For determining the degrees of freedom (DF) required for the testing of fixed effects, one option is to use the “between-within” method, originally proposed by Schluchter and Elashoff (1990) as a small-sample adjustment.

Using this method, the DF are determined by the grouping level at which the term is estimated. Generally, assuming \(G\) levels of grouping:

\(DF_g=N_g-(N_{g-1}+p_g), g=1, ..., G+1\)

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.

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”).

Let’s look at a concrete example and what the “between-within” 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 = "Between-Within")
)
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: Between-Within
#> 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 < 2e-16
#> RACEBlack or African American 1.53050 0.62448 192.00000 2.451 0.015147
#> RACEWhite 5.64357 0.66561 192.00000 8.479 5.98e-15
#> 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 4.19e-09
#> AVISITVIS3 10.34211 0.82269 334.00000 12.571 < 2e-16
#> AVISITVIS4 15.05390 1.31281 334.00000 11.467 < 2e-16
#> 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 non-missing `FEV1`

observation, and 537 non-missing 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 between-subjects effects and belong to level 1; they do not vary within subject across the repeated observations. On the other hand, `AVISIT`

is a within-subject 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.

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 between-subjects degrees of freedom.
- In SAS, the within-subjects degrees of freedom are affected by the number of subjects in which the effect takes different values.
- In SAS, if there are multiple within-subject effects containing classification variables, the within-subject degrees of freedom are partitioned into components corresponding to the subject-by-effect 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 between-within degrees of freedom to the `mmrm`

package are welcome!

Gałecki A, Burzykowski T (2013). “Linear Mixed-Effects Model.” In *Linear mixed-effects models using r* 245–273. Springer.

Schluchter MD, Elashoff JT (1990). “Small-Sample Adjustments to Tests with Unbalanced Repeated Measures Assuming Several Covariance Structures.” *Journal of Statistical Computation and Simulation*, **37**(1-2), 69–87.