Rank tests from partially ordered data using importance and MCMC sampling methods. (English) Zbl 1442.62095

Mondal, Debashis; Hinrichs, Nina

Stat. Sci. 31, No. 3, 325-347 (2016).

Summary: We discuss distribution-free exact rank tests from partially ordered data that arise in various biological and other applications where the primary objective is to conduct testing of significance to assess the linear dependence or to compare different groups. The tests here are obtained by treating the usual rank statistics, based on the completely ordered data as “latent” or missing, and conceptualizing the “latent” \(p\)-value as the random probability under the null hypothesis of a test statistic that is as extreme, or more extreme, than the latent test statistics based on the completely ordered data. The latent \(p\)-value is then predicted by sampling linear extensions or the complete orderings that are consistent with the observed partially ordered data. The sampling methods explored here include importance sampling methods based on randomized topological sorting algorithms, Gibbs sampling methods, random-walk based Metropolis-Hasting sampling methods and random-walk based modern perfect Markov chain Monte Carlo sampling methods. We discuss running times of these sampling methods and their strength and weaknesses. A simulation experiment and three data examples are given. The simulation experiment illustrates how the exact rank tests from partially ordered data work when the desired result is known. The first data example concerns the light preference behavior of fruit flies and tests whether heterogeneity observed in average light-preference behavior can be explained by manipulations in serotonin signaling. The second one is a reanalysis of the lead absorption data in children of employees who worked in a lead battery factory and consolidates the results reported in [P. R. Rosenbaum, Ann. Stat. 19, No. 2, 1091–1097 (1991; Zbl 0729.62045)]. The third one reexamines the breast cosmesis data from D. M. Finkelstein [Biometrics 42, 845–854 (1986; Zbl 0618.62097)].

MSC:

62G10	Nonparametric hypothesis testing
62N01	Censored data models
65C05	Monte Carlo methods

Keywords:

exact tests; fuzzy \(p\)-values; Gibbs sampling; iterval censoring; linear extensions; linear rank statistics; perfect MCMC; proportional hazard model; topological sorting

Citations:

Zbl 0729.62045; Zbl 0618.62097

References:

[1]	Aho, A. V., Garey, M. R. and Ullman, J. D. (1972). The transitive reduction of a directed graph. SIAM J. Comput.1 131-137. · Zbl 0247.05128 · doi:10.1137/0201008
[2]	Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc.12 1119-1178. · Zbl 0932.05001 · doi:10.1090/S0894-0347-99-00307-0
[3]	Bayarri, M. J. and Berger, J. O. (2000). \(p\) values for composite null models. J. Amer. Statist. Assoc.95 1127-1142. · Zbl 1004.62022
[4]	Bayarri, M. J. and Berger, J. O. (2004). The interplay of Bayesian and frequentist analysis. Statist. Sci.19 58-80. · Zbl 1062.62001 · doi:10.1214/088342304000000116
[5]	Besag, J. (2004). Markov Chain Monte Carlo Methods for Statistical Inference. Dept. Statistics, Univ. Washington, Seattle.
[6]	Besag, J. and Clifford, P. (1989). Generalized Monte Carlo significance tests. Biometrika76 633-642. · Zbl 0679.62033 · doi:10.1093/biomet/76.4.633
[7]	Besag, J. and Mondal, D. (2013). Exact goodness-of-fit tests for Markov chains. Biometrics69 488-496. · Zbl 1274.62720 · doi:10.1111/biom.12009
[8]	Besag, J., Green, P., Higdon, D. and Mengersen, K. (1995). Bayesian computation and stochastic systems. Statist. Sci.10 3-41. · Zbl 0955.62552 · doi:10.1214/ss/1177010123
[9]	Brightwell, G. and Winkler, P. (1991). Counting linear extensions. Order8 225-242. · Zbl 0759.06001 · doi:10.1007/BF00383444
[10]	Bubley, R. and Dyer, M. (1999). Faster random generation of linear extensions. Discrete Math.201 81-88. · Zbl 0934.65005 · doi:10.1016/S0012-365X(98)00333-1
[11]	Cormen, T. H., Leiserson, C. E., Rivest, R. L. and Stein, C. (2001). Introduction to Algorithms, 2nd ed. MIT Press, Cambridge, MA. · Zbl 1047.68161
[12]	Cox, D. R. (1972). Regression models and life-tables. J. R. Statist. Soc. Ser. B34 187-220. · Zbl 0243.62041
[13]	Cox, D. R. and Hinkley, D. V. (1979). Theoretical Statistics. Chapman & Hall, London. · Zbl 0334.62003
[14]	Crowley, J. (1974). A note on some recent likelihoods leading to the log rank test. Biometrika61 533-538. · Zbl 0301.62024 · doi:10.1093/biomet/61.3.533
[15]	Fay, M. P. and Shaw, P. A. (2010). Exact and asymptotic weighted log-rank tests for interval censored data: The interval R package. J. Stat. Software36 1-34.
[16]	Ferrenberg, A. M., Landau, D. P. and Swendsen, R. H. (1995). Statistical errors in histogram reweighting. Phys. Rev. E51 5092-5100.
[17]	Finkelstein, D. M. (1986). A proportional hazards model for interval-censored failure time data. Biometrics42 845-854. · Zbl 0618.62097 · doi:10.2307/2530698
[18]	Gelman, A., Meng, X.-L. and Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statist. Sinica6 733-807. · Zbl 0859.62028
[19]	Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell.6 721-741. · Zbl 0573.62030
[20]	Geyer, C. J. and Meeden, G. D. (2005). Fuzzy and randomized confidence intervals and \(P\)-values. Statist. Sci.20 358-366. · Zbl 1130.62319 · doi:10.1214/088342305000000340
[21]	Goggins, W. B. and Finkelstein, D. M. (2000). A proportional hazards model for multivariate interval-censored failure time data. Biometrics56 940-943. · Zbl 1060.62617
[22]	Goggins, W. B., Finkelstein, D. M., Schoenfeld, D. A. and Zaslavsky, A. M. (1998). A Markov chain Monte Carlo EM algorithm for analyzing interval-censored data under the Cox proportional hazards model. Biometrics54 1498-1507. · Zbl 1058.62555
[23]	Gordon, A. D. (1979a). A measure of the agreement between rankings. Biometrika66 7-15. · Zbl 0397.62040 · doi:10.1093/biomet/66.1.7
[24]	Gordon, A. D. (1979b). Another measure of the agreement between rankings. Biometrika66 327-332. · doi:10.1093/biomet/66.2.327
[25]	Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Stat.39 325-346. · Zbl 0187.16401 · doi:10.1214/aoms/1177698394
[26]	Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests, 2nd ed. Academic Press, San Diego, CA. · Zbl 0944.62045
[27]	Huber, M. (2004). Perfect sampling using bounding chains. Ann. Appl. Probab.14 734-753. · Zbl 1052.60057 · doi:10.1214/105051604000000080
[28]	Huber, M. (2006). Fast perfect sampling from linear extensions. Discrete Math.306 420-428. · Zbl 1090.60064 · doi:10.1016/j.disc.2006.01.003
[29]	Hunt, J. W. and Szymanski, T. G. (1977). A fast algorithm for computing longest common subsequences. Commun. ACM20 350-353. · Zbl 0354.68078 · doi:10.1145/359581.359603
[30]	Kahn, A. B. (1962). Topological sorting of large networks. Commun. ACM5 558562. · Zbl 0106.32602
[31]	Kain, J. S., Stokes, C. and de Bivort, B. L. (2012). Phototactic personality in fruit flies and its suppression by serotonin and white. Proc. Natl. Acad. Sci. USA109 19834-19839.
