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The effects of transformations and preliminary tests for non‐linearity in regression
Author(s) -
Grambsch Patricia M.,
O'Brien Peter C.
Publication year - 1991
Publication title -
statistics in medicine
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.996
H-Index - 183
eISSN - 1097-0258
pISSN - 0277-6715
DOI - 10.1002/sim.4780100504
Subject(s) - quadratic equation , type i and type ii errors , mathematics , linearity , variable (mathematics) , variables , transformation (genetics) , linear regression , term (time) , test statistic , statistics , quadratic form (statistics) , statistic , linear model , nominal level , simple (philosophy) , set (abstract data type) , statistical hypothesis testing , computer science , mathematical analysis , confidence interval , quantum mechanics , gene , programming language , biochemistry , physics , geometry , chemistry , epistemology , philosophy
Non‐linear relationships between two variables are often detected as a result of a preliminary statistical test for linearity. Common approaches to dealing with non‐linearity are to (a) make a linearizing transformation in the independent variable or (b) fit a relationship that is non‐linear in the independent variable, such as including a quadratic term. With either approach, the resulting test for association between the two variables can have an inflated type I error. We consider testing the significance of the quadratic term in a quadratic model as a preliminary test for non‐linearity. Using simulation experiments and asymptotic arguments, we quantify the type I error inflation and suggest simple modifications of standard practice to protect the size of the type I error. In the case of quadratic regression, the type I error will be increased by roughly 50 per cent. The simple strategy of appropriately correcting the α‐level is shown to have minimal loss of power if the relationship is truly linear. In the case of a linearizing transformation, the impact on the type I error will depend on the values of the independent variable and on the set of potential linearizing transformations considered. Simulation results suggest that a procedure which adjusts the test statistic according to the results of the preliminary test may offer adequate protection.