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Multinomial goodness‐of‐fit tests for logistic regression models
Author(s) -
Fagerland Morten W.,
Hosmer David W.,
Bofin Anna M.
Publication year - 2008
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.3202
Subject(s) - goodness of fit , statistics , mathematics , contingency table , multinomial distribution , multinomial logistic regression , logistic regression , pearson's chi squared test , degrees of freedom (physics and chemistry) , test statistic , statistic , logistic distribution , null distribution , econometrics , statistical hypothesis testing , physics , quantum mechanics
We examine the properties of several tests for goodness‐of‐fit for multinomial logistic regression. One test is based on a strategy of sorting the observations according to the complement of the estimated probability for the reference outcome category and then grouping the subjects into g equal‐sized groups. A g × c contingency table, where c is the number of values of the outcome variable, is constructed. The test statistic, denoted as C g , is obtained by calculating the Pearson χ 2 statistic where the estimated expected frequencies are the sum of the model‐based estimated logistic probabilities. Simulations compare the properties of C g with those of the ungrouped Pearson χ 2 test ( X 2 ) and its normalized test ( z ). The null distribution of C g is well approximated by the χ 2 distribution with ( g −2) × ( c −1) degrees of freedom. The sampling distribution of X 2 is compared with a χ 2 distribution with n × ( c −1) degrees of freedom but shows erratic behavior. With a few exceptions, the sampling distribution of z adheres reasonably well to the standard normal distribution. Power simulations show that C g has low power for a sample of 100 observations, but satisfactory power for a sample of 400. The tests are illustrated using data from a study of cytological criteria for the diagnosis of breast tumors. Copyright © 2008 John Wiley & Sons, Ltd.