Just to follow-up on what @andrewgelman said, centering changes the interpretation of your main effects because it is a conditional estimate. It’s always helpful to plot the model predictions to test your understanding. I get surprised by this type of things more often than I’d like to admit, but the predictions usually reveal what would have been obvious had I been paying attention.
What you’ll find is that the two models give the exact same predictions (assuming no priors, and in this case practically identical). There is no conditional main effect of sex in the centered model because it just so happens that the male and female prediction lines cross almost exactly at Intervention = 0.5. If you were to change the female success rates to, say, 0.3 and 0.7 (i.e. y = n*c(0.3, 0.70, 0.25, 0.25)), then the centered model would have a significant conditional main effect for sex. And the standard error is smaller in the centered version because you’re comparing the difference at the “center” of Intervention rather than way out on one end. Those estimates on the end will be less stable in most cases.
In the end, centering does impact the results if you don’t also adjust the priors. That is, if you are going to shrink towards zero, then the meaning of zero is relevant. In this case, though, it’s not the reason for different model estimates.
library(dplyr)
library(ggplot2)
library(patchwork)
library(brms)
# Uncentered model
n = 200
dat <- data.frame(Intervention = c(0,1,0,1), SexF = c(0,0,1,1), y = n*c(0.1, 0.50, 0.25, 0.25), n)
Mod <- brm(y|trials(n) ~ Intervention * SexF, prior = prior(cauchy(0, 2.5)), family = binomial, data = dat)
summary(Mod)
pred_df <- expand.grid(Intervention = seq(from = 0, to = 1, length.out = 101),
SexF = c(0, 0.5, 1),
n = 200)
uncentered_plot <- fitted(Mod, newdata = pred_df) %>%
bind_cols(pred_df) %>%
mutate(sex = ordered(SexF, level = c(0, 0.5, 1), labels = c('Male (0)', '"Average (0.5)"', 'Female (1)'))) %>%
ggplot(aes(Intervention, color = sex, fill = sex)) +
geom_vline(xintercept = 0, linetype = 2) +
#geom_ribbon(aes(ymin = `Q2.5`, ymax = `Q97.5`), alpha = 0.5, color = rgb(0,0,0,0)) +
geom_line(aes(y = Estimate), linewidth = 1) +
scale_x_continuous('Intervention', breaks = c(0, 0.5, 1)) +
scale_fill_brewer(palette = 'Set1') +
scale_color_brewer(palette = 'Set1') +
labs(title = 'Predictions from uncentered model')
# Centered model
dat2 <- with(dat, data.frame(Intervention = Intervention-0.5, SexF = SexF-0.5, y, n))
Mod2 <- update(Mod, newdata = dat2)
summary(Mod2)
pred_df2 <- expand.grid(Intervention = seq(from = -0.5, to = 0.5, length.out = 101),
SexF = c(-0.5, 0, 0.5),
n = 200)
centered_plot <- fitted(Mod2, newdata = pred_df2) %>%
bind_cols(pred_df2) %>%
mutate(sex = ordered(SexF, level = c(-0.5, 0, 0.5), labels = c('Male (-0.5)', '"Average (0)"', 'Female (0.5)'))) %>%
ggplot(aes(Intervention, color = sex, fill = sex)) +
geom_vline(xintercept = 0, linetype = 2) +
#geom_ribbon(aes(ymin = `Q2.5`, ymax = `Q97.5`), alpha = 0.5, color = rgb(0,0,0,0)) +
geom_line(aes(y = Estimate), linewidth = 1) +
scale_x_continuous('Intervention - 0.5', breaks = c(-0.5, 0, 0.5)) +
scale_fill_brewer(palette = 'Set1') +
scale_color_brewer(palette = 'Set1') +
labs(title = 'Predictions from centered model')
uncentered_plot + centered_plot
