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result in incorrectly rejecting the null hypothesis that sphericity holds. Collier and colleagues showed this was true when epsilon was extended to as high as .90. The HuynhâFeldt correction, however, is believed to be too liberal and overestimates sphericity. This would result in incorrectly rejecting the alternative hypothesis that sphericity does not hold, when it does. Girden recommended a solution to this problem: when epsilon is > .75, the HuynhâFeldt correction should be applied and when epsilon is < .75 or nothing is known about sphericity, the
GreenhouseâGeisser correction should be applied.
237:
represented on the right-hand side. After obtaining the difference scores for all possible pairs of groups, the variances of each group difference can be contrasted. From the example in Figure 1, the variance of the differences between
Treatment A and B (17) appear to be much greater than the variance of the differences between Treatment A and C (10.3) and between Treatment B and C (10.3). This suggests that the data may violate the assumption of sphericity. To determine whether statistically significant differences exist between the variances of the differences, Mauchly's test of sphericity can be performed.
568:) and can be found on Mauchly's test output in SPSS. Epsilon provides a measure of departure from sphericity. By evaluating epsilon, we can determine the degree to which sphericity has been violated. If the variances of differences between all possible pairs of groups are equal and sphericity is exactly met, then epsilon will be exactly 1, indicating no departure from sphericity. If the variances of differences between all possible pairs of groups are unequal and sphericity is violated, epsilon will be below 1. The further epsilon is from 1, the worse the violation.
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584: + 10 (i.e., the number of levels of the repeated measures factor + 10), then a MANOVA is more powerful; in other cases, repeated measures design should be selected. Additionally, the power of MANOVA is contingent upon the correlations between the dependent variables, so the relationship between the different conditions must also be considered.
557:(1959), the HuynhâFeldt correction (1976), and the lower-bound. Each of these corrections have been developed to alter the degrees of freedom and produce an F-ratio where the Type I error rate is reduced. The actual F-ratio does not change as a result of applying the corrections; only the degrees of freedom.
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Of the three corrections, Huynh-Feldt is considered the least conservative, while
GreenhouseâGeisser is considered more conservative and the lower-bound correction is the most conservative. When epsilon is > .75, the GreenhouseâGeisser correction is believed to be too conservative, and would
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To further illustrate the concept of sphericity, consider a matrix representing data from patients who receive three different types of drug treatments in Figure 1. Their outcomes are represented on the left-hand side of the matrix, while differences between the outcomes for each treatment are
596:
While
Mauchly's test is one of the most commonly used to evaluate sphericity, the test fails to detect departures from sphericity in small samples and over-detects departures from sphericity in large samples. Consequently, the sample size has an influence on the interpretation of the results. In
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since they do not require the assumption of sphericity. However, this procedure can be less powerful than using a repeated measures ANOVA, especially when sphericity violation is not large or sample sizes are small. OâBrien and Kaiser suggested that when you have a large violation of sphericity
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249:, Mauchly's test of sphericity is a popular test to evaluate whether the sphericity assumption has been violated. The null hypothesis of sphericity and alternative hypothesis of non-sphericity in the above example can be mathematically written in terms of difference scores.
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When sphericity has been established, the F-ratio is valid and therefore interpretable. However, if
Mauchly's test is significant then the F-ratios produced must be interpreted with caution as the violations of this assumption can result in an increase in the
514:), sphericity cannot be assumed and we would therefore conclude that there are significant differences between the variances of the differences. Sphericity is always met for two levels of a repeated measure factor and is, therefore, unnecessary to evaluate.
61:. Sphericity can be evaluated when there are three or more levels of a repeated measure factor and, with each additional repeated measures factor, the risk for violating sphericity increases. If sphericity is violated, a decision must be made as to whether a
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commonly being set to .05), we fail to reject the null hypothesis that the variances are equal. Therefore, we could conclude that the assumption has not been violated. However, when the probability of
Mauchly's test statistic is less than or equal to
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SPSS provides an F-ratio from four different methods: Pillai's trace, Wilksâ lambda, Hotelling's trace, and Roy's largest root. In general, Wilksâ lambda has been recommended as the most appropriate multivariate test statistic to use.
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57:). If sphericity is violated (i.e., if the variances of the differences between all combinations of the conditions are not equal), then the variance calculations may be distorted, which would result in an inflated
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Collier, R. O., Jr., Baker, F. B., Mandeville, G. K., & Hayes, T. F. (1967). "Estimates of test size for several test procedures based on conventional variance ratios in the repeated measures design".
