Power and sample size determinations for theWilcoxon signed-rank test

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Power and sample size determinations

for the Wilcoxon signed-rank test

Gwowen Shieh a , Show-Li Jan b & Ronald H. Randles c a

Department of Management Science , National Chiao Tung University , 1001 Ta Hsueh Road, Hsinchu, Taiwan , 30050, Republic of China

b

Department of Applied Mathematics , Chung Yuan Christian University , Chungli, Taiwan , 32023, Republic of China c

Department of Statistics , University of Florida , Gainesville, FL, 32611, USA

Published online: 03 Aug 2007.

To cite this article: Gwowen Shieh , Show-Li Jan & Ronald H. Randles (2007) Power and sample

size determinations for the Wilcoxon signed-rank test, Journal of Statistical Computation and Simulation, 77:8, 717-724, DOI: 10.1080/10629360600635245

To link to this article: http://dx.doi.org/10.1080/10629360600635245

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Journal of Statistical Computation and Simulation Vol. 77, No. 8, August 2007, 717–724

Power and sample size determinations for

the Wilcoxon signed-rank test

GWOWEN SHIEH*†, SHOW-LI JAN‡ and RONALD H. RANDLES§

†Department of Management Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 30050, Republic of China

‡Department of Applied Mathematics, Chung Yuan Christian University, Chungli, Taiwan 32023, Republic of China

§Department of Statistics, University of Florida, Gainesville, FL 32611, USA

(Revised 20 October 2005; in final form 14 February 2006)

The problem of calculating power and sample size for the Wilcoxon signed-rank test is discussed. The exact variance large-sample method is examined and explicit formulas are derived for observations from uniform, normal and Laplace distributions. Numerical results are presented to evaluate the exact variance procedure and compare its performance with two simplified approximations that have been suggested in the statistical literature. From the simulation results, it is evident that the exact variance approach is more accurate than the two approximate methods. To facilitate practical use, tabulated values of the estimated sample sizes are provided.

Keywords: Large-sample approximation; Nonparametric method; One-sample location problem

1. Introduction

The Wilcoxon signed-rank test is one of the most widely used nonparametric methods for the one-sample location problem. It provides an important alternative to the parametric t-test for giving robust results without the restriction of normality assumption in the population. Generally, the power function of the Wilcoxon signed-rank test is very difficult to express and only a few special cases have been examined, see Klotz [1] and Arnold [2] for shifts in normal and t-distributions, respectively. For the purpose of power and sample size calculations for the Wilcoxon signed-rank test, two simplified methods have been proposed in the statistical literature: Lehmann [3, p. 167] and Noether [4]. Both procedures are based on some approxi-mate expressions for the asymptotic normal distribution of the Wilcoxon signed-rank statistic. Despite the extensive applicability in the planning of one-sample study, no research to date has compared these two formulas for their finite-sample properties. In fact, they yield markedly different results according to the findings presented in this article. More importantly, verifica-tion of the accuracy of their methods under a variety of different distribuverifica-tions would be useful.

*Corresponding author. Email: gwshieh@mail.nctu.edu.tw

Journal of Statistical Computation and Simulation

DOI: 10.1080/10629360600635245

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718 G. Shieh et al.

Furthermore, it is important to note that the approximations suggested by Lehmann [3] and Noether [4] rely on the assumption that the alternatives do not differ too much from the null hypothesis. It is of great interest to have a rule of thumb that indicates whether the location shift is small enough so that the results are valid. Unfortunately, no such general guideline is available. This serious disadvantage renders the limited use of their procedures in practical situations.

In order to improve the practical usefulness, this article aims to investigate exact variance large-sample solution to power and sample size determinations for the Wilcoxon signed-rank test. We concentrate on three prominent situations that represent typical light-, standard- and heavy-tailed distributions: uniform, normal and Laplace. For these three cases, analytic forms are derived explicitly and exploited thoroughly to assess the finite-sample adequacy of the exact variance method. Moreover, the characteristics of the simplified procedures considered in equation (4.32) of Lehmann [3] and Noether [4] are also examined in an attempt to provide some guidance in the choice of appropriate method for power and sample size calculations.

The next section presents some of the analytical justification and important details of the exact variance large-sample method, as well as two related simplified approaches. In section 3, Monte Carlo simulation studies are conducted to evaluate the exact variance proce-dure and compare its performance with the approximate approaches under the three prescribed symmetric distributions. The corresponding sample sizes needed to achieve the specified power levels are summarized. Finally, section 4 contains some final remarks.

