STATISTICS IN MEDICINE, VOL. 9, 829-834 (1990)
A STRATIFIED WILCOXON-TYPE TEST FOR TREND DAVID YU-WU PEE Information Management Services Inc., Silver Spring, M D 20910, U.S.A.
AND LAURENCE S. FREEDMAN Biometry Branch, DCPC, National Cancer Instirure. Bethesda. M D 20892, U.S.A
SUMMARY We propose an extension of Cuzick's non-parametric Wilcoxon-like trend test for situations when the observations can be grouped into strata which are thought to be related to the outcome. We illustrate its use in two examples from cancer epidemiology.
INTRODUCTION Cuzick' has described a non-parametric method for testing the trend in a measured variable across a series of ordered groups, giving examples of its use in analysing cancer research data in both animal and human studies. The lack of distributional assumptions makes this test useful in many applications. We have been using this method to analyse data on various panels of serum cancer markers. In these examples (which we will present in more detail later) we were interested in increasing the precision of the test by stratifying the patients into strata which were thought to be more homogeneous with respect to the outcome. For this reason, we wished to extend Cuzick's method to combine the test over several strata. Our paper describes this extension and presents the motivating examples. METHOD Suppose we have G groups indexed by 2. Often Z will be the numbers 1,2, . . . ,G arranged in some natural order. Occasionally a more specific measure may be available, such as a mean exposure level for the group. Let there be N patients, each with a measured variable x i and group index Z i ( i = 1, . . . ,N ) . Then if ri is the rank of x i in the combined sample, Cuzick's test statistic is given by N
T
=
1 Ziri. i= 1
If the distribution function of xi is denoted by Fi(x), whereas Fi(x)= F ( x + ,!?Zi) for some unknown distribution F , then the statistic T yields a test of the null hypothesis that p = 0. Under 0277-671 5/90/07082946$05.00 0 1990 by John Wiley & Sons, Ltd.
Received July 1989 Revised October 1989
830
D. Y-W. PEE AND L. S . FREEDMAN
this null hypothesis, the expected value and variance of T are given by
+ 1)E(Z)/2 var(T) = N 2 ( N + 1) var(Z)/12, E ( T )= N ( N
where E(Z) and var(Z) are the usual mean and variance for a finite population. Treatment of ties can be handled by assigning mid-ranks to the ties. A correction factor for the variance of T is given at the bottom of p. 88 of Reference 1. Now suppose that we divide the patients into S strata. Within each stratum s there are n, patients and they each have measured variable x j s and group index Zjs ( j = 1, . . . ,ns). The sum of n, (s = 1, . . . , S ) is N . We now calculate the Cuzick test statistic for each stratum as
where rjs is the rank if xis within the sth stratum. We then combine these statistics in a weighted linear sum to yield the overall statistic
To =
T,/(ns
+ 1).
S
It may be shown that the weights l/(ns null hypothesis that /?= 0, W O )
+ 1)are statistically optimal (see the appendix).Under the =
c n,E(Zs)/2 c n t var(Zs)/12(ns+ l), S
var(To) =
S
where E(Zs) and var(Z,) are the finite population mean and variance of 2 in stratum s. Ties within each stratum are handled as described for the unstratified test. This simple extension of the statistic can increase the efficiency of the test with judicious choice of strata. A mathematical expression for the estimated gain in efficiency is given in the appendix. In order to facilitate direct comparison of the unstratified test statistic T with the stratified test statistic To, an equivalent form for the unstratified test statistic T is used in the following Ziri/(N + I) instead of on examples. This alternative form of T is based on the expression Ziri. The mean and variance of this alternative statistic are NE(Z)/2 the usual expression and N Z var(Z)/12(N + l), respectively.
