This is the output of a regression of a continuous outcome variable on two categorical predictors: group (placebo vs active) and site (four different study sites.
Estimate Std. Error t value Pr(>|t|)
(Intercept) 11.7494 15.1011 0.778 0.438083
group2 -12.2308 5.3260 -2.296 0.023401 *
site2 24.4840 7.5657 3.236 0.001569 **
site3 20.9089 8.3643 2.500 0.013789 *
site4 25.5829 7.2261 3.540 0.000571 ***
group2:site2 -26.0769 15.0700 -1.730 0.086153 .
group2:site3 -1.3517 16.7772 -0.081 0.935920
group2:site4 -13.4701 14.4273 -0.934 0.352372
When this regression object is passed into an ANOVA via car::Anova(regression, type = 3) I get this output.
Anova Table (Type III tests)
Sum Sq Df F value Pr(>F)
(Intercept) 510 1 0.6054 0.438083
group 4447 1 5.2736 0.023401 *
site 12255 3 4.8446 0.003236 **
group:site 3310 3 1.3086 0.274871
Residuals 100344 119
I need to manually calculate pooled p-values from a multiple imputation for each of the omnibus effects reported in the ANOVA: group, site, and group:site. Why I need to do it manually is not important, but suffice it to say I do. When calculating pooled p-values it is necessary to calculate pooled estimates for the standard errors of each coefficient (see here). This is easy enough with a regression, where there is a standard error reported for each comparison within each factor, but how do I get the equivalent for the omnibus effects of group, site, and their interaction? What is the equivalent of the standard error for an omnibus effect?
