A Comparison of Two Versions of the Hosmer-Lemeshow Goodness-of-Fit Test for Logistic Regression

• Activity Number: 43
• Type: Contributed
• Date/Time: Sunday, August 1, 2010 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistics in Epidemiology
• Author(s): Jana D. Canary*+ and Leigh Blizzard and Ronald Barry and Steve John Quinn and David W. Hosmer
• Companies: University of Tasmania and Menzies Research Institute and University of Alaska, Fairbanks and Menzies Research Institute and University of Massachusetts, Amherst
• Keywords: model fit ; likelihood methods ; score test

Abstract

Hosmer and Lemeshow (1980) introduced a goodness-of-fit test (HL) for logistic regression models that is based on partitioning data into groups of predicted probabilities. They showed via simulation that the distribution of the HL statistic is approximately chi-squared with groups minus two (g-2) d-o-f. Tsiatis (1980) proposed a score test (T) based on fixed groups. Its distribution is asymptotically chi-squared with g-1 d-o-f. We demonstrate that the HL statistic is an approximate form of the T score statistic and explore via simulation the relative performance of the T statistic, the original HL statistic, and the Tsiatis form of the HL statistic (HLT). Interim findings are that the distribution of the HLT statistic is approximately chi-squared with g-2 d-o-f and that the HL and HLT statistics perform similarly. These results suggest that the HL test may have two equivalent forms.