Abstract
For two independent groups, let \(M_j(\mathbf {X})\) be some conditional measure of location for the jth group associated with some random variable Y given \(\mathbf {X}=(X_1, X_2)\). Let \(\Omega =\{\mathbf {X}_1, \ldots , \mathbf {X}_K\}\) be a set of K points to be determined. An extant technique can be used to test \(H_0\): \(M_1(\mathbf {X})=M_2(\mathbf {X})\) for each \(\mathbf {X} \in \Omega \) without making any parametric assumption about \(M_j(\mathbf {X})\). But there are two general reasons to suspect that the method can have relatively low power. The paper reports simulation results on an alternative approach that is designed to test the global hypothesis \(H_0\): \(M_1(\mathbf {X})=M_2(\mathbf {X})\) for all \(\mathbf {X} \in \Omega \). The main result is that the new method offers a distinct power advantage. Using data from the Well Elderly 2 study, it is illustrated that the alternative method can make a practical difference in terms of detecting a difference between two groups.
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References
Bellman RE (1961) Adaptive control processes. Princeton University Press, Princeton
Benjamini Y, Yekutieli D (2001) The control of the false discovery rate in multiple testing under dependency. Ann Stat 29:1165–1188
Bradley JV (1978) Robustness? Br J Math Stat Psychol 31:144–152
Chida Y, Steptoe A (2009) Cortisol awakening response and psychosocial factors: a systematic review and meta-analysis. Biol Psychol 80:265–278
Clark F, Jackson J, Carlson M, Chou C-P, Cherry BJ, Jordan-Marsh M, Knight BG, Mandel D, Blanchard J, Granger DA, Wilcox RR, Lai MY, White B, Hay J, Lam C, Marterella A, Azen SP (2011) Effectiveness of a lifestyle intervention in promoting the well-being of independently living older people: results of the Well Elderly 2 Randomise Controlled Trial. J Epidemiol Community Health 66:782–790. doi:10.1136/jech.2009.099754
Clow A, Thorn L, Evans P, Hucklebridge F (2004) The awakening cortisol response: methodological issues and significance. Stress 7:29–37
Cousins RD (2008) Annotated bibliography of some papers on combining significances or p values. arXiv:0705.2209v2
Donoho DL, Gasko M (1992) Breakdown properties of the location estimates based on halfspace depth and projected outlyingness. Ann Stat 20:1803–1827
Eakman AM, Carlson ME, Clark FA (2010) The meaningful activity participation assessment: a measure of engagement in personally valued activities. Int J Aging Hum Dev 70:299–317
Efromovich S (1999) Nonparametric curve estimation: methods, theory and applications. Springer, New York
Eubank RL (1999) Nonparametric regression and spline smoothing. Marcel Dekker, New York
Foley K, Reed P, Mutran E et al (2002) Measurement adequacy of the CESD among a sample of older African Americans. Psychiatr Res 109:61–69
Fox J (2001) Multiple and generalized nonparametric regression. Sage, Thousands Oaks
Frigge M, Hoaglin DC, Iglewicz B (1989) Some implementations of the boxplot. Am Stat 43:50–54
Györfi L, Kohler M, Krzyzk A, Walk H (2002) A distribution-free theory of nonparametric regression. Springer, New York
Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics. Wiley, New York
Härdle W (1990) Applied nonparametric regression. Econometric society monographs no. 19. Cambridge University Press, Cambridge
Hays RD, Sherbourne CD, Mazel RM (1993) The rand 36-item health survey 1.0. Health Econ 2:217–227
Heritier S, Cantoni E, Copt S, Victoria-Feser M-P (2009) Robust methods in Biostatistics. Wiley, New York
Hoaglin DC (1985) Summarizing shape numerically: the g-and-h distribution. In: Hoaglin D, Mosteller F, Tukey J (eds) Exploring data tables trends and shapes. Wiley, New York, pp 461–515
Hochberg Y (1988) A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75:800–802
Huber PJ, Ronchetti E (2009) Robust statistics, 2nd edn. Wiley, New York
Jackson J, Mandel D, Blanchard J, Carlson M, Cherry B, Azen S, Chou C-P, Jordan-Marsh M, Forman T, White B, Granger D, Knight B, Clark F (2009) Confronting challenges in intervention research with ethnically diverse older adults: the USC Well Elderly II trial. Clin Trials 6:90–101
Lewinsohn PM, Hoberman HM, Rosenbaum M (1988) A prospective study of risk factors for unipolar depression. J Abnorm Psychol 97:251–264
Liu RY, Parelius JM, Singh K (1999) Multivariate analysis by data depth: descriptive statistics, graphics and inference. Ann Stat 27:783–858
Maronna RA, Martin DR, Yohai VJ (2006) Robust statistics: theory and methods. Wiley, New York
McHorney CA, Ware JE, Raozek AE (1993) The MOS 36-item Short-Form Health Survey (SF-36): II. Psychometric and clinical tests of validity in measuring physical and mental health constructs. Med Care 31:247–263
Radloff L (1977) The CESD scale: a self report depression scale for research in the general population. Appl Psychol Meas 1:385–401
Rom DM (1990) A sequentially rejective test procedure based on a modified Bonferroni inequality. Biometrika 77:663–666
Rosenberger JL, Gasko M (1983) Comparing location estimators: trimmed means, medians, and trimean. In: Hoaglin D, Mosteller F, Tukey J (eds) Understanding Robust and exploratory data analysis. Wiley, New York, pp 297–336
Staudte RG, Sheather SJ (1990) Robust estimation and testing. Wiley, New York
Wilcox RR (2012) Introduction to robust estimation and hypothesis testing, 3rd edn. Academic Press, San Diego
Yuen KK (1974) The two sample trimmed t for unequal population variances. Biometrika 61:165–170
Zaykin DV, Zhivotovsky LA, Westfall PH, Weir BS (2002) Truncated product method for combining p values. Gen Epidemiol 22:170–185
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Wilcox, R.R. ANCOVA: a heteroscedastic global test when there is curvature and two covariates. Comput Stat 31, 1593–1606 (2016). https://doi.org/10.1007/s00180-015-0640-4
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DOI: https://doi.org/10.1007/s00180-015-0640-4