How to remove a lower order parameter in a model when the higher order parameters remain?

Question

The problem: I cannot remove a lower order parameter (e.g., a main effects parameter) in a model as long as the higher order parameters (i.e., interactions) rem

Answer 1

Here's a sort of answer; there is no way that I know of to formulate this model directly by the formula ...

Construct data as above:

d <- data.frame(A = rep(c("a1", "a2"), each = 50),
                B = c("b1", "b2"), value = rnorm(100))
options(contrasts=c('contr.sum','contr.poly'))

Confirm original finding that just subtracting the factor from the formula doesn't work:

m1 <- lm(value ~ A * B, data = d)
coef(m1)
## (Intercept)          A1          B1       A1:B1 
## -0.23766309  0.04651298 -0.13019317 -0.06421580 

m2 <- update(m1, .~. - A)
coef(m2)
## (Intercept)          B1      Bb1:A1      Bb2:A1 
## -0.23766309 -0.13019317 -0.01770282  0.11072877 

Formulate the new model matrix:

X0 <- model.matrix(m1)
## drop Intercept column *and* A from model matrix
X1 <- X0[,!colnames(X0) %in% "A1"]

lm.fit allows direct specification of the model matrix:

m3 <- lm.fit(x=X1,y=d$value)
coef(m3)
## (Intercept)          B1       A1:B1 
## -0.2376631  -0.1301932  -0.0642158 

This method only works for a few special cases that allow the model matrix to be specified explicitly (e.g. lm.fit, glm.fit).

More generally:

## need to drop intercept column (or use -1 in the formula)
X1 <- X1[,!colnames(X1) %in% "(Intercept)"]
## : will confuse things -- substitute something inert
colnames(X1) <- gsub(":","_int_",colnames(X1))
newf <- reformulate(colnames(X1),response="value")
m4 <- lm(newf,data=data.frame(value=d$value,X1))
coef(m4)
## (Intercept)          B1   A1_int_B1 
##  -0.2376631  -0.1301932  -0.0642158 

This approach has the disadvantage that it won't recognize multiple input variables as stemming from the same predictor (i.e., multiple factor levels from a more-than-2-level factor).

Answer 2

I think the most straightforward solution is to use model.matrix. Possibly, you could achieve what you want with some fancy footwork and custom contrasts. However, if you want "type 3 esque" p-values, You probably want it for every term in your model, in which case, I think my approach with model.matrix is convenient anyway because you can easily implicitly loop through all models dropping one column at a time. The provision of a possible approach is not an endorsement of the statistical merits of it, but I do think you formulated a clear question and seem to know it may be unsound statistically so I see no reason not to answer it.

## initial data
set.seed(10)
d <- data.frame(
  A = rep(c("a1", "a2"), each = 50),
  B = c("b1", "b2"),
  value = rnorm(100))

options(contrasts=c('contr.sum','contr.poly'))

## create design matrix
X <- model.matrix(~ A * B, data = d)

## fit models dropping one effect at a time
## change from 1:ncol(X) to 2:ncol(X)
## to avoid a no intercept model
m <- lapply(1:ncol(X), function(i) {
  lm(value ~ 0 + X[, -i], data = d)
})
## fit (and store) the full model
m$full <- lm(value ~ 0 + X, data = d)
## fit the full model in usual way to compare
## full and regular should be equivalent
m$regular <- lm(value ~ A * B, data = d)
## extract and view coefficients
lapply(m, coef)

This results in this final output:

[[1]]
   X[, -i]A1    X[, -i]B1 X[, -i]A1:B1 
  -0.2047465   -0.1330705    0.1133502 

[[2]]
X[, -i](Intercept)          X[, -i]B1       X[, -i]A1:B1 
        -0.1365489         -0.1330705          0.1133502 

[[3]]
X[, -i](Intercept)          X[, -i]A1       X[, -i]A1:B1 
        -0.1365489         -0.2047465          0.1133502 

[[4]]
X[, -i](Intercept)          X[, -i]A1          X[, -i]B1 
        -0.1365489         -0.2047465         -0.1330705 

$full
X(Intercept)          XA1          XB1       XA1:B1 
  -0.1365489   -0.2047465   -0.1330705    0.1133502 

$regular
(Intercept)          A1          B1       A1:B1 
 -0.1365489  -0.2047465  -0.1330705   0.1133502 

That is nice so far for models using lm. You mentioned this is ultimately for lmer(), so here is an example using mixed models. I believe it may become more complex if you have more than a random intercept (i.e., effects need to be dropped from the fixed and random parts of the model).

## mixed example
require(lme4)

## data is a bit trickier
set.seed(10)
mixed <- data.frame(
  ID = factor(ID <- rep(seq_along(n <- sample(3:8, 60, TRUE)), n)),
  A = sample(c("a1", "a2"), length(ID), TRUE),
  B = sample(c("b1", "b2"), length(ID), TRUE),
  value = rnorm(length(ID), 3) + rep(rnorm(length(n)), n))

## model matrix as before
X <- model.matrix(~ A * B, data = mixed)

## as before but allowing a random intercept by ID
## becomes trickier if you need to add/drop random effects too
## and I do not show an example of this
mm <- lapply(1:ncol(X), function(i) {
  lmer(value ~ 0 + X[, -i] + (1 | ID), data = mixed)
})

## full model
mm$full <- lmer(value ~ 0 + X + (1 | ID), data = mixed)
## full model regular way
mm$regular <- lmer(value ~ A * B + (1 | ID), data = mixed)

## view all the fixed effects
lapply(mm, fixef)

Which gives us...

[[1]]
   X[, -i]A1    X[, -i]B1 X[, -i]A1:B1 
 0.009202554  0.028834041  0.054651770 

[[2]]
X[, -i](Intercept)          X[, -i]B1       X[, -i]A1:B1 
        2.83379928         0.03007969         0.05992235 

[[3]]
X[, -i](Intercept)          X[, -i]A1       X[, -i]A1:B1 
        2.83317191         0.02058800         0.05862495 

[[4]]
X[, -i](Intercept)          X[, -i]A1          X[, -i]B1 
        2.83680235         0.01738798         0.02482256 

$full
X(Intercept)          XA1          XB1       XA1:B1 
  2.83440919   0.01947658   0.02928676   0.06057778 

$regular
(Intercept)          A1          B1       A1:B1 
 2.83440919  0.01947658  0.02928676  0.06057778