---
title: "Analysis of variance"
editor:
markdown:
wrap: 72
---
## Packages
```{r}
library(tidyverse)
library(smmr)
library(PMCMRplus)
```
## Jumping rats
- Link between exercise and healthy bones (many studies).
- Exercise stresses bones and causes them to get stronger.
- Study (Purdue): effect of jumping on bone density of growing rats.
- 30 rats, randomly assigned to 1 of 3 treatments:
- No jumping (control)
- Low-jump treatment (30 cm)
- High-jump treatment (60 cm)
- 8 weeks, 10 jumps/day, 5 days/week.
- Bone density of rats (mg/cm$^3$) measured at end.
## Jumping rats 2/2
- See whether larger amount of exercise (jumping) went with higher
bone density.
- Random assignment: rats in each group similar in all important ways.
- So entitled to draw conclusions about cause and effect.
## Reading the data
Values separated by spaces:
```{r inference-5-R-9}
my_url <- "http://ritsokiguess.site/datafiles/jumping.txt"
rats <- read_delim(my_url," ")
```
## The data (some random rows)
```{r inference-5-R-10}
rats %>% slice_sample(n=10)
rats
```
## Boxplots
```{r inference-5-R-11, fig.height=3.7}
ggplot(rats, aes(y=density, x=group)) + geom_boxplot()
```
## Or, arranging groups in data (logical) order
```{r inference-5-R-12, fig.height=3.5}
ggplot(rats, aes(y=density, x=fct_inorder(group))) +
geom_boxplot()
```
## Analysis of Variance
- Comparing \> 2 groups of independent observations (each rat only
does one amount of jumping).
- Standard procedure: analysis of variance (ANOVA).
- Null hypothesis: all groups have same mean.
- Alternative: "not all means the same", at least one is different
from others.
## Testing: ANOVA in R
```{r inference-5-R-13}
rats.aov <- aov(density~group,data=rats)
summary(rats.aov)
```
- Usual ANOVA table, small P-value: significant result.
- Conclude that the mean bone densities are not all equal.
- Reject null, but not very useful finding.
## Which groups are different from which?
- ANOVA really only answers half our questions: it says "there are
differences", but doesn't tell us which groups different.
- One possibility (not the best): compare all possible pairs of
groups, via two-sample t.
- First pick out each group:
```{r inference-5-R-14}
rats %>% filter(group=="Control") -> controls
rats %>% filter(group=="Lowjump") -> lows
rats %>% filter(group=="Highjump") -> highs
```
## Control vs. low
```{r inference-5-R-15}
t.test(controls$density, lows$density)
```
No sig. difference here.
## Control vs. high
```{r inference-5-R-16}
t.test(controls$density, highs$density)
```
These are different.
## Low vs. high
```{r inference-5-R-17}
t.test(lows$density, highs$density)
```
These are different too.
## But...
- We just did 3 tests instead of 1.
- So we have given ourselves 3 chances to reject $H_0:$ all means
equal, instead of 1.
- Thus $\alpha$ for this combined test is not 0.05.
## John W. Tukey
::: columns
::: {.column width="40%"}
![](John_Tukey.jpg){width="400"}
:::
::: {.column width="60%"}
- American statistician, 1915--2000
- Big fan of exploratory data analysis
- Popularized boxplot
- Invented "honestly significant differences"
- Invented jackknife estimation
- Coined computing term "bit"
- Co-inventor of Fast Fourier Transform
:::
:::
## Honestly Significant Differences
- Compare several groups with one test, telling you which groups
differ from which.
- Idea: if all population means equal, find distribution of highest
sample mean minus lowest sample mean.
- Any means unusually different compared to that declared
significantly different.
## Tukey on rat data
```{r inference-5-R-18, echo=F}
width <- getOption("width")
options(width = 60)
```
```{r inference-5-R-19}
rats.aov <- aov(density~group, data = rats)
summary(rats.aov)
TukeyHSD(rats.aov)
```
```{r inference-5-R-20, echo=F}
options(width=width)
```
- Again conclude that bone density for highjump group significantly
higher than for other two groups.
## Why Tukey's procedure better than all t-tests
Look at P-values for the two tests:
```
Comparison Tukey t-tests
----------------------------------
Highjump-Control 0.0016 0.0021
Lowjump-Control 0.4744 0.2977
Lowjump-Highjump 0.0298 0.0045
```
- Tukey P-values (mostly) higher.
- Proper adjustment for doing three t-tests at once, not just one in
isolation.
## Checking assumptions
```{r inference-5-R-21}
#| fig.height = 4
ggplot(rats,aes(y = density, x = fct_inorder(group)))+
geom_boxplot()
```
Assumptions:
- Normally distributed data within each group
- with equal group SDs.
## Normal quantile plots by group
```{r inference-5-R-22, fig.height=3.5}
ggplot(rats, aes(sample = density)) + stat_qq() +
stat_qq_line() + facet_wrap( ~ group)
```
## The assumptions
- Normally-distributed data within each group
- Equal group SDs.
- These are shaky here because:
- control group has outliers
- highjump group appears to have less spread than others.
- Possible remedies (in general):
- Transformation of response (usually works best when SD increases
with mean)
- If normality OK but equal spreads not, can use Welch ANOVA.
(Regular ANOVA like pooled t-test; Welch ANOVA like
Welch-Satterthwaite t-test.)
- Can also use Mood's Median Test (see over). This works for any
number of groups.
## Mood's median test here
- Find median of all bone densities, regardless of group
- Count up how many observations in each group above or below overall
median
- Test association between group and being above/below overall median,
using chi-squared test.
- Actually do this using `median_test`:
```{r inference-5-R-27}
median_test(rats, density, group)
```
## Comments
- No doubt that medians differ between groups (not all same).
- This test is equivalent of $F$-test, not of Tukey.
- To determine which groups differ from which, can compare all
possible pairs of groups via (2-sample) Mood's median tests, then
adjust P-values by multiplying by number of 2-sample Mood tests done
(Bonferroni):
```{r inference-5-R-28}
pairwise_median_test(rats, density, group)
```
- Now, lowjump-highjump difference no longer significant.
## Welch ANOVA
- For these data, Mood's median test probably best because we doubt
both normality and equal spreads.
- When normality OK but spreads differ, Welch ANOVA way to go.
- Welch ANOVA done by `oneway.test` as shown (for illustration):
```{r inference-5-R-29}
oneway.test(density~group, data=rats)
```
- P-value very similar, as expected.
- Appropriate Tukey-equivalent here called Games-Howell.
## Games-Howell
- Lives in package `PMCMRplus`. Install first.
```{r games-howell, warning=F}
gamesHowellTest(density ~ factor(group), data = rats)
```
## Deciding which test to do
For two or more samples:
![](testflow.png)