A Pooled Two-Sample Median Test Based on Density Estimation Vadim Y

Journal of Modern Applied Statistical Methods

Volume 10 | Issue 2 Article 28

11-1-2011 A Pooled Two-Sample Median Test Based on Density Estimation Vadim Y. Bichutskiy George Mason University, [email protected]

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Recommended Citation Bichutskiy, Vadim Y. (2011) "A Pooled Two-Sample Median Test Based on Density Estimation," Journal of Modern Applied Statistical Methods: Vol. 10 : Iss. 2 , Article 28. DOI: 10.22237/jmasm/1320121620 Available at: http://digitalcommons.wayne.edu/jmasm/vol10/iss2/28

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Emerging Scholars A Pooled Two-Sample Median Test Based on Density Estimation

Vadim Y. Bichutskiy George Mason University Fairfax, Virginia

A new method based on density estimation is proposed for medians of two independent samples. The test controls the probability of Type I error and is at least as powerful as methods widely used in statistical practice. The method can be implemented using existing libraries in R.

Key words: Sample median, two-sample hypothesis test, adaptive kernel density estimation.

Introduction population median is approximately normal and uses one of several methods for estimating the Let X1n , X2 , … , X be iid having cdf F and pdf f with F(η) = 1/2 so that η is the population standard error of the sample median. Virtually median. Suppose f is continuous at η with f(η) > all methods are very conservative, particularly 0. Denote the sample median by H. It is known for heavy-tailed populations. that H is asymptotically normal with mean η and A new two-sample test is proposed for variance 1/4nf2(η). Estimating the asymptotic comparing medians. When population shapes standard error of the sample median requires an can be assumed to be the same, a pooled test estimate of the population density at the median. statistic, analogous to a pooled two-sample Besides being a challenging problem, density Student’s t statistic for comparing means, is estimation was difficult to apply in practice prior derived. Computer-intensive Monte Carlo to the computer revolution; due to this, several simulations in R (R Development Core Team, alternative methods for estimating the standard 2009) are used to study the properties of the test error of the sample median have been developed and compare it to other methods. The method (Maritz & Jarrett, 1978; McKean & Schrader, offers several additional benefits to practitioners: 1984; Price & Bonett, 2001; Sheather & Maritz, (1) a parameter that controls the trade-off 1983; Sheather, 1986). between making the test conservative and liberal Comparing medians based on two with a suitable value of the parameter producing independent samples is a well-studied problem a test with a nominal significance level; (2) the (see Wilcox & Charlin, 1986; Wilcox, 2005; test is easy to implement in R using the Wilcox, 2006; Wilcox, 2010 also has a good QUANTREG (Koenker, 2009) library. discussion). The methods fall into two main categories. The first uses the bootstrap (Efron, Methodology 1979), and the second assumes the sample Two-Sample Test Statistic for Difference in median or some other estimator of the Medians Let X1n , X2 , … , X and

Y1m , Y2 , … , Y be two independent random Vadim Y. Bichutskiy is a Ph.D. student in the samples of sizes n and m from populations with Department of Statistics. This work was densities fx, fy that are continuous at the medians completed when he was a M.S. student in the ηx, ηy with fx(ηx) > 0, fy(ηy) > 0, respectively. Department of Statistics and Biostatistics at Denote sample medians by Hx, Hy. The test California State University, East Bay hypotheses are: (Hayward). Email him at: [email protected].

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=Δ where H0x:-ηη y 22 vs. nfˆˆ (H )+ mf (H ) H :-ηη≠Δ , ˆ = x xyy 1x y f(H)p nm+ where Δ is a specified difference in medians, and is often 0. is the pooled estimate of the population density For sufficiently large n and m: at the median.

2 H~ N (η , 1/4nf (η ),) Simulations xxxx The software R was used to simulate the power of the pooled test statistic (1). Two cases 2 H~ N (η , 1/4mf (η ),) were considered: (i) population shapes are yy yy assumed to be known, and (ii) population shapes are unknown. The assumption of known population shapes is analogous to the 11 1 HH~−− N ηη , + , assumption of known population variances in xy xy 22 4nf(η )mf(η ) the z-test for comparing the means of two xx yy normal populations since the variance determines the shape of the normal distribution. HH(−−−ηη) xy xy~(0,1). N The goal was to see how the test would perform 11+ 1 for samples of moderate size from symmetric 22 heavy-tailed populations. Parent populations 2nf(xxη )mf( yyη ) investigated were Cauchy, Laplace and Student’s t distributions with 2 and 3 degrees of Assuming the normal approximation freedom. In all settings, the parent populations holds when the standard error of the difference were of the same shape, shifted under the in medians is estimated, then under the null alternative, and a two-sided test H0: ηx = ηy hypothesis, the V statistic is: versus H1: ηx ≠ ηy was performed.

