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## Denominator degrees of freedom for mixed models

### Highlights

• Hypothesis tests and confidence intervals using t and F distributions
• Kenward—Roger
• Satterthwaite
• ANOVA
• Repeated-measures ANOVA
• Residual
• Small-sample inference for linear combinations
• Small-sample inference for linear hypothesis tests
• Small-sample inference for contrasts

In small samples, the sampling distributions of test statistics are known to be t and F in simple cases, and those distributions can be good approximations in other cases. Stata provides five methods for small-sample inference, also known as denominator-degrees-of-freedom (DDF) adjustments, including Satterthwaite and Kenward—Roger. In addition to adjusting the confidence intervals and significance tests reported by Stata's mixed estimation command, small-sample statistics are also provided for subsequent estimation of linear combinations and linear hypothesis tests of fixed effects.

### Let's see it work

Consider a simple random-coefficient model for longitudinal data from Kenward and Roger (1997). There are 24 subjects, identified by the variable id. The subjects can be measured at any of nine time periods, but the outcome y is recorded at only three time periods for each subject, meaning that the subjects are not all seen at the same times.

To study both fixed and random effects of time, we fit the following mixed model using restricted maximum likelihood (REML) with the unstructured covariance between random effects:

. mixed y time || id: time, reml covariance(unstructured)

Performing EM optimization:

Iteration 0:   log likelihood = -109.44372
Iteration 1:   log likelihood = -109.39161
Iteration 2:   log likelihood = -109.39153
Iteration 3:   log likelihood = -109.39153

Computing standard errors:

Mixed-effects REML regression                   Number of obs     =         72
Group variable: id                              Number of groups  =         24

Obs per group:
min =          3
avg =        3.0
max =          3

Wald chi2(1)      =       4.34
Log restricted-likelihood = -109.39153          Prob > chi2       =     0.0372

y       Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]

time    .2765987   .1327319     2.08   0.037     0.164489    .5367485
_cons    1.045034   .2504823     4.17   0.000     .5540973     1.53597

Random-effects Parameters     Estimate   Std. Err.     [95% Conf. Interval]

id: Unstructured
var(time)    .3259698   .1356851      .1441665     .737039
var(_cons)    .4172514   .3432177      .0832198    2.092036
cov(time,_cons)   -.1491218   .1736941      -.489556    .1913124

var(Residual)    .3407946   .0844243      .2097135    .5538077

LR test vs. linear model: chi2(3) = 84.07                 Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.


Our default large-sample inference for time suggests that the fixed time effect is significant at a 5% level (p-value=0.037). Empirical evidence suggests, however, that in small samples, the normal and chi-squared distributions may provide poor approximations to the unknown distributions of the test statistics and may lead to anticonservative results.

In Stata, we can account for small samples by specifying one of the five DDF methods. We use the Kenward—Roger method in this example.

. mixed y time || id: time, reml covariance(unstructured) dfmethod(kroger)

Performing EM optimization:

Iteration 0:   log likelihood = -109.44372
Iteration 1:   log likelihood = -109.39161
Iteration 2:   log likelihood = -109.39153
Iteration 3:   log likelihood = -109.39153

Computing standard errors:

Computing degrees of freedom:

Mixed-effects REML regression                   Number of obs     =         72
Group variable: id                              Number of groups  =         24

Obs per group:
min =          3
avg =        3.0
max =          3
DF method: Kenward-Roger                        DF:           min =      11.68
avg =      17.19
max =      22.69

F(1,    22.69)    =       4.24
Log restricted-likelihood = -109.39153          Prob > F          =     0.0512

y       Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]

time    .2765987     .13434     2.06   0.051    -.0015158    .5547132
_cons    1.045034   .2700712     3.87   0.002     .4548251    1.635242

Random-effects Parameters     Estimate   Std. Err.     [95% Conf. Interval]

id: Unstructured
var(time)    .3259698   .1356851      .1441665     .737039
var(_cons)    .4172514   .3432177      .0832198    2.092036
cov(time,_cons)   -.1491218   .1736941      -.489556    .1913124

var(Residual)    .3407946   .0844243      .2097135    .5538077

LR test vs. linear model: chi2(3) = 84.07                 Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.


After adjusting for a small sample, we do not have sufficient evidence to reject the null hypothesis of no time effect, at least at a 5% significance level.

Our follow-up analyses can also account for small samples, for example, when computing linear combinations,

. lincom _b[_cons] + _b[time], small

( 1)  [y]time + [y]_cons = 0

y       Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]

(1)    1.321632   .2292508     5.77   0.000     .8235855    1.819679


and when performing linear hypothesis tests,

. test (_b[_cons]=1) (_b[time]==0), small

( 1)  [y]_cons = 1
( 2)  [y]time = 0

F(  2, 15.60) =    3.05
Prob > F =    0.0764


### Reference

Kenward, M.G., and J.H. Roger. 1997. Small sample inference for fixed effects from restricted maximum likelihood.
Biometrics 53: 983-997.