seq(from = -2.5, to = 5, length.out = 30))
fitted(b5.6,
newdata = nd,
probs = c(.025, .975, .25, .75)) %>%
as_tibble() %>%
bind_cols(nd) %>%
ggplot(aes(x = log_mass, y = Estimate)) +
geom_ribbon(aes(ymin = Q2.5, ymax = Q97.5),
fill = "firebrick", alpha = 1/5) +
geom_smooth(aes(ymin = Q25, ymax = Q75),
stat = "identity",
fill = "firebrick4", color = "firebrick4", alpha = 1/5, size = 1/2) +
geom_point(data = dcc,
aes(y = kcal.per.g),
size = 2, color = "firebrick4") +
ylab("kcal.per.g") +
coord_cartesian(xlim = range(dcc$log_mass),
ylim = range(dcc$kcal.per.g)) +
theme_bw() +
theme(panel.grid = element_blank())
```

Finally, we’re ready to fit with both predictors included in the “joint model.” Here’s the statistical formula:

\[\begin{align*} \text{kcal.per.g}_i & \sim \operatorname{Normal}(\mu_i, \sigma) \\ \mu_i & = \alpha + \beta_1 \text{neocortex.perc}_i + \beta_2 \log (\text{mass}_i) \\ \alpha & \sim \operatorname{Normal}(0, 100) \\ \beta_1 & \sim \operatorname{Normal}(0, 1) \\ \beta_2 & \sim \operatorname{Normal}(0, 1) \\ \sigma & \sim \operatorname{Uniform}(0, 1). \end{align*}\]

Fit the model.

```
b5.7 <-
brm(data = dcc,
family = gaussian,
kcal.per.g ~ 1 + neocortex.perc + log_mass,
prior = c(prior(normal(0, 100), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 1), class = sigma)),
iter = 2000, warmup = 1000, chains = 4, cores = 4,
control = list(adapt_delta = .9),
seed = 5,
file = "fits/b05.07")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: kcal.per.g ~ 1 + neocortex.perc + log_mass
## Data: dcc (Number of observations: 17)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -1.077 0.583 -2.212 0.119 1.000 2059 1914
## neocortex.perc 0.028 0.009 0.009 0.045 1.000 1999 1858
## log_mass -0.096 0.028 -0.148 -0.038 1.001 1702 1794
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.140 0.030 0.093 0.212 1.000 1199 1171
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

Make Figure 5.7, bottom left.

```
nd <-
tibble(neocortex.perc = 54:80 %>% as.double(),
log_mass = mean(dcc$log_mass))
p1 <-
b5.7 %>%
fitted(newdata = nd,
probs = c(.025, .975, .25, .75)) %>%
as_tibble() %>%
bind_cols(nd) %>%
ggplot(aes(x = neocortex.perc, y = Estimate)) +
geom_ribbon(aes(ymin = Q2.5, ymax = Q97.5),
fill = "firebrick", alpha = 1/5) +
geom_smooth(aes(ymin = Q25, ymax = Q75),
stat = "identity",
fill = "firebrick4", color = "firebrick4", alpha = 1/5, size = 1/2) +
geom_point(data = dcc,
aes(y = kcal.per.g),
size = 2, color = "firebrick4") +
ylab("kcal.per.g") +
coord_cartesian(xlim = range(dcc$neocortex.perc),
ylim = range(dcc$kcal.per.g))
```

Now make Figure 5.7, bottom right, and combine.

```
nd <-
tibble(log_mass = seq(from = -2.5, to = 5, length.out = 30),
neocortex.perc = mean(dcc$neocortex.perc))
p2 <-
b5.7 %>%
fitted(newdata = nd,
probs = c(.025, .975, .25, .75)) %>%
as_tibble() %>%
bind_cols(nd) %>%
ggplot(aes(x = log_mass, y = Estimate)) +
geom_ribbon(aes(ymin = Q2.5, ymax = Q97.5),
fill = "firebrick", alpha = 1/5) +
geom_smooth(aes(ymin = Q25, ymax = Q75),
stat = "identity",
fill = "firebrick4", color = "firebrick4", alpha = 1/5, size = 1/2) +
geom_point(data = dcc,
aes(y = kcal.per.g),
size = 2, color = "firebrick4") +
ylab("kcal.per.g") +
coord_cartesian(xlim = range(dcc$log_mass),
ylim = range(dcc$kcal.per.g))
(p1 | p2) &
theme_bw() &
theme(panel.grid = element_blank())
```

What [this regression model did was] ask if species that have high neocortex percent

for their body masshave higher milk energy. Likewise, the model [asked] if species with high body massfor their neocortex percenthave higher milk energy. Bigger species, like apes, have milk with less energy. But species with more neocortex tend to have richer milk. The fact that these two variables, body size and neocortex, are correlated across species makes it hard to see these relationships, unless we statistically account for both. (pp. 140–141,emphasisin the original)

#### 5.2.0.1 Overthinking: Simulating a masking relationship.

Simulate the data.

```
n <- 100 # number of cases
rho <- .7 # correlation between x_pos and x_neg
set.seed(5) # setting the seed makes the results reproducible
d <-
tibble(x_pos = rnorm(n), # x_pos as a standard Gaussian
x_neg = rnorm(n, rho * x_pos, sqrt(1 - rho^2)), # x_neg correlated with x_pos
y = rnorm(n, x_pos - x_neg)) # y equally associated with x_pos and x_neg
```

Here are the quick `pairs()`

plots.