xr=
z'!=
EXAMPLES Applications of this method are illustrated by two data sets. In the first example, stratification appears from the data to be appropriate. In the second example, stratification is motivated by biological hypothesis but is not supported by the data. Serum foetoacinar pancreatic protein level as a marker for advanced pancreatic cancer Table I presents the serum levels of foetoacinar pancreatic protein (FAP, percentage inhibition) for three groups of subjects: normal controls, patients with chronic pancreatitis, and patients with advanced pancreatic cancer. These groups are ordered considering chronic pancreatitis to be an inflammatory condition related directly or indirectly to the development of malignant disease. Table I1 displays the descriptive statistics by comparison groups as well as by sex. It is clear that the median FAP level in the pancreatic cancer group is higher than in the chronic pancreatitis
831
A STRATIFIED WILCOXON-TYPE TEST FOR TREND
Table I. Levels of serum foetacinar pancreatic protein (percentage inhibition) by sex within each study group Study group
Men
Women
Normal
0, 0, 0, 0, 0, 2, 3, 3, 3, 4, 4, 5, 6, 10, 10, 10, 11, 12, 15, 15, 15, 20, 22, 25, 25, 27, 30
0, 0, 0, 0, 0, 0, 1, 3, 3, 3, 3, 7, 8, 8, 19, 20, 27
Chronic pancreatitis Advanced pancreatic cancer
0, 2, 3, 6, 8,10,10,12,17,20, 20, 21, 27, 36
0, 0, 2, 2, 2, 2,18,20,38
0, 2, 5, 6, 6, 6, I , 8, 9,15, 15, 15, 15, 18, 20, 21, 23, 23, 25, 25, 25, 27, 29, 34, 39,41,46, 50, 52, 68
0, 0, 2, 2, 2, 3, 4, 4, 6, 6, 10, 10, 15, 27, 27, 30, 35, 50
Table 11. Trend analysis for serum foetacinar pancreatic protein (percentage inhibition) Men
All
Study group Normal Chronic pancreatitis Advanced pancreatic cancer
N
1
44
4.5
27
10.0
17
3.0
2
23
10-0
14
11.0
9
20
3
48
15.0
30
205
18
6.0
W)
var(Z) T = TJn,
Women
Group index Z
+ 1)*
var(T)t tS
p-value (two-tailed)
Median
2.03 0.8 1 126.25 117.00 7.61 3.35 00008
N
Median
2.04 0.8 1 79.24 72.50 4.72 3.10 0.0019
N
Median
2.02 0.8 1 47.43 44.50 2.86 1.73 00830
To = 126.67* E(To)= 117.0 var(To)= 7-59 t = 3.51 p = 0.0005
* t
For definition of T, and To see text. Variances are corrected for ties. 1 t = [ T - ~(T)]/[var(T)]”~.
group, which in turn is higher than in the normal control group. This pattern is seen in both men and women. A slight reversal of this ordering is seen for the normal women versus the chronic pancreatitis women comparison. In addition, the median F A P level is consistently higher in men than in women within each patient group. Under these conditions, stratifying by sex and applying the stratified trend test To should result in a more powerful test of the null hypothesis than the unstratified trend test T. The standardized value of To is calculated to be 3.51 compared with a standardized value of Tequal to 3.35. There is an apparent gain in efficiency, with the asymptotic relative efficiency estimated as 1.10 (see equation (2) in the appendix).
Monoclonal antibody CAR-3 as a marker for advanced pancreatic cancer In this example, as in the previous, CAR-3 (CPM) was measured in another three groups of subjects: normal controls, patients with chronic pancreatitis, and patients with advanced pancreatic cancer. The monoclonal antibody CAR-3 is obtained from serum following immunization
832
D. Y-W. PEE AND L. S. FREEDMAN
Table 111. Trend analysis for serum CAR-3 (CPM) monoclonal antibody epitope ~
~~
~
~
All Study group
Group index Z
Normal Chronic pancreatitis Advanced pancreatic cancer
Men
Women
N
Median
N
Median
N
Median
1
45
48.0
23
45.5
22
605
2
24
111.5
15
103.5
9
119.5
3
30
672.25
17
624.0
13
1462.5
E(Z) var(Z)
var(T)t tS
p-value (two-tailed)
1.85 0.74 109.51 91.50 6.06 7.31 < 00001
1.89 0.73 63.19 52.00 3.28 6.18 < 0~0001
1.80 0.77 46.13 39.50 2.76 3.99
To = 10.32* E(To)= 91.5 var(To) = 6.04 t = 1.25
O~OOO1
p < 0~0001
* For definition of T, and
To see text. Variances are corrected for ties. $ t = [ T - E(T)]/[var(T)]*!'.
