I want to compare the following two linear models:

`model 1: y = mean + A + B model 2: y = mean + A + A*B `

Is model 2 equivalent to y = mean + A + B + A*B? Can I use `anova(model1, model2)`

in R to compare the two nested models?

If not, how can I compare them in R?

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#### Best Answer

In R, if you want to test the interaction then you want

`m1 <- lm(y ~ A + B, data = foo) m2 <- lm(y ~ A * B, data = foo) `

then

`anova(m1, m2) `

will give an F test for the interaction.

Note that `m2`

could be written longhand as

`m2 <- lm(y ~ A + B + A:B, data = foo) `

wherein the nesting become clear as `A:B`

is the interaction term and its coefficient would be = 0 in `m1`

. Note the intercept/constant term is implied in these model formulae.

Note that in

`m2 <- lm(y ~ A + A*B) `

the first `A`

is redundant as `A*B`

expands to `A + B + A:B`

. R is clever enough to work out you don't want the following model `A + A + B + A:B`

and only includes `A`

once.

In R, `A`

and `B`

would be factors containing a recording of the label or type of each observation. If by your notation you have 3 levels/types in `A`

and 4 in `B`

, then `A`

and `B`

would be *factor* **vectors** with `length(levels(A))`

and `length(levels(B))`

equal to 3 and 4 respectively. R will create the design matrix for the model from the symbolic representation you provided via the formula. To see what R is doing, you can do `head(model.matrix(~ A + A*B, data = foo))`

to see the first 6 rows of the design matrix. This is coded using treatment contrasts but others are available. See `?contr.sum`

for details on the various contrast types built into R.

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