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analysis is selected. If a univariate method is selected, the repeated-measures ANOVA must be appropriately corrected depending on the degree to which sphericity has been violated.
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practice, the assumption of sphericity is extremely unlikely to be exactly met so it is prudent to correct for a possible violation without actually testing for a violation.
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Huynh, H., & Feldt, L. S. (1976). "Estimation of the Box correction for degrees of freedom from sample data in randomised block and split-plot designs."
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rate, and influence the conclusions drawn from your analysis. In instances where
Mauchly's test is significant, modifications need to be made to the
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356:{\displaystyle H_{0}:\sigma _{{\text{Tx A}}-{\text{Tx B}}}^{2}=\sigma _{{\text{Tx A}}-{\text{Tx C}}}^{2}=\sigma _{{\text{Tx B}}-{\text{Tx C}}}^{2}}
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Statistical software should not provide output for a test of sphericity for two levels of a repeated measure factor; however, some versions of
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Interpreting
Mauchly's test is fairly straightforward. When the probability of Mauchly's test statistic is greater than or equal to
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O'Brien, R. G. & Kaiser, M. K. (1985). "The MANOVA approach for analyzing repeated measures designs: An extensive primer".
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Sphericity is an important assumption of a repeated-measures ANOVA. It is the condition of equal
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Greenhouse, S. W., & Geisser, S. (1959). "On methods in the analysis of profile data."
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produce an output table with degrees of freedom equal to 0, and a period in place of a numeric
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Mauchly, J. W. (1940). "Significance test for sphericity of a normal
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Designing experiments and analyzing data: A model comparison perspective
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395:{\displaystyle H_{1}:{\text{The variances are not all equal}}.}
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Hinton, P. R., Brownlow, C., & McMurray, I. (2004).
734:"Sphericity in Repeated Measures Analysis of Variance"
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The test statistic for these estimates is denoted by
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793:: CS1 maint: multiple names: authors list (
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575:Another alternative procedure is using the
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674:The Annals of Mathematical Statistics
577:multivariate test statistics (MANOVA)
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2943:Accelerated failure time (AFT) model
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2538:Analysis of variance (ANOVA, anova)
2633:CochranâMantelâHaenszel statistics
1259:Pearson product-moment correlation
840:Statistical Methods for Psychology
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930:Journal of Educational Statistics
651:Discovering Statistics Using SPSS
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2892:Least-squares spectral analysis
386:The variances are not all equal
1873:Mean-unbiased minimum-variance
37:. It was developed in 1940 by
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3186:Geographic information system
2402:Simultaneous equations models
555:GreenhouseâGeisser correction
2369:Coefficient of determination
1980:Uniformly most powerful test
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2938:Proportional hazards models
2882:Spectral density estimation
2864:Vector autoregression (VAR)
2298:Maximum posterior estimator
1530:Randomized controlled trial
914:Girden, E. R. (1992).
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2698:Multivariate distributions
1118:Average absolute deviation
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918:. Newbury Park, CA: Sage.
827:. Newbury Park, CA: Sage.
73:Measurement of sphericity
20:Mauchly's sphericity test
3181:Environmental statistics
2703:Elliptical distributions
2496:Generalized linear model
2425:Simple linear regression
2195:HodgesâLehmann estimator
1652:Probability distribution
1561:Stochastic approximation
1123:Coefficient of variation
939:-variate distribution."
916:ANOVA: repeated measures
825:ANOVA: Repeated measures
529:Violations of sphericity
2841:Cross-correlation (XCF)
2449:Non-standard predictors
1883:LehmannâScheffĂ© theorem
1556:Adaptive clinical trial
842:. Wadsworth Publishing.
687:10.1214/aoms/1177731915
664:Mauchly, J. W. (1940).
537:Violation of sphericity
507:{\displaystyle \alpha }
483:{\displaystyle \alpha }
462:{\displaystyle \alpha }
442:{\displaystyle \alpha }
418:{\displaystyle \alpha }
3237:Mathematics portal
3058:Engineering statistics
2966:NelsonâAalen estimator
2543:Analysis of covariance
2430:Ordinary least squares
2354:Pearson product-moment
1758:Statistical functional
1669:Empirical distribution
1502:Controlled experiments
1231:Frequency distribution
1009:Descriptive statistics
886:Psychological Bulletin
838:Howell, D. C. (2009).
