Example of Analyze Binary Response for Factorial Design

A food scientist is studying factors that affect food spoilage. The scientist uses a 2-level factorial experiment to assess several factors that could impact the rate of food spoiling.

The scientist analyzes a 2-level factorial design to determine how preservative type, vacuum packaging pressure, contamination level, and cooling temperature affect the spoilage of fruit. The response is binary—whether spoilage is detected or not—in a sample of 500 containers of fruit.

  1. Open the sample data, FoodSpoilage.MTW.
  2. Choose Stat > DOE > Factorial > Analyze Binary Response.
  3. In Event name, enter Event.
  4. In Number of events, enter Spoilage.
  5. In Number of trials, enter Containers.
  6. Click Terms.
  7. Under Include terms in the model up through order, choose 2.
  8. Click OK in each dialog box.

Interpret the results

In the Deviance table, the p-values for three of the main effect terms—Preservative, VacuumPress, and ContaminationLevel—are significant. Because the p-values are less than the significance level of 0.05, the scientist concludes that these factors are statistically significant. None of the two-way interactions are significant. The scientist can consider reducing the model.

The Deviance R2 value shows that the model explains 97.95% of the total deviance in the response, which indicates that the model fits the data well.

Most of the VIFs are small, which indicates that the terms in the model are not correlated.

The Pareto plot of the effects allow you to visually identify the important effects and compare the relative magnitude of the various effects. In these results, three main effects are statistically significant (α = 0.05) - preservative type (A), vacuum seal pressure (B), and contamination level (C). In addition, you can see that the largest effect is preservative type (A) because it extends the farthest. The effect for the preservative by cooling temperature interaction (AD) is the smallest because it extends the least.

Factorial Binary Logistic Regression: Spoilage versus Preservative, VacuumPress, ...

Method Link function Logit Rows used 16
Response Information Event Variable Value Count Name Spoilage Event 506 Event Non-event 7482 Containers Total 7988
Coded Coefficients Term Effect Coef SE Coef VIF Constant -2.7370 0.0479 Preservative 0.4497 0.2249 0.0477 1.03 VacuumPress 0.2574 0.1287 0.0477 1.06 ContaminationLevel 0.2954 0.1477 0.0478 1.06 CoolTemp -0.1107 -0.0554 0.0478 1.07 Preservative*VacuumPress -0.0233 -0.0117 0.0473 1.05 Preservative*ContaminationLevel 0.0722 0.0361 0.0474 1.06 Preservative*CoolTemp 0.0067 0.0034 0.0472 1.05 VacuumPress*ContaminationLevel -0.0430 -0.0215 0.0469 1.04 VacuumPress*CoolTemp -0.0115 -0.0058 0.0465 1.02 ContaminationLevel*CoolTemp 0.1573 0.0786 0.0467 1.02
Odds Ratios for Continuous Predictors Unit of Odds 95% Change Ratio CI VacuumPress 10.0 * (*, *) ContaminationLevel 22.5 * (*, *) CoolTemp 5.0 * (*, *) Odds ratios are not calculated for predictors that are included in interaction terms because these ratios depend on values of the other predictors in the interaction terms.
Odds Ratios for Categorical Predictors Odds 95% Level A Level B Ratio CI Preservative Any level Any level * (*, *) Odds ratio for level A relative to level B Odds ratios are not calculated for predictors that are included in interaction terms because these ratios depend on values of the other predictors in the interaction terms.
Model Summary Deviance Deviance R-Sq R-Sq(adj) AIC AICc BIC 97.95% 76.75% 105.98 171.98 114.48
Goodness-of-Fit Tests Test DF Chi-Square P-Value Deviance 5 0.97 0.965 Pearson 5 0.97 0.965 Hosmer-Lemeshow 6 0.10 1.000
Analysis of Variance Source DF Adj Dev Adj Mean Chi-Square P-Value Model 10 46.2130 4.6213 46.21 0.000 Preservative 1 22.6835 22.6835 22.68 0.000 VacuumPress 1 7.3313 7.3313 7.33 0.007 ContaminationLevel 1 9.6209 9.6209 9.62 0.002 CoolTemp 1 1.3441 1.3441 1.34 0.246 Preservative*VacuumPress 1 0.0608 0.0608 0.06 0.805 Preservative*ContaminationLevel 1 0.5780 0.5780 0.58 0.447 Preservative*CoolTemp 1 0.0051 0.0051 0.01 0.943 VacuumPress*ContaminationLevel 1 0.2106 0.2106 0.21 0.646 VacuumPress*CoolTemp 1 0.0153 0.0153 0.02 0.902 ContaminationLevel*CoolTemp 1 2.8475 2.8475 2.85 0.092 Error 5 0.9674 0.1935 Total 15 47.1804
Regression Equation in Uncoded Units P(Event) = exp(Y')/(1 + exp(Y')) Y' = -2.721 + 0.188 Preservative + 0.0172 VacuumPress - 0.00249 ContaminationLevel - 0.0286 CoolTemp - 0.00117 Preservative*VacuumPress + 0.00160 Preservative*ContaminationLevel + 0.00067 Preservative*CoolTemp - 0.000096 VacuumPress*ContaminationLevel - 0.000115 VacuumPress*CoolTemp + 0.000699 ContaminationLevel*CoolTemp
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