Nonparametric Tests of Interaction in Experimental Design

Review of Educational Research - Tập 60 Số 1 - Trang 91-126 - 1990
Shlomo S. Sawilowsky1
1Wayne State University

Tóm tắt

Until recently the design of experiments in the behavioral and social sciences that focused on interaction effects demanded the use of the parametric analysis of variance. Yet, researchers have been concerned by the presence of nonnormally distributed variables. Although nonparametric statistics are recommended in these situations, researchers often rely on the robustness of parametric tests. Further, often it is assumed that nonparametric methods lack statistical power and that there is a paucity of techniques in more complicated research designs, such as in testing for interaction effects. This paper reviewed (a) research in the past decade and a half that addressed concerns in selecting parametric and nonparametric statistics and (b) 10 recently developed nonparametric techniques for the testing of interactions in experimental design. The review shows that these new techniques are robust, powerful, versatile, and easy to compute. An application of selected nonparametric techniques on fabricated data is provided.

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Tài liệu tham khảo

Adichie JN, 1978, Annals of Statistics, 5, 1012

Agresti A, 1986, Communications in Statistics, 15, 1417, 10.1080/03610928608829193

Akritas MG, 1988, The rank transform method in some two factor designs

Anderson NH, 1961, Psychological Bulletin, 55, 305, 10.1037/h0042576

Andrews FC, 1954, Annals of Mathematical Statistics, 25, 724, 10.1214/aoms/1177728658

Ary D, 1976, Introduction to statistics

Aubuchon JC, 1984, Handbook of statistics, 4, 259

10.1177/001316446602600204

Barlow RE, 1972, Statistical inference under order restrictions: The theory and application of isotonic regression

BartzAEBasic statistical concepts19812nd edMinneapolis, MNBurgess

Bell CB, 1965, Annals of Mathematical Statistics, 36, 203, 10.1214/aoms/1177700283

Berry KJ, 1983, British Journal of Mathematical and Statistical Psychology, 36, 202, 10.1111/j.2044-8317.1983.tb01125.x

BestJResearch in education19814th edEnglewood Cliffs, NJPrentice-Hall

BestJWKahnJVResearch in education19896th edEnglewood Cliffs, NJPrentice-Hall

Bishop T, 1976, Heteroscedastic ANOVA, MANOVA and multiple comparisons

Blair RC, 1980, A comparison of the power of the two independent means t test to that of the Wilcoxon‘s rank-sum test for samples of various populations

10.3102/00346543051004499

Blair RC, 1985, Paper presented at the annual meeting of the American Educational Research Association and the National Council on Measurement in Education

10.1177/0193841X8000400506

10.3102/10769986005004309

Blair RC, 1981, British Journal of Mathematical and Statistical Psychology, 31, 124, 10.1111/j.2044-8317.1981.tb00623.x

10.3102/10769986010004368

Blair RC, 1985, Psychological Bulletin, 97, 119, 10.1037/0033-2909.97.1.119

Blair RC, 1980, British Journal of Mathematical and Statistical Psychology, 33, 114, 10.1111/j.2044-8317.1980.tb00783.x

Blair RC, 1990, Paper presented at the annual meeting of the American Educational Research Association

Blair RC, 1987, Communications in Statistics: Computation and Simulation, B16, 1133, 10.1080/03610918708812642

Bloom BS, 1984, Educational Leadership, 41, 4

Blum JR, 1954, Review of Educational Research, 24, 467

Boik RJ, 1987, British Journal of Mathematical and Statistical Psychology, 40, 26, 10.1111/j.2044-8317.1987.tb00865.x

Boneau CA, 1960, Psychological Bulletin, 57, 49, 10.1037/h0041412

Boneau CA, 1962, Psychological Review, 69, 246, 10.1037/h0047269

Borg WR, 1987, Applying educational research: A guide for teachers

Box GEP, 1953, Biometrika, 40, 318, 10.1093/biomet/40.3-4.318

Box GEP, 1954, Annals of Mathematical Statistics, 25, 290, 10.1214/aoms/1177728786