Here we fit the models with a little help from the `update()`

function.

```
b5.0_both <-
brm(data = d, family = gaussian,
y ~ 1 + x_pos + x_neg,
prior = c(prior(normal(0, 100), class = Intercept),
prior(normal(0, 1), class = b),
prior(cauchy(0, 1), class = sigma)),
seed = 5,
file = "fits/b05.00_both")
b5.0_pos <-
update(b5.0_both,
formula = y ~ 1 + x_pos,
seed = 5,
file = "fits/b05.00_pos")
b5.0_neg <-
update(b5.0_both,
formula = y ~ 1 + x_neg,
seed = 5,
file = "fits/b05.00_neg")
```

Compare the coefficients.

```
## Estimate Est.Error Q2.5 Q97.5
## Intercept -0.02 0.12 -0.26 0.21
## x_pos 0.26 0.13 0.02 0.51
```

```
## Estimate Est.Error Q2.5 Q97.5
## Intercept 0.01 0.12 -0.23 0.24
## x_neg -0.29 0.11 -0.51 -0.07
```

```
## Estimate Est.Error Q2.5 Q97.5
## Intercept -0.01 0.10 -0.20 0.19
## x_pos 0.96 0.15 0.66 1.25
## x_neg -0.89 0.13 -1.15 -0.63
```

If you move the value of

`rho`

closer to zero, this masking phenomenon will diminish. I you make`rho`

closer to 1 or -1, it will magnify. But if`rho`

gets very close to 1 or -1, then the two predictors contain exactly the same information, and there’s no hope for any statistical model to tease out the true underlying association used in the simulation. (p. 141)

## 5.3 Multicollinearity

Multicollinearity means very strong correlation between two or more predictor variables. The consequence of it is that the posterior distribution will say that a very large range of parameter values are plausible, from tiny associations to massive ones, even if all of the variables are in reality strongly associated with the outcome. This frustrating phenomenon arises from the details of how statistical control works. So once you understand multicollinearity, you will better understand [multivariable] models in general. (pp. 141–142)

### 5.3.1 Multicollinear legs.

Let’s simulate some leg data.

```
n <- 100
set.seed(5)
d <-
tibble(height = rnorm(n, mean = 10, sd = 2),
leg_prop = runif(n, min = 0.4, max = 0.5)) %>%
mutate(leg_left = leg_prop * height + rnorm(n, mean = 0, sd = 0.02),
leg_right = leg_prop * height + rnorm(n, mean = 0, sd = 0.02))
```

`leg_left`

and `leg_right`

are **highly** correlated.

```
## leg_left leg_right
## leg_left 1.0000 0.9996
## leg_right 0.9996 1.0000
```

Have you ever even seen a \(\rho = .9996\) correlation, before? Here it is in a plot.

```
d %>%
ggplot(aes(x = leg_left, y = leg_right)) +
geom_point(alpha = 1/2, color = "firebrick4") +
theme_bw() +
theme(panel.grid = element_blank())
```

Here’s our attempt to predict `height`

with both legs in the model.

```
b5.8 <-
brm(data = d,
family = gaussian,
height ~ 1 + leg_left + leg_right,
prior = c(prior(normal(10, 100), class = Intercept),
prior(normal(2, 10), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.08")
```

Let’s inspect the damage.

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: height ~ 1 + leg_left + leg_right
## Data: d (Number of observations: 100)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.79 0.29 1.21 2.36 1.00 5295 3792
## leg_left 0.80 2.15 -3.31 5.01 1.00 1982 2348
## leg_right 1.05 2.15 -3.18 5.16 1.00 1978 2329
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.60 0.04 0.52 0.70 1.00 3479 3162
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

That ‘Est.Error’ column isn’t looking too good. But it’s easy to miss that, which is why McElreath suggested “a graphical view of the [output] is more useful because it displays the posterior [estimates] and [intervals] in a way that allows us with a glance to see that something has gone wrong here” (p. 143).

Here’s our coefficient plot using `brms::mcmc_plot()`

.

```
mcmc_plot(b5.8,
type = "intervals",
prob = .5,
prob_outer = .95,
point_est = "median") +
labs(title = "The coefficient plot for the two-leg model",
subtitle = "Holy smokes; look at the widths of those betas!") +
theme_bw() +
theme(text = element_text(size = 14),
axis.text.y = element_text(hjust = 0),
axis.ticks.y = element_blank(),
panel.grid = element_blank())
```

This is perhaps the simplest way to plot the bivariate posterior of our two predictor coefficients, Figure 6.2.a.

