Example 7.1: Dose-Response Study

The LOGISTIC Procedure

Example 7.1: Dose-Response Study

Researchers are interested in analyzing how mortality rates change with respect to dosage of a drug. The data contains life/death outcomes for six levels of drug dosage (0 to 5). One subject in each of three different research centers (A to C) is given a specific dose of the drug, and the Response is recorded as 0 for survival and 1 for death; the Z variable will be used in a later analysis:

   data dose;
      input Center $ Dose Response @@;
      Z = 2 - Response;
      datalines;
   A 0 0    B 0 0    C 0 0 
   A 1 0    B 1 0    C 1 0 
   A 2 0    B 2 0    C 2 0 
   A 3 0    B 3 0    C 3 0 
   A 4 1    B 4 0    C 4 0 
   A 5 1    B 5 0    C 5 1 
   ;

Since all of the cells have counts that are less than 5, the applicability of large sample theory is questionable. For each subject i receiving dosage x_i, i=1,...,18, let Y_i=1 if the subject died, Y_i=0 otherwise, and $\pi_i=\Pr(Y_i=1| x_i)$ . Then the linear logistic model for this problem is ${logit}(\pi_i)=\log(\frac{\pi_i}{1-\pi_i})=\alpha + x_i\beta$ , which fits a common intercept and slope for the i subjects. In the PROC LOGISTIC invocation below, the EXACT statement requests an exact analysis and the ESTIMATE option produces exact parameter estimates:

   proc logistic data=dose descending;
      model Response = Dose;
      exact Dose / estimate=both;
   run;

Output 7.1.1 displays some of the unconditional asymptotic results that are produced by default. The likelihood ratio and score tests reject the null hypothesis that $\beta$ is zero. However, the Wald test does not reject this null hypothesis. The conflicting conclusions of these tests are a telltale sign that the large sample approximation is unreliable. The estimates for the intercept $\alpha$ and the slope $\beta$ both have p-values greater than 0.05, indicating marginal influence. The confidence limits for the odds ratio of the dose parameter contains the value 1, from which you could conclude, if you accept the model, that there is no change in mortality with a change in dosage.

Output 7.1.1: Output from Asymptotic Analysis

The LOGISTIC Procedure

Testing Global Null Hypothesis: BETA=0
Test	Chi-Square	DF	Pr > ChiSq
Likelihood Ratio	8.1478	1	0.0043
Score	5.7943	1	0.0161
Wald	2.7249	1	0.0988

Analysis of Maximum Likelihood Estimates
Parameter	DF	Estimate	Standard Error	Chi-Square	Pr > ChiSq
Intercept	1	-9.4745	5.5677	2.8958	0.0888
Dose	1	2.0804	1.2603	2.7249	0.0988

Odds Ratio Estimates
Effect	Point Estimate	95% Wald Confidence Limits
Dose	8.007	0.677	94.679

Output 7.1.2 shows the results from the exact conditional analysis. The p-values in the "Conditional Exact Tests" table lead to rejecting the null hypothesis that $\beta$ is zero (no conclusions can be made about $\alpha$ since it is "conditioned" away). The "Exact Parameter Estimates" table shows that the slope $\beta$ is estimated to be $\hat\beta=1.8$ , and since the 95% confidence interval for the exponential of $\hat\beta$ does not contain 1, the odds of death increase significantly with dosage. Note that the exact tests do not produce standard errors for the estimates.

Output 7.1.2: Output from EXACT Analysis

The LOGISTIC Procedure

Exact Conditional Analysis

Conditional Exact Tests
Effect	Test	Statistic	p-Value
Effect	Test	Statistic	Exact	Mid
Dose	Score	5.4724	0.0245	0.0190
	Probability	0.0110	0.0245	0.0190

Exact Parameter Estimates
Parameter	Estimate	95% Confidence Limits		p-Value
Dose	1.7999	0.1157	5.8665	0.0245

Exact Odds Ratios
Parameter	Estimate	95% Confidence Limits		p-Value
Dose	6.049	1.123	353.000	0.0245

The unconditional asymptotic and conditional exact results produce conflicting conclusions for this example. Stokes, Davis, and Koch (1995) recommend looking at the exact results when sample sizes are small and the approximate p-values are less than 0.10. For this example, the small sample size and the conflicting results for the asymptotic hypothesis tests indicate that an exact analysis is appropriate.

