Hosmer–Lemeshow test
Encyclopedia
The Hosmer–Lemeshow test is a statistical test for goodness of fit
for logistic regression
models. It is used frequently in risk prediction
models. The test assesses whether or not the observed event rates match expected event rates in subgroups of the model population. The Hosmer–Lemeshow test specifically identifies subgroups as the deciles of fitted risk values. Models for which expected and observed event rates in subgroups are similar are called well calibrated.
Here Og, Eg, Ng, and πg denote the observed events, expected events, observations, predicted risk for the gth risk decile group, and n is the number of groups. The test statistic asymptotically follows a
distribution with n-2 degrees of freedom. The number of risk groups may be adjusted depending on how many fitted risks are determined by the model. This helps to avoid singular decile groups.
Goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g...
for logistic regression
Logistic regression
In statistics, logistic regression is used for prediction of the probability of occurrence of an event by fitting data to a logit function logistic curve. It is a generalized linear model used for binomial regression...
models. It is used frequently in risk prediction
Predictive analytics
Predictive analytics encompasses a variety of statistical techniques from modeling, machine learning, data mining and game theory that analyze current and historical facts to make predictions about future events....
models. The test assesses whether or not the observed event rates match expected event rates in subgroups of the model population. The Hosmer–Lemeshow test specifically identifies subgroups as the deciles of fitted risk values. Models for which expected and observed event rates in subgroups are similar are called well calibrated.
Hosmer–Lemeshow test
The Hosmer–Lemeshow test statistic is given by:Here Og, Eg, Ng, and πg denote the observed events, expected events, observations, predicted risk for the gth risk decile group, and n is the number of groups. The test statistic asymptotically follows a
distribution with n-2 degrees of freedom. The number of risk groups may be adjusted depending on how many fitted risks are determined by the model. This helps to avoid singular decile groups.