Comparing nested models with NAs in R - r

Question

I am trying to compare nested regression models using the anova() function in R, but am running into problems because the level 1 and level 2 models differ in the number of observations due to missing cases. Here is a simple example:
# Create dataframe with multiple predictors with different number of NAs
dep <- c(45,46,45,48,49)
basevar <- c(10,12,10,16,17)
pred1 <- c(NA,20,NA,19,21)
dat <- data.frame(dep,basevar,pred1)
# Define level 1 of the nested models
basemodel <- lm(dep ~ basevar, data = dat)
# Add level 2
model1 <- lm(dep ~ basevar + pred1, data = dat)
# Compare the models (uh oh!)
anova(basemodel, model1)
I have seen 2 suggestions to similar problems, but both are problematic.
Suggestion 1: Impute the missing data. The problem with this is that the missing cases in my data were removed because they were outliers, and thus are not "missing at random," and imputing may overfit the data.
Suggestion 2: Make a separate data frame containing only the complete cases for the variable with missing cases, and use that for regressions. This is also problematic if you are creating multiple nested models sharing the same level 1 variable, but in which the level 2 variables differ in the number of missing cases. Here is an example of this:
# Create a new predictor variable with a different number of NAs from pred1
pred2 <- c(23,21,NA,10,11)
dat <- cbind(dat,pred2)
# Create dataframe containing only completed cases of pred1
nonadat1 <- subset(dat, subset = !is.na(pred1))
# Do the same for pred2
nonadat2 <- subset(dat, subset = !is.na(pred2))
# Define level 1 of the nested models within dataframe of pred1 complete cases
basemodel1 <- lm(dep ~ basevar, data = nonadat1)
# Check values of the model
summary(basemodel1)
# Add level 2
model1 <- lm(dep ~ basevar + pred1, data = nonadat1)
# Compare the models (yay it runs!)
anova(basemodel1, model1)
# Define level 1 of the nested models within dataframe of pred2 complete cases
basemodel2 <- lm(dep ~ basevar, data = nonadat2)
# Values are different from those in basemodel1
summary(basemodel2)
# Add level 2
model2 <- lm(dep ~ basevar + pred2, data = nonadat2)
# Compare the models
anova(basemodel2, model2)
As you can see, creating individual data frames creates differences at level 1 of the nested models, which makes interpretation problematic.
Does anyone know how I can compare these nested models while circumventing these problems?

Answer 1

Could this work? See here for more information. It doesn't exactly deal with the fact that models are fitted on different datasets, but it does allow for a comparison.
A<-logLik(basemodel)
B<-logLik(model1)
(teststat <- -2 * (as.numeric(A)-as.numeric(B)))
(p.val <- pchisq(teststat, df = 1, lower.tail = FALSE))