with the human epidermoid carcinoma line A 431. Table 111 displays the median CAR-3 in the three groups stratified by sex. There is a clearly increasing trend in the median of CAR-3 over the three groups. For biological reasons, there was interest in examining the trend among men and women separately. However, on examination of the data, it is not clear that men and women (except perhaps among the cancer group) have different median levels of CAR-3. The standardized trend test Tapplied to the pooled data resulted in a value of 7.31. The standardized stratified trend test To was calculated as 7.25. In this case, the estimated asymptotic relative efficiency was 0.992. DISCUSSION When studying treatment effects in clinical trials or effects of exposure to risk factors in epidemiology, adjustment for covariates is often used to reduce bias and increase For non-parametric testing, methods of adjustment by stratification have been described for the Wilcoxon rank-sum test,4, the Gehan test6 and the logrank test.' This paper extends the stratification method to Cuzick's test. The stratified test should be of use when important covariates exist allowing identification of prognostic strata, particularly when these covariates are not well balanced across the comparison groups and when there are ample data to support stratification. ACKNOWLEDGEMENT
This research was supported by NCI contract NO1-CB-51010-54 for the first author. The authors thank Dr. I. Masnyk of NCT for suggesting the topic, Dr. E. Slud of the University of Maryland for helpful discussions, and Mrs. Sandra Kline of IMS for excellent secretarial support. The data presented in the examples were kindly made available by Dr. M. Prat of the Universita di Torino and Dr. M. Escribano of the Centre National de la Recherche Scientifique.
833
A STRATIFIED WILCOXON-TYPE TEST FOR TREND
APPENDIX For the unstratified case, Cuzick's test statistic T may be rewritten as
where g indexes the groups (g = 1, . . . ,G), S,,, is the absolute difference in Zs between group g and group g', W,,, is the Mann-Whitney statistic for comparing groups g and g', and Sj = Zi, assuming that subjects are labelled 1 to N in ascending order of Zi. Since S j is a constant for the data set, the first term on the right hand side of equation (1) is a trend statistic which is equivalent to Cuzick's T. We denote this alternative statistic CC,
A STRATIFIED WILCOXON-TYPE TEST FOR TREND DAVID YU-WU PEE Information Management Services Inc., Silver Spring, M D 20910, U.S.A.
AND LAURENCE S. FREEDMAN Biometry Branch, DCPC, National Cancer Instirure. Bethesda. M D 20892, U.S.A
SUMMARY We propose an extension of Cuzick's non-parametric Wilcoxon-like trend test for situations when the observations can be grouped into strata which are thought to be related to the outcome. We illustrate its use in two examples from cancer epidemiology.
INTRODUCTION Cuzick' has described a non-parametric method for testing the trend in a measured variable across a series of ordered groups, giving examples of its use in analysing cancer research data in both animal and human studies. The lack of distributional assumptions makes this test useful in many applications. We have been using this method to analyse data on various panels of serum cancer markers. In these examples (which we will present in more detail later) we were interested in increasing the precision of the test by stratifying the patients into strata which were thought to be more homogeneous with respect to the outcome. For this reason, we wished to extend Cuzick's method to combine the test over several strata. Our paper describes this extension and presents the motivating examples. METHOD Suppose we have G groups indexed by 2. Often Z will be the numbers 1,2, . . . ,G arranged in some natural order. Occasionally a more specific measure may be available, such as a mean exposure level for the group. Let there be N patients, each with a measured variable x i and group index Z i ( i = 1, . . . ,N ) . Then if ri is the rank of x i in the combined sample, Cuzick's test statistic is given by N
T
=
1 Ziri. i= 1
If the distribution function of xi is denoted by Fi(x), whereas Fi(x)= F ( x + ,!?Zi) for some unknown distribution F , then the statistic T yields a test of the null hypothesis that p = 0. Under 0277-671 5/90/07082946$05.00 0 1990 by John Wiley & Sons, Ltd.
