STA 610L: Module 2.4

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# STA 610L: Module 2.4
## Multi-way ANOVA and interactions
### Dr. Olanrewaju Michael Akande

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## Multi-way ANOVA and interactions

ANOVA can be easily extended to accommodate any number of categorical variables.

Variables may each contribute independently to a response, or they may work together to influence response values.

*Interaction effects* are important when the association between one independent variable and the response may depend on the level of another independent variable.

[Click this link for insight on what interactions imply in terms of group means](https://courses.washington.edu/smartpsy/interactions.htm)

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## Interaction example

For example, suppose that we are interested in a two-way ANOVA model in which the response `$y$` is a measure of headache pain, and the independent variables include the type of pill taken `$j$`, with `$j=1$` for placebo and `$j=2$` for ibuprofen, and the number of pills taken `$k$`, where `$k=1,2$`.

While we may expect lower pain if multiple ibuprofen pills are taken, we would not expect the same decrease in pain if multiple placebo pills were taken.

Consider the model
`$$y_{ijk}=\mu + \alpha I(j=2) + \beta I(k=2) + \gamma I(j=k=2)+\varepsilon_{ijk}.$$`

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## Interaction example

`$$y_{ijk}=\mu + \alpha I(j=2) + \beta I(k=2) + \gamma I(j=k=2)+\varepsilon_{ijk}$$`

In this model, the mean is parameterized as follows.

| Drug | \# of Pills | Mean |
| ---- | :-----------: | :----: |
| Placebo | 1 | `$\mu$` |
| Ibuprofen | 1 | `$\mu+\alpha$` |
| Placebo | 2 | `$\mu+\beta$` |
| Ibuprofen | 2 | `$\mu +\alpha+\beta+\gamma$` |

What types of parameter values would we expect to see if there is an interaction in which there is a dose effect for Ibuprofen but not for placebo?

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## Interaction example

`$$y_{ijk}=\mu + \alpha I(j=2) + \beta I(k=2) + \gamma I(j=k=2)+\varepsilon_{ijk}$$`

In this model,

- the expected difference in pain level moving from 1 to 2 ibuprofen pills is `$\mu+\alpha - \mu - \alpha - \beta - \gamma$`
  - the expected difference in pain level moving from 1 to 2 placebo pills is `$\mu - \mu - \beta$`
  - the expected drug effect for those taking one pill is `$\mu+\alpha-\mu=\alpha$`
  - the expected drug effect for those taking two pills is `$\mu+\alpha+\beta+\gamma - \mu - \beta=\alpha+\gamma$`
  
  
So no interaction `$(\gamma=0)$` means that the drug effect is the same regardless of the number of pills taken.

For there to be no drug effect at all, we need `$\gamma=0$` and `$\alpha=0$`.

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## R's most exciting data

We are going to explore R's most thrilling data -- the famous tooth growth in Guinea pigs data!

Ahh, how cute!  Our Dickensian guinea pig has a mystery to solve -- which type of Vitamin C supplement is best for tooth growth!

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## R's most exciting data

Guinea pig dental problems are NOT fun.

Our dataset (Crampton, 1947) contains, as a response, the length of odontoblasts (cells responsible for tooth growth) in 60 guinea pigs, each of which receives one dose of vitamin C (0.5, 1, or 2 mg/day) via one of two delivery methods (orange juice (OJ) or ascorbic acid (VC)).

Researchers wanted to know if the odontoblast length could be used as a marker of Vitamin C uptake, for the purposes of providing better nutritional supplementation to members of the Canadian armed forces (alas, the first of many injustices for Oliver Twisted Teeth -- the study was not done to help little Guinea piggies).

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## R's most exciting data

```r
library(ggplot2)
gp=ToothGrowth
gp$dose=as.factor(gp$dose)
# Default bar plot
p<- ggplot(gp, aes(x=dose, y=len, fill=supp)) + 
  geom_bar(stat="identity", position=position_dodge()) 
# Finished bar plot
p+labs(title="Odontoblast length by dose", x="Dose (mg)", y = "Length")+
   theme_classic() +
   scale_fill_manual(values=c('#999999','#E69F00'))
```

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## R's most exciting data

Looking at the boxplot of the growth data, what type of ANOVA model may be most appropriate?

We will revisit this in the class discussion sesssion.

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# What's next?

### Move on to the readings for the next module!