If you’d like a nicer and more focused attempt, you might have to revert to the `posterior_samples()`

function and a little ggplot2 code.

```
post <- posterior_samples(b5.8)
post %>%
ggplot(aes(x = b_leg_left, y = b_leg_right)) +
geom_point(color = "firebrick", alpha = 1/10, size = 1/3) +
theme_bw() +
theme(panel.grid = element_blank())
```

While we’re at it, you can make a similar plot with the `mcmc_scatter()`

function.

```
post %>%
mcmc_scatter(pars = c("b_leg_left", "b_leg_right"),
size = 1/3,
alpha = 1/10) +
theme_bw() +
theme(panel.grid = element_blank())
```

But wow, those coefficients look about as highly correlated as the predictors, just with the reversed sign.

```
## b_leg_left b_leg_right
## b_leg_left 1.0000000 -0.9995765
## b_leg_right -0.9995765 1.0000000
```

On page 165, McElreath clarified that “from the computer’s perspective, this model is simply:”

\[\begin{align*} y_i & \sim \operatorname{Normal}(\mu_i, \sigma) \\ \mu_i & = \alpha + (\beta_1 + \beta_2) x_i. \end{align*}\]

Accordingly, here’s the posterior of the sum of the two regression coefficients, Figure 6.2.b. We’ll use `tidybayes::stat_halfeye()`

to both plot the density and mark off the posterior median and percentile-based 95% probability intervals at its base.

```
library(tidybayes)
post %>%
ggplot(aes(x = b_leg_left + b_leg_right, y = 0)) +
stat_halfeye(point_interval = median_qi, .width = .95,
fill = "firebrick") +
scale_y_continuous(NULL, breaks = NULL) +
labs(title = "Sum the multicollinear coefficients",
subtitle = "Marked by the median and 95% PIs") +
theme_bw() +
theme(panel.grid = element_blank())
```

Now we fit the model after ditching one of the leg lengths.

```
b5.9 <-
brm(data = d,
family = gaussian,
height ~ 1 + leg_left,
prior = c(prior(normal(10, 100), class = Intercept),
prior(normal(2, 10), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.09")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: height ~ 1 + leg_left
## Data: d (Number of observations: 100)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.79 0.29 1.23 2.35 1.00 6544 4807
## leg_left 1.84 0.06 1.72 1.96 1.00 6592 4788
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.60 0.04 0.52 0.70 1.00 4792 4086
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

That posterior \(SD\) looks much better. Compare this density to the one in Figure 6.1.b.

```
posterior_samples(b5.9) %>%
ggplot(aes(x = b_leg_left, y = 0)) +
stat_halfeye(point_interval = median_qi, .width = .95,
fill = "firebrick") +
scale_y_continuous(NULL, breaks = NULL) +
labs(title = "Just one coefficient needed",
subtitle = "Marked by the median and 95% PIs",
x = "only b_leg_left, this time") +
theme_bw() +
theme(panel.grid = element_blank())
```

We might also use `tidybayes::stat_pointinterval()`

to compare the posterior of `b_leg_left`

from `b5.9`

to the joint posterior `b_leg_left + b_leg_right`

from `b5.8`

. For kicks, we’ll depict the posteriors with `tidybayes::stat_gradientinterval()`

and use three levels of posterior intervals.

```
bind_cols(post %>% transmute(`b_leg_left + b_leg_right` = b_leg_left + b_leg_right),
posterior_samples(b5.9) %>% transmute(`only b_leg_left` = b_leg_left)) %>%
gather() %>%
ggplot(aes(x = value, y = key)) +
stat_gradientinterval(.width = c(.5, .8, .95),
fill = "firebrick") +
labs(x = NULL, y = NULL) +
theme_bw() +
theme(axis.text.y = element_text(hjust = 0),
axis.ticks.y = element_blank(),
panel.grid = element_blank())
```

The results are within simulation variance of one another.

When two predictor variables are very strongly correlated, including both in a model may lead to confusion.The posterior distribution isn’t wrong, in such a case. It’s telling you that the question you asked cannot be answered with these data. And that’s a great thing for a model to say, that it cannot answer your question. (p. 145,emphasisin the original)

### 5.3.2 Multicollinear `milk`

.

Multicollinearity arises in real data, too.

Unload rethinking and load brms.

We’ll follow the text and fit the two univariable models, first. Note our use of the `update()`

function.

```
# `kcal.per.g` regressed on `perc.fat`
b5.10 <-
brm(data = d,
family = gaussian,
kcal.per.g ~ 1 + perc.fat,
prior = c(prior(normal(.6, 10), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.10")
# `kcal.per.g` regressed on `perc.lactose`
b5.11 <-
update(b5.10,
newdata = d,
formula = kcal.per.g ~ 1 + perc.lactose,
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.11")
```

Compare the coefficient summaries.

```
## Estimate Est.Error Q2.5 Q97.5
## b_Intercept 0.301 0.039 0.226 0.380
## b_perc.fat 0.010 0.001 0.008 0.012
## sigma 0.080 0.012 0.061 0.107
```

```
## Estimate Est.Error Q2.5 Q97.5
## b_Intercept 1.167 0.046 1.078 1.259
## b_perc.lactose -0.011 0.001 -0.012 -0.009
## sigma 0.067 0.010 0.051 0.088
```

If you’d like to get just the 95% intervals similar to the way McElreath reported them in the prose on page 146, you might use the handy `posterior_interval()`

function.

```
## 2.5% 97.5%
## 0.008 0.012
```

```
## 2.5% 97.5%
## -0.012 -0.009
```

Now “watch what happens when we place both predictor variables in the same regression model” (p. 146).

```
b5.12 <-
update(b5.11,
newdata = d,
formula = kcal.per.g ~ 1 + perc.fat + perc.lactose,
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.12")
```

The posteriors for coefficients, especially for `perc.fat`

, shrank to zero.

