I want to get the difference of differences from a limma voom analysis. In a simplified example, I have 3 genotypes: Parent1, Parent2 and the Hybrid (of the two parents). For this example, I want to find whether the Hybrid expression is closer to the expression level of Parent1 or Parent2 (or equally close to both parents). In the analysis, we already compare both parents to the Hybrid in individual contrasts.

y <- DGEList(countdata)
y <- calcNormFactors(y)
design <- model.matrix(~ 0 + sampleinfo$Genotype + sampleinfo$REP.BAT)
v <- voomWithQualityWeights(y,design)
fit <- lmFit(v, design)

makeContrasts(Parent1-Hybrid, Parent2-Hybrid, levels=design)->mat.a

fit.a <- contrasts.fit(fit, mat.a)
fit.a <- eBayes(fit.a)
fit.treat.a <- treat(fit.a)

By subtracting the absolute values of the coefficients from those contrasts, we should get the difference of differences. What I am looking for, is to get the corresponding statistical test for these results.

abs(fit.treat.a[["coefficients"]][,1])-abs(fit.treat.a[["coefficients"]][,2])

Is there a way, to also extract the corresponding standard deviation and calculate the corresponding t-tests? Or is there a way, to build a contrast directly, that compares the difference of Parent1vsHybrid and Parent2vsHybrid directly?

makeContrasts((Parent1-Hybrid) - (Parent2-Hybrid), levels=design)->mat.a

Like this, but it the Hybrid is being cut out in this formulation, resulting in Parent1-Parent2, which is not what I want.

Determining which parent the hybrid is closest to is not equivalent to any linear model hypothesis test and so can't be tested in limma.

You can however test whether Parent1 is equal to Parent2 and whether either is different from the Hybrid. That requires three separate contrasts, and those three comparisons tell you everything you need to know to determine which can be distinguished from which.

You can also use plotMDS() to explore the relationships between the samples in terms of leading-fold-change distances. That will show visually whether hybrid samples tend to cluster closer to one parent or the other.