Received July 1989 Revised October 1989
830
D. Y-W. PEE AND L. S . FREEDMAN
this null hypothesis, the expected value and variance of T are given by
+ 1)E(Z)/2 var(T) = N 2 ( N + 1) var(Z)/12, E ( T )= N ( N
where E(Z) and var(Z) are the usual mean and variance for a finite population. Treatment of ties can be handled by assigning mid-ranks to the ties. A correction factor for the variance of T is given at the bottom of p. 88 of Reference 1. Now suppose that we divide the patients into S strata. Within each stratum s there are n, patients and they each have measured variable x j s and group index Zjs ( j = 1, . . . ,ns). The sum of n, (s = 1, . . . , S ) is N . We now calculate the Cuzick test statistic for each stratum as
where rjs is the rank if xis within the sth stratum. We then combine these statistics in a weighted linear sum to yield the overall statistic
To =
T,/(ns
+ 1).
S
It may be shown that the weights l/(ns null hypothesis that /?= 0, W O )
+ 1)are statistically optimal (see the appendix).Under the =
c n,E(Zs)/2 c n t var(Zs)/12(ns+ l), S
var(To) =
S
where E(Zs) and var(Z,) are the finite population mean and variance of 2 in stratum s. Ties within each stratum are handled as described for the unstratified test. This simple extension of the statistic can increase the efficiency of the test with judicious choice of strata. A mathematical expression for the estimated gain in efficiency is given in the appendix. In order to facilitate direct comparison of the unstratified test statistic T with the stratified test statistic To, an equivalent form for the unstratified test statistic T is used in the following Ziri/(N + I) instead of on examples. This alternative form of T is based on the expression Ziri. The mean and variance of this alternative statistic are NE(Z)/2 the usual expression and N Z var(Z)/12(N + l), respectively.
xr=
z'!=
EXAMPLES Applications of this method are illustrated by two data sets. In the first example, stratification appears from the data to be appropriate. In the second example, stratification is motivated by biological hypothesis but is not supported by the data. Serum foetoacinar pancreatic protein level as a marker for advanced pancreatic cancer Table I presents the serum levels of foetoacinar pancreatic protein (FAP, percentage inhibition) for three groups of subjects: normal controls, patients with chronic pancreatitis, and patients with advanced pancreatic cancer. These groups are ordered considering chronic pancreatitis to be an inflammatory condition related directly or indirectly to the development of malignant disease. Table I1 displays the descriptive statistics by comparison groups as well as by sex. It is clear that the median FAP level in the pancreatic cancer group is higher than in the chronic pancreatitis
831
A STRATIFIED WILCOXON-TYPE TEST FOR TREND
Table I. Levels of serum foetacinar pancreatic protein (percentage inhibition) by sex within each study group Study group
Men
Women
Normal
0, 0, 0, 0, 0, 2, 3, 3, 3, 4, 4, 5, 6, 10, 10, 10, 11, 12, 15, 15, 15, 20, 22, 25, 25, 27, 30
0, 0, 0, 0, 0, 0, 1, 3, 3, 3, 3, 7, 8, 8, 19, 20, 27
Chronic pancreatitis Advanced pancreatic cancer
0, 2, 3, 6, 8,10,10,12,17,20, 20, 21, 27, 36
0, 0, 2, 2, 2, 2,18,20,38
0, 2, 5, 6, 6, 6, I , 8, 9,15, 15, 15, 15, 18, 20, 21, 23, 23, 25, 25, 25, 27, 29, 34, 39,41,46, 50, 52, 68
0, 0, 2, 2, 2, 3, 4, 4, 6, 6, 10, 10, 15, 27, 27, 30, 35, 50
Table 11. Trend analysis for serum foetacinar pancreatic protein (percentage inhibition) Men
All
Study group Normal Chronic pancreatitis Advanced pancreatic cancer
N
1
44
4.5
27
10.0
17
3.0
2
23
10-0
14
11.0
9
20
3
48
15.0
30
205
18
6.0
W)
var(Z) T = TJn,
Women
Group index Z
+ 1)*
var(T)t tS
p-value (two-tailed)
Median
2.03 0.8 1 126.25 117.00 7.61 3.35 00008
N
Median
2.04 0.8 1 79.24 72.50 4.72 3.10 0.0019
N
Median
2.02 0.8 1 47.43 44.50 2.86 1.73 00830
To = 126.67* E(To)= 117.0 var(To)= 7-59 t = 3.51 p = 0.0005
* t
For definition of T, and To see text. Variances are corrected for ties. 1 t = [ T - ~(T)]/[var(T)]”~.
