linear mixed effect models: random intercept, no fixed intercept interpretation

Question

I'm learning about linear mixed effect models and I'm having a hard time interpreting a repeated measures model that has a random intercept, but no fixed intercept.

As an example:

lmer(DV ~ IV1 + IV1:Time + (1|Group/Subject), data = data)

In such a scenario, Time is a fixed time point with a pre, post, and follow up 1-month post-measurement, and the IV is a factor (e.g., gender). Subjects are nested within groups.

I have only been able to find one discussion about this, but there was no answer.

Answer 1

Your statement about "a repeated measures model that has a random intercept, but no random intercept" contradicts itself and does not make sense. Do you mean "a repeated measures model that has a random intercept, but no fixed intercept"?

Imagine a simple situation where you measure a dependent variable (DV) just once for each subject in 3 distinct treatment groups: A, B and C. If you fit a model like this in R:

lm(DV ~ -1 + TreatmentGroup) 

R will report an output which would look like this:

Term               Estimate       etc.
          
TreatmentGroupA       100         etc.
TreatmentGroupB       210
TreatmentGroupC       250         etc. 

The estimates reported by R are simply the mean values of the dependent variable DV for each treatment group. Thus, the mean value of DV for treatment group A is 100, etc.

If you fit a model using a different parametrization like so:

lm(DV ~ 1 + TreatmentGroup) 

then R will report a different output:

this:

Term                  Estimate                        etc.
          
TreatmentGroupB       110 (= 210 - 100 = 110) 
TreatmentGroupC       150 (= 250 - 100 = 150)         etc. 

With this second parametrization, R sets aside treatment group A as a reference group and then proceeds to compare the remaining treatment groups against the reference group in terms of the mean value of the dependent variable DV. So the output reports 110 as the difference in estimated mean values of DV between treatment group B and treatment group A, and 150 as the difference in estimated mean values of DV between treatment group C and treatment group A. You can think of 110 as the amount you need to add to the mean value of DV for people in Treatment Group A in order to get the mean value of DV for people in Treatment Group B: 100 + 110 = 210. Similarly, you can think of 150 as the amount you need to add to the mean value of DV for people in Treatment Group A in order to get the mean value of DV for people in Treatment Group C: 100 + 150 = 250.

Note that, when R encounters the -1 notation for the intercept, it only applies it to the first categorical predictor it encounters in the model (in this case, TreatmentGroup, which must be declared as a factor in R). If the model includes a second categorical predictor, say Gender (with categories Female, Male, Other), R uses the -1 parameteization for the first categorical variable it encounters and the +1 parametrization for the second (and any subsequent other) categorical variable. So a model formula using the -1 syntax like:

lm(DV ~ -1 + TreatmentGroup + Gender) 

would result in output like this:

Term               Estimate       
          
TreatmentGroupA       100   <- mean value of DV for Females in Treatment Group A
TreatmentGroupB       210   <- mean value of DV for Females in Treatment Group B 
TreatmentGroupC       250   <- mean value of DV for Females in Treatment Group C  

GenderMale             30   <- amount we need to add to mean value of 
                               DV for Females in a treatment group to get the 
                               mean value of DV for Males in that treatment group 

GenderOther            15   <- amount we need to add to mean value of 
                               DV for Females in a treatment group to get the 
                               mean value of DV for people of Other gender in that treatment group 

This ouput treats includes all 3 categories of TreatmentGroup (because of the -1 applying to it) but sets aside the Female gender.

The same principle applies in a linear mixed effects model context. Let's say that you have 3 treatment groups but now each subject in these groups yields 4 measurements on the DV. A model formula like:

lmer(DV ~ -1 + TreatmentGroup + (1|Subject), data = data)

would prompt R to return the mean value of the DV in each treatment group. If you interact this variable with a continuous predictor variable, Predictor:

lmer(DV ~ -1 + TreatmentGroup + TreatmentGroup:Predictor + (1|Subject), data = data)

I suspect R will produce output like:

this:

Fixed effects:

Term                       Estimate       etc.
          
TreatmentGroupA              100          etc.
TreatmentGroupB              210
TreatmentGroupC              250          etc. 
TreatmentGroupA:Predictor     10
TreatmentGroupB:Predictor     25
TreatmentGroupC:Predictor     30          etc.

(I'm not in front of a computer, so can't verify that R omits the main effect of the Predictor from its output.) Given these are dummy variables (e.g., TreatmentGroupA = 1 if subject is in treatment group A and 0 otherwise), you can easily work out from the output reported by R - presuming it looks like the one above - that:

mean value of DV for subjects in TreatmentGroup A = 100 + 10*Predictor
mean value of DV for subjects in TreatmentGroup B = 210 + 25*Predictor
mean value of DV for subjects in TreatmentGroup C = 250 + 30*Predictor

So the mean value in each treatment group is affected by the value of the Predictor.

If your subjects are nested in larger entities (e.g., Subjects nested in Schools), the same principles apply.