Physicians’ Health Study – Regular Aspirin Intake v. Myocardial Infarction
The Physicians’ Health Study Research Group at Harvard Medical School conducted a 5-year
randomized study to test whether regular aspirin intake reduces the likelihood of mortality from
cardiovascular disease. There were 22,071 physicians who participated in the study, with 11,037
being randomly assigned to take an aspirin tablet every other day, and the remaining physicians
assigned to take a placebo tablet every other day. During the 5 years, the frequency of
occurrence of myocardial infarction (heart attack) was tabulated for all physicians. Some of the
physicians had fatal attacks; others had nonfatal attacks. The results are presented in the
contingency table below.
Fatal Attack
Placebo
Aspirin
18
5
Myocardial Infarction
Nonfatal
No Attack
Attack
171
10845
99
10933
The SAS program below is used to analyze the results of the study. The Myocardial Infarction
variable is dichotomized, with the categories Fatal Attack and Nonfatal Attack being combined.
proc format;
value drugfmt
value hrt2fmt
1
2
1
2
=
=
=
=
"Placebo"
"Aspirin";
"MI"
"No MI";
;
data three;
input drug heart count;
format drug drugfmt. heart hrt2fmt.;
cards;
1 1
189
1 2 10845
2 1
104
2 2 10933
;
proc freq;
weight count;
tables drug*heart / all;
title "Test of Association Between Aspirin Intake";
title2 "And Myocardial Infarction Events";
title3 "Fatal and Non-Fatal Categories Collapsed";
;
proc corr;
weight count;
var drug heart;
title "Pearson Correlation Coefficient Between";
title2 "Dichotomous Variables Drug and";
title3 "Myocardial Infarction Events";
;
run;
The format procedure assigns labels to two created formats – drugfmt and hrt2fmt. The data
statement creates a data set called “one.” The input statement tells SAS to read three variable
values from each line of data – two dichotomous variables (drug and heart) and a third variable
called count that lists the cell frequencies from the table. The format statement assigns the
created formats to their respective variables. The cards statement marks the beginning of the
data list. The frequency procedure analyzes the two-way contingency table that is produced by
the tables statement. The weight statement in the frequency procedure weights each combination
of values of drug and heart by the frequency in that cell of the table. The all option in the tables
statement tells SAS to produce all possible statistical results; we will examine only a few of
them. Since both variables are dichotomous, we may also calculate the Pearson correlation
coefficient as a measure of ordinal association. The correlation procedure provided this result.
The output of the SAS program is listed below.
Test of Association Between Aspirin Intake
And Myocardial Infarction Events 14:29 Thursday, October 9, 2008
Fatal and Non-Fatal Categories Collapsed
The FREQ Procedure
Table of drug by heart
drug
heart
Frequency‚
Percent ‚
Row Pct ‚
Col Pct ‚MI
‚No MI
‚ Total
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
Placebo ‚
189 ‚ 10845 ‚ 11034
‚
0.86 ‚ 49.14 ‚ 49.99
‚
1.71 ‚ 98.29 ‚
‚ 64.51 ‚ 49.80 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
Aspirin ‚
104 ‚ 10933 ‚ 11037
‚
0.47 ‚ 49.54 ‚ 50.01
‚
0.94 ‚ 99.06 ‚
‚ 35.49 ‚ 50.20 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
Total
293
21778
22071
1.33
98.67
100.00
Statistics for Table of drug by heart
Statistic
DF
Value
Prob
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Chi-Square
1
25.0139
<.0001
Likelihood Ratio Chi-Square
1
25.3720
<.0001
Continuity Adj. Chi-Square
1
24.4291
<.0001
Mantel-Haenszel Chi-Square
1
25.0128
<.0001
Phi Coefficient
0.0337
Contingency Coefficient
0.0336
Cramer's V
0.0337
1
Fisher's Exact Test
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Cell (1,1) Frequency (F)
189
Left-sided Pr <= F
1.0000
Right-sided Pr >= F
3.253E-07
Table Probability (P)
Two-sided Pr <= P
1.516E-07
5.033E-07
Test of Association Between Aspirin Intake
And Myocardial Infarction Events 14:29 Thursday, October 9, 2008