670:-Variate Distribution"
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3095:System identification
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2757:Exponential smoothing
2671:Discriminant analysis
2666:Canonical correlation
2530:Partition of variance
2392:Regression validation
2236:(JonckheereâTerpstra)
2135:Likelihood-ratio test
1824:Frequentist inference
1736:Locationâscale family
1657:Sampling distribution
1622:Statistical inference
1589:Cross-sectional study
1576:Observational studies
1535:Randomized experiment
1364:Stem-and-leaf display
1166:Central limit theorem
812:. Belmont: Wadsworth.
649:Field, A. P. (2005).
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245:Developed in 1940 by
3283:Analysis of variance
3076:Probabilistic design
2661:Principal components
2504:Exponential families
2456:Nonlinear regression
2435:General linear model
2397:Mixed effects models
2387:Errors and residuals
2364:Confounding variable
2266:Bayesian probability
2244:Van der Waerden test
2234:Ordered alternative
1999:Multiple comparisons
1878:RaoâBlackwellization
1841:Estimating equations
1797:Statistical distance
1515:Factorial experiment
1048:Arithmetic-Geometric
653:. Sage Publications.
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55:independent variable
3148:Official statistics
3071:Methods engineering
2752:Seasonal adjustment
2520:Poisson regressions
2440:Bayesian regression
2379:Regression analysis
2359:Partial correlation
2331:Regression analysis
1930:Prediction interval
1925:Likelihood interval
1915:Confidence interval
1907:Interval estimation
1868:Unbiased estimators
1686:Model specification
1566:Up-and-down designs
1254:Partial correlation
1210:Index of dispersion
1128:Interquartile range
823:Girden, E. (1992).
719:. Laerd Statistics.
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33:used to validate a
3168:Spatial statistics
3048:Medical statistics
2948:First hitting time
2902:Whittle likelihood
2553:Degrees of freedom
2548:Multivariate ANOVA
2481:Heteroscedasticity
2293:Bayesian estimator
2258:Bayesian inference
2107:KolmogorovâSmirnov
1992:Randomization test
1962:Testing hypotheses
1935:Tolerance interval
1846:Maximum likelihood
1741:Exponential family
1674:Density estimation
1634:Statistical theory
1594:Natural experiment
1540:Scientific control
1457:Survey methodology
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2009:Parametric tests
1987:Permutation test
1970:1- & 2-tails
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859:. Archived from
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254:
253:
247:John W. Mauchly
243:
75:
47:
17:
12:
11:
5:
3301:
3299:
3291:
3290:
3285:
3275:
3274:
3268:
3267:
3265:
3264:
3252:
3240:
3226:
3213:
3210:
3209:
3206:
3205:
3202:
3201:
3199:
3198:
3193:
3188:
3183:
3178:
3172:
3170:
3164:
3163:
3161:
3160:
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3109:
3107:
3101:
3100:
3098:
3097:
3092:
3087:
3078:
3073:
3068:
3062:
3060:
3054:
3053:
3051:
3050:
3045:
3040:
3031:
3029:Bioinformatics
3025:
3023:
3013:
3012:
3007:
3000:
2999:
2996:
2995:
2992:
2991:
2988:
2987:
2985:
2984:
2978:
2976:
2972:
2971:
2969:
2968:
2962:
2960:
2954:
2953:
2951:
2950:
2945:
2940:
2935:
2929:
2927:
2918:
2912:
2911:
2908:
2907:
2905:
2904:
