Complete the following steps to interpret a Poisson regression model. Key output includes the p-value, coefficients, model summary statistics, and the residual plots.

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 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 have the same mean number of events.
- If an interaction term is significant, you can conclude that the relationship between the predictor and the number of events 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 number of events depends on the magnitude of the predictor.

Wald Test | |||
---|---|---|---|

Source | DF | Chi-Square | P-Value |

Regression | 3 | 56.29 | 0.000 |

Hours Since Cleanse | 1 | 4.74 | 0.029 |

Temperature | 1 | 38.46 | 0.000 |

Size of Screw | 1 | 13.09 | 0.000 |

Term | Coef | SE Coef | Z-Value | P-Value | VIF |
---|---|---|---|---|---|

Constant | 4.3982 | 0.0628 | 70.02 | 0.000 | |

Hours Since Cleanse | 0.01798 | 0.00826 | 2.18 | 0.029 | 1.00 |

Temperature | -0.001974 | 0.000318 | -6.20 | 0.000 | 1.00 |

Size of Screw | |||||

small | -0.1546 | 0.0427 | -3.62 | 0.000 | 1.00 |

In these results, all three predictors are statistically significant at the 0.05 level. You can conclude that changes in these variables are associated with changes in the response variable.

Use the coefficient to determine whether a change in a predictor variable makes the event more likely or less likely. The estimated coefficient for a predictor represents the change in the link function for each unit change in the predictor, while the other predictors in the model are held constant. The relationship between the coefficient and the number of events depends on several aspects of the analysis, including the link function and the reference levels for categorical predictors that are in the model. Generally, positive coefficients make the event more likely and negative coefficients make the event less likely. An estimated coefficient near zero implies that the effect of the predictor is small, or nonexistent.

The interpretation of the estimated coefficients for categorical predictors is relative to the reference level of the predictor. Positive coefficients indicate that the event is more likely at that level of the predictor than at the reference level of the factor. Negative coefficients indicate that the event is less likely at that level of the predictor than at the reference level.

The coefficient for Hours Since Cleanse is positive, which suggests that longer hours are associated with higher values of the response. The coefficient for temperature is negative, which suggests that higher temperatures are associated with lower values of the response.

The size of the screw is a categorical variable with one coefficient, which indicates that the variable has 2 levels and the variable uses 0, 1 coding. The coefficient for the small screw is negative, so the small screw is associated with lower values of the response than the reference level.

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.

Use the goodness-of-fit tests to determine whether the predicted numbers of events deviate from the observed numbers of events in a way that the Poisson distribution does not predict. If the p-value for the goodness-of-fit test is lower than your chosen significance level, you can reject the null hypothesis that the Poisson distribution provides a good fit. This list provides common reasons for deviations:

- 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.

Test | DF | Estimate | Mean | Chi-Square | P-Value |
---|---|---|---|---|---|

Deviance | 32 | 31.60722 | 0.98773 | 31.61 | 0.486 |

Pearson | 32 | 31.26713 | 0.97710 | 31.27 | 0.503 |

In these results, the goodness-of-fit tests both have p-values higher than the usual significance level of 0.05. There is insufficient evidence to conclude that the predicted numbers of events deviate from the observed numbers of events.

Use AIC, AICc, and BIC to compare different models. For each statistic, smaller values are desirable. However, the model with the smallest value 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.

Deviance R-Sq | Deviance R-Sq(adj) | AIC | AICc | BIC |
---|---|---|---|---|

64.20% | 60.80% | 253.29 | 254.58 | 259.62 |

Term | Coef | SE Coef | Z-Value | P-Value | VIF |
---|---|---|---|---|---|

Constant | 4.3982 | 0.0628 | 70.02 | 0.000 | |

Hours Since Cleanse | 0.01798 | 0.00826 | 2.18 | 0.029 | 1.00 |

Temperature | -0.001974 | 0.000318 | -6.20 | 0.000 | 1.00 |

Size of Screw | |||||

small | -0.1546 | 0.0427 | -3.62 | 0.000 | 1.00 |

In the first set of results, the AIC is approximately 253. The AICc is approximately 255. The BIC is approximately 260. This model does not include the interaction between temperature and the size of the screw. The information criteria from an individual model do not indicate how well the model fits the data because the values depend on the sample size.

Deviance R-Sq | Deviance R-Sq(adj) | AIC | AICc | BIC |
---|---|---|---|---|

85.99% | 81.46% | 236.05 | 238.05 | 243.97 |

Term | Coef | SE Coef | Z-Value | P-Value | VIF |
---|---|---|---|---|---|

Constant | 4.5760 | 0.0736 | 62.15 | 0.000 | |

Hours Since Cleanse | 0.01798 | 0.00826 | 2.18 | 0.029 | 1.00 |

Temperature | -0.003285 | 0.000441 | -7.46 | 0.000 | 1.92 |

Size of Screw | |||||

small | -0.5444 | 0.0990 | -5.50 | 0.000 | 5.37 |

Temperature*Size of Screw | |||||

small | 0.002804 | 0.000640 | 4.38 | 0.000 | 6.64 |

In the second set of results, the AIC is approximately 236. The AICc is approximately 238. The BIC is approximately 244. This model includes the interaction between temperature and the size of the screw. The smaller values indicates that the model with the interaction performs better.