Bradley JV, 1968, Distribution-free statistical tests

Bradley JV, 1977, American Statistician, 31, 147, 10.1080/00031305.1977.10479224

Bradley JV, 1978, British Journal of Mathematical and Statistical Psychology, 31, 144, 10.1111/j.2044-8317.1978.tb00581.x

Bradley JV, 1979, Journal of Quality Technology, 11, 177, 10.1080/00224065.1979.11980909

Bradley JV, 1980, Bulletin of the Psychonomic Society, 15, 275, 10.3758/BF03334531

Bradley JV, 1980, Bulletin of the Psychonomic Society, 15, 29, 10.3758/BF03329752

Bradley JV, 1980, Bulletin of the Psychonomic Society, 16, 333, 10.3758/BF03329558

Bradley JV, 1982, Bulletin of the Psychometrics Society, 20, 85, 10.3758/BF03330089

Brown MB, 1974, Technometrics, 16, 129, 10.1080/00401706.1974.10489158

Brunner E, 1984, Metrika, 31, 233, 10.1007/BF01915206

Capon JA, 1988, Elementary statistics for the social sciences

ChaseCElementary statistical procedures19762nd edNew YorkMcGraw-Hill

Chernoff H, 1958, Annals of Mathematical Statistics, 29, 972, 10.1214/aoms/1177706436

Christensen HB, 1977, Statistics step by step

Clayton KN, 1984, An introduction to statistics for psychology and education

Cochran WG, 1947, Biometrics, 3, 27, 10.2307/3001535

Cochran WG, 1950, Experimental designs

Conover WJ, 1971, Practical nonparametric statistics

ConoverWJPractical nonparametric statistics19802nd edNew YorkJohn Wiley

Conover WJ, 1976, Communications in Statistics, A5, 1349, 10.1080/03610927608827447

Conover WJ, 1980, Communications in Statistics, 9, 465, 10.1080/03610928008827895

10.1080/00031305.1981.10479327

Conover WJ, 1982, Biometrics, 38, 715, 10.2307/2530051

Couch JV, 1982, Fundamentals of statistics for the behavioral sciences

CouchJVFundamentals of statistics for the behavioral sciences19872nd edSt. Paul, MNSt. Martin’s Press

Dinham SM, 1976, Exploring statistics

Dixon WJ, 1954, Annals of Mathematical Statistics, 25, 610, 10.1214/aoms/1177728732

DixonWJMasseyFJIntroduction to statistical analysis19693rd edNew YorkMcGraw-Hill

DownieNMHeathRBasic statistical methods19703rd edNew YorkHarper and Row

Downie NM, 1977, Descriptive and inferential statistics

Edgington ES, 1980, Randomization tests

Englehart M, 1972, Methods of educational research

Fallik F, 1983, Statistics for behavioral sciences

Fawcett RF, 1984, Communications in Statistics, BI3, 213, 10.1080/03610918408812368

10.1177/001316447403400406

Fienberg SE, 1985, The analysis of cross-classified categorical data, 2

Fisher RA, 1922, Philosophical Transactions of the Royal Society of London, 309

Fisher RA, 1949, Statistical tables for biological, agricultural, and medical research, 3

Fligner MA, 1981, Comment American Statistician, 35, 131

Freeman DM, 1987, Applied categorical data analysis

Freund JE, 1970, Statistics: A first course

Friedman M, 1937, Journal of the American Statistical Association, 32, 675, 10.1080/01621459.1937.10503522

10.2466/pr0.1959.5.g.115

10.3102/00346543045001043

GarrettHStatistical methods in psychology and education19666th edNew YorkDavid McKay

Gatta LA, 1973, Journal of Research in Science Teaching, 10, 3, 10.1002/tea.3660100103

Gay LR, 1976, Educational research: Competencies for analysis and application

GayLREducational research: Competencies for analysis and applications19883rd edColumbus, OHCharles E. Merrill

