Hypothesis tests and confidence intervals using t and F distributions
Five denominator-degrees-of-freedom (DDF) adjustments
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's mixed command 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.
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 ... Performing gradient-based optimization: Iteration 0: Log restricted-likelihood = -109.44372 Iteration 1: Log restricted-likelihood = -109.39161 Iteration 2: Log restricted-likelihood = -109.39153 Iteration 3: Log restricted-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 | Coefficient Std. err. z P>|z| [95% conf. interval] | |
time | .2765987 .1327319 2.08 0.037 .0164489 .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 | |
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 ... Performing gradient-based optimization: Iteration 0: Log restricted-likelihood = -109.44372 Iteration 1: Log restricted-likelihood = -109.39161 Iteration 2: Log restricted-likelihood = -109.39153 Iteration 3: Log restricted-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 | Coefficient 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 | |
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 | Coefficient 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
Kenward, M.G., and J.H. Roger. 1997. Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53: 983-997.
Read more about small-sample adjustments in the Stata Multilevel Mixed-Effects Reference Manual, see [ME] Mixed