kidiq_tv.Rmd

---
title: "Regression and Other Stories: KidIQ"
author: "Andrew Gelman, Jennifer Hill, Aki Vehtari"
date: "`r Sys.Date()`"
output:
  github_document:
    toc: true
---
Tidyverse version by Bill Behrman.

Linear regression with multiple predictors. See Chapters 10, 11, and
12 in Regression and Other Stories.

-------------

```{r, message=FALSE}
# Packages
library(tidyverse)
library(rstanarm)

# Parameters
  # Kid test score data
file_kids <- here::here("KidIQ/data/kidiq.csv")
  # Common code
file_common <- here::here("_common.R")

#===============================================================================

# Run common code
source(file_common)
```

# 10 Linear regression with multiple predictors

## 10.1 Adding predictors to a model

### Starting with a binary predictor

Data

```{r, message=FALSE}
kids <- read_csv(file_kids)

kids
```

Fit linear regression with binary predictor.

The option `refresh = 0` suppresses the default Stan sampling progress output. This is useful for small data with fast computation. For more complex models and bigger data, it can be useful to see the progress.

```{r}
set.seed(765)

fit_1 <- stan_glm(kid_score ~ mom_hs, data = kids, refresh = 0)

fit_1
```

Child test score vs. mother high school completion.

```{r}
intercept <- coef(fit_1)[["(Intercept)"]]
slope <- coef(fit_1)[["mom_hs"]]

kids %>% 
  ggplot(aes(mom_hs, kid_score)) +
  geom_count() +
  geom_abline(slope = slope, intercept = intercept) +
  scale_x_continuous(
    breaks = 0:1,
    minor_breaks = NULL,
    labels = c("No", "Yes")
  ) +
  scale_y_continuous(breaks = scales::breaks_width(20)) +
  labs(
    title = "Child test score vs. mother high school completion",
    x = "Mother completed high school",
    y = "Child test score",
    size = "Count"
  )
```

### A single continuous predictor

Fit linear regression with a single continuous predictor.

```{r}
set.seed(765)

fit_2 <- stan_glm(kid_score ~ mom_iq, data = kids, refresh = 0)

fit_2
```

Child test score vs. mother IQ score.

```{r}
intercept <- coef(fit_2)[["(Intercept)"]]
slope <- coef(fit_2)[["mom_iq"]]

kids %>% 
  ggplot(aes(mom_iq, kid_score)) +
  geom_point() +
  geom_abline(slope = slope, intercept = intercept) +
  scale_x_continuous(breaks = scales::breaks_width(10)) +
  scale_y_continuous(breaks = scales::breaks_width(20)) +
  labs(
    title = "Child test score vs. mother IQ score",
    x = "Mother IQ score",
    y = "Child test score"
  )
```

### Including both predictors

Fit linear regression with binary predictor and continuous predictor.

```{r}
set.seed(765)

fit_3 <- stan_glm(kid_score ~ mom_hs + mom_iq, data = kids, refresh = 0)

fit_3
```

Child test score vs. mother IQ score and high school completion.

```{r}
lines <- 
  tribble(
    ~mom_hs, ~intercept, ~slope,
    0, coef(fit_3)[["(Intercept)"]], coef(fit_3)[["mom_iq"]],
    1, 
      coef(fit_3)[["(Intercept)"]] + coef(fit_3)[["mom_hs"]],
      coef(fit_3)[["mom_iq"]]
  )

kids %>% 
  ggplot(aes(mom_iq, kid_score, color = factor(mom_hs))) +
  geom_point() +
  geom_abline(
    aes(slope = slope, intercept = intercept, color = factor(mom_hs)),
    data = lines
  ) +
  scale_x_continuous(breaks = scales::breaks_width(10)) +
  scale_y_continuous(breaks = scales::breaks_width(20)) +
  scale_color_discrete(breaks = 0:1, labels = c("No", "Yes")) +
  theme(legend.position = "bottom") +
  labs(
    title = "Child test score vs. mother IQ score and high school completion",
    subtitle = "Without interaction",
    x = "Mother IQ score",
    y = "Child test score",
    color = "Mother completed high school"
  )
```

## 10.3 Interactions

Fit linear regression with binary predictor, continuous predictor, and their interaction.

```{r}
set.seed(765)

fit_4 <- 
  stan_glm(
    kid_score ~ mom_hs + mom_iq + mom_hs:mom_iq,
    data = kids,
    refresh = 0
  )

fit_4
```