```
## Estimate Est.Error Q2.5 Q97.5
## b_Intercept 1.006 0.223 0.584 1.451
## b_perc.fat 0.002 0.003 -0.003 0.007
## b_perc.lactose -0.009 0.003 -0.014 -0.004
## sigma 0.068 0.010 0.052 0.091
```

You can make also pairs plots with GGally (Schloerke et al., 2020), which will also compute the point estimates for the bivariate correlations. Here’s a default plot.

```
#install.packages("GGally", dependencies = T)
library(GGally)
ggpairs(data = d, columns = c(3:4, 6)) +
theme(panel.grid = element_blank())
```

But you can customize these, too. E.g.,

```
my_diag <- function(data, mapping, ...) {
ggplot(data = data, mapping = mapping) +
geom_density(fill = "firebrick4", size = 0)
}
my_lower <- function(data, mapping, ...) {
ggplot(data = data, mapping = mapping) +
geom_smooth(method = "lm", color = "firebrick4", size = 1/3,
fill = "firebrick", alpha = 1/5) +
geom_point(color = "firebrick", alpha = .8, size = 1/4)
}
# then plug those custom functions into `ggpairs()`
ggpairs(data = d, columns = c(3:4, 6),
diag = list(continuous = my_diag),
lower = list(continuous = my_lower)) +
theme_bw() +
theme(axis.text = element_blank(),
axis.ticks = element_blank(),
panel.grid = element_blank(),
strip.background = element_rect(fill = "white", color = "white"))
```

Our two predictor “variables are negatively correlated, and so strongly so that they are nearly redundant. Either helps in predicting `kcal.per.g`

, but neither helps much *once you already know the other*” (p. 148, *emphasis* in the original). You can really see that on the lower two scatter plots. You’ll note the `ggpairs()`

plot also showed the Pearson’s correlation coefficients, se we don’t need to use the `cor()`

function like McElreath did in the text.

In the next section, we’ll run the simulation necessary for our version of Figure 5.10.

#### 5.3.2.1 Overthinking: Simulating collinearity.

First we’ll get the data and define the functions. You’ll note I’ve defined my `sim_coll()`

a little differently from `sim.coll()`

in the text. I’ve omitted `rep.sim.coll()`

as an independent function altogether, but computed similar summary information with the `summarise()`

code at the bottom of the block.

```
sim_coll <- function(seed, rho) {
set.seed(seed)
d <-
d %>%
mutate(x = rnorm(n(),
mean = perc.fat * rho,
sd = sqrt((1 - rho^2) * var(perc.fat))))
m <- lm(kcal.per.g ~ perc.fat + x, data = d)
sqrt(diag(vcov(m)))[2] # parameter SD
}
# how many simulations per `rho`-value would you like?
n_seed <- 100
# how many `rho`-values from 0 to .99 would you like to evaluate the process over?
n_rho <- 30
d <-
tibble(seed = 1:n_seed) %>%
expand(seed, rho = seq(from = 0, to = .99, length.out = n_rho)) %>%
mutate(parameter_sd = purrr::map2_dbl(seed, rho, sim_coll)) %>%
group_by(rho) %>%
# we'll `summarise()` our output by the mean and 95% intervals
summarise(mean = mean(parameter_sd),
ll = quantile(parameter_sd, prob = .025),
ul = quantile(parameter_sd, prob = .975))
```

We’ve added 95% interval bands to our version of Figure 5.10.

```
d %>%
ggplot(aes(x = rho, y = mean)) +
geom_smooth(aes(ymin = ll, ymax = ul),
stat = "identity",
fill = "firebrick", color = "firebrick4", alpha = 1/5, size = 1/2) +
labs(x = expression(rho),
y = "parameter SD") +
coord_cartesian(ylim = c(0, .0072)) +
theme_bw() +
theme(panel.grid = element_blank())
```

Did you notice we used the base R `lm()`

function to fit the models? As McElreath rightly pointed out, `lm()`

presumes flat priors. Proper Bayesian modeling could improve on that. But then we’d have to wait for a whole lot of HMC chains to run and until our personal computers or the algorithms we use to fit our Bayesian models become orders of magnitude faster, we just don’t have time for that.

### 5.3.3 Post-treatment bias.

It helped me understand the next example by mapping out the sequence of events McElreath described in the second paragraph:

- seed and sprout plants
- measure heights
- apply different antifungal soil treatments (i.e., the experimental manipulation)
- measure (a) the heights and (b) the presence of fungus

Based on the design, let’s simulate our data.

```
n <- 100
set.seed(5)
d <-
tibble(treatment = rep(0:1, each = n / 2),
fungus = rbinom(n, size = 1, prob = .5 - treatment * 0.4),
h0 = rnorm(n, mean = 10, sd = 2),
h1 = h0 + rnorm(n, mean = 5 - 3 * fungus, sd = 1))
```

We’ll use `head()`

to peek at the data.

```
## # A tibble: 6 x 4
## treatment fungus h0 h1
## <int> <int> <dbl> <dbl>
## 1 0 0 12.9 18.9
## 2 0 1 10.4 13.1
## 3 0 1 12.0 13.5
## 4 0 0 8.82 14.6
## 5 0 0 9.78 14.2
## 6 0 1 8.15 11.4
```

Fit the model containing `fungus`

as a predictor.

