---
title: "Issues in the interpretation of 2-way ANOVA model outputs"
output: slidy_presentation
---

## The coke data:
```{r}
library("ISwR")
data(coking)
attach(coking)
boxplot(time~temp*width,data=coking, 
        col=(c("gold","darkgreen")),
        main="Time to process coke",
        xlab="Temperature . Oven Size",
        ylab="Time")
```

## Experimental design
```{r}
table(coking[,c('width','temp')])
```

Notation:

- $n_{ij}$ number of observation for level $i$ of Temperature and level $i$ of Temperature

- $n_{i.}$ number of observation for level $i$ of Temperature

- $n_{.j}$ number of observation for  level $j$ of Width

- $n$ total nulber of observations

The experimental design is full, balanced with replications. 

## Orthogonality

We have also:

$$ n_{ij} = \frac{n_{i.}n_{.j}}{n} $$

This property is known as orthogonality of the design.

## Testing effects of factors in an orthogonal experimental design

We compare below the output of various model fitting

##  time~temp
```{r}
summary(lm(time~temp))
anova(lm(time~temp))
```

## Boxplot of time~temp ignoring the effect of width
```{r}
boxplot(time~temp,data=coking, 
        col="red",
        main="Time to process coke",
        xlab="Temperature",
        ylab="Time")
```
  Disregardnig the potential effect of width makes us blind to the effect of `temp`.      
        

##  time~width
```{r}
summary(lm(time~width))
anova(lm(time~width))
```

## Boxplot of time~width ignoring the effect of temp
```{r}
boxplot(time~width,data=coking, 
        col="red",
        main="Time to process coke",
        xlab="Width",
        ylab="Time")
```

## Take-home message \# 1:

It can be dangerous to draw conclusions from separate analyses of the various factors.

##  time~temp+width
```{r}
summary(lm(time~temp+width))
```

##  time~temp+width
```{r}
anova(lm(time~temp+width))
```
The estimated effect and p-values relative to temperature are not the same under the models `time~temp` and 
`time~temp+width`

##
```{r}
anova(lm(time~temp))
```

## Order of factors in the formula

Perhaps not suprisingly `lm(time~temp+width)` and  `lm(time~width+temp)` return striclty the same numerical output:
```{r}
anova(lm(time~temp+width))

anova(lm(time~width+temp))
```


## Analysis in un-balanced design:
Let us create an artifially un-balnced deisgn by dropping the first observation:
```{r}
coking.unb = coking[-1,]
attach(coking.unb)
table(coking.unb[,c('width','temp')])
```
Doing so, we have lost the orthogonality property. 

## Order of factors in the formula for unbalanced design

Perhaps not suprisingly `lm(time~temp+width)` and  `lm(time~width+temp)` return striclty the same numerical output:
```{r}
anova(lm(time~temp+width,data=coking.unb))

anova(lm(time~width+temp,data=coking.unb))
```

## Take-home message \# 2: 
The lack of orthogonlality complicates the interpretation of analyses of variance.