×

Analysis of unbalanced factorial designs with heteroscedastic data. (English) Zbl 1185.62139

Summary: The present study investigates the operating characteristics of several Box-type [see G.E.P. Box, Ann. Math. Stat. 25, 290–302 (1954; Zbl 0055.37305)] and Welch-James (WJ) modifications of factorial designs lacking homogeneity, normality, and orthogonality. For comparison purposes the behaviours of Proc Mixed and Proc GLM, available from the SAS program, were also examined. When the shape of the distribution was symmetric, the Box-type, WJ, and Proc Mixed approaches consistently controlled the rates of error; however, when the distribution was moderately skewed only the Box-type approach limited the number of errors to the nominal value. In distributions with extreme skewness, the procedure was predominantly conservative but showed improved rates of Type-I error control using the Box-Cox method of power transformation. The execution of Proc GLM was considerably influenced by the presence of heterogeneity and scarcely affected by the absence of normality. With regard to test sensitivity, the WJ and Proc Mixed approaches were substantially more powerful than the Box-type approach when variances and cells sizes were negatively paired. However, they were equally powerful when this relationship was positive. When the population variances were homogeneous, the differences in power slightly favoured the Proc GLM approach.

MSC:

62K15 Factorial statistical designs
62H15 Hypothesis testing in multivariate analysis
65C05 Monte Carlo methods
62J05 Linear regression; mixed models
65C60 Computational problems in statistics (MSC2010)

Citations:

Zbl 0055.37305

Software:

SAS; MIXED; PROC GLM; SAS/STAT
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1093/biomet/63.3.581 · Zbl 0344.62034 · doi:10.1093/biomet/63.3.581
[2] DOI: 10.1037/0033-2909.101.3.464 · doi:10.1037/0033-2909.101.3.464
[3] DOI: 10.1007/BF02294383 · Zbl 0863.62069 · doi:10.1007/BF02294383
[4] DOI: 10.1093/biomet/67.1.85 · Zbl 0422.62066 · doi:10.1093/biomet/67.1.85
[5] DOI: 10.1007/BF02294772 · Zbl 0905.62028 · doi:10.1007/BF02294772
[6] DOI: 10.1214/aoms/1177728786 · Zbl 0055.37305 · doi:10.1214/aoms/1177728786
[7] DOI: 10.2307/2965420 · Zbl 0921.62096 · doi:10.2307/2965420
[8] Richter S. J., J. Modern Appl. Stat. Meth. 2 pp 152– (2003)
[9] SAS Institute. 2005.The MIXED procedure, in SAS/STAT User’s Guide, SAS On-Line Documentation. SAS Institute Inc., Cary, NC
[10] DOI: 10.2307/2533558 · Zbl 0890.62042 · doi:10.2307/2533558
[11] DOI: 10.1177/0013164403260196 · doi:10.1177/0013164403260196
[12] DOI: 10.1207/s15327906mbr4002_2 · doi:10.1207/s15327906mbr4002_2
[13] DOI: 10.1207/s15327906mbr4104_6 · doi:10.1207/s15327906mbr4104_6
[14] Searle, S. R. 1987. ”Linear Models for Unbalanced Data”. New York: Wiley. · Zbl 1095.62080
[15] DOI: 10.2307/2682884 · Zbl 0341.62060 · doi:10.2307/2682884
[16] Timm N. H., Multivar. Behav. Res. Monogr. 75 pp 1– (1975)
[17] DOI: 10.2307/2684275 · doi:10.2307/2684275
[18] Milliken, G. A. and Johnson, D. E. 1992. ”Analysis of Messy Data, Volume 1: Designed Experiments”. London: Chapman & Hall. · Zbl 0799.62001
[19] DOI: 10.1111/1469-8986.00060 · doi:10.1111/1469-8986.00060
[20] Box G. E.P., J. Roy. Stat. Soc. Ser. B. 26 pp 211– (1964)
[21] DOI: 10.1198/108571102816 · doi:10.1198/108571102816
[22] Micceri T., Psychol. Bull. 92 pp 778– (1989)
[23] DOI: 10.1007/BF02293811 · Zbl 0388.62023 · doi:10.1007/BF02293811
[24] Bradley J., Br. J. Math. Stat. Psychol. 31 pp 144– (1978) · doi:10.1111/j.2044-8317.1978.tb00581.x
[25] DOI: 10.1002/sim.1537 · doi:10.1002/sim.1537
[26] DOI: 10.1177/0013164406294777 · doi:10.1177/0013164406294777
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.