Example 28.7: Computing the Cochran-Armitage Trend Test
The data set Pain contains hypothetical data for a
clinical trial of a drug therapy to control pain. The
clinical trial investigates whether adverse responses
increase with larger drug doses. Subjects receive either a
placebo or one of four drug doses. An adverse response is
recorded as Adverse='Yes'; otherwise, it is recorded as
Adverse='No'. The number of subjects for each drug
dose and response combination is contained in the variable
Count.
data Pain;
input Dose Adverse $ Count @@;
datalines;
0 No 26 0 Yes 6
1 No 26 1 Yes 7
2 No 23 2 Yes 9
3 No 18 3 Yes 14
4 No 9 4 Yes 23
;
The TABLES statement in the following program produces a
two-way table. The MEASURES option produces measures of
association, and the CL option produces confidence limits
for these measures. The TREND option tests for a trend
across the ordinal values of the Dose variable
with the Cochran-Armitage test.
The EXACT statement produces exact p-values for this test,
and the MAXTIME= option terminates the exact
computations if they do not complete within 60 seconds.
The TEST statement computes
an asymptotic test for Somer's D(C|R).
These statements produce Output 28.7.1 through Output 28.7.3.
proc freq data=Pain;
weight Count;
tables Dose*Adverse / trend measures cl;
test smdcr;
exact trend / maxtime=60;
title1 'Clinical Trial for Treatment of Pain';
run;
Output 28.7.1: Contingency Table
|
| Clinical Trial for Treatment of Pain |
Frequency
Percent
Row Pct
Col Pct |
|
| Table of Dose by Adverse |
| Dose |
Adverse |
Total |
| No |
Yes |
| 0 |
26
16.15
81.25
25.49 |
6
3.73
18.75
10.17 |
32
19.88
|
| 1 |
26
16.15
78.79
25.49 |
7
4.35
21.21
11.86 |
33
20.50
|
| 2 |
23
14.29
71.88
22.55 |
9
5.59
28.13
15.25 |
32
19.88
|
| 3 |
18
11.18
56.25
17.65 |
14
8.70
43.75
23.73 |
32
19.88
|
| 4 |
9
5.59
28.13
8.82 |
23
14.29
71.88
38.98 |
32
19.88
|
| Total |
102
63.35 |
59
36.65 |
161
100.00 |
|
|
The "Row Pct" values in Output 28.7.1 show the expected
increasing trend in the proportion of adverse effects due to
increasing dosage (from 18.75% to 71.88%).
Output 28.7.2: Measures of Association
|
| Clinical Trial for Treatment of Pain |
| Statistics for Table of Dose by Adverse |
| Statistic |
Value |
ASE |
95%
Confidence Limits |
| Gamma |
0.5313 |
0.0935 |
0.3480 |
0.7146 |
| Kendall's Tau-b |
0.3373 |
0.0642 |
0.2114 |
0.4631 |
| Stuart's Tau-c |
0.4111 |
0.0798 |
0.2547 |
0.5675 |
| Somers' D C|R |
0.2569 |
0.0499 |
0.1592 |
0.3547 |
| Somers' D R|C |
0.4427 |
0.0837 |
0.2786 |
0.6068 |
| Pearson Correlation |
0.3776 |
0.0714 |
0.2378 |
0.5175 |
| Spearman Correlation |
0.3771 |
0.0718 |
0.2363 |
0.5178 |
| Lambda Asymmetric C|R |
0.2373 |
0.0837 |
0.0732 |
0.4014 |
| Lambda Asymmetric R|C |
0.1250 |
0.0662 |
0.0000 |
0.2547 |
| Lambda Symmetric |
0.1604 |
0.0621 |
0.0388 |
0.2821 |
| Uncertainty Coefficient C|R |
0.1261 |
0.0467 |
0.0346 |
0.2175 |
| Uncertainty Coefficient R|C |
0.0515 |
0.0191 |
0.0140 |
0.0890 |
| Uncertainty Coefficient Symmetric |
0.0731 |
0.0271 |
0.0199 |
0.1262 |
| Somers' D C|R |
| Somers' D C|R |
0.2569 |
| ASE |
0.0499 |
| 95% Lower Conf Limit |
0.1592 |
| 95% Upper Conf Limit |
0.3547 |
| Test of H0: Somers' D C|R = 0 |
| ASE under H0 |
0.0499 |
| Z |
5.1511 |
| One-sided Pr > Z |
<.0001 |
| Two-sided Pr > |Z| |
<.0001 |
|
Output 28.7.2 displays the measures of association produced
by the MEASURES option. Somer's D(C|R) measures the
association treating the column variable (Adverse) as
the response and the row variable (Dose) as a
predictor. Because the asymptotic 95% confidence limits
do not contain zero, this indicates a strong positive association.
Similarly, the Pearson and Spearman correlation coefficients
show evidence of a strong positive association, as
hypothesized.
Output 28.7.3: Tests
|
| Clinical Trial for Treatment of Pain |
| Statistics for Table of Dose by Adverse |
| Cochran-Armitage Trend Test |
| Statistic (Z) |
-4.7918 |
| |
|
| Asymptotic Test |
|
| One-sided Pr < Z |
<.0001 |
| Two-sided Pr > |Z| |
<.0001 |
| |
|
| Exact Test |
|
| One-sided Pr <= Z |
7.237E-07 |
| Two-sided Pr >= |Z| |
1.324E-06 |
|
The Cochran-Armitage test (Output 28.7.3) supports the trend
hypothesis. The
small left-sided p-values for the Cochran-Armitage test
indicate that the probability of the Column 1 level (
Adverse='No') decreases as Dose increases or,
equivalently, that the probability of the Column 2 level
(Adverse='Yes') increases as Dose increases.
The two-sided p-value tests against either an increasing or
decreasing alternative. This is an appropriate hypothesis
when you want to determine whether the drug has progressive
effects on the probability of adverse effects but the
direction is unknown.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
