---
title: "Analysis of Covariance"
editor: 
  markdown: 
    wrap: 72
---

## Analysis of covariance

-   ANOVA: explanatory variables categorical (divide data into groups)

-   traditionally, analysis of covariance has categorical $x$'s plus one
    numerical $x$ ("covariate") to be adjusted for.

-   `lm` handles this too.

-   Simple example: two treatments (drugs) (`a` and `b`), with before
    and after scores.

-   Does knowing before score and/or treatment help to predict after
    score?

-   Is after score different by treatment/before score?

## Data

Treatment, before, after:

```         
a 5 20
a 10 23
a 12 30
a 9 25
a 23 34
a 21 40
a 14 27
a 18 38
a 6 24
a 13 31
b 7 19
b 12 26
b 27 33
b 24 35
b 18 30
b 22 31
b 26 34
b 21 28
b 14 23
b 9 22
```

\normalsize

## Packages

```{r bAncova-1}
library(tidyverse)
library(broom)
library(marginaleffects)
```

the last of these for predictions.

## Read in data

```{r bAncova-2, message=F}
url <- "http://ritsokiguess.site/datafiles/ancova.txt"
prepost <- read_delim(url, " ")
prepost
```

## Making a plot

```{r ancova-plot, fig.height=4.5}
ggplot(prepost, aes(x = before, y = after, colour = drug)) +
  geom_point() + geom_smooth(method = "lm")
```

## Comments

-   As before score goes up, after score goes up.

-   Red points (drug A) generally above blue points (drug B), for
    comparable before score.

-   Suggests before score effect *and* drug effect.

## The means

```{r bAncova-3 }
prepost %>%
  group_by(drug) %>%
  summarize(
    before_mean = mean(before),
    after_mean = mean(after)
  )
```

-   Mean "after" score slightly higher for treatment A.

-   Mean "before" score much higher for treatment B.

-   Greater *improvement* on treatment A.

## Testing for interaction

```{r bAncova-4 }
prepost.1 <- lm(after ~ before * drug, data = prepost)
anova(prepost.1)
summary(prepost.1)
```

-   Interaction not significant. Will remove later.

## Predictions

Set up values to predict for:

```{r}
summary(prepost)
```

```{r}
new <- datagrid(before = c(9.75, 14, 21.25), 
                drug = c("a", "b"), model = prepost.1)
new
```

## and then

```{r}
cbind(predictions(prepost.1, newdata = new)) %>% 
  select(drug, before, estimate, conf.low, conf.high)
```

\normalsize

## Predictions (with interaction included), plotted

```{r, fig.height=4}
plot_predictions(model = prepost.1, condition = c("before", "drug"))
```

Lines almost parallel, but not quite.

## Taking out interaction

\small

```{r bAncova-8 }
prepost.2 <- update(prepost.1, . ~ . - before:drug)
summary(prepost.2)
anova(prepost.2)
```

\normalsize

-   Take out non-significant interaction.

-   `before` and `drug` strongly significant.

-   Do predictions again and plot them.

## Predictions

```{r}
cbind(predictions(prepost.2, newdata = new)) %>% 
  select(drug, before, estimate)
```

## Plot of predicted values

```{r, fig.height=4}
plot_predictions(prepost.2, condition = c("before", "drug"))
```

This time the lines are *exactly* parallel. No-interaction model forces
them to have the same slope.

## Different look at model output

-   `anova(prepost.2)` tests for significant effect of before score and
    of drug, but doesn't help with interpretation.

-   `summary(prepost.2)` views as regression with slopes:

\scriptsize

```{r bAncova-11 }
summary(prepost.2)
```

\normalsize

## Understanding those slopes

\footnotesize

```{r bAncova-12}
tidy(prepost.2)
```

\normalsize

-   `before` ordinary numerical variable; `drug` categorical.

-   `lm` uses first category `druga` as baseline.

-   Intercept is prediction of after score for before score 0 and *drug
    A*.

-   `before` slope is predicted change in after score when before score
    increases by 1 (usual slope)

-   Slope for `drugb` is *change* in predicted after score for being on
    drug B rather than drug A. Same for *any* before score (no
    interaction).

## Summary

-   ANCOVA model: fits different regression line for each group,
    predicting response from covariate.

-   ANCOVA model with interaction between factor and covariate allows
    different slopes for each line.

-   Sometimes those lines can cross over!

-   If interaction not significant, take out. Lines then parallel.

-   With parallel lines, groups have consistent effect regardless of
    value of covariate.