The following data, taken from Cox and Snell (1989, pp. 10–11), consists of the number, Notready
, of ingots that are not ready for rolling, out of Total
tested, for several combinations of heating time and soaking time:
data ingots; input Heat Soak Notready Total @@; lnTotal= log(Total); datalines; 7 1.0 0 10 14 1.0 0 31 27 1.0 1 56 51 1.0 3 13 7 1.7 0 17 14 1.7 0 43 27 1.7 4 44 51 1.7 0 1 7 2.2 0 7 14 2.2 2 33 27 2.2 0 21 51 2.2 0 1 7 2.8 0 12 14 2.8 0 31 27 2.8 1 22 51 4.0 0 1 7 4.0 0 9 14 4.0 0 19 27 4.0 1 16 ;
The following invocation of PROC GENMOD fits an asymptotic (unconditional) Poisson regression model to the data. The variable
Notready
is specified as the response variable, and the continuous predictors Heat
and Soak
are defined in the CLASS statement as categorical predictors that use reference coding. Specifying the offset variable as
lnTotal
enables you to model the ratio Notready/Total
.
proc genmod data=ingots; class Heat Soak / param=ref; model Notready=Heat Soak / offset=lnTotal dist=poisson link=log; exact Heat Soak / joint estimate; exactoptions statustime=10; run;
The EXACT statement is specified to additionally fit an exact conditional Poisson regression model. Specifying the lnTotal
offset variable models the ratio Notready/Total
; in this case, the Total
variable contains the largest possible response value for each observation. The JOINT option produces a joint test for the significance of the covariates, along with the usual marginal tests. The ESTIMATE option produces exact parameter estimates for the covariates. The STATUSTIME=10 option is specified in the EXACTOPTIONS statement for monitoring the progress of the results; this example can take several minutes to complete due to the JOINT option. If you run out of memory, see the SAS Companion for your system for information about how to increase the available
memory.
The “Criteria For Assessing Goodness Of Fit” table is displayed in Output 40.11.1. Comparing the deviance of 10.9363 to an asymptotic chi-square distribution with 11 degrees of freedom, you find that the p-value is 0.449. This indicates that the specified model fits the data reasonably well.
Output 40.11.1: Unconditional Goodness of Fit Criteria
Criteria For Assessing Goodness Of Fit | |||
---|---|---|---|
Criterion | DF | Value | Value/DF |
Deviance | 11 | 10.9363 | 0.9942 |
Scaled Deviance | 11 | 10.9363 | 0.9942 |
Pearson Chi-Square | 11 | 9.3722 | 0.8520 |
Scaled Pearson X2 | 11 | 9.3722 | 0.8520 |
Log Likelihood | -7.2408 | ||
Full Log Likelihood | -12.9038 | ||
AIC (smaller is better) | 41.8076 | ||
AICC (smaller is better) | 56.2076 | ||
BIC (smaller is better) | 49.3631 |
From the “Analysis Of Parameter Estimates” table in Output 40.11.2, you can see that only two of the Heat parameters are deemed significant. Looking at the standard errors, you can see that
the unconditional analysis had convergence difficulties with the Heat=7 parameter (Standard Error=264324.6), which means you
cannot fit this unconditional Poisson regression model to this data.
Output 40.11.2: Unconditional Maximum Likelihood Parameter Estimates
Analysis Of Maximum Likelihood Parameter Estimates | ||||||||
---|---|---|---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error | Wald 95% Confidence Limits | Wald Chi-Square | Pr > ChiSq | ||
Intercept | 1 | -1.5700 | 1.1657 | -3.8548 | 0.7147 | 1.81 | 0.1780 | |
Heat | 7 | 1 | -27.6129 | 264324.6 | -518094 | 518039.0 | 0.00 | 0.9999 |
Heat | 14 | 1 | -3.0107 | 1.0025 | -4.9756 | -1.0458 | 9.02 | 0.0027 |
Heat | 27 | 1 | -1.7180 | 0.7691 | -3.2253 | -0.2106 | 4.99 | 0.0255 |
Soak | 1 | 1 | -0.2454 | 1.1455 | -2.4906 | 1.9998 | 0.05 | 0.8304 |
Soak | 1.7 | 1 | 0.5572 | 1.1217 | -1.6412 | 2.7557 | 0.25 | 0.6193 |
Soak | 2.2 | 1 | 0.4079 | 1.2260 | -1.9951 | 2.8109 | 0.11 | 0.7394 |
Soak | 2.8 | 1 | -0.1301 | 1.4234 | -2.9199 | 2.6597 | 0.01 | 0.9272 |
Scale | 0 | 1.0000 | 0.0000 | 1.0000 | 1.0000 |
Note: | The scale parameter was held fixed. |
Following the output from the asymptotic analysis, the exact conditional Poisson regression results are displayed, as shown in Output 40.11.3.
Output 40.11.3: Exact Tests
Exact Conditional Tests | ||||
---|---|---|---|---|
Effect | Test | Statistic | p-Value | |
Exact | Mid | |||
Joint | Score | 18.3665 | 0.0137 | 0.0137 |
Probability | 1.294E-6 | 0.0471 | 0.0471 | |
Heat | Score | 15.8259 | 0.0023 | 0.0022 |
Probability | 0.000175 | 0.0063 | 0.0062 | |
Soak | Score | 1.4612 | 0.8683 | 0.8646 |
Probability | 0.00735 | 0.8176 | 0.8139 |
The Joint test in the “Conditional Exact Tests” table in Output 40.11.3 is produced by specifying the JOINT option in the EXACT statement. The p-values for this test indicate that the parameters for Heat
and Soak
are jointly significant as explanatory effects in the model. If the Heat
variable is the only explanatory variable in your model, then the rows of this table labeled as “Heat” show the joint significance of all the Heat
effect parameters in that reduced model. In this case, a model that contains only the Heat
parameters still explains a significant amount of the variability; however, you can see that a model that contains only the
Soak
parameters would not be significant.
The “Exact Parameter Estimates” table in Output 40.11.4 displays parameter estimates and tests of significance for the levels of the CLASS variables. Again, the Heat=7 parameter
has some difficulties; however, in the exact analysis, a median unbiased estimate is computed for the parameter instead of a maximum likelihood estimate. The confidence limits show that the Heat
variable contains some explanatory power, while the categorical Soak
variable is insignificant and can be dropped from the model.
Output 40.11.4: Exact Parameter Estimates
Exact Parameter Estimates | |||||||
---|---|---|---|---|---|---|---|
Parameter | Estimate | Standard Error | 95% Confidence Limits | Two-sided p-Value | |||
Heat | 7 | -2.7552 | * | . | -Infinity | -0.7864 | 0.0199 |
Heat | 14 | -3.0255 | 1.0128 | -5.7450 | -0.6194 | 0.0113 | |
Heat | 27 | -1.7846 | 0.8065 | -3.6779 | 0.2260 | 0.0844 | |
Soak | 1 | -0.3231 | 1.1717 | -2.8673 | 3.6754 | 1.0000 | |
Soak | 1.7 | 0.5375 | 1.1284 | -1.8056 | 4.4588 | 1.0000 | |
Soak | 2.2 | 0.4035 | 1.2347 | -2.5785 | 4.5054 | 1.0000 | |
Soak | 2.8 | -0.1661 | 1.4214 | -4.5490 | 4.2168 | 1.0000 |
Note: | * indicates a median unbiased estimate. |
Note: If you want to make predictions from the exact results, you can obtain an estimate for the intercept parameter by specifying the INTERCEPT keyword in the EXACT statement. You should also remove the JOINT option to reduce the amount of time and memory consumed.