Exercise 6

Author

Instructor of PSYC 573

The analyses in this exercise are from the example in the note “9 Multiple Predictors” (https://marklhc.quarto.pub/psyc573-2024fall/docs/06b-multiple-predictors.html#conditional-effectssimple-slopes)

S = 0 for non-southern states; S = 1 for southern states

Consider the interaction model

\[ \begin{aligned} D_i & \sim N(\mu_i, \sigma) \\ \mu_i & = \beta_0 + \beta_1 S_i + \beta_2 A_i + \beta_3 S_i \times A_i \end{aligned} \]

Q1

Express, in terms of the model parameters (e.g., $\beta_0$, $\beta_1$),

the predicted divorce rate ($\mu$) for a southern state with MedianAgeMarriage = 2.5: _____________________
the predicted $\mu$ for a non-southern state with MedianAgeMarriage = 2.5: _____________________
the difference between (a) and (b): ____________________________

Q2

The following shows the estimated coefficients (from brms)

	Estimate	Est.Error	Q2.5	Q97.5
Intercept	2.79	0.46	1.91	3.73
Southsouth	3.15	1.50	0.18	6.12
MedianAgeMarriage	-0.71	0.17	-1.07	-0.37
Southsouth:MedianAgeMarriage	-1.20	0.58	-2.36	-0.03

and the interaction plot:

Label $\beta_0$, $\beta_1$, $\beta_2$, $\beta_3$, and $\sigma$ in the graph above (or describe where they are in your words).

--- title: "Exercise 6" author: "Instructor of PSYC 573" echo: false format: pdf --- The analyses in this exercise are from the example in the note "9 Multiple Predictors" (<https://marklhc.quarto.pub/psyc573-2024fall/docs/06b-multiple-predictors.html#conditional-effectssimple-slopes>) ```{r} #| message: false library(tidyverse) library(brms) options(brms.backend = "cmdstanr") library(rstan) ``` ```{r} #| include: false waffle_divorce <- read_delim( # read delimited files "https://raw.githubusercontent.com/rmcelreath/rethinking/master/data/WaffleDivorce.csv", delim = ";" ) # Rescale Marriage and Divorce by dividing by 10 waffle_divorce$Marriage <- waffle_divorce$Marriage / 10 waffle_divorce$Divorce <- waffle_divorce$Divorce / 10 waffle_divorce$MedianAgeMarriage <- waffle_divorce$MedianAgeMarriage / 10 # Recode `South` to a factor variable waffle_divorce$South <- factor(waffle_divorce$South, levels = c(0, 1), labels = c("non-south", "south") ) # See data description at https://rdrr.io/github/rmcelreath/rethinking/man/WaffleDivorce.html ``` S = 0 for non-southern states; S = 1 for southern states Consider the interaction model $$ \begin{aligned} D_i & \sim N(\mu_i, \sigma) \\ \mu_i & = \beta_0 + \beta_1 S_i + \beta_2 A_i + \beta_3 S_i \times A_i \end{aligned} $$ ## Q1 Express, in terms of the model parameters (e.g., $\beta_0$, $\beta_1$), (a) the predicted divorce rate ($\mu$) for a southern state with `MedianAgeMarriage` = 2.5: _____________________ (b) the predicted $\mu$ for a non-southern state with `MedianAgeMarriage` = 2.5: _____________________ (c) the difference between (a) and (b): ____________________________ ## Q2 ```{r} #| include: false m_inter <- brm( Divorce ~ South * MedianAgeMarriage, data = waffle_divorce, prior = prior(normal(0, 2), class = "b") + prior(normal(0, 10), class = "b", coef = "Southsouth") + prior(normal(0, 10), class = "Intercept") + prior(student_t(4, 0, 3), class = "sigma"), seed = 941, file = "ex6_inter" ) ``` The following shows the estimated coefficients (from `brms`) ```{r} fixef(m_inter) |> knitr::kable(digits = 2) ``` and the interaction plot: ```{r} plot( conditional_effects(m_inter, effects = "MedianAgeMarriage", conditions = data.frame(South = c("south", "non-south"), cond__ = c("South", "Non-South")) ), points = TRUE ) ``` Label $\beta_0$, $\beta_1$, $\beta_2$, $\beta_3$, and $\sigma$ in the graph above (or describe where they are in your words).