For determining the degrees of freedom (DF) required for the testing of fixed effects, one option is to use the “betweenwithin” method, originally proposed by Schluchter and Elashoff (1990) as a smallsample adjustment.
General definition
Using this method, the DF are determined by the grouping level at which the term is estimated. Generally, assuming \(G\) levels of grouping:
\(DF_g=N_g(N_{g1}+p_g), g=1, ..., G+1\)
where \(N_g\) is the number of groups at the \(g\)th grouping level and \(p_g\) is the number of parameters estimated at that level.
\(N_0=1\) if the model includes an intercept term and \(N_0=0\) otherwise. Note however that the DF for the intercept term itself (when it is included) are calculated at the \(G+1\) level, i.e. for the intercept we use \(DF_{G+1}\) degrees of freedom.
We note that general contrasts \(C\beta\) have not been considered in the literature so far. Here we therefore use a pragmatic approach and define that for a general contrast matrix \(C\) we take the minimum DF across the involved coefficients as the DF.
MMRM special case
In our case of an MMRM (with only fixed effect terms), there is only a single grouping level (subject), so \(G=1\). This means there are 3 potential “levels” of parameters (Gałecki and Burzykowski (2013)):
 Level 0: The intercept term, assuming the model has been fitted with
one.
 We use \(DF_2\) degrees of freedom as defined below.
 Level 1: Effects that change between subjects, but not across
observations within subjects.
 These are the “between parameters”.
 The corresponding degrees of freedom are \(DF_1 = N_1  (N_0 + p_1)\).
 In words this can be read as:
“Between” DF = “number of subjects”  (“1 if intercept otherwise 0” + “number of between parameters”).
 Level 2: Effects that change within subjects.
 These are the “within parameters”.
 The corresponding degrees of freedom are \(DF_2 = N_2  (N_1 + p_2)\).
 In words this can be read as:
“Within” DF = “number of observations”  (“number of subjects” + “number of within parameters”).
Example
Let’s look at a concrete example and what the “betweenwithin” degrees of freedom method gives as results:
fit < mmrm(
formula = FEV1 ~ RACE + SEX + ARMCD * AVISIT + us(AVISIT  USUBJID),
data = fev_data,
control = mmrm_control(method = "BetweenWithin")
)
summary(fit)
#> mmrm fit
#>
#> Formula: FEV1 ~ RACE + SEX + ARMCD * AVISIT + us(AVISIT  USUBJID)
#> Data: fev_data (used 537 observations from 197 subjects with maximum 4
#> timepoints)
#> Covariance: unstructured (10 variance parameters)
#> Method: BetweenWithin
#> Vcov Method: Asymptotic
#> Inference: REML
#>
#> Model selection criteria:
#> AIC BIC logLik deviance
#> 3406.4 3439.3 1693.2 3386.4
#>
#> Coefficients:
#> Estimate Std. Error df t value Pr(>t)
#> (Intercept) 30.77748 0.88656 334.00000 34.715 < 2e16
#> RACEBlack or African American 1.53050 0.62448 192.00000 2.451 0.015147
#> RACEWhite 5.64357 0.66561 192.00000 8.479 5.98e15
#> SEXFemale 0.32606 0.53195 192.00000 0.613 0.540631
#> ARMCDTRT 3.77423 1.07415 192.00000 3.514 0.000551
#> AVISITVIS2 4.83959 0.80172 334.00000 6.037 4.19e09
#> AVISITVIS3 10.34211 0.82269 334.00000 12.571 < 2e16
#> AVISITVIS4 15.05390 1.31281 334.00000 11.467 < 2e16
#> ARMCDTRT:AVISITVIS2 0.04193 1.12932 334.00000 0.037 0.970407
#> ARMCDTRT:AVISITVIS3 0.69369 1.18765 334.00000 0.584 0.559558
#> ARMCDTRT:AVISITVIS4 0.62423 1.85085 334.00000 0.337 0.736129
#>
#> (Intercept) ***
#> RACEBlack or African American *
#> RACEWhite ***
#> SEXFemale
#> ARMCDTRT ***
#> AVISITVIS2 ***
#> AVISITVIS3 ***
#> AVISITVIS4 ***
#> ARMCDTRT:AVISITVIS2
#> ARMCDTRT:AVISITVIS3
#> ARMCDTRT:AVISITVIS4
#> 
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Covariance estimate:
#> VIS1 VIS2 VIS3 VIS4
#> VIS1 40.5537 14.3960 4.9747 13.3867
#> VIS2 14.3960 26.5715 2.7855 7.4745
#> VIS3 4.9747 2.7855 14.8979 0.9082
#> VIS4 13.3867 7.4745 0.9082 95.5568
Let’s try to calculate the degrees of freedom manually now.
In fev_data
there are 197 subjects with at least one
nonmissing FEV1
observation, and 537 nonmissing
observations in total. Therefore we obtain the following numbers of
groups \(N_g\) at the levels \(g=1,2\):
 \(N_1 = 197\)
 \(N_2 = 537\)
And we note that \(N_0 = 1\) because we use an intercept term.
Now let’s look at the design matrix:
head(model.matrix(fit), 1)
#> (Intercept) RACEBlack or African American RACEWhite SEXFemale ARMCDTRT
#> 2 1 1 0 1 1
#> AVISITVIS2 AVISITVIS3 AVISITVIS4 ARMCDTRT:AVISITVIS2 ARMCDTRT:AVISITVIS3
#> 2 1 0 0 1 0
#> ARMCDTRT:AVISITVIS4
#> 2 0
Leaving the intercept term aside, we therefore have the following number of parameters for the corresponding effects:

RACE
: 2 
SEX
: 1 
ARMCD
: 1 
AVISIT
: 3 
ARMCD:AVISIT
: 3
In the model above, RACE
, SEX
and
ARMCD
are betweensubjects effects and belong to level 1;
they do not vary within subject across the repeated observations. On the
other hand, AVISIT
is a withinsubject effect; it
represents study visit, so naturally its value changes over repeated
observations for each subject. Similarly, the interaction of
ARMCD
and AVISIT
also belongs to level 2.
Therefore we obtain the following numbers of parameters \(p_g\) at the levels \(g=1,2\):
 \(p_1 = 2 + 1 + 1 = 4\)
 \(p_2 = 3 + 3 = 6\)
And we obtain therefore the degrees of freedom \(DF_g\) at the levels \(g=1,2\):
 \(DF_1 = N_1  (N_0 + p_1) = 197  (1 + 4) = 192\)
 \(DF_2 = N_2  (N_1 + p_2) = 537  (197 + 6) = 334\)
So we can finally see that those degrees of freedom are exactly as displayed in the summary table above.
Differences compared to SAS
The implementation described above is not identical to that of SAS. Differences include:
 In SAS, when using an unstructured covariance matrix, all effects are assigned the betweensubjects degrees of freedom.
 In SAS, the withinsubjects degrees of freedom are affected by the number of subjects in which the effect takes different values.
 In SAS, if there are multiple withinsubject effects containing classification variables, the withinsubject degrees of freedom are partitioned into components corresponding to the subjectbyeffect interactions.
 In SAS, the final effect you list in the
CONTRAST
/ESTIMATE
statement is used to define the DF for general contrasts.
Code contributions for adding the SAS version of betweenwithin
degrees of freedom to the mmrm
package are welcome!