---
title: "Problem Set 9 Answer Key"
author: "WHL"
output: html_document
---

```{r setup, results = 'hide', message = FALSE, include = FALSE}
options(digits = 4, scipen = 999)
knitr::opts_chunk$set(echo = TRUE, fig.width = 4, fig.height = 4)
library(tidyverse)
library(lmSupport)
library(lme4)
library(ez)
options(knitr.kable.NA = '')
```

# 1

```{r, message = FALSE, results='hide'}

# import data
d <- read.csv("http://whlevine.hosted.uark.edu/psyc5143/ps10.csv")

# make sure factors are factors
d$id <- as.factor(d$id)
d$A <- as.factor(d$A)

# what coding is R using for factor A?
contrasts(d$A)
# dummy coding

# useful info: condition means
d %>% group_by(A) %>% summarise(M = mean(DV))

# fitting the model treating the data as between-subjects
modelBetween <- lm(DV ~ A, d)

# getting some SS values
anova(modelBetween)

SS_between <- 154.8
SS_within <- 158.8
SS_total <- SS_between + SS_within

# another way to get these
modelC <- lm(DV ~ 1, d)
modelEffectSizes(modelC)

modelA <- lm(DV ~ 1 + A, d)
modelEffectSizes(modelA)

modelCompare(modelC, modelA)

```

$SS_{between}$ = `r SS_between`. $SS_{within}$ = `r SS_within`. $SS_{total}$ = `r SS_total`.

# 2

```{r, message=FALSE, results='hide'}
# a & b

# probably the easiest way to do the ANOVA
ezANOVA(data = d,
				dv = DV,
				wid = id,
				within = A,
				detailed = TRUE)$ANOVA

# the SSs
SS_persons <- 141.6   # this is SSd for the intercept
SS_A <- 154.8         # this is SSn for the IV
SS_resid <- 17.2      # this is SSd for the Iv

# checking equality
all.equal(SS_persons + SS_resid, SS_within) # this function checks for the
                                            # approximate equality of two things
```

a) $SS_{A}$ = `r SS_A`. $SS_{persons}$ = `r SS_persons`. $SS_{error}$ = `r SS_resid`.

b) $SS_{between}$ = $SS_{A}$. $SS_{within}$ from #1 = $SS_{persons}$ + $SS_{error}$ from #2.

c) The denominator for $F$ in #2 is much smaller than in #1 because $SS_{persons}$ has been removed from it. It cost a few $df$, but the $F$-ratio increased by more than a factor of 6!

# 3

```{r, results='hide'}
modelLMEM <- lmer(DV ~ A + (1|id), d)
anova(modelLMEM)
```

$F$ = 36 again!