Gibbons JD, 1971, Nonparametric statistical inference

Gibbons JD, 1985, Nonparametric methods for qualitative analysis

GibbonsJDNonparametric statistical inference1985b2nd edNew YorkMarcel Dekker

10.3102/00346543042003237

Goodard RH, 1940, Psychometrica, 5, 263, 10.1007/BF02287972

Gravetter FJ, 1985, Statistics for the behavioral sciences

Groggel DJ, 1987, Communications in Statistics, 16, 601, 10.1080/03610918708812607

GuilfordJPFruchterBFundamental statistics in psychology and education19786th edNew YorkMcGraw-Hill

Haber A, 1969, General statistics

Habib AR, 1989, Communications in Statistics, 18, 793, 10.1080/03610918908812791

Hack HRB, 1958, Biometrika, 45, 260, 10.1093/biomet/45.1-2.260

Hajek J, 1968, Annals of Mathematical Statistics, 39, 325, 10.1214/aoms/1177698394

Hajek J, 1967, Theory of rank tests

Harwell MR, 1988, Journal of Counseling and Development, 67, 35, 10.1002/j.1556-6676.1988.tb02007.x

Harwell MR, 1990, Journal of Experimental Education, 58, 143, 10.1080/00220973.1990.10806530

Harwell MR, British Journal of Mathematical and Statistical Psychology

Harwell MR, 1988, Psychological Bulletin, 104, 268, 10.1037/0033-2909.104.2.268

10.3102/10769986014004351

Harwell MR, 1989, Paper presented at the annual meeting of the American Educational Research Association

10.2466/pr0.1974.34.3c.1095

Hays WL, 1963, Statistics

Hays WL, 1988, Statistics

Hedgeman V, 1976, Technometrics, 18, 273, 10.1080/00401706.1976.10489447

Hemelrijk J, 1961, Quantitative methods in psychology

Hettmansperger TP, 1984, Statistical inference based on ranks

Hettmansperger TP, 1983, Journal of the American Statistical Association, 78, 885, 10.1080/01621459.1983.10477036

Hildebrand DK, 1986, Statistical thinking for behavioral scientists

HinkleDEWiersmaWJursSGApplied statistics for the behavioral sciences19882nd edBostonHoughton Mifflin

Hodges JC, 1956, Annals of Mathematical Statistics, 27, 324, 10.1214/aoms/1177728261

Hodges JC, 1962, Annals of Mathematical Statistics, 33, 482, 10.1214/aoms/1177704575

Hoeffding W, 1951, Proceedings of the second Berkeley symposium on mathematics, statistics, and probability, 83, 10.1525/9780520411586-008

Hoeffding W, 1952, Annals of Mathematical Statistics, 23, 169, 10.1214/aoms/1177729436

Hollander M, 1973, Nonparametric methods

Hora SC, 1984, Journal of the American Statistical Association, 79, 668

Hora SC, 1988, Journal of the American Statistical Association, 83, 462, 10.1080/01621459.1988.10478618

Hornsnell G, 1953, Biometrika, 40, 128, 10.1093/biomet/40.1-2.128

Howell DC, 1989, Fundamental statistics for the behavioral sciences

Huber PS, 1981, Robust statistics, 10.1002/0471725250

Iman RL, 1974, Canadian Journal of Statistics, 2, 227, 10.2307/3314695

Iman RL, 1976, A comparison of several rank tests for the two-way layout

Iman RL, 1979, Technometrics, 21, 499, 10.1080/00401706.1979.10489820

Iman RL, 1984, Journal of the American Statistical Association, 79, 674, 10.1080/01621459.1984.10478096

International Mathematical and Statistical LibrariesIMSL library, reference manual198710th edHouston, TXAuthor

Ito PK, 1980, Handbook of statistics, 1, 199

10.2466/pms.1984.59.1.31

10.2466/pms.1984.58.3.979

Johnson M, 1977, A review of research methods in education

Johnson RR, 1973, Elementary statistics

KachiganSKStatistical analysis: An interdisciplinary introduction to univariate and multivariate methods19862nd edNew YorkRadius Press

Kennedy JJ, 1983, Analyzing qualitative data: Introducing log-linear analysis for behavioral research

Kepner JL, 1988, Journal of the American Statistical Association, 83, 456, 10.1080/01621459.1988.10478617

