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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?

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