Fatal and Non-Fatal Categories Collapsed
The FREQ Procedure
Statistics for Table of drug by heart
Statistic
Value
ASE
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Gamma
0.2938
0.0561
Kendall's Tau-b
0.0337
0.0065
Stuart's Tau-c
0.0077
0.0015
Somers' D C|R
Somers' D R|C
0.0077
0.1471
0.0015
0.0282
Pearson Correlation
Spearman Correlation
0.0337
0.0337
0.0065
0.0065
Lambda Asymmetric C|R
Lambda Asymmetric R|C
Lambda Symmetric
0.0000
0.0077
0.0075
0.0000
0.0015
0.0015
Uncertainty Coefficient C|R
Uncertainty Coefficient R|C
Uncertainty Coefficient Symmetric
0.0081
0.0008
0.0015
0.0032
0.0003
0.0006
Estimates of the Relative Risk (Row1/Row2)
Type of Study
Value
95% Confidence Limits
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Case-Control (Odds Ratio)
1.8321
1.4400
2.3308
Cohort (Col1 Risk)
1.8178
1.4330
2.3059
Cohort (Col2 Risk)
0.9922
0.9892
0.9953
Sample Size = 22071
2
Test of Association Between Aspirin Intake
And Myocardial Infarction Events 14:29 Thursday, October 9, 2008
Fatal and Non-Fatal Categories Collapsed
The FREQ Procedure
Summary Statistics for drug by heart
Cochran-Mantel-Haenszel Statistics (Based on Table Scores)
Statistic
Alternative Hypothesis
DF
Value
Prob
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1
Nonzero Correlation
1
25.0128
<.0001
2
Row Mean Scores Differ
1
25.0128
<.0001
3
General Association
1
25.0128
<.0001
3
Estimates of the Common Relative Risk (Row1/Row2)
Type of Study
Method
Value
95% Confidence Limits
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Case-Control
Mantel-Haenszel
1.8321
1.4400
2.3308
(Odds Ratio)
Logit
1.8321
1.4400
2.3308
Cohort
(Col1 Risk)
Mantel-Haenszel
Logit
1.8178
1.8178
1.4330
1.4330
2.3059
2.3059
Cohort
(Col2 Risk)
Mantel-Haenszel
Logit
0.9922
0.9922
0.9892
0.9892
0.9953
0.9953
Total Sample Size = 22071
Pearson Correlation Coefficient Between
Dichotomous Variables Drug and 14:29 Thursday, October 9, 2008
Myocardial Infarction Events
The CORR Procedure
2
Variables:
drug
heart
Weight Variable:
count
Variable
drug
heart
N
4
4
Mean
1.50007
1.98672
Simple Statistics
Std Dev
42.88648
9.81683
Sum
33108
43849
Pearson Correlation Coefficients, N = 4
Prob > |r| under H0: Rho=0
drug
heart
drug
1.00000
0.03367
0.9663
heart
0.03367
0.9663
1.00000
Minimum
1.00000
1.00000
Maximum
2.00000
2.00000
4
The point estimate of the odds ratio is
n n
18910933  1.8321 .
ˆ  11 22 
n12 n21 10845104
As derived in class, the 95% large-sample confidence interval for the log of the odds ratio is
n n 
 18910933
1
1
1
1
1
1
1
1
ln  11 22   z 



 ln 
 1.96




n11 n22 n12 n21
189 10933 10845 104
 10845104
 n12 n22 
2
 0.6054  0.2408  0.3647, 0.8462 .
Then the 95% large-sample confidence interval for the odds ratio is
exp 0.3647, exp 0.8462  1.4400, 2.3308.
These may be found in both the last table on page 2 and the last table on page 3 of the SAS
output. We are 95% confident that the odds of a myocardial infarction for those taking a placebo
are between 1.4400 and 2.3308 times as great as the odds of a myocardial infarction for those
taking aspirin.
Since both variables are dichotomous, they may also be considered to be ordinal, and we may
use the gamma measure of ordinal association. Since the formulae for the M.L.E. and the A.S.E.
for gamma are rather complicated, I will just note that the 95% large-sample C.I. for gamma is
given in the first table on page 2 of the output as
ˆ  z  A.S .E.ˆ   0.2938  1.960.0561  0.1838, 0.4038 . We can say with confidence that
2
there is a positive association between the two variables – those who take aspirin are less likely
to have a myocardial infarction than those who take a placebo.
The Pearson correlation coefficient, given in the last table on page 4 of the SAS output, is found
to be ˆ  r  0.03367 , denoting a weak positive relationship between the two variables. Note
that this result is also given in the first table on page 2 of the output, both as the Pearson
correlation and as the Spearman correlation. These two numbers are the same, since both
variables are dichotomous.