```
b5.13 <-
brm(data = d,
family = gaussian,
h1 ~ 1 + h0 + treatment + fungus,
prior = c(prior(normal(0, 100), class = Intercept),
prior(normal(0, 10), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.13")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: h1 ~ 1 + h0 + treatment + fungus
## Data: d (Number of observations: 100)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 4.12 0.56 3.02 5.22 1.00 5611 3814
## h0 1.09 0.05 0.98 1.20 1.00 6088 3741
## treatment 0.01 0.21 -0.42 0.38 1.01 785 657
## fungus -2.93 0.23 -3.37 -2.48 1.00 2665 2922
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.00 0.08 0.87 1.17 1.01 2826 2286
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

Now fit the model after excluding `fungus`

, our post-treatment variable.

```
b5.14 <-
update(b5.13,
formula = h1 ~ 1 + h0 + treatment,
iter = 2000, warmup = 1000, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.14")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: h1 ~ h0 + treatment
## Data: d (Number of observations: 100)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 3.61 0.91 1.82 5.46 1.00 4259 2775
## h0 1.00 0.09 0.83 1.18 1.00 4347 2998
## treatment 0.87 0.33 0.21 1.53 1.00 4358 2791
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.67 0.12 1.45 1.93 1.00 4308 2808
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

“Now the impact of treatment is strong and positive, as it should be” (p. 152). In this case, there were really two outcomes. The first was the one we modeled, the height at the end of the experiment (i.e., `h1`

). The second outcome, which was clearly related to `h1`

, was the presence of fungus, captured by our binomial variable `fungus`

. If you wanted to model that, you’d fit a logistic regression model, which we’ll learn about in Chapter 10.

## 5.4 Categorical variables

Many readers will already know that variables like this, routinely called

factors, can easily be included in linear models. But what is not widely understood is how these variables are included in a model… Knowing how the machine works removes a lot of this difficulty. (p. 153,emphasisin the original)

### 5.4.1 Binary categories.

Reload the `Howell1`

data.

Unload rethinking and load brms.

Just in case you forgot what these data were like:

```
## Rows: 544
## Columns: 4
## $ height <dbl> 151.7650, 139.7000, 136.5250, 156.8450, 145.4150, 163.8300, 149.2250, 168.9100, 14…
## $ weight <dbl> 47.82561, 36.48581, 31.86484, 53.04191, 41.27687, 62.99259, 38.24348, 55.47997, 34…
## $ age <dbl> 63.0, 63.0, 65.0, 41.0, 51.0, 35.0, 32.0, 27.0, 19.0, 54.0, 47.0, 66.0, 73.0, 20.0…
## $ male <int> 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0…
```

Let’s fit the first `height`

model with the `male`

dummy.

Note. The uniform prior McElreath used in the text in conjunction with the `brms::brm()`

function seemed to cause problems for the HMC chains, here. After experimenting with start values, increasing `warmup`

, and increasing `adapt_delta`

, switching out the uniform prior did the trick. Anticipating Chapter 8, I recommend you use a weakly-regularizing half Cauchy for \(\sigma\).

```
b5.15 <-
brm(data = d,
family = gaussian,
height ~ 1 + male,
prior = c(prior(normal(178, 100), class = Intercept),
prior(normal(0, 10), class = b),
prior(cauchy(0, 2), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.15")
```

“To interpret these estimates, you have to note that the parameter \(\alpha\) ([`Intercept`

]) is now the average height *among females*” (p. 154).

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: height ~ 1 + male
## Data: d (Number of observations: 544)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 134.83 1.57 131.72 137.86 1.00 5895 4258
## male 7.26 2.24 2.85 11.73 1.00 6051 4641
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 27.37 0.81 25.81 29.03 1.00 5223 4457
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

Our samples from the posterior are already in the HMC iterations. All we need to do is put them in a data frame and we’ll be well-positioned to compute the intervals for the average height among men.

```
post <- posterior_samples(b5.15)
post %>%
transmute(male_height = b_Intercept + b_male) %>%
mean_qi(.width = .89)
```

```
## male_height .lower .upper .width .point .interval
## 1 142.0856 139.3451 144.7568 0.89 mean qi
```

You can also do this with `fitted()`

.

```
## Estimate Est.Error Q2.5 Q97.5
## [1,] 142.0856 1.698185 138.7722 145.4056
```

And you could even plot.

```
fitted(b5.15,
newdata = nd,
summary = F) %>%
as_tibble() %>%
ggplot(aes(x = V1, y = 0)) +
stat_halfeye(point_interval = median_qi, .width = .95,
fill = "firebrick4") +
scale_y_continuous(NULL, breaks = NULL) +
labs(subtitle = "Model-implied male heights",
x = expression(alpha + beta["male"])) +
theme_bw() +
theme(panel.grid = element_blank())
```

#### 5.4.1.1 Overthinking: Re-parameterizing the model.

The reparameterized model follows the form

\[\begin{align*} \text{height}_i & \sim \operatorname{Normal}(\mu_i, \sigma) \\ \mu_i & = \alpha_\text{female} (1 - \text{male}_i) + \alpha_\text{male} \text{male}_i. \end{align*}\]

So then a `female`

dummy would satisfy the condition \(\text{female}_i = (1 - \text{male}_i)\). Let’s make that dummy.

Everyone has their own idiosyncratic way of coding. One of my quirks is I always explicitly specify a model’s intercept following the form `y ~ 1 + x`

, where `y`

is the criterion, `x`

stands for the predictors, and `1`

is the intercept. You don’t have to do this, of course. You could just code `y ~ x`

to get the same results. The `brm()`

function assumes you want that intercept. One of the reasons I like the verbose version is it reminds me to think about the intercept and to include it in my priors. Another nice feature is that is helps me make sense of the code for this model: `height ~ 0 + male + female`

. When we replace `… ~ 1 + …`

with `… ~ 0 + …`

, we tell `brm()`

to remove the intercept. Removing the intercept allows us to include ALL levels of a given categorical variable in our model. In this case, we’ve expressed sex as two dummies, `female`

and `male`

. Taking out the intercept lets us put both dummies into the formula.

