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Sequential versus adjusted sums of squares

Sums of squares



A.k.a Type I.


A.k.a. Type II.

Sequential sums of squares

For a multiple linear regression, the sequential sum of squares is,

  • Increase in SSR when one or more variables are added to the model.

  • Decrease in SSE when one or more variables are added to the model.

In other words, it is the improvement in the amount of variation in the response that is accounted for by the response variables, when we add a particular variable or variables to the model.

It has a particular notation,



We have a model with only .

We have a model with only and .

We add to a model containing . What is the corresponding change in SSR or SSE? For example, we may analyze what change happens to SSR or SSE when adding a second variable to a linear regression model that only contained .

The increase in the regression sum of squares when we add to a model containing and . In other words,


Remember that ,


We never need to clarify which variables are included in SSTO because it is independent of which variables are being used. SSR and SSE are variable-depending proportions of SSTO.

Adjusted sums of squares

When conducting a multiple linear regression, this will be in an ANOVA tabla that conveys the SSR and SSE, adjusted for the other variables.

Sequential versus adjusted

A major difference is that for a sequential sums of squares,

However, for adjusted sums of squares,