Consider an RBD with V treatments and b blocks. Let SST, SSB and SSE denote treatment, block and error sum of squares respectively. If a same constant C ≠ 0 is added to every observation in one particular block leaving all other observations unchanged, then which of the following is necessarily true ?
SST and SSE remain unchanged while SSB changes
In an RBD every treatment appears exactly once in each block. Adding C to all observations of one block adds C to every treatment total, so every treatment mean rises by C/b and the grand mean rises by C/b.
So the treatment deviations \(\bar y_{i.} - \bar y_{..}\) are unchanged and SST is unchanged.
The altered block mean rises by C while the other block means do not, so the block deviations \(\bar y_{.j} - \bar y_{..}\) change and SSB changes (in general).
The residual \(y_{ij} - \bar y_{i.} - \bar y_{.j} + \bar y_{..}\) changes by \(C - C/b - C + C/b = 0\) in the altered block and by \(0 - C/b - 0 + C/b = 0\) elsewhere, so SSE is unchanged; any claim that SST or SSE changes fails.
Check: the change is exactly a block effect added to the model, which is absorbed entirely by the block term and leaves treatment comparisons and residuals untouched.
Hence, SST and SSE remain unchanged while SSB changes.
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