2. Power and sample size calculations

Consider a random sample X1, . . . , XN from an arbitrary continuous and symmetric

cumulative distribution F (x− θ), where θ is the unique median and mean if it exists. It is desirable to test the hypothesis H0: θ = 0 versus the alternative H1: θ > 0. We focus on the Wilcoxon signed-rank statistic W in the context of nonparametric methods defined as follows:

W =

N

i=1

ϕ(Xi)R(|Xi|),

where ϕ(Xi)= 1 if Xi >0 and 0 otherwise, and R(|Xi|) is the rank of |Xi| among

|X1|, . . . , |XN|. It was shown in Theorem 2.5.1 of Hettmansperger [5, p. 47] that the mean μ

and variance σ2_{of W are}

μ= Np1+N (N− 1) 2 p2 and σ2= Np1(1− p1)+N (N− 1) 2 p2(1− p2)+ 2N(N − 1)(p3− p1p2) + N(N − 1)(N − 2)(p4− p2 2), (1) respectively, where p1= P (X1 >0)= F (θ), p2= P (X1+ X2>0)= F (2θ+ x)f (x) dx, p3= P (X1+ X2>0, X1 >0)= (p12+ p2)/2, p4= P (X1+ X2>0, X1+ X3>0)= {F (2θ + x)}2_{f (x)}_dx,

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Power and sample size determinations for the Wilcoxon signed-rank test 719

and F (·) and f (·) are the cumulative distribution function and probability density function of X1 under the null hypothesis H0: θ = 0, respectively. Furthermore, Theorem 2.5.4 of Hettmansperger [5, p. 56] shows that (W − μ)/σ has an asymptotic standard normal distribution. Note that both μ and σ2 _{are functions of (p1}_{, p}

2, p3, p4)and consequently depend on the value of θ . For the case of θ = 0, it can be easily obtained that μ and σ2 given in equation (1) reduce to

μ0= N (N+ 1) 4 and σ 2 0 = N (N+ 1)(2N + 1) 24 ,

respectively. Therefore, the aforementioned test of location shift is carried out by rejecting

H0: θ = 0 if the standardized value (W − μ0)/σ0is greater than zα, where α is the specified

significance level and zαis the 100(1− α)th percentile of the standard normal distribution.

For the purpose of power and sample size calculations, we proceed to consider that W˙∼N(μ, σ2)under H1. Hence, given distribution F (x− θ) with location θ(>0) and sample size N , the statistical power achieved for testing hypothesis H0: θ = 0 versus the alternative

H1: θ > 0 with specified significance level α is approximated by the probability

P{W > μ0+ zασ0} . = 1 −  zασ0+ μ0− μ σ , (2)

where (·) is the cumulative distribution function of the standard normal distribution. This process can be reversed to calculate the sample size needed to test the null hypothesis with specified significance level α and power 1− β . However, it usually involves an iterative process to find the solution because all the parameter values (μ0, σ0, μ, σ )depend on the sample size N . More specifically, the resulting sample size, denoted by NE, is the minimum

N which satisfies the inequality

(μ− μ0)≥ (zασ0+ zβσ ).

In general, there are no simple closed-form expressions for the preceding equations given in equation (1) except in some special cases. Essentially, the numerical computation requires the one-dimensional integration with respect to the probability density function f (x). However, it is prudent to examine the exact variance large-sample method described earlier for some special F (x− θ) distributions that possesses potentially important implications. Hence, the following three cases ranging from light-tailed to heavy-tailed distributions are presented. 1. Uniform (−1/2, 1/2): p1= 1 2+ θ, p2= 1 2 + 2θ(1 − θ) and p4= 1 3 + 2θ − 8θ3 3 for θ≥ 1 2. 2. Standard normal N (0, 1): p1 = (θ), p2= ( √

2θ ) and p4= E[{(2θ + Z)}2], where Z ∼ N(0, 1). 3. Laplace (0, 1): p₁= 1 − 1 2e −θ_{, p} 2 = 1 − 1 2(1+ θ)e −2θ _{and p} 4= 1 − 7 12+ θ e−2θ − 1 12e −4θ_.

These explicit expressions are employed to illustrate the distinct features of the exact variance method in the subsequent section.

