## 10.5 When the moderator is multicategorical

### 10.5.1 An example.

Just as a refresher, here’s the `print()`

output for `model2`

.

`print(model2, digits = 3)`

```
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: liking ~ D1 + D2 + sexism + D1:sexism + D2:sexism
## Data: protest (Number of observations: 129)
## 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 Eff.Sample Rhat
## Intercept 7.709 1.055 5.721 9.828 1502 1.001
## D1 -4.110 1.522 -7.057 -1.093 1551 1.000
## D2 -3.498 1.429 -6.246 -0.722 1415 1.001
## sexism -0.473 0.206 -0.889 -0.077 1485 1.001
## D1:sexism 0.897 0.291 0.319 1.462 1524 1.001
## D2:sexism 0.778 0.279 0.239 1.318 1406 1.001
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
## sigma 1.006 0.064 0.890 1.143 2865 1.000
##
## Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
## is a crude measure of effective sample size, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

The Bayesian \(R^2\):

`bayes_R2(model2) %>% round(digits = 3)`

```
## Estimate Est.Error Q2.5 Q97.5
## R2 0.155 0.049 0.064 0.256
```

And the \(R^2\) difference between this and the model excluding the interaction terms:

```
bayes_R2(model1, summary = F) %>%
as_tibble() %>%
rename(`Model 1` = R2) %>%
bind_cols(
bayes_R2(model2, summary = F) %>%
as_tibble() %>%
rename(`Model 2` = R2)
) %>%
transmute(difference = `Model 2` - `Model 1`) %>%
summarize(mean = mean(difference),
ll = quantile(difference, probs = .025),
ul = quantile(difference, probs = .975)) %>%
mutate_if(is.double, round, digits = 3)
```

```
## # A tibble: 1 x 3
## mean ll ul
## <dbl> <dbl> <dbl>
## 1 0.084 -0.039 0.202
```

Much like in the text, our Figure 10.7 is just a little different from what we did with Figure 10.3.

```
# This will help us with the `geom_text()` annotation
slopes <-
tibble(slope = c(fixef(model2)["sexism", "Estimate"] + fixef(model2)["D1:sexism", "Estimate"],
fixef(model2)["sexism", "Estimate"] + fixef(model2)["D2:sexism", "Estimate"],
fixef(model2)["sexism", "Estimate"]),
x = c(4.8, 4.6, 5),
y = c(6.37, 6.25, 4.5),
condition = c("Individual Protest", "Collective Protest", "No Protest")) %>%
mutate(label = str_c("This slope is about ", slope %>% round(digits = 3)),
condition = factor(condition, levels = c("No Protest", "Individual Protest", "Collective Protest")))
# Here we plot
model2_fitted %>%
ggplot(aes(x = sexism)) +
geom_ribbon(aes(ymin = Q2.5, ymax = Q97.5),
alpha = 1/3) +
geom_ribbon(aes(ymin = Q25, ymax = Q75),
alpha = 1/3) +
geom_line(aes(y = Estimate)) +
geom_text(data = slopes,
aes(x = x,
y = y,
label = label)) +
coord_cartesian(xlim = 4:6) +
labs(x = expression(paste("Perceived Pervasiveness of Sex Discrimination in Society (", italic(X), ")")),
y = "Evaluation of the Attorney") +
facet_wrap(~condition) +
theme_minimal()
```

### 10.5.2 Probing the interaction and interpreting the regression coefficients.

We computed the posterior means for the slopes when prepping for the figure, above. Here’s how we might get more complete posterior summaries. Much like in the text, our Figure 10.7 is just a little different from what we did with Figure 10.3.

```
post <-
posterior_samples(model2) %>%
transmute(`No Protest` = b_sexism + `b_D1:sexism`*0 + `b_D2:sexism`*0,
`Individual Protest` = b_sexism + `b_D1:sexism`*1 + `b_D2:sexism`*0,
`Collective Protest` = b_sexism + `b_D1:sexism`*0 + `b_D2:sexism`*1)
post %>%
gather() %>%
mutate(key = factor(key, levels = c("No Protest", "Individual Protest", "Collective Protest"))) %>%
group_by(key) %>%
summarise(mean = mean(value),
sd = sd(value),
ll = quantile(value, probs = .025),
ul = quantile(value, probs = .975)) %>%
mutate_if(is.double, round, digits = 3)
```

```
## # A tibble: 3 x 5
## key mean sd ll ul
## <fct> <dbl> <dbl> <dbl> <dbl>
## 1 No Protest -0.473 0.206 -0.889 -0.077
## 2 Individual Protest 0.424 0.204 0.007 0.824
## 3 Collective Protest 0.306 0.182 -0.063 0.655
```

Here are the differences among the three protest groups.

```
post %>%
transmute(`Individual Protest - No Protest` = `Individual Protest` - `No Protest`,
`Collective Protest - No Protest` = `Collective Protest` - `No Protest`,
`Individual Protest - Collective Protest` = `Individual Protest` - `Collective Protest`) %>%
gather() %>%
# again, not necessary, but useful for reordering the summaries
mutate(key = factor(key, levels = c("Individual Protest - No Protest", "Collective Protest - No Protest", "Individual Protest - Collective Protest"))) %>%
group_by(key) %>%
summarise(mean = mean(value),
sd = sd(value),
ll = quantile(value, probs = .025),
ul = quantile(value, probs = .975)) %>%
mutate_if(is.double, round, digits = 3)
```

```
## # A tibble: 3 x 5
## key mean sd ll ul
## <fct> <dbl> <dbl> <dbl> <dbl>
## 1 Individual Protest - No Protest 0.897 0.291 0.319 1.46
## 2 Collective Protest - No Protest 0.778 0.279 0.239 1.32
## 3 Individual Protest - Collective Protest 0.119 0.277 -0.431 0.643
```