Interpret the key results for Fit Binary Logistic Model

Complete the following steps to interpret a binary logistic model. Key output includes the p-value, the coefficients, R2, and the goodness-of-fit tests.

Step 1: Determine whether the association between the response and the term is statistically significant

To determine whether the association between the response and each term in the model is statistically significant, compare the p-value for the term to your significance level to assess the null hypothesis. The null hypothesis is that there is no association between the term and the response. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association exists when there is no actual association.
P-value ≤ α: The association is statistically significant
If the p-value is less than or equal to the significance level, you can conclude that there is a statistically significant association between the response variable and the term.
P-value > α: The association is not statistically significant
If the p-value is greater than the significance level, you cannot conclude that there is a statistically significant association between the response variable and the term. You may want to refit the model without the term.
If there are multiple predictors without a statistically significant association with the response, you can reduce the model by removing terms one at a time. For more information on removing terms from the model, go to Model reduction.
If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows:
  • If a continuous predictor is significant, you can conclude that the coefficient for the predictor is different from zero.
  • If a categorical predictor is significant, you can conclude that not all of the levels of the factor have the same probability.
  • If an interaction term is significant, you can conclude that the relationship between a predictor and the probability of the event depends on the other predictors in the term.
  • If a polynomial term is significant, you can conclude that the relationship between a predictor and the probability of the event depends on the magnitude of the predictor.
Deviance Table Source DF Adj Dev Adj Mean Chi-Square P-Value Regression 1 22.7052 22.7052 22.71 0.000 Dose (mg) 1 22.7052 22.7052 22.71 0.000 Error 4 0.9373 0.2343 Total 5 23.6425
Coefficients Term Coef SE Coef VIF Constant -5.25 1.99 Dose (mg) 3.63 1.30 1.00
Odds Ratios for Continuous Predictors Odds Ratio 95% CI Dose (mg) 37.5511 (2.9645, 475.6528)
Key Results: P-Value, Coefficients

In these results, the dosage is statistically significant at the significance level of 0.05. You can conclude that changes in the dosage are associated with changes in the probability that the event occurs.

Assess the coefficient to determine whether a change in a predictor variable makes the event more likely or less likely. The relationship between the coefficient and the probability depends on several aspects of the analysis, including the link function. Generally, positive coefficients indicate that the event becomes more likely as the predictor increases. Negative coefficients indicate that the event becomes less likely as the predictor increases. For more information, go to Coefficients and regression equation for Fit Binary Logistic Model.

The coefficient for Dose is 3.63, which suggests that higher dosages are associated with higher probabilities that the event will occur.

If an interaction term is statistically significant, the relationship between a predictor and the response differs by the level of the other predictor. In this case, you should not interpret the main effects without considering the interaction effect. To obtain a better understanding of the main effects, interaction effects, and curvature in your model, go to Factorial Plots and Response Optimizer.

Step 2: Understand the effects of the predictors

Use the odds ratio to understand the effect of a predictor. The interpretation of the odds ratio depends on whether the predictor is categorical or continuous. Minitab calculates odds ratios when the model uses the logit link function.
Odds Ratios for Continuous Predictors

Odds ratios that are greater than 1 indicate that the event is more likely to occur as the predictor increases. Odds ratios that are less than 1 indicate that the event is less likely to occur as the predictor increases.

Binary Logistic Regression: No Bacteria versus Dose (mg)

Odds Ratios for Continuous Predictors Unit of Change Odds Ratio 95% CI Dose (mg) 0.5 6.1279 (1.7218, 21.8095)
Key Result: Odds Ratio

In these results, the model uses the dosage level of a medicine to predict the presence or absence of bacteria in adults. Each pill contains a 0.5 mg dose, so the researchers use a unit change of 0.5 mg. The odds ratio is approximately 6. For each additional pill that an adult takes, the odds that a patient does not have the bacteria increase by about 6 times.

Odds Ratios for Categorical Predictors

For categorical predictors, the odds ratio compares the odds of the event occurring at 2 different levels of the predictor. Minitab sets up the comparison by listing the levels in 2 columns, Level A and Level B. Level B is the reference level for the factor. Odds ratios that are greater than 1 indicate that the event is less likely at level B. Odds ratios that are less than 1 indicate that the event is more likely at level B. For information on how to select the reference level for the analysis, go to Specify the coding scheme for Fit Binary Logistic Model.

Binary Logistic Regression: Cancellation versus Month

Odds Ratios for Categorical Predictors Level A Level B Odds Ratio 95% CI Month 2 1 1.1250 (0.0600, 21.0867) 3 1 3.3750 (0.2897, 39.3222) 4 1 7.7143 (0.7460, 79.7712) 5 1 2.2500 (0.1107, 45.7226) 6 1 6.0000 (0.5322, 67.6495) 3 2 3.0000 (0.2547, 35.3340) 4 2 6.8571 (0.6556, 71.7201) 5 2 2.0000 (0.0976, 41.0034) 6 2 5.3333 (0.4679, 60.7972) 4 3 2.2857 (0.4103, 12.7323) 5 3 0.6667 (0.0514, 8.6389) 6 3 1.7778 (0.2842, 11.1200) 5 4 0.2917 (0.0252, 3.3719) 6 4 0.7778 (0.1464, 4.1326) 6 5 2.6667 (0.2124, 33.4861) Odds ratio for level A relative to level B
Key Result: Odds Ratio

In these results, the categorical predictor is the month from the start of a hotel's busy season. The response is whether or not a guest cancels a reservation. The largest odds ratio is approximately 8, when level A is month 4 and level B is month 1. This indicates that the odds that a guest cancels a reservation in month 4 is approximately 8 times higher than the odds that a guest cancels a reservation in month 1.