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Along the same line of power and sample size calculations within the framework of Wilcoxon signed-rank test, two simplified approximations to the exact variance large-sample result have been proposed. Based on the asymptotic normal distribution described earlier, Lehmann [3, p. 167] and Hettmansperger [5, p. 60] suggested to approximate the power by

1− zα− N θf (0)+ N(N − 1)θf∗(0) σ0 , (3)

where f (0) is f (x) evaluated at x = 0, and f∗(0)=f2_(x)_{dx. Correspondingly, Noether [4]} proposed the power function

1− zα− (3N)1/2 p2− 1 2 . (4)

Just as in the case of the exact variance method, both (3) and (4) can be applied to construct respective sample size estimates NLand NNrequired for testing the specified hypothesis with

significance level α and power 1− β. Nevertheless, it is worthwhile to note that these two simplified procedures are valid for small values of θ . Failing to account for this nature may distort power analysis and lead to a poor choice of sample size. This phenomenon will be demonstrated in the following numerical illustrations.

3. Simulation study

As all the three approaches considered here use large-sample justifications, simulation studies are conducted to assess their adequacy for finite-sample and robustness for various configu-rations. For illustration, the three distributions of uniform, normal and Laplace are exploited as the bases for the numerical examinations.

To help clarify similarities and differences for the competing procedures in performing power and sample size calculations, the sample sizes (NE, NL, NN)needed to achieve the

power levels: 0.80, 0.90 and 0.95 are computed for the three methods defined in equations (2)–(4). We assume throughout the demonstration that type I error rate α= 0.05. In each case, a total of six values of location shift are evaluated for F (x− θ) in terms of = θ/σ . The results for the three power levels of 0.80, 0.90 and 0.95 are summarized in tables 1–3, respectively. SAS codes for the calculation of the exact variance large-sample method are available upon request.

Table 1. Sample size required to attain power level 0.80.

= θ/σ 0.1 0.2 0.4 0.6 0.8 1.0 Uniform (−1/2, 1/2) Exact variance 653 172 47 23 14 10 Lehmann 620 157 41 19 12 8 Noether 656 175 50 26 17 13 Standard normal Exact variance 649 164 42 20 12 9 Lehmann 649 163 42 19 11 8 Noether 652 167 45 23 15 12 Laplace (0, 1) Exact variance 419 109 31 16 11 8 Lehmann 412 103 26 11 6 4 Noether 422 112 34 19 14 12

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Power and sample size determinations for the Wilcoxon signed-rank test 721

Table 2. Sample size required to attain power level 0.90.

Preliminary inspection of the tables yield the expected general relations: the computed sample size increases as the location shift decreases for all three approaches. Also, it is inter-esting to note that the ordering of sample size estimates is consistently NN > NE≥ NLexcept

in the case of standard normal distribution for power level 0.95 where NN > NL≥ NE in

table 3. However, the discrepancy between NLand NEin the last situation is never larger than

1. Although these values of sample sizes allow comparison of relative efficiencies of the meth-ods, the magnitude of the sample size affects the accuracy of the asymptotic distribution and the resulting formula. A fair comparison among these approaches must adjust for this factor.

In order to identify the most reliable method, we need to evaluate their actual or simulated power with the nominal power for a given sample size. Hence, we unify the sample sizes in the following simulations by choosing the sample size NEin tables 1–3 as the benchmark

to re-calculate the nominal powers for all competing approaches. Accordingly, the nominal powers for the exact variance method are slightly greater than 0.8, 0.90 and 0.95 in tables 4–6, respectively.

Estimates of the true power associated with the given sample size and distribution configuration are then computed through Monte Carlo simulation of 10,000 independent data sets. For each replicate, NEobservations are generated from the selected distribution. Then

the Wilcoxon signed-rank test statistic is computed and the simulated power is the proportion of the 10,000 replicates whose standardized test statistic values exceed the critical value zα.

The simulation results are presented in tables 4–6. The adequacy of the sample size formula Table 3. Sample size required to attain power level 0.95.

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Table 4. Simulated power at specified sample size when nominal power of exact variance method is about 0.80.