Kerlinger FN, 1964, Foundations of behavioral research

KerlingerFNFoundations of behavioral research19732nd edNew YorkHolt, Rinehart, and Winston

Keselman HJ, 1977, British Journal of Mathematical and Statistical Psychology, 30, 213, 10.1111/j.2044-8317.1977.tb00742.x

Kirk RE, 1968, Experimental design: Procedures for the behavioral sciences

Kirk RE, 1972, Statistical issues: A reader for the behavioral sciences

Kleijnen PC, 1987, Statistical tools for simulation practitioners

Klugh HE, 1970, Statistics: The essentials for research

KlughHEStatistics: The essentials for research19742nd edNew YorkJohn Wiley

Kossack CF, 1975, Introduction to statistics and computer programming

Krauth J, 1988, Techniques in behavioral and neural sciences, 2

Kreyszig E, 1970, Introductory mathematical statistics: Principles and methods

Kruskal WH, 1952, Annals of Mathematical Statistics, 23, 525, 10.1214/aoms/1177729332

Kruskal WH, 1952, Journal of the American Statistical Association, 47, 583, 10.1080/01621459.1952.10483441

Kurtz A, 1979, Statistical methods in education and psychology, 10.1007/978-1-4612-6129-2

Kurtz A, 1983, Introduction to social statistics

Lapin L, 1975, Statistics: Meaning and method

Lapin L, 1980, Statistics: Meaning and method, 2

LeedyPDPractical research: Planning and design19894th edNew YorkMacmillan

Lehmann EL, 1975, Nonparametrics

Lehmann EL, 1975, Nonparametrics: Statistical methods based on ranks

Lehmann EL, 1959, Testing statistical hypotheses

Lemmer HH, 1980, Communications in Statistics, 9, 1427, 10.1080/03610928008827972

Li JCR, 1964, Statistical inference I

Lindquist EF, 1953, Design and analysis of experiments in psychology and education

Lu HT, 1979, Journal of the American Statistical Association, 74, 715, 10.1080/01621459.1979.10481676

Lunney GH, 1970, Journal of Educational Measurement, 7, 263, 10.1111/j.1745-3984.1970.tb00727.x

Lutz GM, 1983, Understanding social statistics

Lynch MD, 1976, Elements of statistical inference for education and psychology

10.3102/00028312009002311

Mann HB, 1947, American Mathematical Society, 18, 50

Mansfield E, 1986, Basic statistics with applications

Marascuilo LA, 1977, Nonparametric and distribution-free methods for the social sciences

Marascuilo LA, 1988, Statistical methods for the social and behavioral sciences

MasonEJBrambleWJUnderstanding and conducting research: Applications in education and the behavioral sciences19892nd edNew YorkMcGraw-Hill

McCallRBFundamental statistics for psychology19752nd edNew YorkHarcourt Brace Jovanovich

McCallRBFundamental statistics for psychology19803rd edNew YorkHarcourt Brace Jovanovich

McCallRBFundamental statistics for psychology19864th edNew YorkHarcourt Brace Jovanovich

McMillanJHSchumacherSResearch in education: A conceptual introduction19892nd edGlenview, ILScott, Foresman

McNemarQPsychological statistics19623rd edNew YorkJohn Wiley

McSweeney M, 1967, An empirical study of two proposed nonparametric tests for main effects and interaction

McSweeney M, 1969, British Journal of Mathematical and Statistical Psychology, 22, 177, 10.1111/j.2044-8317.1969.tb00429.x

Meddis R, 1984, Statistics using ranks: A unified approach

MendenhallWIntroduction to probability and statistics19682nd edBelmonte, CAWadsworth

MendenhallWIntroduction to probability and statistics19713rd edBelmonte, CAWadsworth

Mendenhall W, 1973, Mathematical statistics with applications

Micceri T, 1986, Paper presented at the annual meeting of the Florida Educational Research Association

