SSE can never be

Question: SSE can never be

a. larger than SST

b. smaller than SST

c. equal to 1

d. equal to zero

Answer:- The correct option is b. smaller than SST

SSE can never be less than SST. SST includes the complete variance in data, while SSE shows the variation not accounted by model. Since the model will always account for some of the variance, SSE must be smaller.

In more detail, SST accounts for the total variability in the response variable y and it is obtained by summing up squared deviations of each observation y from their overall mean. SSE, on the other hand, captures all of that left after fitting the model and being variability. It is calculated by adding the squares of residuals, which are basically differences between observed yi and fitted values ŷi.

The main point is that ŷ i will be closer to yi than the average one. This is because the fitted values are estimates that have been adjusted to each observation due to predictor variables used in this model. Thus the residuals yi – ŷ i will be of a smaller magnitude than that their deviations, namely, mean(y). When squared, the smaller residuals will reach SSE that is lower than total sum of larger deviations in English abbreviated as SST.

Overall, SST constitutes the upper limit on variability. SSE must be less because this model cannot explain more variance than is there at startup. Once again the only case where SSE could equal to SST is when the model fits nothing but that it will not be useful. Therefore, we are able to state that SSE must be less than SST for any sensible model.