For more information, go to Odds Ratios for Fit Binary Logistic Model.

Step 3: Determine how well the model fits your data

To determine how well the model fits your data, examine the statistics in the Model Summary table.

For binary logistic regression, the data format affects most of the model summary and goodness-of-fit statistics. The AIC and the Hosmer-Lemeshow test are unaffected by the data format and are, therefore, comparable between formats. For more information, go to How data formats affect goodness-of-fit in binary logistic regression.

Deviance R-sq

The higher the deviance R2, the better the model fits your data. Deviance R2 is always between 0% and 100%.

Deviance R2 always increases when you add additional predictors to a model. For example, the best 5-predictor model will always have an R2 that is at least as high as the best 4-predictor model. Therefore, deviance R2 is most useful when you compare models of the same size.

For binary logistic regression, the format of the data affects the deviance R2 value. The deviance R2 is usually higher for data in Event/Trial format. Deviance R2 values are comparable only between models that use the same data format.

Deviance R2 is just one measure of how well the model fits the data. Even when a model has a high R2, you should check the residual plots and goodness-of-fit tests to assess how well a model fits the data.

Deviance R-sq (adj)

Use adjusted deviance R2 to compare models that have different numbers of predictors. Deviance R2 always increases when you add a predictor to the model. The adjusted deviance R2 value incorporates the number of predictors in the model to help you choose the correct model.

AIC

Use AIC to compare different models. The smaller the AIC, the better the model fits the data. However, the model with the smallest AIC for a set of predictors does not necessarily fit the data well. Also use goodness-of-fit tests and residual plots to assess how well a model fits the data.

Model Summary Deviance Deviance R-Sq R-Sq(adj) AIC 96.04% 91.81% 21.68
Key Results: Deviance R-Sq, Deviance R-Sq (adj), AIC

In these results, the model explains 96.04% of the deviance in the response variable. For these data, the Deviance R2 value indicates the model provides a good fit to the data. If additional models are fit with different predictors, use the adjusted Deviance R2 value and the AIC value to compare how well the models fit the data.

Step 4: Determine whether the model does not fit the data

Use the goodness-of-fit tests to determine whether the predicted probabilities deviate from the observed probabilities in a way that the binomial distribution does not predict. 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. This list provides common reasons for the deviation:
  • Incorrect link function
  • Omitted higher-order term for variables in the model
  • Omitted predictor that is not in the model
  • Overdispersion

If the deviation is statistically significant, you can try a different link function or change the terms in the model.

For binary logistic regression, the format of the data affects the p-value because it changes the number of trials per row.

  • Deviance: The p-value for the deviance test tends to be lower for data that are in the Binary Response/Frequency format compared to data in the Event/Trial format. For data in Binary Response/Frequency format, the Hosmer-Lemeshow results are more trustworthy.
  • Pearson: The approximation to the chi-square distribution that the Pearson test uses is inaccurate when the expected number of events per row in the data is small. Thus, the Pearson goodness-of-fit test is inaccurate when the data are in Binary Response/Frequency format.
  • Hosmer-Lemeshow: The Hosmer-Lemeshow test does not depend on the number of trials per row in the data as the other goodness-of-fit tests do. When the data have few trials per row, the Hosmer-Lemeshow test is a more trustworthy indicator of how well the model fits the data.
Response Information Event Variable Value Count Name Event Event 160 Event Non-event 340 Trial Total 500
Goodness-of-Fit Tests Test DF Chi-Square P-Value Deviance 2 3.78 0.151 Pearson 2 3.76 0.152 Hosmer-Lemeshow 3 3.76 0.288
Key Results for Event/Trial Format: Response Information, Deviance Test, Pearson Test, Hosmer-Lemeshow Test

In these results, the Response Information table shows Event and Trial in the Variable column. These labels indicate that the data are in Event/Trial format. All of the goodness-of-fit tests have p-values higher than the usual significance level of 0.05. The tests do not provide evidence that the predicted probabilities deviate from the observed probabilities in a way that the binomial distribution does not predict.

Response Information Variable Value Count Y Event 160 (Event) Non-event 340 Total 500
Goodness-of-Fit Tests Test DF Chi-Square P-Value Deviance 497 552.03 0.044 Pearson 497 504.42 0.399 Hosmer-Lemeshow 3 3.76 0.288
Key Results for Binary Response/Frequency Format: Response Information, Deviance Test, Pearson Test, Hosmer-Lemeshow Test

In these results for the same data, the Response Information table shows Y in the variable column. This label indicates that the data are in Binary Response/Frequency format. The deviance test has a p-value less than the usual significance level of 0.05, but the Hosmer-Lemeshow test is the most trustworthy test. The Hosmer-Lemeshow test does not provide evidence that the predicted probabilities deviate from the observed probabilities in a way that the binomial distribution does not predict.

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