Nature of the Relationship between Predictors and Dependent in Regression

Given the interpretation of regression coefficients for continuous predictors is of the form: a one unit increase in the predictor leads to a "coefficient" unit increase in the:

• dependent (linear regression)
• logit of the dependent (logistic regression)

holding all other predictor values constant, my question is whether prior to fitting the regression, you must ensure (via some transformation) that each predictor (in isolation) has the appropriate linear relationship with the dependent:

• straight-line relationship with the dependent (linear regression)
• straight-line relationship with the logit of the dependent (logistic regression)

?

1 Answer

If you attempt to solve for linearizing transformations before the formal model fitting you will not recognize the true number of unknown regression coefficients that are in play. This type of model uncertainty causes standard errors, confidence limits, and $P$-values to be incorrect in the direction of being anti-conservative.

Instead think of transformation estimation as an intrinsic part of model fitting, and you will get not only a better model (because how you transform one variable depends on how interactions are handled and how you transform other variables) but also correct statistical inference. Regression splines are your friends here. For detailed examples of this approach using R see http://biostat.mc.vanderbilt.edu/rms - look for "Handouts".