2899:
2894:
2889:
2884:
2878:
2876:
2870:
2869:
2867:
2866:
2861:
2856:
2848:
2843:
2838:
2837:
2836:
2834:partial (PACF)
2825:
2823:
2817:
2816:
2814:
2813:
2808:
2803:
2795:
2790:
2784:
2782:
2781:Specific tests
2778:
2777:
2775:
2774:
2769:
2764:
2759:
2754:
2749:
2744:
2739:
2733:
2731:
2724:
2718:
2717:
2715:
2714:
2713:
2712:
2711:
2710:
2695:
2694:
2693:
2683:
2681:Classification
2678:
2673:
2668:
2663:
2658:
2653:
2647:
2645:
2639:
2638:
2636:
2635:
2630:
2628:McNemar's test
2625:
2620:
2615:
2610:
2604:
2602:
2592:
2591:
2574:
2567:
2566:
2563:
2562:
2559:
2558:
2556:
2555:
2550:
2545:
2540:
2534:
2532:
2526:
2525:
2523:
2522:
2506:
2500:
2498:
2492:
2491:
2489:
2488:
2483:
2478:
2473:
2468:
2466:Semiparametric
2463:
2458:
2452:
2450:
2446:
2445:
2443:
2442:
2437:
2432:
2427:
2421:
2419:
2413:
2412:
2410:
2409:
2404:
2399:
2394:
2389:
2383:
2381:
2375:
2374:
2372:
2371:
2366:
2361:
2356:
2350:
2348:
2338:
2337:
2334:
2333:
2328:
2322:
2321:
2314:
2313:
2310:
2309:
2306:
2305:
2303:
2302:
2301:
2300:
2290:
2285:
2280:
2279:
2278:
2273:
2262:
2260:
2254:
2253:
2250:
2249:
2247:
2246:
2241:
2240:
2239:
2231:
2223:
2207:
2204:(MannâWhitney)
2199:
2198:
2197:
2184:
2183:
2182:
2171:
2169:
2163:
2162:
2160:
2159:
2158:
2157:
2152:
2147:
2137:
2132:
2129:(ShapiroâWilk)
2124:
2119:
2114:
2109:
2104:
2096:
2090:
2088:
2082:
2081:
2079:
2078:
2070:
2061:
2049:
2043:
2041:Specific tests
2037:
2036:
2033:
2032:
2030:
2029:
2024:
2019:
2013:
2011:
2005:
2004:
2002:
2001:
1996:
1995:
1994:
1984:
1983:
1982:
1972:
1966:
1964:
1958:
1957:
1955:
1954:
1953:
1952:
1947:
1937:
1932:
1927:
1922:
1917:
1911:
1909:
1903:
1902:
1900:
1899:
1894:
1893:
1892:
1887:
1886:
1885:
1880:
1865:
1864:
1863:
1858:
1853:
1848:
1837:
1835:
1826:
1820:
1819:
1817:
1816:
1811:
1806:
1805:
1804:
1794:
1789:
1788:
1787:
1777:
1776:
1775:
1770:
1765:
1755:
1750:
1745:
1744:
1743:
1738:
1733:
1717:
1716:
1715:
1710:
1705:
1695:
1694:
1693:
1688:
1678:
1677:
1676:
1666:
1665:
1664:
1654:
1649:
1644:
1638:
1636:
1626:
1625:
1620:
1613:
1612:
1609:
1608:
1605:
1604:
1602:
1601:
1596:
1591:
1586:
1580:
1578:
1572:
1571:
1569:
1568:
1563:
1558:
1552:
1550:
1546:
1545:
1543:
1542:
1537:
1532:
1527:
1522:
1517:
1512:
1506:
1504:
1498:
1497:
1495:
1494:
1492:Standard error
1489:
1484:
1479:
1478:
1477:
1472:
1461:
1459:
1453:
1452:
1450:
1449:
1444:
1439:
1434:
1429:
1424:
1422:Optimal design
1419:
1414:
1408:
1406:
1396:
1395:
1390:
1383:
1382:
1379:
1378:
1375:
1374:
1372:
1371:
1366:
1361:
1356:
1351:
1346:
1341:
1336:
1331:
1326:
1321:
1316:
1311:
1306:
1301:
1295:
1293:
1287:
1286:
1284:
1283:
1278:
1277:
1276:
1271:
1261:
1256:
1250:
1248:
1242:
1241:
1239:
1238:
1233:
1228:
1222:
1220:
1219:Summary tables
1216:
1215:
1213:
1212:
1206:
1204:
1198:
1197:
1194:
1193:
1191:
1190:
1189:
1188:
1183:
1178:
1168:
1162:
1160:
1154:
1153:
1151:
1150:
1145:
1140:
1135:
1130:
1125:
1120:
1114:
1112:
1106:
1105:
1103:
1102:
1097:
1092:
1091:
1090:
1085:
1080:
1075:
1070:
1065:
1060:
1055:
1053:Contraharmonic
1050:
1045:
1034:
1032:
1023:
1013:
1012:
1007:
1000:
999:
997:
996:
991:
985:
982:
981:
976:
974:
973:
966:
959:
951:
945:
944:
943:, 11, 204â209.
933:
926:
919:
910:
907:
904:
903:
876:
854:"Mauchly Test"
845:
830:
815:
800:
759:(3): 339â353.