Micceri T, 1989, Psychological Bulletin, 105, 156, 10.1037/0033-2909.105.1.156

Michael WB, 1963, Review of Educational Research, 33, 474

Miller I, 1965, Probability and statistics for engineers

Mills RL, 1977, Statistics for applied economics and business

Minium EW, 1970, Statistical reasoning in psychology and education

Neave HR, 1968, Technometrics, 10, 509, 10.1080/00401706.1968.10490598

Noether GE, 1955, Annals of Mathematical Statistics, 26, 64, 10.1214/aoms/1177728593

Noether GE, 1967, Elements of nonparametric statistics

Noether GE, 1981, American Statistician, 35, 129

Noether GE, 1984, American Statistician, 38, 173

Nunnally J, 1975, Introduction to statistics for psychology and education

NunnallyJPsychometric theory19782nd edNew YorkMcGraw-Hill

10.3102/10769986009002129

Olson CL, 1987, Essentials of statistics: Making sense of data

Pagano RR, 1981, Understanding statistics in the behavioral sciences

Page EB, 1966, Review of Educational Research, 36, 517

Palumbo DJ, 1969, Statistics in political and behavioral science

PalumboDJStatistics in political and behavioral science19772nd edNew YorkColumbia University Press

Parket IR, 1974, Statistics for business decision making

Patel KM, 1973, Journal of the American Statistical Association, 68, 615, 10.1080/01621459.1973.10481394

Pearson ES, 1929, Biometrika, 23, 259, 10.1093/biomet/21.1-4.259

Penfield DA, 1985, Paper presented at the annual meeting of the American Educational Research Association

Pitman EJG, 1948, Lecture notes on non-parametric statistics

Puri ML, 1964, Annals of Mathematical Statistics, 35, 102, 10.1214/aoms/1177703733

Puri ML, 1969, Annals of Mathematical Statistics, 40, 1325, 10.1214/aoms/1177697505

Puri ML, 1971, Nonparametric methods in multivariate analysis

Puri ML, 1973, Annals of Statistics, 1, 553

Puri ML, 1985, Nonparametric methods in general linear models

Quade D, 1979, Journal of the American Statistical Association, 74, 680, 10.1080/01621459.1979.10481670

Randles RH, 1979, Introduction to the theory of nonparametric tests

Randolph EA, 1989, Paper presented at the 11th annual meeting of the Mid-Western Educational Research Association

Rao CR, 1951, Bulletin of the International Statistics Institute, 33, 177

10.1177/0193841X8500900407

Rasmussen JL, 1986, British Journal of Mathematical and Statistical Psychology, 39, 213, 10.1111/j.2044-8317.1986.tb00858.x

Reinach SG, 1965, South African Journal of Agricultural Science, 8, 941

Reinach SG, 1966, Distribution-free methods in experimental design

Rider PR, 1929, Biometrika, 21, 124, 10.1093/biomet/21.1-4.124

Rineman WC, 1983, Journal of the American Statistical Association, 78, 655, 10.1080/01621459.1983.10478025

10.3102/00028312014004493

Roscoe JT, 1969, Fundamental research statistics for the behavioral sciences

RoscoeJTFundamental research statistics for the behavioral sciences19752nd edNew YorkHolt, Rinehart, and Winston

Runyon RP, 1968, Fundamentals of behavioral statistics

Runyon RP, 1971, Fundamentals of behavioral statistics, 2

RunyonRPHaberAFundamentals of behavioral statistics19763rd edReading, MAAddison-Wesley

RunyonRPHaberAFundamentals of behavioral statistics19804th edReading, MAWesley

RunyonRPHaberAFundamentals of behavioral statistics19845th edNew YorkRandom House

RunyonRPHaberAFundamentals of behavioral statistics19886th edNew YorkRandom House

SAS Institute, 1985, SAS user’s guide: Statistics, 5

SAS Institute, 1987, SAS/stat guide for personal computers, 6

Sawilowsky SS, 1985, Florida Journal of Educational Research, 27, 83

Sawilowsky SS, 1985, Robust and power analysis of the 2 × 2 × 2 ANOVA, rank transformation, random normal scores, and expected normal scores transformation tests

Sawilowsky SS, 1989, Paper presented at the annual meeting of the American Educational Research Association

SawilowskySSBlairRCA more realistic look at the robustness of the independent and dependent samples t tests to departures from population normality1990Manuscript submitted for publication

