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
Deviance Table Source DF Adj Dev Adj Mean Chi-Square P-Value Regression 3 56.670 18.8900 56.67 0.000 Hours Since Cleanse 1 4.744 4.7444 4.74 0.029 Temperature 1 38.800 38.8000 38.80 0.000 Size of Screw 1 13.126 13.1256 13.13 0.000 Error 32 31.607 0.9877 Total 35 88.277
Model Summary Deviance Deviance R-Sq R-Sq(adj) AIC 64.20% 60.80% 253.29
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
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
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
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
Deviance Table Source DF Adj Dev Adj Mean Chi-Square P-Value Regression 4 75.911 18.9778 75.91 0.000 Hours Since Cleanse 1 4.744 4.7444 4.74 0.029 Temperature 1 56.970 56.9703 56.97 0.000 Size of Screw 1 30.518 30.5182 30.52 0.000 Temperature*Size of Screw 1 19.241 19.2412 19.24 0.000 Error 31 12.366 0.3989 Total 35 88.277
Model Summary Deviance Deviance R-Sq R-Sq(adj) AIC 85.99% 81.46% 236.05
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
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
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
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