(H−−Δ H ) Adaptive Kernel Density Estimation VN= xy ~(0,1) When population shapes are unknown, 11 1 ˆ + fx(ηx) and fy(ηy) are estimated with fHxx() and 22 2nf(H)mf(H)ˆˆ ˆ xx yy fHy ()y , respectively, using adaptive kernel density estimation (AKDE). ˆ ˆ ∈ d where fHxx() and fHyy() are respective Let X1n , X2 , … , X  be a sample population density estimates at the median. from unknown density f. The AKDE is a three Further, if it is assumed that the two step procedure: populations have the same shape, possibly with  1. Find a pilot estimate f(X) that satisfies a difference in location, then fx(ηx) = fy(ηy), and i =  > 1, 2,… , n. the density estimates can be pooled to obtain a f(X)i 0, pooled test statistic: 2. Define local bandwidth factors (H−−Δ H ) ={f(X )/ g} = xy λ i  i -γ where g is the geometric VNp ~(0,1) 111+  ≤≤ mean of the f(X)i and 0 γ 1 is the ˆ 2fp (H) n m sensitivity parameter. (1) 3. The adaptive kernel estimate is defined by

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n (Efron & Tibshirani, 1993. p. 221); and (iv) f(X)ˆ = n-1 h -d -d K{h -1 -1 (X-X )}  λλiii permutation test. Figure 3 shows the receiver i=1 operating characteristic (ROC) curves for a balanced design with n = m = 30. The parent where K(.) is a kernel function and h is the populations were of the same shape in each case bandwidth. and the difference in population medians was set to 1. For the bootstrap and the permutation test, The AKDE method varies the the difference in medians was used as the metric. bandwidth among data points and is better suited Each point on the curves is based on 10,000 for heavy-tailed populations than ordinary KDE simulated samples. (Silverman, 1998, pp. 100-110). Intuitively, the AKDE is based on the idea that for heavy-tailed Conclusion populations a larger bandwidth is needed for Tests for comparing medians tend to be very data points in the tails of the distribution (i.e., conservative. The proposed test is able to control for outliers). In R, function AKJ in library the probability of Type I error. It is as powerful QUANTREG implements AKDE. Obtaining the as the permutation test and the bootstrap and is pilot estimate requires the use of another density more powerful than the MWW test for heavy- estimation method, such as ordinary KDE. The tailed populations. The more heavy-tailed the general view in the literature is that AKDE is parent population, the greater the power fairly robust to the method used for the pilot advantage of the proposed test over the MWW estimate (Silverman, 1998) and that the choice test; when the parent population is light-tailed, of the sensitivity parameter γ is more critical. the MWW test is more powerful than the When using AKDE with Gaussian kernel, if the proposed test. parent population has tails close to normal then A key precept of the method is that γ <.5 should be used, however, if the parent AKDE provides a better estimate of the population density at the median, especially for population is heavy-tailed then γ >.5 should be heavy-tailed populations, than ordinary KDE. As =. used. Thus, γ 5 is a good choice and has expected, using ordinary KDE makes the test been shown to reduce bias (Abramson, 1982). very conservative where the Type I error rate can be as low as 0.02 at the 5% significance Results level. Case 1: Known Population Shapes These experiments show that the Figure 1 shows the power curves for the sensitivity parameter γ in AKDE controls the pooled test when population shapes are assumed trade-off between making the test conservative to be known at the 5% level of significance. and liberal, with a suitable value of γ producing Each point on the curves is based on 10,000 a test with a nominal significance level. The simulated samples. The Type I error rate is Type I error rate of the test can be increased controlled very well. (decreased) by increasing (decreasing) γ .