10.3102/10769986014003255

Scheffe H, 1959, The analysis of variance

Scheirer GJ, 1976, Biometrics, 32, 429, 10.2307/2529511

Schrader RM, 1980, Biometrika, 67, 93, 10.1093/biomet/67.1.93

Sen PK, 1980, Biometrika, 69, 245

Sen PK, 1977, Zeitschriftfuer Wahrscheinlich-Keitstheorie und Verwandte Gebiete, 39, 175, 10.1007/BF00535470

Shoemaker LH, 1986, Communications in Statistics, 15, 609, 10.1080/03610918608812528

Siegel S, 1956, Nonparametric statistics for the behavioral sciences

SiegelSCastellanNJJrNonparametric statistics for the behavioral sciences19882nd edNew YorkMcGraw-Hill

Smitley WDS, 1981, A comparison of the power of the two independent means t test and the Mann-Whitney U test

SnedecorGWCochranWGStatistical methods19807th edAmes, IAIowa University Press

Sowell EJ, 1982, Analyzing educational research

Spatz C, 1989, Basic statistics: Tales of distributions

SpenceJTUnderwoodBJDuncanCPCottonJWElementary statistics19682nd edNew YorkAppleton-Century-Crofts

Sprent P, 1989, Applied nonparametric statistical methods

Sprinthall RC, 1982, Basic statistical analysis

Srisukho D, 1974, Monte Carlo study of the power of H-test compared to F-test when population distributions are different inform

Stephenson WR, 1988, Communications in Statistics, 17, 451, 10.1080/03610918808812673

Stevens SS, 1946, Science, 103, 677, 10.1126/science.103.2684.677

Still AW, 1981, British Journal of Mathematical and Statistical Psychology, 34, 243, 10.1111/j.2044-8317.1981.tb00634.x

Stuart A, 1956, Bulletin of the International Statistics Institute, 36, 79

SummersGWPetersWSArmstrongCPBasic statistics in business and economics19772nd edBelmont, CAWadsworth

Tan WY, 1982, Communications in Statistics, 11, 2485

Tate MW, 1975, Nonparametric and shortcut statistics: In the social, biological, and medical sciences

Terry ME, 1952, Annals of Mathematical Statistics, 23, 346, 10.1214/aoms/1177729381

Thompson GL, 1989, Journal of the American Statistical Association, 84, 325, 10.1080/01621459.1989.10478773

Thompson GL, Journal of the American Statistical Association

Tomarkin AJ, 1986, Psychological Bulletin, 99, 90, 10.1037/0033-2909.99.1.90

10.3102/10769986005002169

Van Elteren P, 1959, Biometrika, 46, 465, 10.1093/biomet/46.3-4.475

10.3102/00346543054001087

Wampold BE, 1990, Theory and application of statistics

Welch BL, 1937, Biometrika, 29, 350, 10.1093/biomet/29.3-4.350

Wilcox RR, 1987, New statistical procedures for the social sciences

Wilcox RR, 1986, Communications in Statistics—Simulations and Computation, 15, 933, 10.1080/03610918608812553

Wilcoxon F, 1945, Biometrics, 1, 80, 10.2307/3001968

Wilcoxon F, 1947, Biometrics, 3, 119, 10.2307/3001946

Wilcoxon F, 1949, Some rapid approximate statistical procedures

WilliamsFReasoning with statistics19792nd edNew YorkHolt, Rinehart, and Winston

Wilson KV, 1956, Psychological Bulletin, 53, 96, 10.1037/h0047975

Winer BJ, 1962, Statistical principles in experimental design, 10.1037/11774-000

WinerBJStatistical principles in experimental design19712nd edNew YorkMcGraw-Hill

Winn PR, 1978, Business statistics

Wolfowitz J, 1949, Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 93

WonnacottTHIntroductory statistics for business and economics19772nd edSanta Barbara, CAJohn Wiley

Worsley KJ, 1977, Biometrics, 33, 532, 10.2307/2529369

Wright RLD, 1976, Understanding statistics: An informal introduction for the behavioral sciences