group, which in turn is higher than in the normal control group. This pattern is seen in both men and women. A slight reversal of this ordering is seen for the normal women versus the chronic pancreatitis women comparison. In addition, the median F A P level is consistently higher in men than in women within each patient group. Under these conditions, stratifying by sex and applying the stratified trend test To should result in a more powerful test of the null hypothesis than the unstratified trend test T. The standardized value of To is calculated to be 3.51 compared with a standardized value of Tequal to 3.35. There is an apparent gain in efficiency, with the asymptotic relative efficiency estimated as 1.10 (see equation (2) in the appendix).
Monoclonal antibody CAR-3 as a marker for advanced pancreatic cancer In this example, as in the previous, CAR-3 (CPM) was measured in another three groups of subjects: normal controls, patients with chronic pancreatitis, and patients with advanced pancreatic cancer. The monoclonal antibody CAR-3 is obtained from serum following immunization
832
D. Y-W. PEE AND L. S. FREEDMAN
Table 111. Trend analysis for serum CAR-3 (CPM) monoclonal antibody epitope ~
~~
~
~
All Study group
Group index Z
Normal Chronic pancreatitis Advanced pancreatic cancer
Men
Women
N
Median
N
Median
N
Median
1
45
48.0
23
45.5
22
605
2
24
111.5
15
103.5
9
119.5
3
30
672.25
17
624.0
13
1462.5
E(Z) var(Z)
var(T)t tS
p-value (two-tailed)
1.85 0.74 109.51 91.50 6.06 7.31 < 00001
1.89 0.73 63.19 52.00 3.28 6.18 < 0~0001
1.80 0.77 46.13 39.50 2.76 3.99
To = 10.32* E(To)= 91.5 var(To) = 6.04 t = 1.25
O~OOO1
p < 0~0001
* For definition of T, and
To see text. Variances are corrected for ties. $ t = [ T - E(T)]/[var(T)]*!'.
with the human epidermoid carcinoma line A 431. Table 111 displays the median CAR-3 in the three groups stratified by sex. There is a clearly increasing trend in the median of CAR-3 over the three groups. For biological reasons, there was interest in examining the trend among men and women separately. However, on examination of the data, it is not clear that men and women (except perhaps among the cancer group) have different median levels of CAR-3. The standardized trend test Tapplied to the pooled data resulted in a value of 7.31. The standardized stratified trend test To was calculated as 7.25. In this case, the estimated asymptotic relative efficiency was 0.992. DISCUSSION When studying treatment effects in clinical trials or effects of exposure to risk factors in epidemiology, adjustment for covariates is often used to reduce bias and increase For non-parametric testing, methods of adjustment by stratification have been described for the Wilcoxon rank-sum test,4, the Gehan test6 and the logrank test.' This paper extends the stratification method to Cuzick's test. The stratified test should be of use when important covariates exist allowing identification of prognostic strata, particularly when these covariates are not well balanced across the comparison groups and when there are ample data to support stratification. ACKNOWLEDGEMENT
This research was supported by NCI contract NO1-CB-51010-54 for the first author. The authors thank Dr. I. Masnyk of NCT for suggesting the topic, Dr. E. Slud of the University of Maryland for helpful discussions, and Mrs. Sandra Kline of IMS for excellent secretarial support. The data presented in the examples were kindly made available by Dr. M. Prat of the Universita di Torino and Dr. M. Escribano of the Centre National de la Recherche Scientifique.
833
A STRATIFIED WILCOXON-TYPE TEST FOR TREND
APPENDIX For the unstratified case, Cuzick's test statistic T may be rewritten as
where g indexes the groups (g = 1, . . . ,G), S,,, is the absolute difference in Zs between group g and group g', W,,, is the Mann-Whitney statistic for comparing groups g and g', and Sj = Zi, assuming that subjects are labelled 1 to N in ascending order of Zi. Since S j is a constant for the data set, the first term on the right hand side of equation (1) is a trend statistic which is equivalent to Cuzick's T. We denote this alternative statistic CC,
