Logistic regression
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The document discusses three statistical tests used for logistic regression models: 1) The Wald test calculates a z-statistic for each model coefficient and uses it to test the statistical significance, but can be unreliable for small sample sizes. 2) The likelihood-ratio test compares the maximized likelihood of the full model to a simpler model, and is recommended for stepwise model building. 3) The Hosmer-Lemshow goodness-of-fit test groups subjects by predicted probability and compares observed to predicted outcomes, with a non-significant result indicating good model fit.
Logistic regression
- 1. Wald Test: A Wald test is used to test the statistical significance of each coefficient (b) in the model. A Wald test calculates a Z statistic, which is: This z value is then squared, yielding a Wald statistic with a chi-square distribution. However, several authors have identified problems with the use of the Wald statistic. Menard (1995) warns that for large coefficients, standard error is inflated, lowering the Wald statistic (chi-square) value. Agresti (1996) states that the likelihood-ratio test is more reliable for small sample sizes than the Wald test. Likelihood-Ratio Test: The likelihood-ratio test uses the ratio of the maximized value of the likelihood function for the full model (L1) over the maximized value of the likelihood function for the simpler model (L0). The likelihood-ratio test statistic equals: This log transformation of the likelihood functions yields a chi-squared statistic. This is the recommended test statistic to use when building a model through backward stepwise elimination. Hosmer-Lemshow Goodness of Fit Test: The Hosmer-Lemshow statistic evaluates the goodness-of-fit by creating 10 ordered groups of subjects and then compares the number actually in the each group (observed) to the number predicted by the logistic regression model (predicted). Thus, the test statistic is a chi-square statistic with a desirable outcome of non-significance, indicating that the model prediction does not significantly differ from the observed. The 10 ordered groups are created based on their estimated probability; those with estimated probability below 0.1 form one group, and so on, up to those with probability 0.9 to 1.0. Each of these categories is further divided into two groups based on the actual observed outcome variable (success, failure). The expected frequencies for each of the cells are obtained from the model.If the model is good, then most of the subjects with success are classified in the higher deciles of risk and those with failure in the lower deciles of risk.