```
b5.15b <-
brm(data = d,
family = gaussian,
height ~ 0 + male + female,
prior = c(prior(normal(178, 100), class = b),
prior(cauchy(0, 2), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.15b")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: height ~ 0 + male + female
## Data: d (Number of observations: 544)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## male 142.31 1.69 139.00 145.51 1.00 6293 4459
## female 134.65 1.63 131.50 137.82 1.00 5765 4463
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 27.37 0.84 25.81 29.08 1.00 6293 4228
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

If we wanted the formal difference score from such a model, we’d subtract.

```
posterior_samples(b5.15b) %>%
transmute(dif = b_male - b_female) %>%
ggplot(aes(x = dif, y = 0)) +
stat_halfeye(point_interval = median_qi, .width = .95,
fill = "firebrick4") +
scale_y_continuous(NULL, breaks = NULL) +
labs(subtitle = "Model-implied difference score",
x = expression(alpha["male"] - alpha["female"])) +
theme_bw() +
theme(panel.grid = element_blank())
```

### 5.4.2 Many categories.

When there are more than two categories, you’ll need more than one dummy variable. Here’s the general rule: To include \(k\) categories in a linear model, you require \(k - 1\) dummy variables. Each dummy variable indicates, with the value 1, a unique category. The category with no dummy variable assigned to it ends up again as the “intercept” category. (p. 155)

We’ll practice with `milk`

.

Unload rethinking and load brms.

With the tidyverse, we can peek at `clade`

with `distinct()`

in the place of base R `unique()`

.

```
## clade
## 1 Strepsirrhine
## 2 New World Monkey
## 3 Old World Monkey
## 4 Ape
```

As `clade`

has 4 categories, let’s use `if_else()`

to convert these to 4 dummy variables.

```
d <-
d %>%
mutate(clade_nwm = if_else(clade == "New World Monkey", 1, 0),
clade_owm = if_else(clade == "Old World Monkey", 1, 0),
clade_s = if_else(clade == "Strepsirrhine", 1, 0),
clade_ape = if_else(clade == "Ape", 1, 0))
```

Now we’ll fit the model with three of the four dummies. In this model, `clade_ape`

is the reference category captured by the intercept.

```
b5.16 <-
brm(data = d,
family = gaussian,
kcal.per.g ~ 1 + clade_nwm + clade_owm + clade_s,
prior = c(prior(normal(.6, 10), class = Intercept),
prior(normal(0, 1), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.16")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: kcal.per.g ~ 1 + clade_nwm + clade_owm + clade_s
## Data: d (Number of observations: 29)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 0.55 0.04 0.46 0.63 1.00 4814 4231
## clade_nwm 0.17 0.06 0.04 0.29 1.00 5162 4390
## clade_owm 0.24 0.07 0.11 0.38 1.00 5282 4581
## clade_s -0.04 0.07 -0.18 0.11 1.00 5671 4637
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.13 0.02 0.10 0.17 1.00 5019 4573
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

Here we grab the chains, our draws from the posterior.

```
## b_Intercept b_clade_nwm b_clade_owm b_clade_s sigma lp__
## 1 0.5023688 0.19347314 0.2374078 -0.0304667634 0.1417314 9.256221
## 2 0.5353170 0.16257762 0.3546575 -0.0469542473 0.1464990 8.347143
## 3 0.6134532 0.10425840 0.2044839 0.0004440653 0.1502873 7.414177
## 4 0.6393792 0.02331188 0.1558937 -0.0720549047 0.1247856 7.226460
## 5 0.5407044 0.20503726 0.1963322 -0.0916914179 0.1113628 9.453499
## 6 0.5148228 0.14641295 0.2395858 0.0274923717 0.1412166 9.133430
```

You might compute averages for each category and summarizing the results with the transpose of base R’s `apply()`

function, rounding to two digits of precision.

```
post$mu_ape <- post$b_Intercept
post$mu_nwm <- post$b_Intercept + post$b_clade_nwm
post$mu_owm <- post$b_Intercept + post$b_clade_owm
post$mu_s <- post$b_Intercept + post$b_clade_s
round(t(apply(post[ ,7:10], 2, quantile, c(.5, .025, .975))), digits = 2)
```

```
## 50% 2.5% 97.5%
## mu_ape 0.55 0.46 0.63
## mu_nwm 0.71 0.63 0.80
## mu_owm 0.79 0.68 0.89
## mu_s 0.51 0.39 0.62
```

Here’s a more tidyverse sort of way to get the same thing, but this time with means and HPDIs via the `tidybayes::mean_hdi()`

function.

```
post %>%
transmute(mu_ape = b_Intercept,
mu_nwm = b_Intercept + b_clade_nwm,
mu_owm = b_Intercept + b_clade_owm,
mu_s = b_Intercept + b_clade_s) %>%
gather() %>%
group_by(key) %>%
mean_hdi() %>%
mutate_if(is.double, round, digits = 2)
```

```
## # A tibble: 4 x 7
## key value .lower .upper .width .point .interval
## <chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 mu_ape 0.55 0.46 0.63 0.95 mean hdi
## 2 mu_nwm 0.71 0.63 0.8 0.95 mean hdi
## 3 mu_owm 0.79 0.68 0.89 0.95 mean hdi
## 4 mu_s 0.51 0.39 0.62 0.95 mean hdi
```

You could also summarize with `fitted()`

.

```
nd <- tibble(clade_nwm = c(1, 0, 0, 0),
clade_owm = c(0, 1, 0, 0),
clade_s = c(0, 0, 1, 0),
primate = c("New World Monkey", "Old World Monkey", "Strepsirrhine", "Ape"))
fitted(b5.16,
newdata = nd,
summary = F) %>%
as_tibble() %>%
gather() %>%
mutate(primate = rep(c("New World Monkey", "Old World Monkey", "Strepsirrhine", "Ape"),
each = n() / 4)) %>%
ggplot(aes(x = value, y = reorder(primate, value))) +
stat_halfeye(point_interval = median_qi, .width = .95,
fill = "firebrick4") +
labs(x = "kcal.per.g",
y = NULL) +
theme_bw() +
theme(axis.text.y = element_text(hjust = 0),
axis.ticks.y = element_blank(),
panel.grid = element_blank())
```

And there are multiple ways to compute summary statistics for the difference between `NWM`

and `OWM`

, too.

```
## 50% 2.5% 97.5%
## -0.07428075 -0.20879251 0.06399374
```

```
## dif .lower .upper .width .point .interval
## 1 -0.07428075 -0.2087925 0.06399374 0.95 median qi
```

### 5.4.3 Adding regular predictor variables.

If we wanted to fit the model including `perc.fat`

as an additional predictor, the basic statistical formula would be

\[ \mu_i = \alpha + \beta_\text{clade_nwm} \text{clade_nwm}_i + \beta_\text{clade_owm} \text{clade_owm}_i + \beta_\text{clade_s} \text{clade_s}_i + \beta_\text{perc.fat} \text{perc.fat}_i. \]

The corresponding `formula`

argument within `brm()`

would be `kcal.per.g ~ 1 + clade_nwm + clade_owm + clade_s + perc.fat`

.

### 5.4.4 Another approach: Unique intercepts.

“Another way to conceptualize categorical variables is to construct a vector of intercept parameters, one parameter for each category” (p. 158). Using the code below, there’s no need to transform `d$clade`

into `d$clade_id`

. The advantage of this approach is the indices in the model summary are more descriptive than `a[1]`

through `a[4]`

.

```
b5.16_alt <-
brm(data = d,
family = gaussian,
kcal.per.g ~ 0 + clade,
prior = c(prior(normal(.6, 10), class = b),
prior(uniform(0, 10), class = sigma)),
iter = 2000, warmup = 500, chains = 4, cores = 4,
seed = 5,
file = "fits/b05.16_alt")
```

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: kcal.per.g ~ 0 + clade
## Data: d (Number of observations: 29)
## Samples: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup samples = 6000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## cladeApe 0.55 0.04 0.46 0.63 1.00 7217 4233
## cladeNewWorldMonkey 0.71 0.04 0.63 0.80 1.00 7744 4293
## cladeOldWorldMonkey 0.79 0.05 0.68 0.89 1.00 6551 4201
## cladeStrepsirrhine 0.51 0.06 0.39 0.62 1.00 6887 4287
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.13 0.02 0.10 0.18 1.00 5011 4378
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

See? This is much easier than trying to remember which one was which in an arbitrary numeric index.

## 5.5 ~~Ordinary least squares and ~~`lm()`

`lm()`

Since this section centers on the frequentist `lm()`

function, I’m going to largely ignore it. A couple things, though. You’ll note how the brms package uses the `lm()`

-like design formula syntax. Although not as pedagogical as the more formal rethinking syntax, it has the advantage of cohering with the popular lme4 (Bates et al., 2015, 2020) syntax for multilevel models.

Also, on page 161 McElreath clarified that one cannot use the `I()`

syntax with his rethinking package. Not so with brms. The `I()`

syntax works just fine with `brms::brm()`

. We’ve already made use of it in the “Polynomial regression” section of Chapter 4.

## Session info