742:
722:
701:
680:(2): 204â209.
656:
638:
619:SPSS Explained
605:
604:
602:
599:
593:
590:
530:
527:
503:
479:
458:
438:
414:
403:
402:
391:
383:
378:
374:
363:
350:
340:
331:
327:
322:
312:
303:
299:
294:
284:
275:
271:
266:
262:
242:
241:Interpretation
239:
232:
231:
228:
225:
222:
218:
217:
214:
211:
208:
205:
202:
199:
195:
194:
191:
188:
185:
182:
179:
176:
172:
171:
168:
165:
162:
159:
156:
153:
149:
148:
145:
142:
139:
136:
133:
130:
126:
125:
122:
119:
116:
113:
110:
107:
103:
102:
99:
96:
93:
90:
87:
84:
74:
71:
46:
43:
15:
13:
10:
9:
6:
4:
3:
2:
3300:
3289:
3286:
3284:
3281:
3280:
3278:
3263:
3262:
3253:
3251:
3250:
3241:
3239:
3238:
3233:
3227:
3225:
3224:
3215:
3214:
3211:
3197:
3194:
3192:
3191:Geostatistics
3189:
3187:
3184:
3182:
3179:
3177:
3174:
3173:
3171:
3169:
3165:
3159:
3158:Psychometrics
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3110:
3108:
3106:
3102:
3096:
3093:
3091:
3088:
3086:
3082:
3079:
3077:
3074:
3072:
3069:
3067:
3064:
3063:
3061:
3059:
3055:
3049:
3046:
3044:
3041:
3039:
3035:
3032:
3030:
3027:
3026:
3024:
3022:
3021:Biostatistics
3018:
3014:
3010:
3005:
3001:
2983:
2982:Log-rank test
2980:
2979:
2977:
2973:
2967:
2964:
2963:
2961:
2959:
2955:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2930:
2928:
2926:
2922:
2919:
2917:
2913:
2903:
2900:
2898:
2895:
2893:
2890:
2888:
2885:
2883:
2880:
2879:
2877:
2875:
2871:
2865:
2862:
2860:
2857:
2855:
2853:(BoxâJenkins)
2849:
2847:
2844:
2842:
2839:
2835:
2832:
2831:
2830:
2827:
2826:
2824:
2822:
2818:
2812:
2809:
2807:
2806:DurbinâWatson
2804:
2802:
2796:
2794:
2791:
2789:
2788:DickeyâFuller
2786:
2785:
2783:
2779:
2773:
2770:
2768:
2765:
2763:
2762:Cointegration
2760:
2758:
2755:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2737:Decomposition
2735:
2734:
2732:
2728:
2725:
2723:
2719:
2709:
2706:
2705:
2704:
2701:
2700:
2699:
2696:
2692:
2689:
2688:
2687:
2684:
2682:
2679:
2677:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2657:
2654:
2652:
2649:
2648:
2646:
2644:
2640:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2614:
2611:
2609:
2608:Cohen's kappa
2606:
2605:
2603:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2572:
2568:
2554:
2551:
2549:
2546:
2544:
2541:
2539:
2536:
2535:
2533:
2531:
2527:
2521:
2517:
2513:
2507:
2505:
2502:
2501:
2499:
2497:
2493:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2461:Nonparametric
2459:
2457:
2454:
2453:
2451:
2447:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2422:
2420:
2418:
2414:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2384:
2382:
2380:
2376:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2351:
2349:
2347:
2343:
2339:
2332:
2329:
2327:
2324:
2323:
2319:
2315:
2299:
2296:
2295:
2294:
2291:
2289:
2286:
2284:
2281:
2277:
2274:
2272:
2269:
2268:
2267:
2264:
2263:
2261:
2259:
2255:
2245:
2242:
2238:
2232:
2230:
2224:
2222:
2216:
2215:
2214:
2211:
2210:Nonparametric
2208:
2206:
2200:
2196:
2193:
2192:
2191:
2185:
2181:
2180:Sample median
2178:
2177:
2176:
2173:
2172:
2170:
2168:
2164:
2156:
2153:
2151:
2148:
2146:
2143:
2142:
2141:
2138:
2136:
2133:
2131:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2101:
2097:
2095:
2092:
2091:
2089:
2087:
2083:
2077:
2075:
2071:
2069:
2067:
2062:
2060:
2055:
2051:
2050:
2047:
2044:
2042:
2038:
2028:
2025:
2023:
2020:
2018:
2015:
2014:
2012:
2010:
2006:
2000:
1997:
1993:
1990:
1989:
1988:
1985:
1981:
1978:
1977:
1976:
1973:
1971:
1968:
1967:
1965:
1963:
1959:
1951:
1948:
1946:
1943:
1942:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1913:
1912:
1910:
1908:
1904:
1898:
1895:
1891:
1888:
1884:
1881:
1879:
1876:
1875:
1874:
1871:
1870:
1869:
1866:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1843:
1842:
1839:
1838:
1836:
1834:
1830:
1827:
1825:
1821:
1815:
1812:
1810:
1807:
1803:
1800:
1799:
1798:
1795:
1793:
1790:
1786:
1785:loss function
1783:
1782:
1781:
1778:
1774:
1771:
1769:
1766:
1764:
1761:
1760:
1759:
1756:
1754:
1751:
1749:
1746:
1742:
1739:
1737:
1734:
1732:
1726:
1723:
1722:
1721:
1718:
1714:
1711:
1709:
1706:
1704:
1701:
1700:
1699:
1696:
1692:
1689:
1687:
1684:
1683:
1682:
1679:
1675:
1672:
1671:
1670:
1667:
1663:
1660:
1659:
1658:
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1639:
1637:
1635:
1631:
1627:
1623:
1618:
1614:
1600:
1597:
1595:
1592:
1590:
1587:
1585:
1582:
1581:
1579:
1577:
1573:
1567:
1564:
1562:
1559:
1557:
1554:
1553:
1551:
1547:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1507:
1505:
1503:
1499:
1493:
1490:
1488:
1487:Questionnaire
1485:
1483:
1480:
1476:
1473:
1471:
1468:
1467:
1466:
1463:
1462:
1460:
1458:
1454:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1409:
1407:
1405:
1401:
1397:
1393:
1388:
1384:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1345:
1342:
1340:
1337:
1335:
1332:
1330:
1327:
1325:
1322:
1320:
1317:
1315:
1314:Control chart
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1296:
1294:
1292:
1288:
1282:
1279:
1275:
1272:
1270:
1267:
1266:
1265:
1262:
1260:
1257:
1255:
1252:
1251:
1249:
1247:
1243:
1237:
1234:
1232:
1229:
1227:
1224:
1223:
1221:
1217:
1211:
1208:
1207:
1205:
1203:
1199:
1187:
1184:
1182:
1179:
1177:
1174:
1173:
1172:
1169:
1167:
1164:
1163:
1161:
1159:
1155:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1115:
1113:
1111:
1107:
1101:
1098:
1096:
1093:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1069:
1066:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1040:
1039:
1036:
1035:
1033:
1031:
1027:
1024:
1022:
1018:
1014:
1010:
1005:
1001:
995:
992:
990:
987:
986:
983:
979:
972:
967:
965:
960:
958:
953:
952:
949:
942:
938:
934:
931:
927:
925:, 24, 95â112.
924:
923:Psychometrika
920:
917:
913:
912:
908:
899:
895:
891:
887:
880:
877:
866:on 2013-05-11
862:
855:
849:
846:
841:
834:
831:
826:
819:
816:
811:
804:
801:
796:
790:
782:
778:
774:
770:
766:
762:
758:
754:
753:Psychometrika
746:
743:
735:
729:
727:
723:
718:
712:
710:
708:
706:
702:
697:
693:
688:
683:
679:
675:
671:
669:
660:
657:
652:
645:
643:
639:
634:
628:
620:
613:
611:
607:
600:
598:
591:
589:
585:
583:
578:
573:
569:
567:
563:
558:
556:
551:
549:
545:
535:
528:
526:
524:
520:
515:
501:
493:
477:
456:
436:
428:
412:
389:
381:
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3133:Econometrics
3083: /
3066:Chemometrics
3043:Epidemiology
3036: /
3009:Applications
2851:ARIMA model
2798:Q-statistic
2747:Stationarity
2643:Multivariate
2586: /
2582: /
2580:Multivariate
2578: /
2518: /
2514: /
2288:Bayes factor
2187:Signed rank
2099:
2073:
2065:
2053:
1748:Completeness
1584:Cohort study
1482:Opinion poll
1417:Missing data
1404:Study design
1359:Scatter plot
1281:Scatter plot
1274:Spearman's Ï
1236:Grouped data
940:
936:
929:
922:
915:
889:
885:
879:
868:. Retrieved
861:the original
848:
839:
833:
824:
818:
809:
803:
789:cite journal
756:
752:
745:
717:"Sphericity"
677:
673:
667:
659:
650:
621:. Routledge.