= θ/σ 0.1 0.2 0.4 0.6 0.8 1.0 Uniform (−1/2, 1/2) Sample size 653 172 47 23 14 10 Nominal power Exact variance 0.8001 0.8012 0.8022 0.8111 0.8171 0.8458 Lehmann 0.8180 0.8332 0.8541 0.8733 0.8843 0.9028 Noether 0.7986 0.7958 0.7825 0.7686 0.7445 0.7273 Simulated power Error 0.8055 0.8064 0.8091 0.8123 0.8360 0.8593 Exact variance −0.0054 −0.0052 −0.0069 −0.0012 −0.0189 −0.0135 Lehmann 0.0125 0.0268 0.0450 0.0610 0.0483 0.0435 Noether −0.0069 −0.0106 −0.0266 −0.0437 −0.0915 −0.1320 Standard normal Sample size 649 164 42 20 12 9 Nominal power Exact variance 0.8001 0.8015 0.8001 0.8088 0.8074 0.8478 Lehmann 0.8005 0.8031 0.8074 0.8251 0.8395 0.8823 Noether 0.7985 0.7953 0.7762 0.7561 0.7195 0.7070 Simulated power Error 0.8054 0.7970 0.7990 0.8084 0.8235 0.8468 Exact variance −0.0053 0.0045 0.0011 0.0004 −0.0161 0.0010 Lehmann −0.0049 0.0061 0.0084 0.0167 0.0160 0.0355 Noether −0.0069 −0.0017 −0.0228 −0.0523 −0.1040 −0.1398 Laplace (0, 1) Sample size 419 109 31 16 11 8 Nominal power Exact variance 0.8002 0.8003 0.8047 0.8079 0.8308 0.8185 Lehmann 0.8061 0.8208 0.8653 0.9097 0.9530 0.9726 Noether 0.7976 0.7907 0.7708 0.7409 0.7225 0.6755 Simulated power Error 0.8050 0.7954 0.8099 0.8117 0.8265 0.8196 Exact variance −0.0048 0.0049 −0.0052 −0.0038 0.0043 −0.0011 Lehmann 0.0011 0.0254 0.0554 0.0980 0.1265 0.1530 Noether −0.0074 −0.0047 −0.0391 −0.0708 −0.1040 −0.1441

= θ/σ 0.1 0.2 0.4 0.6 0.8 1.0 Uniform (−1/2, 1/2) Sample size 903 236 63 30 17 12 Nominal power Exact variance 0.9002 0.9008 0.9014 0.9090 0.9011 0.9290 Lehmann 0.9127 0.9219 0.9322 0.9408 0.9351 0.9470 Noether 0.8986 0.8944 0.8777 0.8582 0.8138 0.7937 Simulated power Error 0.9047 0.8982 0.8928 0.8986 0.8862 0.9203 Exact variance −0.0045 0.0026 0.0086 0.0104 0.0149 0.0087 Lehmann 0.0080 0.0237 0.0394 0.0422 0.0489 0.0267 Noether −0.0061 −0.0038 −0.0151 −0.0404 −0.0724 −0.1266 Standard normal Sample size 898 225 57 26 15 11 Nominal power Exact variance 0.9003 0.9005 0.9024 0.9054 0.9036 0.9355 Lehmann 0.9002 0.9001 0.9013 0.9045 0.9064 0.9337 Noether 0.8986 0.8940 0.8762 0.8465 0.8007 0.7810 (continued)

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Power and sample size determinations for the Wilcoxon signed-rank test 723 Table 5. Continued. = θ/σ 0.1 0.2 0.4 0.6 0.8 1.0 Simulated power Error 0.9027 0.9011 0.8988 0.8940 0.8988 0.9163 Exact variance −0.0024 −0.0006 0.0036 0.0114 0.0048 0.0192 Lehmann −0.0025 −0.0010 0.0025 0.0105 0.0076 0.0174 Noether −0.0041 −0.0071 −0.0226 −0.0475 −0.0981 −0.1353 Laplace (0, 1) Sample size 580 150 42 21 14 10 Nominal power Exact variance 0.9004 0.9005 0.9053 0.9073 0.9236 0.9212 Lehmann 0.9042 0.9131 0.9393 0.9609 0.9814 0.9898 Noether 0.8980 0.8913 0.8714 0.8364 0.8097 0.7590 Simulated power Error 0.8971 0.8937 0.9002 0.8990 0.9101 0.9059 Exact variance 0.0033 0.0068 0.0051 0.0083 0.0135 0.0153 Lehmann 0.0071 0.0194 0.0391 0.0619 0.0713 0.0839 Noether 0.0009 −0.0024 −0.0288 −0.0626 −0.1004 −0.1469