You can also perform a stratified analysis to control for the research centers. The strata are treated as nuisance parameters and a conditional likelihood removes them from the analysis. Your model contains a different intercept term for each stratum:

${logit}(\pi_{hi})=\alpha_h + x_{hi}{\beta}$

where h indexes the strata, $\alpha_h$ are the strata intercepts, and i indexes the subjects within the strata.

With PROC LOGISTIC, you can specify a stratification variable by including it in the CLASS statement. For example, a stratification variable that has three levels can be parameterized as

Stratum Level 1 Level 2

1 1 0

2 0 1

3 0 0

where the usual intercept term represents the last strata level, and the other strata levels are a combination of the intercept and the appropriate level term. This is defined in the CLASS statement with the PARAM=REF option.

You can perform a stratified analysis by using the following statements, where Center is defined to be a classification variable and is conditioned out of the analysis by specifying only Dose in the EXACT statement.

   proc logistic data=dose descending;
      class Center / param=ref;
      model Response = Center Dose;
      exact Dose / estimate=both;
   run;

The usual asymptotic analysis indicates that there is complete separation of the data. (This means that unique maximum likelihood estimates do not exist.) You can see that the parameter estimates do not seem to converge if you specify both the ITPRINT and NOCHECK options in the MODEL statement. However, the exact analysis is still valid in this case, and exact tests and estimates for the conditional analysis are computed and displayed in Output 7.1.3.

Output 7.1.3: Stratified Output from EXACT Analysis

The LOGISTIC Procedure

Exact Conditional Analysis

Conditional Exact Tests
Effect	Test	Statistic	p-Value
Effect	Test	Statistic	Exact	Mid
Dose	Score	5.5714	0.0222	0.0167
	Probability	0.0111	0.0222	0.0167

Exact Parameter Estimates
Parameter	Estimate		95% Confidence Limits		p-Value
Dose	1.3204	*	0.1354	Infinity	0.0222

NOTE:

* indicates a median unbiased estimate.

Exact Odds Ratios
Parameter	Estimate		95% Confidence Limits		p-Value
Dose	3.745	*	1.145	Infinity	0.0222

NOTE:

* indicates a median unbiased estimate.

The median unbiased estimate is created instead of the conditional MLE because the value of the observed sufficient statistic lies at an extreme of the derived distribution, implying that the conditional MLE does not exist. The confidence interval for the exact odds ratio does not include 1, so you can conclude that the odds of death increases significantly with dosage. Even though the asymptotic results are unreliable, the exact analysis allows you to conclude that there is a significant effect due to Dose.

This exact analysis should be compared to an asymptotic conditional likelihood analysis, which is available with the PHREG procedure. First, define a variable Z to be 1 if the response is an event and 2 if the response is a nonevent. This variable is used as the time variable as well as the censoring indicator (with 2 as the censored value) in the MODEL statement of PROC PHREG. Also specify the TIES=DISCRETE option to request the discrete logistic model, and the STRATA statement to specify the strata on which to condition:

   proc phreg data=dose;
      strata Center;
      model Z*Z(2)=Dose / ties=discrete;
   run;

For these data, the PHREG procedure does not converge and the maximum likelihood estimates are not valid. Generally, the conditional score statistics for testing the overall null hypothesis should be the same for both the asymptotic conditional analysis in PROC PHREG and the exact analysis in PROC LOGISTIC. However, PROC PHREG computes the p-value by comparing the value of the conditional score statistic to a chi-squared distribution, while PROC LOGISTIC derives its p-value from the exact conditional distribution. Also, inference on individual parameters is often not the same.

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