618:
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586:
581:
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570:
565:
559:
552:
540:
522:
516:
491:
426:
404:
244:
235:
67:multivariate
48:
39:John Mauchly
25:
23:
19:
18:
3261:WikiProject
3176:Cartography
3138:Jurimetrics
3090:Reliability
2821:Time domain
2800:(LjungâBox)
2722:Time-series
2600:Categorical
2584:Time-series
2576:Categorical
2511:(Bernoulli)
2346:Correlation
2326:Correlation
2122:JarqueâBera
2094:Chi-squared
1856:M-estimator
1809:Asymptotics
1753:Sufficiency
1520:Interaction
1432:Replication
1412:Effect size
1369:Violin plot
1349:Radar chart
1329:Forest plot
1319:Correlogram
1269:Kendall's Ï
932:, 1, 69â82.
892:: 316â333.
3277:Categories
3128:Demography
2846:ARMA model
2651:Regression
2228:(Friedman)
2189:(Wilcoxon)
2127:Normality
2117:Lilliefors
2064:Student's
1940:Resampling
1814:Robustness
1802:divergence
1792:Efficiency
1730:(monotone)
1725:Likelihood
1642:Population
1475:Stratified
1427:Population
1246:Dependence
1202:Count data
1133:Percentile
1110:Dispersion
1043:Arithmetic
978:Statistics
870:2012-04-29
601:References
592:Criticisms
63:univariate
45:Sphericity
24:Mauchly's
2509:Logistic
2276:posterior
2202:Rank sum
1950:Jackknife
1945:Bootstrap
1763:Bootstrap
1698:Parameter
1647:Statistic
1442:Statistic
1354:Run chart
1339:Pie chart
1334:Histogram
1324:Fan chart
1299:Bar chart
1181:L-moments
1068:Geometric
627:cite book
502:α
478:α
457:α
437:α
413:α
339:−
330:σ
311:−
302:σ
283:−
274:σ
221:Variance:
78:Figure 1
51:variances
3223:Category
2916:Survival
2793:Johansen
2516:Binomial
2471:Isotonic
2058:(normal)
1703:location
1510:Blocking
1465:Sampling
1344:QâQ plot
1309:Box plot
1291:Graphics
1186:Skewness
1176:Kurtosis
1148:Variance
1078:Heronian
1073:Harmonic
781:42325937
210:−3
164:−5
3249:Commons
3196:Kriging
3081:Process
3038:studies
2897:Wavelet
2730:General
1897:Plug-in
1691:L space
1470:Cluster
1171:Moments
989:Outline
773:5234710
696:2235878
562:epsilon
525:value.
490:(i.e.,
449:, with
425:(i.e.,
83:Patient
59:F-ratio
3118:Census
2708:Normal
2656:Manova
2476:Robust
2226:2-way
2218:1-way
2056:-test
1727:
1304:Biplot
1095:Median
1088:Lehmer
1030:Center
779:
771:
694:
2742:Trend
2271:prior
2213:anova
2102:-test
2076:-test
2068:-test
1975:Power
1920:Pivot
1713:shape
1708:scale
1158:Shape
1138:Range
1083:Heinz
1058:Cubic
994:Index
864:(PDF)
857:(PDF)
777:S2CID
737:(PDF)
692:JSTOR
494:<
429:>
230:10.3
29:is a
2975:Test
2175:Sign
2027:Wald
1100:Mode
1038:Mean
795:link
769:PMID
633:link
519:SPSS
343:Tx C
335:Tx B
315:Tx C
307:Tx A
287:Tx B
279:Tx A
227:10.3
92:Tx C
89:Tx B
86:Tx A
2155:BIC
2150:AIC
894:doi
761:doi
682:doi
170:10
65:or
22:or
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2100:G
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684::
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566:Δ
564:(
523:p
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382::
377:1
373:H
349:2
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321:2
298:=
293:2
270::
265:0
261:H
213:2
207:7
201:9
198:5
190:3
187:0
175:4
167:5
152:3
144:7
141:5
129:2
118:3
106:1
26:W
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