Case 2: Unknown Population Shapes The asymptotic distribution of the Figure 2 shows the power curves for the sample median has been known for over 50 pooled test when population shapes are years (Chu, 1955; Chu & Hotelling, 1955), but it unknown at the 5% level of significance and is only now with the improvement in computing using AKDE with γ =.5 . Each point on the power that this theory can be practically employed to derive useful statistical curves is based on 10,000 simulated samples. methodology, illustrating the interplay between The Type I error rate is controlled very well. st theory, methodology and computation in the 21

century. Comparisons with Other Methods

The test was compared to the following methods: (i) Student’s t-test; (ii) Mann-Whitney-

Wilcoxon (MWW) rank sum test; (iii) bootstrap

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Figure 1: Power Curves for Known Population Shapes (10,000 Simulated Samples)

Cauchy Laplace

Student’s t (df = 2) Student’s t (df = 3)

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Figure 2: Power Curves for Unknown Population Shapes

(10,000 Simulated Samples, AKDE with γ =.5 )

Cauchy Laplace

Student’s t (df = 2) Student’s t (df = 3)

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Figure 3: ROC Curves. Balanced Design with n = m = 30 (10,000 Simulated Samples)

(The curves for the permutation test coincide closely with the curves for the proposed test and have been omitted for clarity.)

Cauchy Laplace

Student’s t (df = 2) Student’s t (df = 3)

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Acknowledgements McKean, J. W., & Schrader, R. M. The author thanks Professor Emeritus Bruce E. (1984). A comparison of methods for Trumbo, Professor Eric A. Suess and Professor Studentizing the sample median. Joshua D. Kerr at California State University, Communications in Statistics – Simulation and East Bay (Hayward), for helpful discussions and Computation, 13, 751-773. suggestions. The journal staff improved the Price, R. M., & Bonett, D. G. (2001). prose. Earlier versions of this work were Estimating the variance of the sample median. contributed at Joint Statistical Meetings Journal of Statistical Computation and (Bichutskiy, Kerr & Trumbo, 2009; Bichutskiy, Simulation, 68, 295-305. et al., 2010). R Development Core Team. (2009). R: A language and environment for statistical References computing. R Foundation for Statistical Abramson, I. S. (1982). On bandwidth Computing, Vienna, Austria. ISBN 3-900051- variation in kernel estimates: A square root law. 07-0, URL http://www.R-project.org. The Annals of Statistics, 10, 1217-1223. Sheather, S. J., & Maritz, J. S. (1983). Bichutskiy, V. Y., Kerr, J., & Trumbo, An estimate of the asymptotic standard error of B. E. (2009). Classroom simulation: the sample median. Australian Journal of Investigation of the asymptotic distribution of Statistics, 25(1), 109-122. the sample median. In JSM Proceedings, Sheather, S. J. (1986). A finite sample Statistical Education Section, Alexandria, VA: estimate of the variance of the sample median. American Statistical Association, 3715-3728. Statistics and Probability Letters, 4, 337-342. Bichutskiy, V. Y., Kerr, J. D., Suess, E. Silverman, B. W. (1998). Density A., & Trumbo, B. E. (2010). Classroom estimation for statistics and data Analysis. Boca derivation and simulation: An asymptotic two- Raton, FL: Chapman & Hall/CRC. sample test for comparing population medians. Wilcox, R. R., & Charlin, V. L. (1986). In JSM Proceedings, Statistical Education Comparing medians: A Monte Carlo study. Section, Alexandria, VA: American Statistical Journal of Educational Statistics, 11(4), 263- Association, 4531-4545. 274. Chu, J. T. (1955). On the distribution of Wilcox, R. R. (2005). Comparing the sample median. The Annals of Mathematical medians: An overview plus new results on Statistics, 26, 112-116. dealing with heavy-tailed distributions. The Chu, J. T., & Hotelling, H. (1955). The Journal of Experimental Education, 73(3), 249- moments of the sample median. The Annals of 263. Mathematical Statistics, 26, 593–606. Wilcox, R. R. (2006). Comparing Efron, B. (1979). Bootstrap methods: medians. Computational Statistics & Data Another look at the jackknife. The Annals of Analysis, 51, 1934-1943. Statistics, 7(1), 1-26. Wilcox, R. R. (2010). Fundamentals of Efron, B., & Tibshirani, R. J. (1993). An modern statistical methods: Substantially introduction to the bootstrap. Boca Raton, FL: improving power and accuracy, 2nd Edition. Chapman & Hall/CRC. New York, NY: Springer. Koenker, R. (2009). quantreg: Quantile regression, R package version 4.44. http://CRAN.R-project.org/package=quantreg. Maritz, J. S., & Jarrett, R. G. (1978). A note on estimating the variance of the sample median. Journal of the American Statistical Association, 73, 194-196.

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