Figure 4.12.5: Hosmer and Lemeshow Test

Here the null hypothesis is that the true probabilities are those specified by the model.

Contingency table for Hosmer and Lemeshow goodness-of-fit test

The Hosmer–Lemeshow test is a for for models. It is used frequently in 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 s of fitted risk values. Models for which expected and observed event rates in subgroups are similar are called well calibrated.

For data in Binary Response/Frequency format, the Hosmer-Lemeshow results are more trustworthy.

5.2.2 The Hosmer-Lemeshow Tests

proposed a two degree-of-freedom test to ascertain whether a generalized logistic model is better than a standard model fit to the data. Her test determines whether two parameters in a generalized logistic model are equal to zero. Briefly, the two additional parameters allow the tails of the logistic regression model that is the small and large probabilities to be either heavier or lighter than the standard logistic regression model. This test is not actually a goodness-of-fit test since, it does not compare observed and fitted values. However, it does provide a test of the basic logistic regression model assumption and in that sense it may be considered as a useful adjunct to the Hosmer-Lemeshow and Osius-Rojek goodness of fit tests. The test has not been implemented in any package but it can be easily obtained from four steps procedure as:

I have never heard of the Hosmer and Lemeshow test, but Google refersme to

The binary logistic regression model has been fitted under the assumption that we are at least preliminarily satisfied with our efforts at the model building stage. By this we mean that, to the best of our knowledge, the model contains those variables that should be in the model and the variables have been entered in the correct functional form. Hence, we may like to know how effectively the model has described the outcome variable. Once, the particular multiple logistic regression model has been fitted, we begin the process of model assessment. The likelihood ratio test is performed to test the overall significance of all coefficients in the model on the basis of test statistic:

Lemeshow S, Hosmer DW Jr, Klar J, & Lwanga SK (1990) Adequacy of sample size in health studies.


Hosmer lemeshow null hypothesis | scholarly search

If the p-value for the goodness-of-fit test is lower than your chosen significance level, the predicted probabilities deviate from the observed probabilities in a way that the binomial distribution does not predict.

design patterns - Hosmer-Lemeshow goodness of Fit test …

"The Hosmer-Lemeshow test is for overall calibration error, not for any particular lack of fit such as quadratic effects. It does not properly take overfitting into account, is arbitrary to choice of bins and method of computing quantiles, and often has power that is too low."

Hosmer-Lemeshow Test for Logistic Regression | …

They recommended that overall assessment of fit be examined using a combination of Hosmer-Lemeshow goodness-of-fit test, Osius and Rojek normal approximation test and Stukel test as adjunct of their tests. The large sample normal approximation to the distribution of the Pearson chi-square statistic derived by may be easily computed in any package that has the option to save the fitted values from the logistic regression model and do a weighted linear regression.

Trust the Hosmer-Lemeshow Test for Logistic Regression March 5, ..

where, g denotes the number of groups, n'k is the number of observations in the kth group, ok is the sum of the Y values for the kth group and is the average of the ordered for the kth group. demonstrated that under the null hypothesis that the fitted logistic regression model is the correct model, the distribution of the statistic Ĉ is well approximated by the chi-square distribution with g-2 degrees of freedom. This test is more reliable and robust than the traditional chi-square test (). The value of the Hosmer-Lemeshow goodness-of-fit statistic computed from the frequencies () is Ĉ = 5.28 and the corresponding p-value computed from the chi-square distribution with 8 degrees of freedom is 0.73. The large p-value signifies that there is no significant difference between the observed and the predicted values of the outcome. This indicates that the model seems to fit quite reasonable. A comparison of the observed and expected frequencies in each of the 20 cells () also shows close agreement within each decile. examined the distributional properties of their test via simulations.

The older Hosmer-Lemeshow test requires binning and ..

Hosmer-Lemeshow goodness-of-fit test divides subjects into deciles based on predicted probabilities and computes a chi-square from observed and expected frequencies (). Using this grouping strategy, the Hosmer-Lemeshow goodness-of-fit statistic, Ĉ is obtained by calculating the Pearson chi-square statistic from the gx2 table of observed and estimated expected frequencies. A formula defining the calculation of Ĉ is as follows: