understanding_parameters_lm_anova.Rmd

Interpreting the parameters of a linear model in a simple 2x2 anova design
===========================================================================
# Time-stamp: <2012-03-13 09:57 christophe@pallier.org>

Comparison of two groups
========================

```{r}
meang1 <- 100
meang2 <- 110
g1 <- meang1 + scale(rnorm(30,))*10
g2 <- meang2 + scale(rnorm(30))*10
```

```

boxplot(cbind(g1, g2))

t.test(g1, g2, var.equal=TRUE)
```

```{r}
```

First, let us create a 2x2 design, resulting from the crossing of binary factors 'a' and 'b': 

```{r}  
a <- gl(2, 1, 4)
b <- gl(2, 2, 4)
means <- c(4, 6, 10, 15)
interaction.plot(a,b,means,ylim=c(0,17),legend=F,type='b',pch=16,bty="l")
```

First, we fit an additive model without the interaction term.

```{r}
tapply(means, list(a=a, b=b), mean)
diff(tapply(means, list(a), mean)) # main effect of a
diff(tapply(means, list(b), mean)) # main effect of b

summary(mod2<-lm(means~a+b))
model.matrix(mod2)
model.tables(aov(means~a+b))
model.tables(aov(means~a*b))
```

Then, we fit a model including the interaction:

```{r, fig=TRUE}
par(las=1)

interaction.plot(a,b,means,ylim=c(0,17),legend=F,type='b',pch=16,bty="l")

eps=.9
text(1,4+eps,"a1b1")
text(2,6+eps,"a2b1")
text(1,10+eps,"a1b2")
text(2,15+eps,"a2b2")

lines(c(1,2),c(10,12),lty=3)

arrows(1,0,1,4, code=3,length=.1)
arrows(2,4,2,6, code=3, length=.1)
arrows(1,4,1,10, code=3, length=.1)
arrows(2,12,2,15, code=3, length=.1)


mod1 = lm(means~a*b)
model.matrix(mod1)
summary(mod1)

text(1.08,2,"Intercept") # intercept
text(2.08,5,"a2") # a2
text(1.08,7.5,"b2") # b2
text(2.08,13,"a2:b2") # a2:b2
```