Centered, variable-wise Centered, term-wise Standardized, variable-wise Standardized, term-wise As it turns out, these models are all equivalent. Although they have different values of , they produce the same predicted values for , and the residuals are the same, that is . This means that the overall fit, , of the models is the same. Additionally, all of the t-statistics and p-values for s representing the same effect are equal for the highest order effects, i.e. centering and scaling do not change the statistical significance of highest order effects (but will in general change t-values and p-values for lower order, included effects, and intercepts). Given the correct centering and scaling constants, each set of s can be calculated as a linear transformation of any of the other sets (without any need to transform the data). The real differences, then, are in ease of calculation, use, and interpretation. For ease of calculation it is hard to beat term-wise centering. Except for the intercept, each , and the intercept is . More explicitly This is transformation is like the transformation in an additive model. The one wrinkle here is to note that . Because of this, unless there is a constant effect hidden in the interaction term, and there is no simple interpretation of the first-order effects. Another way to say this is when there is no easily interpreted point in the data space where both and cancel out leaving only the effect of . Suppose we seek the point at which both and cancel out. For to cancel this implies . Then the product term is And for this to be zero In other words, the parameter is the effect of at the point . Rewritten in terms of our centered variables, this is the point . Because this is a specific point, we can further calculate the predicted value of A parallel derivation isolates the parameter .
Centered, variable-wise Centered, term-wise Standardized, variable-wise Standardized, term-wise As it turns out, these models are all equivalent. Although they have different values of , they produce the same predicted values for , and the residuals are the same, that is . This means that the overall fit, , of the models is the same. Additionally, all of the t-statistics and p-values for s representing the same effect are equal for the highest order effects, i.e. centering and scaling do not change the statistical significance of highest order effects (but will in general change t-values and p-values for lower order, included effects, and intercepts). Given the correct centering and scaling constants, each set of s can be calculated as a linear transformation of any of the other sets (without any need to transform the data). The real differences, then, are in ease of calculation, use, and interpretation. For ease of calculation it is hard to beat term-wise centering. Except for the intercept, each , and the intercept is . More explicitly This is transformation is like the transformation in an additive model. The one wrinkle here is to note that . Because of this, unless there is a constant effect hidden in the interaction term, and there is no simple interpretation of the first-order effects. Another way to say this is when there is no easily interpreted point in the data space where both and cancel out leaving only the effect of . Suppose we seek the point at which both and cancel out. For to cancel this implies . Then the product term is And for this to be zero In other words, the parameter is the effect of at the point . Rewritten in terms of our centered variables, this is the point . Because this is a specific point, we can further calculate the predicted value of A parallel derivation isolates the parameter .
Centering%20and%20Standardizing%20Data%20and%20the%20Change%20of%20Coordinates%20for%20Model%20Parameters.docx