Child test score vs. mother IQ score and high school completion: With interaction.

```{r}
lines <- 
  tribble(
    ~mom_hs, ~intercept, ~slope,
    0, coef(fit_4)[["(Intercept)"]], coef(fit_4)[["mom_iq"]],
    1, 
      coef(fit_4)[["(Intercept)"]] + coef(fit_4)[["mom_hs"]],
      coef(fit_4)[["mom_iq"]] + coef(fit_4)[["mom_hs:mom_iq"]]
  )

kids %>% 
  ggplot(aes(mom_iq, kid_score, color = factor(mom_hs))) +
  geom_point() +
  geom_abline(
    aes(slope = slope, intercept = intercept, color = factor(mom_hs)),
    data = lines
  ) +
  scale_x_continuous(breaks = scales::breaks_width(10)) +
  scale_y_continuous(breaks = scales::breaks_width(20)) +
  scale_color_discrete(breaks = 0:1, labels = c("No", "Yes")) +
  theme(legend.position = "bottom") +
  labs(
    title = "Child test score vs. mother IQ score and high school completion",
    subtitle = "With interaction",
    x = "Mother IQ score",
    y = "Child test score",
    color = "Mother completed high school"
  )
```

Child test score vs. mother IQ score and high school completion. With interaction.

```{r, fig.asp=0.475}
kids %>% 
  ggplot(aes(mom_iq, kid_score, color = factor(mom_hs))) +
  geom_point(size = 0.75) +
  geom_abline(
    aes(slope = slope, intercept = intercept, color = factor(mom_hs)),
    data = lines
  ) +
  coord_cartesian(xlim = c(0, NA), ylim = c(-20, NA)) +
  scale_x_continuous(breaks = scales::breaks_width(10)) +
  scale_y_continuous(breaks = scales::breaks_width(20)) +
  scale_color_discrete(breaks = 0:1, labels = c("No", "Yes")) +
  theme(legend.position = "bottom") +
  labs(
    title = "Child test score vs. mother IQ score and high school completion",
    subtitle = "With interaction",
    x = "Mother IQ score",
    y = "Child test score",
    color = "Mother completed high school"
  )
```

# 11 Assumptions, diagnostics, and model evaluation

## 11.2 Plotting the data and fitted model

### Displaying uncertainty in the fitted regression

Child test score vs. mother IQ score: With 50% and 90% predictive intervals.

```{r}
v <- 
  tibble(mom_iq = seq_range(kids$mom_iq)) %>% 
  predictive_intervals(fit = fit_2)

v %>% 
  ggplot(aes(mom_iq)) +
  geom_ribbon(aes(ymin = `5%`, ymax = `95%`), alpha = 0.25) +
  geom_ribbon(aes(ymin = `25%`, ymax = `75%`), alpha = 0.5) +
  geom_line(aes(y = .pred)) +
  geom_point(aes(y = kid_score), data = kids) +
  scale_x_continuous(breaks = scales::breaks_width(10)) +
  scale_y_continuous(breaks = scales::breaks_width(20)) +
  labs(
    title = "Child test score vs. mother IQ score",
    subtitle = "With 50% and 90% predictive intervals",
    x = "Mother IQ score",
    y = "Child test score"
  )
```

### Displaying using one plot for each input variable

Child test score vs. mother IQ score and mean high school completion: With 50% and 90% predictive intervals.

```{r}
v <- 
  tibble(
    mom_hs = mean(kids$mom_hs),
    mom_iq = seq_range(kids$mom_iq)
  ) %>% 
  predictive_intervals(fit = fit_3)

v %>% 
  ggplot(aes(mom_iq)) +
  geom_ribbon(aes(ymin = `5%`, ymax = `95%`), alpha = 0.25) +
  geom_ribbon(aes(ymin = `25%`, ymax = `75%`), alpha = 0.5) +
  geom_line(aes(y = .pred)) +
  geom_point(aes(y = kid_score), data = kids) +
  scale_x_continuous(breaks = scales::breaks_width(10)) +
  scale_y_continuous(breaks = scales::breaks_width(20)) +
  labs(
    title = 
      "Child test score vs. mother IQ score and mean high school completion",
    subtitle = "With 50% and 90% predictive intervals",
    x = "Mother IQ score",
    y = "Child test score"
  )
```

