I don't understand why the sum of adjusted sum squares of each predictor(0.0979+9.08723=9.1851) don't equal the total regression sum of square(11.7778)? And I know how to calculate sum of adjusted sum squares of a simple linear regression model as there is only one predictor, so the only one predictor has it all. But how to calculate sum squares for each predictor in a multiple linear regression model?
The sequential sum of squares tells us how much the SSE declines after we add another variable to the model that contains only the variables preceding it. By contrast, the adjusted sum of squares tells us how much the SSE declines after we add another variable to the model that contains every other variable. In your specific example, the logic would be as follows:
If we add Vocab to the model that already contains SDMT, the SSE would decline by 9.0872 If we add SDMT to the model that already contains Vocab, the SSE would decline by 0.0979.
In the Sequential Sum of Squares Analysis, the logic would look like this:
If we add Vocab to the model that contains no predictors, the SSE would decline by 11.6799. If we add SDMT to the model that already contains Vocab, the SSE would decline by 0.0979.
The Adj SS for each predictor tells you how much the SS Regression would be increased (or, equivalently, the SSE would be reduced) by adding that predictor to the model that already has all the other predictors.
If Vocab is already in the model (that would already increase the SS Regression from 0 up to some amount) and then you add SDMI to the model, that addition of SDMI would increase the SS Regression by an additional 9.0872. Since the total SS Regression is 11.7778, you can subtract to find that by itself, Vocab would give a SS Regression of 11.7778 - 9.0872 = 2.6906. If you do the Regression with just the one predictor Vocab, you'll see that's true.
So if you start with zero predictors and add Vocab, SS Regression increases by 2.6906. Then if you also add SDMI, the SS Regression increases by an additional 9.0872. These are the sequential SS, which add up to the total SS Regression of 11.7778.
