# Satterthwaite degrees of freedom in a mixed model change drastically depending on the DV

I have a couple of MLM models created using lme4:

y1 ~ x1 + x2 + x3 + x4 + (1+x4|id)
y2 ~ x1 + x2 + x3 + x4 + (1+x4|id)


notice that the only difference between them is the DV.

When I use lmerTest to get p-values I notice that my degrees of freedom for some of the predictors changes quite drastically between the 2 models. For example, in model 1 x4 df might be 38.50, while in model 2 df for the same predictor might be 260.50

Is that expected behavior?

Given that my predictor variables are identical in both cases (i.e. this can't be a case of one model having more missing data than the other), why is there such a difference in the degrees of freedom when only the DV is changed?

Is there something about the Satterthwaite approximation that takes into account the DV, and hence degrees of freedom are expected to be so different?

EDIT

comparing Satterthwaite and Kenward-Roger (Id prefer to use regular summary(model) as it gives me more information, like random effects estimates and beta estimates)

Im not sure why there are minor fluctuations in df across the board between the 2 models (for both Satterthwaite and Kenward-Roger), but more importantly, notice how the x4 df is 10x larger in model 2 when using Satterthwaite

model 1:

model <- lmer(step_mean ~ x1 + x2 + x3 + x4 + (1+x4|id), data=df, REML=T)

summary(model)

Fixed effects:
Estimate Std. Error        df t value Pr(>|t|)
(Intercept)          -0.06003    0.12845  35.07528  -0.467 0.643161
x1                    0.49117    0.12548  35.12842   3.914 0.000398 ***
x2                   -0.01394    0.01225 259.84143  -1.138 0.256368
x3                    0.01414    0.28512  34.47940   0.050 0.960745
x4                   -0.04091    0.01086  25.53492  -3.767 0.000874 ***

anova(model, ddf='Kenward-Roger')

Analysis of Variance Table of type III  with  Kenward-Roger
approximation for degrees of freedom
Sum Sq Mean Sq NumDF   DenDF F.value    Pr(>F)
x1                  0.47355 0.47355     1  35.119 14.5328 0.0005336 ***
x2                  0.04145 0.04145     1 259.955  1.2721 0.2604083
x3                  0.00007 0.00007     1  34.435  0.0023 0.9621706
x4                  0.43630 0.43630     1  27.832 13.3899 0.0010463 **


model 2:

model <- lmer(stride ~ x1 + x2 + x3 + x4 + (1+x4|id), data=df, REML=T)

summary(model)

Fixed effects:
Estimate Std. Error        df t value Pr(>|t|)
(Intercept)          -0.05924    0.09010  35.35792  -0.657    0.515
x1                    0.08257    0.08865  34.98204   0.931    0.358
x2                   -0.03555    0.05087 295.62573  -0.699    0.485
x3                    0.08774    0.20271  35.43835   0.433    0.668
x4                    0.02290    0.04407 260.86367   0.520    0.604

anova(model, ddf='Kenward-Roger')

Analysis of Variance Table of type III  with  Kenward-Roger
approximation for degrees of freedom
Sum Sq Mean Sq NumDF   DenDF F.value Pr(>F)
x1                  0.52223 0.52223     1  34.736 0.82341 0.3704
x2                  0.29974 0.29974     1 294.418 0.47260 0.4923
x3                  0.11041 0.11041     1  35.005 0.17409 0.6791
x4                  0.15516 0.15516     1  24.510 0.24464 0.6253

• See my edit. Satterthwaite df increases drastically for one of the predictors between the 2 models. but kenward-roger remains somewhat consistent, but there are still minor fluctuations in df across models Apr 28 '18 at 21:14

The Satterthwaite method depends on the dependent variable through the Hessian of the (reml) log-likelihood/deviance function with respect to the variance-parameters (there is also a gradient involved), so it is not surprising if the denominator degrees of freedom change with a change of the dependent variable.

In this concrete case x4 enters in the random-effects as well as fixed-effects, which explains why the denominator df for the fixed effect of x4 changes between the models while the denominator df for the other fixed effects are not (as) affected (but also note the denominator df for x2).

Three things inhibits further exploration of this concrete example:

1. The contrasts may or may not differ between summary and this type III anova so we may not be comparing like with like if we compare the concrete examples of Satterthwaite and Kenward-Roger (KR) methods. Use summary(model, ddf="Kenward-Roger") if you want to use KR df in summary. Also note that while the Satterthwaite implementation is the same for t and 1 df F tests, they actually do differ for KR which can lead to additional differences between summary and anova outputs.
2. The implementation was completely rewritten in lmerTest version >= 3.0-0 so knowing which version is being used is relevant (please include a sessionInfo()). On closer inspection of the output I think you are actually using an old version of lmerTest, so I suggest you upgrade as a first move (cf. https://github.com/runehaubo/lmerTestR).
3. Seeing the estimates of the variance parameters is relevant here (in addition to potential convergence warnings), so please include the complete outputs of summary.

Cheers Rune

• +1. Welcome to CV. It's great to see one of the authors of lmerTest here; we sometimes have questions about it and it would be great to have additional expertise. I remember some interesting questions that remained unanswered, can try to find them and give you some links later. Apr 30 '18 at 8:11
• Thanks. I’m happy to help so just let me know if there are unanswered questions where I can assist. I suppose there is a way to receive notifications about (new) questions on selected topics- I’ll see if I can turn that on. Apr 30 '18 at 16:04
• There is this "fundamental" question stats.stackexchange.com/questions/108161 that would really benefit from having an authoritative and comprehensive answer. I am happy to put a bounty there if you decide to post one. Apart from that, I remember a bunch of questions about analyzing balanced designs suitable for RM-ANOVA where Satterthwaite dfs were giving seemingly very wrong numbers, while K-R gave sensible results (similar to RM-ANOVA). E.g. stats.stackexchange.com/questions/84268 (see the end of Ben Bolker's answer and my last comment underneath about his model2). Apr 30 '18 at 21:25