I have a linear model with several predictors (only main effects). One of them is a catagorical variable X with 4 levels (A...D). One level (A) is used as the reference level, so the model gives coefficients for the other levels that refer to the differences between these levels versus the reference (B-A, C-A and D-A).
Given I have the standard errors of these coefficients and the numbers of each level of X, and I also have the total number of coefficinets in the model as well as R², SEy, SSreg and SSres: how can I calculate the SE for the other pair-wise differences (B-C, B-D, C-D)?
Example: As a test I set up a model with 5 coefs plus intercept and get
Coef "B" "C" "D"
SE 0.323 0.302 0.233
N 7 16 12
total N = 55 (so there are 20 values for level "A")
R² = 0.9796
SEy = 0.0752
SSreg = 6.809
SSres = 0.1415
Thus, the SE of C-A is 0.302. How could I use the above results to get the SE of C-B?
If I calculate the same model on the same data but with "B" as a reference I get
Coef "A" "C" "D"
SE 0.323 0.267 0.301
N 20 16 12
So this way I get the SE of C-B as 0.267. But if I don't have the possibility to re-calculate the model, if I only have the results for a reference level being "A", how can I get this SE of C-B?
If you really can't run the analysis again (and you don't have the covariance matrix already), then there is no way to calculate the answers you want.
If you can do the analysis again but Simen's answer seems too much like hard work for you, simply relabel the levels of the factor so that B (or C) is the baseline. That will give you two more standard errors of differences, and then you will need one more relabelling to get the final s.e.d. (However, it seems that you have already considered but rejected this.)
Thank you for your answers. Actually, this is all I have. I do not have any covariance matrix, and I don't have the possibility to re-calculate the model. The example I gave was just something self-created to try out some ideas and crosscheck if they would work. Now at least I know that there is no possibility.
I believe that statistical questions of this type are better handled through stats.stackexchange.com
If you know, whether the variables are positively or negatively correlated, you can at least get un upper boundary for the SD and hence for the precision. As VAR(Y-X)=VAR(X)+VAR(Y)+2COV(X,Y).
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