```
## R version 3.6.3 (2020-02-29)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS Catalina 10.15.3
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] parallel stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] GGally_2.0.0 patchwork_1.0.1.9000 tidybayes_2.1.1 bayesplot_1.7.1
## [5] urbnmapr_0.0.0.9002 ggrepel_0.8.2 forcats_0.5.0 stringr_1.4.0
## [9] dplyr_1.0.1 purrr_0.3.4 readr_1.3.1 tidyr_1.1.1
## [13] tibble_3.0.3 tidyverse_1.3.0 brms_2.13.5 Rcpp_1.0.5
## [17] dagitty_0.2-2 rstan_2.19.3 ggplot2_3.3.2 StanHeaders_2.21.0-1
##
## loaded via a namespace (and not attached):
## [1] readxl_1.3.1 backports_1.1.9 plyr_1.8.6 igraph_1.2.5
## [5] svUnit_1.0.3 splines_3.6.3 crosstalk_1.1.0.1 TH.data_1.0-10
## [9] rstantools_2.1.1 inline_0.3.15 digest_0.6.25 htmltools_0.5.0
## [13] rsconnect_0.8.16 fansi_0.4.1 magrittr_1.5 modelr_0.1.6
## [17] matrixStats_0.56.0 xts_0.12-0 sandwich_2.5-1 prettyunits_1.1.1
## [21] colorspace_1.4-1 rvest_0.3.5 ggdist_2.1.1 haven_2.2.0
## [25] xfun_0.13 callr_3.4.4 crayon_1.3.4 jsonlite_1.7.0
## [29] survival_3.1-12 zoo_1.8-7 glue_1.4.2 gtable_0.3.0
## [33] emmeans_1.4.5 V8_3.0.2 pkgbuild_1.1.0 shape_1.4.4
## [37] abind_1.4-5 scales_1.1.1 mvtnorm_1.1-0 DBI_1.1.0
## [41] miniUI_0.1.1.1 xtable_1.8-4 HDInterval_0.2.0 units_0.6-6
## [45] stats4_3.6.3 DT_0.13 htmlwidgets_1.5.1 httr_1.4.1
## [49] threejs_0.3.3 RColorBrewer_1.1-2 arrayhelpers_1.1-0 ellipsis_0.3.1
## [53] reshape_0.8.8 pkgconfig_2.0.3 loo_2.3.1 farver_2.0.3
## [57] dbplyr_1.4.2 utf8_1.1.4 tidyselect_1.1.0 labeling_0.3
## [61] rlang_0.4.7 reshape2_1.4.4 later_1.1.0.1 munsell_0.5.0
## [65] cellranger_1.1.0 tools_3.6.3 cli_2.0.2 generics_0.0.2
## [69] broom_0.5.5 ggridges_0.5.2 evaluate_0.14 fastmap_1.0.1
## [73] yaml_2.2.1 processx_3.4.4 knitr_1.28 fs_1.4.1
## [77] nlme_3.1-144 mime_0.9 xml2_1.3.1 compiler_3.6.3
## [81] shinythemes_1.1.2 rstudioapi_0.11 curl_4.3 e1071_1.7-3
## [85] reprex_0.3.0 stringi_1.4.6 highr_0.8 ps_1.3.4
## [89] Brobdingnag_1.2-6 lattice_0.20-38 Matrix_1.2-18 classInt_0.4-3
## [93] markdown_1.1 shinyjs_1.1 vctrs_0.3.4 pillar_1.4.6
## [97] lifecycle_0.2.0 bridgesampling_1.0-0 estimability_1.3 httpuv_1.5.4
## [101] R6_2.4.1 bookdown_0.18 promises_1.1.1 KernSmooth_2.23-16
## [105] gridExtra_2.3 codetools_0.2-16 boot_1.3-24 colourpicker_1.0
## [109] MASS_7.3-51.5 gtools_3.8.2 assertthat_0.2.1 withr_2.2.0
## [113] shinystan_2.5.0 multcomp_1.4-13 mgcv_1.8-31 hms_0.5.3
## [117] grid_3.6.3 coda_0.19-3 class_7.3-15 rmarkdown_2.1
## [121] sf_0.9-1 shiny_1.5.0 lubridate_1.7.8 base64enc_0.1-3
## [125] dygraphs_1.1.1.6
```

### References

Bates, D., Maechler, M., Bolker, B., & Walker, S. (2020). *lme4: Linear mixed-effects models using Eigen’ and S4*. https://CRAN.R-project.org/package=lme4

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. *Journal of Statistical Software*, *67*(1), 1–48. https://doi.org/10.18637/jss.v067.i01

Bürkner, P.-C. (2020b). *brms reference manual, Version 2.13.5*. https://CRAN.R-project.org/package=brms/brms.pdf

Gabry, J. (2020b). *Plotting MCMC draws using the bayesplot package*. https://CRAN.R-project.org/package=bayesplot/vignettes/plotting-mcmc-draws.html

Gabry, J., Simpson, D., Vehtari, A., Betancourt, M., & Gelman, A. (2019). Visualization in Bayesian workflow. *Journal of the Royal Statistical Society: Series A (Statistics in Society)*, *182*(2), 389–402. https://doi.org/10.1111/rssa.12378

Hinde, K., & Milligan, L. A. (2011). Primate milk: Proximate mechanisms and ultimate perspectives. *Evolutionary Anthropology: Issues, News, and Reviews*, *20*(1), 9–23. https://doi.org/10.1002/evan.20289

McElreath, R. (2015). *Statistical rethinking: A Bayesian course with examples in R and Stan*. CRC press. https://xcelab.net/rm/statistical-rethinking/

McElreath, R. (2020b). *Statistical rethinking: A Bayesian course with examples in R and Stan* (Second Edition). CRC Press. https://xcelab.net/rm/statistical-rethinking/

Schloerke, B., Crowley, J., Di Cook, Briatte, F., Marbach, M., Thoen, E., Elberg, A., & Larmarange, J. (2020). *GGally: Extension to ’ggplot2’*. https://CRAN.R-project.org/package=GGally

Slowikowski, K. (2020). *ggrepel: Automatically position non-overlapping text labels with ’ggplot2’*. https://CRAN.R-project.org/package=ggrepel

Urban Institute. (2020). *urbnmapr: State and county maps with Alaska and Hawaii*. https://github.com/UrbanInstitute/urbnmapr