Example of Fit Poisson Model

A quality engineer is concerned about two types of defects in molded resin parts: discoloration and clumping. Discolored streaks in the final product can result from contamination in hoses and from abrasions to resin pellets. Clumping can occur when the process is run at higher temperatures and faster rates of transfer. The engineer identifies three possible predictor variables for the responses (defects). The engineer records the number of each type of defect in hour long sessions, while varying the predictor levels.

The engineer wants to study how several predictors affect discoloration defects in resin parts. Because the response variable describes the number of times that an event occurs in a finite observation space, the engineer fits a Poisson model.

  1. Enter the sample data, ResinDefects.MTW.
  2. Choose Stat > Regression > Poisson Regression > Fit Poisson Model.
  3. In Response, enter 'Discoloration Defects'.
  4. In Continuous predictors, enter 'Hours Since Cleanse' Temperature.
  5. In Categorical predictors, enter 'Size of Screw'.
  6. Click Graphs.
  7. In Residuals for plots, select Standardized.
  8. Under Residuals plots, select Four in one.
  9. Click OK in each dialog box.

Interpret the results

The plot of the standardized deviance residuals versus the fitted values shows a distinct curve. In the plot of the residuals versus order, the residuals in the middle tend to be higher than the residuals at the beginning and end of the data set. For these data, both patterns are because of a missing interaction term between the size of the screw and the temperature. The pattern is visible on the residuals versus order plot because the engineer did not collect the data in random order. The engineer refits the model with the interaction between temperature and the size of the screw to model the defects more accurately.

Poisson Regression Analysis: Discoloratio versus Hours Since , Temperature, ...

Method Link function Natural log Categorical predictor coding (1, 0) Rows used 36
Regression Equation Discoloration Defects = exp(Y') Size of Screw large Y' = 4.398 + 0.01798 Hours Since Cleanse - 0.001974 Temperature small Y' = 4.244 + 0.01798 Hours Since Cleanse - 0.001974 Temperature
Coefficients Term Coef SE Coef VIF Constant 4.3982 0.0628 Hours Since Cleanse 0.01798 0.00826 1.00 Temperature -0.001974 0.000318 1.00 Size of Screw small -0.1546 0.0427 1.00
Model Summary Deviance Deviance R-Sq R-Sq(adj) AIC AICc BIC 64.20% 60.80% 253.29 254.58 259.62
Goodness-of-Fit Tests 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
Analysis of Variance 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
Fits and Diagnostics for Unusual Observations Discoloration Obs Defects Fit Resid Std Resid 33 43.00 58.18 -2.09 -2.18 R R Large residual

For the model with the interaction, the AIC is approximately 236, which is lower than the model without the interaction. The AIC criterion indicates that the model with the interaction is better than the model without the interaction. The curvature in the residuals versus fits plot is gone. The engineer decides to interpret this model rather than the model without the interaction.

Poisson Regression Analysis: Discoloratio versus Hours Since , Temperature, ...

Method Link function Natural log Categorical predictor coding (1, 0) Rows used 36
Regression Equation Discoloration Defects = exp(Y') Size of Screw large Y' = 4.576 + 0.01798 Hours Since Cleanse - 0.003285 Temperature small Y' = 4.032 + 0.01798 Hours Since Cleanse - 0.000481 Temperature
Coefficients Term Coef SE Coef VIF Constant 4.5760 0.0736 Hours Since Cleanse 0.01798 0.00826 1.00 Temperature -0.003285 0.000441 1.92 Size of Screw small -0.5444 0.0990 5.37 Temperature*Size of Screw small 0.002804 0.000640 6.64
Model Summary Deviance Deviance R-Sq R-Sq(adj) AIC AICc BIC 85.99% 81.46% 236.05 238.05 243.97
Goodness-of-Fit Tests Test DF Estimate Mean Chi-Square P-Value Deviance 31 12.36598 0.39890 12.37 0.999 Pearson 31 12.31611 0.39729 12.32 0.999
Analysis of Variance Wald Test Source DF Chi-Square P-Value Regression 4 78.77 0.000 Hours Since Cleanse 1 4.74 0.029 Temperature 1 55.60 0.000 Size of Screw 1 30.21 0.000 Temperature*Size of Screw 1 19.17 0.000
    By using this site you agree to the use of cookies for analytics and personalized content.  Read our policy