Suppose you have tumor and normal samples treated with compound A with 0, 1, 2 mM for 0, 1, 2 hours, and you want to do DGE analysis using edgeR. You want to answer the question "which genes are differentially expressed in response to compound A": after taking into account of the impact of disease condition, time of treatment, and concentration of A (noted as [A]), you want to focus on the relationship of treatment by A and gene expression, and don't care about disease condition, time of treatment, not even [A].

How do you do it in edgeR? 

In R, assuming the log(expression) being normally distributed, you could simply do glm(log(expression) ~ disease_condition + [A] + time. Then you only need to check the slope of [A] and its p-value, right? But since log(expression) is not normally distributed, we cannot use glm in R. 

While in edgeR, things seem to be much more complicated. First, both time and [A] cannot be considered to be continuous (correct me if I am wrong); Second, I don't know how to do the DGE analysis.

For simplicity, let's forget about time in the problem -- just consider disease condition and [A] (0 & 1 mM), each with a replicate. Your gene count file looks like the following (T for tumor, N for normal; A0 for [A] = 0, A1 for [A] = 1; column 2~9 are gene counts for 8 samples, one column for one sample):

    GENE T_A0 T_A0 T_A1 T_A1 N_A0 N_A0 N_A1 N_A1
    GENE_A ...
    GENE_B ...
    ...

You will have four levels: T_A0, T_A1, N_A0, N_A1.

Your code snippet looks like the following:

    y <- DGEList(counts=d, group=grp);
    y <- calcNormFactors(y); # global normalization
    dsgn <- model.matrix(~0+grp); # now allowing intercept
    y <- estimateDisp(y, dsgn);
    fit <-glmFit(y, dsgn)

The question is how to construct your test (or contrast). Assuming the levels are ordered as above, what would be your contrast? c(-0.5, 0.5, -0.5, 0.5) or something else? Or is there a better approach? Or am I terribly wrong?

Thanks in advance!

You are wrong on a number of counts.

1. If log-expression were normally distributed, you should be using linear models with lm. And if you were to do that anyway, it would be better to use limma via voom.
2. edgeR is no more complicated than glm. Whatever formula you want to use in the latter, you can use in the former. The only additional work required is to perform empirical Bayes shrinkage, and that is completely necessary in the presence of limited replicates.
3. Time and concentration can be continuous. Why makes you think that they can't be?

Moving onto your simple example, the contrast between levels would indeed be:

con <- makeContrasts((T_A1 + N_A1)/2 - (T_A0 + N_A0)/2, levels=design)

... which will test if the average of A1 groups is different from the average of A0 groups. If you want to be more stringent, you might do:

con <- makeContrasts(T_A1 - T_A0, N_A1 - N_A0, levels=design)

... and select for genes that are significant and have the same sign of the log-fold change in both contrasts.

Obviously, you can also use an additive model in edgeR by changing the design matrix (~disease + A). In this case, just dropping the treated/untreated factor would tell you the effect of A. However, this involves making some additional assumptions regarding the lack of interaction between disease and treatment effects - regardless of whether you use edgeR or glm.