Use a Pareto chart of the standardized effects to compare the relative magnitude and the statistical significance of main, square, and interaction effects.
Minitab plots the standardized effects in the decreasing order of their absolute values. The reference line on the chart indicates which effects are significant. By default, Minitab uses a significance level of 0.05 to draw the reference line.
Term | Coef | SE Coef | VIF |
---|---|---|---|
Constant | 2.394 | 0.145 | |
Bake Time | 0.7349 | 0.0538 | 1.11 |
Bake Temperature 2 | 0.5451 | 0.0541 | 1.20 |
Bake Time*Bake Time | -0.384 | 0.153 | 1.04 |
Bake Time*Bake Temperature 2 | -0.5106 | 0.0562 | 1.24 |
In these results, the coefficients for Bake Time and Bake Temperature 2 are positive numbers. The coefficient for the squared term of Bake Time and the coefficient for the interaction term between Bake Time and Bake Temperature 2 are negative numbers. Generally, positive coefficients make the event more likely and negative coefficients make the event less likely as the value of the term increases.
Source | DF | Adj Dev | Adj Mean | Chi-Square | P-Value |
---|---|---|---|---|---|
Model | 4 | 737.452 | 184.363 | 737.45 | 0.000 |
Bake Time | 1 | 203.236 | 203.236 | 203.24 | 0.000 |
Bake Temperature 2 | 1 | 100.432 | 100.432 | 100.43 | 0.000 |
Bake Time*Bake Time | 1 | 6.770 | 6.770 | 6.77 | 0.009 |
Bake Time*Bake Temperature 2 | 1 | 80.605 | 80.605 | 80.61 | 0.000 |
Error | 45 | 32.276 | 0.717 | ||
Total | 49 | 769.728 |
In these results, the main effects for Bake Time and Bake Temperature 2 are statistically significant at the 0.05 level. You can conclude that changes in these variables are associated with changes in the response variable. Because higher-order terms are in the model, the coefficients for the main effects do not completely describe the effect of these factors.
The square term for Bake Time is significant. You can conclude that changes in this variable are associated with changes in the response variable, but the association is not linear.
The interaction effect between Bake Time and Bake Temperature 2 is significant. You can conclude that the effect on color of changes in Bake Time depends on the level of Bake Temperature 2. Equivalently, you can conclude that the effect on color of changes in Bake Temperature 2 depends on the level of Bake Time.
Unit of Change | Odds Ratio | 95% CI | |
---|---|---|---|
Bake Time | 2 | * | (*, *) |
Bake Temperature 2 | 15 | 2.1653 | (1.9652, 2.3858) |
In these results, the model has 3 terms to predict whether the color of pretzels meets quality standards has 3 terms: Bake Time, Bake Temperature 2, and the square term for Bake Time. In this example, an acceptable color is the Event.
The unit of change shows the difference in natural units for a coded unit in the design. For example, in natural units, the low level of Bake Temperature 2 is 127. The high level is 157 degrees. The distance from the low level to the midpoint represents a change of 1 coded unit. In this case, that distance is 15 degrees.
The odds ratio for Bake Temperature 2 is approximately 2.17. For each 15 degrees that the temperature rises, the odds that a the pretzel color is acceptable increases by about 2.17 times.
The odds ratio for Bake Time is missing because the model includes the squared term for Bake Time. The odds ratio does not have a fixed value because the value depends on the value of Bake Time.
Level A | Level B | Odds Ratio | 95% CI |
---|---|---|---|
Month | |||
2 | 1 | 1.1250 | (0.0600, 21.0834) |
3 | 1 | 3.3750 | (0.2897, 39.3165) |
4 | 1 | 7.7143 | (0.7461, 79.7592) |
5 | 1 | 2.2500 | (0.1107, 45.7172) |
6 | 1 | 6.0000 | (0.5322, 67.6397) |
3 | 2 | 3.0000 | (0.2547, 35.3325) |
4 | 2 | 6.8571 | (0.6556, 71.7169) |
5 | 2 | 2.0000 | (0.0976, 41.0019) |
6 | 2 | 5.3333 | (0.4679, 60.7946) |
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) |
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. In this example, a cancellation is the Event. The largest odds ratio is approximately 7.71, 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.
To determine how well the model fits your data, examine the goodness-of-fit statistics in the Model Summary table.
Many of the model summary and goodness-of-fit statistics are affected by how the data are arranged in the worksheet and whether there is one trial per row or multiple trials per row. The Hosmer-Lemeshow test is unaffected by how the data are arranged and is comparable between one trial per row and multiple trials per row. For more information, go to How data formats affect goodness-of-fit in binary logistic regression.
The higher the deviance R^{2} value, the better the model fits your data. Deviance R^{2} is always between 0% and 100%.
Deviance R^{2} always increases when you add additional terms to a model. For example, the best five-term model will always have a deviance R^{2} that is at least as high as the best four-predictor model. Therefore, deviance R^{2} is most useful when you compare models of the same size.
The data arrangement affects the deviance R^{2} value. The deviance R ^{2} is usually higher for data with multiple trials per row than for data with a single trial per row. Deviance R ^{2} values are comparable only between models that use the same data format.
Goodness-of-fit statistics are just one measure of how well the model fits the data. Even when a model has a desirable value, you should check the residual plots and goodness-of-fit tests to assess how well a model fits the data.
Use adjusted deviance R^{2} to compare models that have different numbers of terms. Deviance R^{2} always increases when you add a term to the model. The adjusted deviance R^{2} value incorporates the number of terms in the model to help you choose the correct model.
Deviance R-Sq | Deviance R-Sq(adj) | AIC | AICc | BIC |
---|---|---|---|---|
95.81% | 95.16% | 243.85 | 245.80 | 255.32 |
In these results, the model explains 95.81% of the total deviance in the response variable. For these data, the deviance R^{2} value indicates the model provides a good fit to the data. If additional models are fit with different terms, use the adjusted deviance R^{2} value, the AIC value, the AICc value, and the BIC value to compare how well the model fits the data.
If the deviation is statistically significant, you can try a different link function or change the terms in the model.
Test | DF | Chi-Square | P-Value |
---|---|---|---|
Deviance | 44 | 32.26 | 0.905 |
Pearson | 44 | 31.98 | 0.911 |
Hosmer-Lemeshow | 7 | 4.18 | 0.758 |
In these results, alll 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.