```{r setup, include=FALSE}
knitr::opts_chunk$set(echo=TRUE, fig.align="center")
```

#### Statistics for Laboratory Scientists ( 140.615 )

## Analysis of Variance - Non-Parametric Methods

#### Example from class: IL10 cytokines

Read the data, and include log10 of IL10 as a column.

```{r}
library(SPH.140.615)
il10 <- cbind(il10, logIL10=log10(il10$IL10))
```

The analysis of variance tables.

```{r}
aov.il10 <- aov(IL10 ~ Strain, data=il10)
anova(aov.il10)
aov.logil10 <- aov(logIL10 ~ Strain, data=il10)
anova(aov.logil10)
```

The Kruskal-Wallis test.

```{r}
kruskal.test(IL10 ~ Strain, data=il10)
kruskal.test(logIL10 ~ Strain, data=il10)
```

#### Example from class: diets and blood coagulation times 

```{r}
coag <- c(62, 60, 63, 59,
          63, 67, 71, 64, 65, 66,
          68, 66, 71, 67, 68, 68,
          56, 62, 60, 61, 63, 64, 63, 59)
ttt <- factor(rep(LETTERS[1:4],c(4,6,6,8)))
```

The analysis of variance table.

```{r}
summary(aov(coag~ttt)) 
```

The Kruskal-Wallis test by brute-force.

```{r}
ranks <- rank(coag)
ranks
rbar <- tapply(ranks, ttt, mean)
rbar
nt <- tapply(ranks, ttt, length)
nt
N <- sum(nt)
N
e <- (N+1)/2
e
H <- 12/(N*(N+1))*sum(nt*(rbar-e)^2)
H
ties <- table(coag)
ties
D <- 1-sum(ties^3-ties)/(N^3-N)
D
kw <- H/D
kw
pchisq(kw, length(nt)-1, lower.tail=FALSE)
```

The Kruskal-Wallis test with the built-in function.

```{r}
kruskal.test(coag ~ ttt)
```

#### Example from class: the fake data with the outlier

Read and plot the data.

```{r, fig.width=8}
datA <- data.frame(x=c(33,34,35,36,39,  30,31,32,32,33,  31,33,33,34,36),
                   g=factor(rep(c("A","B","C"), rep(5,3))))
datB <- datA
datB$x[5] <- 50

set.seed(1)
par(las=1, mfrow=c(1,2))
stripchart(x ~ g, data=datA, pch=1, method="jitter", jitter=0.05, xlim=c(30,50))
abline(h=1:3, lty="dotted", col="lightgrey")
stripchart(x ~ g, data=datB, pch=1, method="jitter", jitter=0.05, xlim=c(30,50))
abline(h=1:3, lty="dotted", col="lightgrey")
```



The ANOVA with dataset A.

```{r}
anova(aov(x ~ g, data=datA))
```

The ANOVA with dataset B.

```{r}
anova(aov(x ~ g, data=datB))
```

The Kruskal-Wallis test.

```{r}
kruskal.test(x ~ g, data=datA)
kruskal.test(x ~ g, data=datB)
```