Box-Cox Transformations for Generalized Linear Models

• Activity Number: 186
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 10:30 AM to 12:20 PM
• Sponsor: Biometrics Section
• Author(s): Patrick Johnston*+
• Keywords: Box-Cox transformation ; generalized linear models ; blink rate ; interblink interval

Abstract

The Box-Cox method attempts to power-transform the outcome variable to follow a homogeneous normal linear model (LM). Because this is not always possible, we extend the Box-Cox approach to exponential family generalized linear models (GLMs). Thus a power transformation is sought such that the transformed outcome follows a GLM. Candidate distributions for GLMs include the gamma and Wald in addition to the normal, and candidate link functions include the reciprocal and logarithm in addition to the identity. Comparisons between models are assessed by AIC.

We apply the method to studies in which the ith subject is observed to have N(i) events over time T(i). This gives two interpretable outcomes: rates y(i) = N(i)/T(i) and paces 1/y(i) = T(i)/N(i). For example, running performance can be measured in miles per minute (speed) or minutes per mile (pace), and ocular blinking activity can be measured in blinks per minute (blink rate) or minutes per blink (interblink interval). The original Box-Cox method chooses the outcome more closely approximating a LM, although in this setting both approximations may be poor. Instead we propose choosing the outcome more closely approximating a GLM.