= θ/σ 0.1 0.2 0.4 0.6 0.8 1.0 Uniform (−1/2, 1/2) Sample size 1139 296 78 36 21 13 Nominal power Exact variance 0.9501 0.9502 0.9509 0.9538 0.9602 0.9537 Lehmann 0.9580 0.9631 0.9681 0.9701 0.9710 0.9612 Noether 0.9487 0.9449 0.9306 0.9085 0.8799 0.8212 Simulated power Error 0.9496 0.9496 0.9481 0.9431 0.9433 0.9277 Exact variance 0.0005 0.0006 0.0028 0.0107 0.0169 0.0260 Lehmann 0.0084 0.0135 0.0200 0.0270 0.0277 0.0335 Noether −0.0009 −0.0047 −0.0175 −0.0346 −0.0634 −0.1065 Standard normal Sample size 1133 283 71 32 18 12 Nominal power Exact variance 0.9501 0.9503 0.9522 0.9559 0.9552 0.9599 Lehmann 0.9499 0.9494 0.9488 0.9493 0.9466 0.9507 Noether 0.9488 0.9452 0.9307 0.9055 0.8603 0.8114 Simulated power Error 0.9504 0.9459 0.9489 0.9455 0.9361 0.9449 Exact variance −0.0003 0.0044 0.0033 0.0104 0.0191 0.0150 Lehmann −0.0005 0.0035 −0.0001 0.0038 0.0105 0.0058 Noether −0.0016 −0.0007 −0.0182 −0.0400 −0.0758 −0.1335 Laplace (0, 1) Sample size 731 189 52 25 16 11 Nominal power Exact variance 0.9500 0.9506 0.9531 0.9505 0.9571 0.9503 Lehmann 0.9523 0.9579 0.9716 0.9805 0.9902 0.9939 Noether 0.9482 0.9437 0.9259 0.8885 0.8533 0.7932 Simulated power Error 0.9491 0.9509 0.9491 0.9390 0.9405 0.9283 Exact variance 0.0009 −0.0003 0.0040 0.0115 0.0166 0.0220 Lehmann 0.0032 0.0070 0.0225 0.0415 0.0497 0.0656 Noether −0.0009 −0.0072 −0.0232 −0.0505 −0.0872 −0.1351

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is determined by the difference between the nominal power and simulated power. For ease of comparison, the discrepancy between nominal power and simulated power is also reported and is termed as error= nominal power − simulated power. In general, the absolute errors increase with decreasing nominal powers for all competing methods. The results suggest that the exact variance method performs consistently well with respect to all distribution and location speci-fications. For the two approximate procedures, the nominal powers of Lehmann’s [3] approach tend to be higher than the simulated powers while Noether’s [4] method shows the opposite pattern that the nominal powers are generally lower than the simulated powers. In addition, both approximations give accurate results for small values of = 0.1 and 0.2. However, Lehmann’s method is slightly inferior. As expected, larger values of appear to degrade the two approximate approaches, especially, this phenomenon is more pronounced for Noether’s approximation. Their performances also vary with the structure of distribution. In the case of normal distribution, it is interesting to note that Lehmann’s approximation maintains a rea-sonable agreement between the simulated power and the nominal power for all values of . Nevertheless, Lehmann’s approximation incurs substantially larger errors for heavy-tailed Laplace distribution than the light-tailed uniform distribution. Also, Noether’s approximation seems to deteriorate progressively from light- to heavy-tailed cases. Overall, the exact variance large-sample method has a clear advantage over the two approximate counterparts.

4. Conclusion

The purpose of this article is to present power and sample size determinations for the Wilcoxon signed-rank test that have not previously been discussed in literature, especially, the exact variance large-sample method for obtaining accurate results. Analytical formulas are provided for the three prominent situations of light-, standard- and heavy-tailed distributions: uniform, normal and Laplace that researchers are likely to encounter with real data. In addition, we exam-ine two approximations that are valid only for small values of location shift. Particular emphasis is devoted to the demonstration of their differences that arise in power function considerations. According to our findings, the accuracy of the two approximate approaches not only varies with the underlying distributions but also decreases considerably for = θ/σ > 0.2. One clear advantage of the exact variance approach is that it circumvents the restriction of small values of location shift, however, the formulation and computation is slightly more involved. In sum-mary, the exact variance large-sample method is recommended according to its remarkable accuracy under the range of distributions and location configurations considered here.

Acknowledgement

The authors wish to thank the associate editor and a referee for their suggestions for improving the clarity of the exposition. The research of the second author was partially supported by National Science Council grant NSC-93-2118-M-033-002.

References

[1] Klotz, J., 1963, Small sample power and efficiency for the one sample Wilcoxon and Normal scores tests. Annals

of Mathematical Statistics, 34, 624–632.

[2] Arnold, H.J., 1965, Small sample power of the one sample Wilcoxon test for non-normal shift alternatives. Annals

of Mathematical Statistics, 36, 1767–1778.

[3] Lehmann, E.L., 1998, Nonparametrics: Statistical Methods Based on Ranks (Upper Saddle River, NJ: Prentice-Hall).

[4] Noether, G.E., 1987, Sample size determination for some common nonparametric tests. Journal of the American

Statistical Association, 82, 645–647.

[5] Hettmansperger, T.P., 1984, Statistical Inference Based on Ranks (New York: Wiley).