A partial sum of squares, also known as a Type II sum of squares or a deleted sum of squares, represents the additional variation explained by adding a specific variable to a statistical model that already includes other variables. It's calculated iteratively, assessing the unique contribution of each variable after accounting for the others. This contrasts with a sequential sum of squares (Type I), which assesses the contribution of each variable in the order they are added to the model.
Understanding Partial Sum of Squares
Imagine you're building a model to predict crop yield. You might include variables like rainfall, fertilizer use, and sunlight. A partial sum of squares for fertilizer use would tell you how much additional variation in crop yield is explained by fertilizer after considering the effects of rainfall and sunlight. It isolates the unique impact of fertilizer.
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Iterative Calculation: The calculation involves comparing the residual sum of squares (the unexplained variation) of a model with and without the variable in question. The difference is the partial sum of squares for that variable. As mentioned in the provided text from Mason Gunter Hunter, this iterative process is key to understanding how partial sums of squares are determined.
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Significance Testing: Partial sums of squares are crucial for testing the statistical significance of individual variables in multiple regression and ANOVA models. They allow you to determine if a variable makes a unique contribution to the model above and beyond the other variables already included.
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Contrast with Sequential Sums of Squares: Unlike sequential sums of squares, which depend on the order variables are entered into the model, partial sums of squares are independent of this order. Each variable's contribution is assessed relative to all other variables in the model. The provided text from Reed College highlights this key difference, stating that unlike partial sums of squares, sequential sums of squares build the model variable-by-variable.
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Example: In a multiple linear regression analysis with variables X1, X2, and X3, the partial sum of squares for X2 represents the reduction in the residual sum of squares when X2 is added to a model already containing X1 and X3 (or any other combination).
The references clearly state that partial sums of squares are calculated iteratively and are also known as deleted sums of squares because the value is calculated by subtracting the residual SS with and without the parameter. They are also referred to as Type II sums of squares and are useful in determining the significance of each independent variable in a model.