[32]	Kalbfleisch, J. D. and Prentice, R. L. (1980). The Statistical Analysis of Failure Time Data. Wiley, New York. · Zbl 0504.62096
[33]	Karzanov, A. and Khachiyan, L. (1991). On the conductance of order Markov chains. Order8 7-15. · Zbl 0736.06002 · doi:10.1007/BF00385809
[34]	Kruskal, W. H. and Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. J. Amer. Statist. Assoc.47 583-621. · Zbl 0048.11703
[35]	Lerche, D., Brüggemann, R., Sørensen, P., Carlsen, L. and Nielsen, O. J. (2002). A comparison of partial order technique with three methods of multi-criteria analysis for ranking of chemical substances. J. Chem. Inf. Comput. Sci.42 1086-1098.
[36]	Liu, J. S. (2008). Monte Carlo Strategies in Scientific Computing. Springer, New York. · Zbl 1132.65003
[37]	Matthews, P. (1991). Generating a random linear extension of a partial order. Ann. Probab.19 1367-1392. · Zbl 0728.60009 · doi:10.1214/aop/1176990349
[38]	Meng, X.-L. (1994). Posterior predictive \(p\)-values. Ann. Statist.22 1142-1160. · Zbl 0820.62027 · doi:10.1214/aos/1176325622
[39]	Mondal, D. and Hinrichs, N. (2016). Supplement to “Rank tests from partially ordered data using importance and MCMC sampling methods.” DOI:10.1214/16-STS549SUPP. · Zbl 1442.62095
[40]	Morton, D. E., Saah, A. J., Silberg, S. L., Owens, W. L., Roberts, M. A. and Saah, M. D. (1982). Lead absorption in children of employees in a lead-related industry. Am. J. Epidemiol.115 549-555.
[41]	Page, E. B. (1963). Ordered hypotheses for multiple treatments: A significance test for linear ranks. J. Amer. Statist. Assoc.58 216-230. · Zbl 0114.11102 · doi:10.1080/01621459.1963.10500843
[42]	Patil, G. P. and Taillie, C. (2004). Multiple indicators, partially ordered sets, and linear extensions: Multi-criterion ranking and prioritization. Environ. Ecol. Stat.11 199-228. · doi:10.1023/B:EEST.0000027209.93218.d9
[43]	Prentice, R. L. (1978). Linear rank tests with right censored data. Biometrika65 167-179. · Zbl 0377.62024 · doi:10.1093/biomet/65.1.167
[44]	Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms9 223-252. · Zbl 0859.60067
[45]	Puri, M. L. and Sen, P. K. (1971). Nonparametric Methods in Multivariate Analysis. Wiley, New York. · Zbl 0237.62033
[46]	Riggle, J. (2009). The complexity of ranking hypotheses in optimality theory. Comput. Linguist.35 47-59. · doi:10.1162/coli.07-031-R2-06-98
[47]	Rosenbaum, P. R. (1991). Some poset statistics. Ann. Statist.19 1091-1097. · Zbl 0729.62045 · doi:10.1214/aos/1176348141
[48]	Rosenbaum, P. R. (2002). Observational Studies, 2nd ed. Springer, New York. · Zbl 0985.62091
[49]	Rubin, D. B. (1987). A noniterative sampling/importance resampling alternative to the data augmentation algorithm for creating a few imputations when fractions of missing information are modest: The SIR algorithm. J. Amer. Statist. Assoc.82 543-546.
[50]	Satten, G. A. (1996). Rank-based inference in the proportional hazards model for interval censored data. Biometrika83 355-370. · Zbl 0864.62079
[51]	Schensted, C. (1961). Longest increasing and decreasing subsequences. Canad. J. Math.13 179-191. · Zbl 0097.25202 · doi:10.4153/CJM-1961-015-3
[52]	Self, S. G. and Grossman, E. A. (1986). Linear rank tests for interval-censored data with application to PCB levels in adipose tissue of transformer repair workers. Biometrics42 521-530.
[53]	Skare, Ø., Bølviken, E. and Holden, L. (2003). Improved sampling-importance resampling and reduced bias importance sampling. Scand. J. Stat.30 719-737. · Zbl 1055.65019 · doi:10.1111/1467-9469.00360
[54]	Smith, A. F. M. and Gelfand, A. E. (1992). Bayesian statistics without tears: A sampling-resampling perspective. Amer. Statist.46 84-88.
[55]	Tarjan, R. E. (1976). Edge-disjoint spanning trees and depth-first search. Acta Inform.6 171-185. · Zbl 0307.05104 · doi:10.1007/BF00268499
[56]	Thompson, E. A. and Geyer, C. J. (2007). Fuzzy \(p\)-values in latent variable problems. Biometrika94 49-60. · Zbl 1143.62087 · doi:10.1093/biomet/asm001
[57]	Vandal, A. C., Conder, M. D. E. and Gentleman, R. (2009). Minimal covers of maximal cliques for interval graphs. Ars Combin.92 97-129. · Zbl 1224.05426
[58]	Wilson, D. B. (2004). Mixing times of Lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab.14 274-325. · Zbl 1040.60063 · doi:10.1214/aoap/1075828054
[59]	Zar, J. H. (1972). Significance testing of the Spearman rank correlation coefficient. J. Amer. Statist. Assoc.67 578-580. · Zbl 0248.62016