Child test score vs. mother high school completion and mean IQ score.

```{r}
v <- 
  tibble(
    mom_hs = 0:1,
    mom_iq = mean(kids$mom_iq)
  ) %>% 
  predictive_intervals(fit = fit_3)

v %>% 
  ggplot(aes(mom_hs)) +
  geom_ribbon(aes(ymin = `5%`, ymax = `95%`), alpha = 0.25) +
  geom_ribbon(aes(ymin = `25%`, ymax = `75%`), alpha = 0.5) +
  geom_line(aes(y = .pred)) +
  geom_count(aes(y = kid_score), data = kids) +
  scale_x_continuous(
    breaks = 0:1,
    minor_breaks = NULL,
    labels = c("No", "Yes")
  ) +
  scale_y_continuous(breaks = scales::breaks_width(20)) +
  labs(
    title =
      "Child test score vs. mother high school completion and mean IQ score",
    x = "Mother completed high school",
    y = "Child test score",
    size = "Count"
  )
```

## 11.3 Residual plots

Residual vs. mother IQ score.

```{r}
kids %>% 
  mutate(resid = residuals(fit_2)) %>% 
  ggplot(aes(mom_iq, resid)) +
  geom_hline(yintercept = 0, color = "white", size = 2) +
  geom_point() +
  scale_x_continuous(breaks = scales::breaks_width(10)) +
  labs(
    title = "Residual vs. mother IQ score",
    x = "Mother IQ score",
    y = "Residual"
  )
```

# 12 Transformations and regression

## 12.2 Centering and standardizing for models with interactions

The linear model with binary predictor, continuous predictor, and their interaction from [Section 10.3](#103-interactions).

```{r}
fit_4
```

### Centering by subtracting the mean of the data

We can simplify the interpretation of the regression model by first subtracting the mean of each input variable:

```{r}
kids <- 
  kids %>% 
  mutate(
    mom_hs_c1 = mom_hs - mean(mom_hs),
    mom_iq_c1 = mom_iq - mean(mom_iq)
  )
```

Each main effect now corresponds to a predictive difference with the other input at its average value:

```{r}
set.seed(765)

fit_4c1 <- 
  stan_glm(
    kid_score ~ mom_hs_c1 + mom_iq_c1 + mom_hs_c1:mom_iq_c1,
    data = kids,
    refresh = 0
  )

fit_4c1
```

### Using a conventional centering point

Another option is to center based on an understandable reference point, for example, the midpoint of the range for `mom_hs` and the population average IQ.

```{r}
kids <- 
  kids %>% 
  mutate(
    mom_hs_c2 = mom_hs - 0.5,
    mom_iq_c2 = mom_iq - 100
  )
```

In this parameterization, the coefficient of `mom_hs_c2` is the average predictive difference between a child with `mom_hs` = 1 and `mom_hs` = 0, among those children with `mom_iq` = 100. Similarly, the coefficient of `mom_iq_c2` corresponds to a comparison under the condition `mom_hs` = 0.5, which includes no actual data but represents a midpoint of the range.

```{r}
set.seed(765)

fit_4c2 <- 
  stan_glm(
    kid_score ~ mom_hs_c2 + mom_iq_c2 + mom_hs_c2:mom_iq_c2,
    data = kids,
    refresh = 0
  )

fit_4c2
```

### Standardizing by subtracting the mean and dividing by 2 standard deviations

A natural step is to scale the predictors by dividing by 2 standard deviations:

```{r}
kids <- 
  kids %>% 
  mutate(
    mom_hs_z = (mom_hs - mean(mom_hs)) / (2 * sd(mom_hs)),
    mom_iq_z = (mom_iq - mean(mom_iq)) / (2 * sd(mom_iq))
  )
```

We can now interpret all the coefficients on a roughly common scale (except the intercept, which now corresponds to the average predicted outcome with all inputs at their mean):

```{r}
set.seed(765)

fit_4z <- 
  stan_glm(
    kid_score ~ mom_hs_z + mom_iq_z + mom_hs_z:mom_iq_z,
    data = kids,
    refresh = 0
  )

fit_4z
```

## 12.5 Other transformations

### Using discrete rather than continuous predictors

Another input variable that can be used in these models is maternal employment, which is defined on a four-point ordered scale:

```{r}
kids <- 
  kids %>% 
  mutate(mom_work = as.factor(mom_work))
```

Fitting a simple model using descrete predictors yields:

```{r}
set.seed(765)

fit_5 <- stan_glm(kid_score ~ mom_work, data = kids, refresh = 0)

fit_5
```