At first glance this may seem to be a very silly question. Everyone knows that ** mixed**
reports chi-square and that chi-square does not have denominator degrees of freedom. Certainly,

**with its chi-square works very well on large datasets. But, what about with small experimental design type data? The problem with chi-square in small datasets is that the p-values are on the optimistic side. Anova with their F-ratios adjust for the small sample size by adjusting the denominator degrees of freedom.**

`mixed`

Rescaling chi-square as an F-ratio is easy, just divide the chi-square value by its degrees of freedom. So a chi-square value of 6.9 with 3 df rescales to an F-ratio of 2.3 with 2 degrees of freedom. The trick is to estimate a reasonable value for the denominator degrees of freedom.

Consider the following two-group

design in which each subject receives four
treatments (**(a)**** b**) in a counterbalanced order.

| b s | 1 2 3 4 | Total -----------+--------------------------------------------+---------- 1 | 1 1 1 1 | 4 2 | 1 1 1 1 | 4 3 | 1 1 0 1 | 3 4 | 1 1 1 1 | 4 5 | 1 0 1 1 | 3 6 | 1 1 1 1 | 4 7 | 0 1 1 1 | 3 8 | 1 1 0 1 | 3 -----------+--------------------------------------------+---------- Total | 7 7 6 8 | 28`use https://stats.idre.ucla.edu/stat/data/repeated_missing, clear`

`tab s b`

Due to random instrument failure one observation on each of four subjects is missing.
If we were to run this as a traditional repeated measures anova we would have to
drop all of the data for subjects 3, 5, 7 and 8. By running the analysis using
** mixed** we can retain all of the observations.

Performing EM optimization: Performing gradient-based optimization: Iteration 0: log restricted-likelihood = -31.286348 Iteration 1: log restricted-likelihood = -31.28616 Iteration 2: log restricted-likelihood = -31.28616 Computing standard errors: Mixed-effects REML regression Number of obs = 28 Group variable: s Number of groups = 8 Obs per group: min = 3 avg = 3.5 max = 4 Wald chi2(7) = 224.12 Log restricted-likelihood = -31.28616 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 2.a | -.4684204 .703144 -0.67 0.505 -1.846557 .9097166 | b | 2 | .25 .5749169 0.43 0.664 -.8768165 1.376816 3 | 3.263198 .6279667 5.20 0.000 2.032406 4.49399 4 | 4.25 .5749169 7.39 0.000 3.123184 5.376816 | a#b | 2 2 | 1.861466 .8920747 2.09 0.037 .1130314 3.6099 2 3 | .7925351 .9271515 0.85 0.393 -1.024648 2.609719 2 4 | 2.71842 .8515934 3.19 0.001 1.049328 4.387513 | _cons | 3.75 .4638207 8.09 0.000 2.840928 4.659072 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ s: Identity | sd(_cons) | .4466089 .2491581 .1496413 1.332917 -----------------------------+------------------------------------------------ sd(Residual) | .8130553 .1495124 .567013 1.165862 ------------------------------------------------------------------------------ LR test vs. linear regression: chibar2(01) = 1.45 Prob >= chibar2 = 0.1139`mixed y a##b || s:, var reml`

We can checkout what is and is not significant according to ** mixed** using the

**command.**

`contrast`

Contrasts of marginal linear predictions Margins : asbalanced ------------------------------------------------ | df chi2 P>chi2 -------------+---------------------------------- y | a | 1 3.88 0.0489 | b | 3 207.84 0.0000 | a#b | 3 11.56 0.0091 ------------------------------------------------`contrast a##b`

These results imply that the interaction and both main effects are statistically
significant. However, there are only four subjects nested in each level of variable ** b**.
If there were no missing observations across time a repeated measures anova be our best bet.
But since there are missing observations we will rescale the chi-square values to F-ratios
and try to estimate the denominator degrees of freedom that can used with the F-distribution.

The way we will do this is to run ** anova** to obtain the between and within degrees
of freedom. Although we are running

**we won’t look at the anova results but only at the degrees of freedom. We also won’t bother with the**

`anova`

**option for**

`repeated`

**. We will boldface the degrees of freedom of interest.**

`anova`

Number of obs = 28 R-squared = 0.9446 Root MSE = .825177 Adj R-squared = 0.8932 Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 162.574315 13 12.5057165 18.37 0.0000 | a | 5.12395138 1 5.12395138 3.86 0.0971 s|a | 7.96717172`anova y a / s|a b a#b`

61.32786195 -----------+---------------------------------------------------- b | 136.083668 3 45.3612228 66.62 0.0000 a#b | 7.95033498 3 2.65011166 3.89 0.0325 | Residual | 9.5328282814.680916306 -----------+---------------------------------------------------- Total | 172.107143 27 6.37433862

You didn’t look at the F-ratios, did you? Just look at the two bolded degrees of freedom.

So now we know that the denominator degrees of freedom are 6 and 14. We can now rescale
the chi-square vales for ** mixed** as F-ratios and obtain p-values.

First the ** a#b** interaction.

chi-square = 11.56 df = 3 F = 11.56/3 = 3.8533333 df = 3 & 14 p-value = Ftail(3,14,3.8533333) = .03348207

The main effect for ** b** has the same denominator degrees of freedom as the interaction.

chi-square = 207.84 df = 3 F = 207.84/3 = 69.28 df = 3 & 14 p-value = Ftail(3,14,69.28) = 1.218e-08

Finally, the ** a** main effect which has six degrees of freedom in the denominator.

chi-square = 3.88 df = 3 F = 3.88/1 = 1 df = 1 & 6 p-value = Ftail(1,6,3.88) = .09638074

While the conclusions for ** b** and

**do not change the F-ratio for the**

`a#b`

**main effect is not significant even thought the**

`a`

**chi-square suggested that it was.**

`mixed`

See also

xtmixed & denominator degrees of freedom: myth or magic — 2011 Chicago Stata Conference