Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

History of statistics wikipedia, lookup

Student's t-test wikipedia, lookup

Taylor's law wikipedia, lookup

Bootstrapping (statistics) wikipedia, lookup

Resampling (statistics) wikipedia, lookup

Misuse of statistics wikipedia, lookup

Psychometrics wikipedia, lookup

Transcript

The Assumptions of ANOVA Dennis Monday Gary Klein Sunmi Lee May 10, 2005 Major Assumptions of Analysis of Variance • The Assumptions – Independence – Normally distributed – Homogeneity of variances • Our Purpose – Examine these assumptions – Provide various tests for these assumptions • Theory • Sample SAS code (SAS, Version 8.2) – Consequences when these assumptions are not met – Remedial measures Normality • Why normal? – ANOVA is an Analysis of Variance – Analysis of two variances, more specifically, the ratio of two variances – Statistical inference is based on the F distribution which is given by the ratio of two chi-squared distributions – No surprise that each variance in the ANOVA ratio come from a parent normal distribution • Calculations can always be derived no matter what the distribution is. Calculations are algebraic properties separating sums of squares. Normality is only needed for statistical inference. Normality Tests • Wide variety of tests we can perform to test if the data follows a normal distribution. • Mardia (1980) provides an extensive list for both the univariate and multivariate cases, categorizing them into two types – Properties of normal distribution, more specifically, the first four moments of the normal distribution • Shapiro-Wilk’s W (compares the ratio of the standard deviation to the variance multiplied by a constant to one) – Goodness-of-fit tests, • Kolmogorov-Smirnov D • Cramer-von Mises W2 • Anderson-Darling A2 Normality Tests proc univariate data=temp normal plot; var expvar; run; proc univariate data=temp normal plot; var normvar; run; Tests for Normality Tests for Normality Test --Statistic--- -----p Value------ Test --Statistic--- -----p Value------ Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling W D W-Sq A-Sq Pr Pr Pr Pr Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling W D W-Sq A-Sq Pr Pr Pr Pr 0.731203 0.206069 1.391667 7.797847 < > > > W D W-Sq A-Sq <0.0001 <0.0100 <0.0050 <0.0050 0.989846 0.057951 0.03225 0.224264 < > > > W D W-Sq A-Sq 0.6521 >0.1500 >0.2500 >0.2500 Normal Probability Plot 8.25+ | * | | | * | | * | + 4.25+ ** | ++++ ** +++ | *+++ | +++* | ++**** | ++++ ** | ++++***** | ++****** 0.25+* * ****************** +----+----+----+----+----+----+----+----+----+----+ Stem 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 Leaf 0 # 1 Boxplot * 1 1 * 2 5 4 588 3 59 00112234 56688 00011122223444 55555566667777778999999 000011111111111112222222233333334444444 ----+----+----+----+----+----+----+---- 1 1 1 3 1 2 8 5 14 23 39 * 0 0 0 0 | | | +--+--+ *-----* +-----+ Normal Probability Plot 2.3+ ++ * | ++* | +** | +** | **** | *** | **+ | ** | *** | **+ | *** 0.1+ *** | ** | *** | *** | ** | +*** | +** | +** | **** | ++ | +* -2.1+*++ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 Stem 22 20 18 16 14 12 10 8 6 4 2 0 -0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 Leaf 1 7 90 047 6779 469002 2368 005546 228880077 5233446 3458447 366904459 52871 884318651 98619 60 98557220 963 584 853 0 4 8 ----+----+----+----+ Multiply Stem.Leaf by 10**-1 # 1 1 2 3 4 6 4 6 9 7 7 9 5 9 5 2 8 3 3 3 1 1 1 Boxplot | | | | | | | +-----+ | | | | *-----* | + | | | | | +-----+ | | | | | | | | Consequences of Non-Normality • F-test is very robust against non-normal data, especially in a fixed-effects model • Large sample size will approximate normality by Central Limit Theorem (recommended sample size > 50) • Simulations have shown unequal sample sizes between treatment groups magnify any departure from normality • A large deviation from normality leads to hypothesis test conclusions that are too liberal and a decrease in power and efficiency Remedial Measures for NonNormality • Data transformation • Be aware - transformations may lead to a fundamental change in the relationship between the dependent and the independent variable and is not always recommended. • Don’t use the standard F-test. – Modified F-tests • Adjust the degrees of freedom • Rank F-test (capitalizes the F-tests robustness) – Randomization test on the F-ratio – Other non-parametric test if distribution is unknown – Make up our own test using a likelihood ratio if distribution is known Independence • Independent observations – No correlation between error terms – No correlation between independent variables and error • Positively correlated data inflates standard error – The estimation of the treatment means are more accurate than the standard error shows. Independence Tests • If we have some notion of how the data was collected, we can check if there exists any autocorrelation. • The Durbin-Watson statistic looks at the correlation of each value and the value before it – Data must be sorted in correct order for meaningful results – For example, samples collected at the same time would be ordered by time if we suspect results could depend on time Independence Tests proc glm data=temp; class trt; model y = trt / p; output out=out_ds r=resid_var; run; quit; proc glm data=temp; class trt; model y = trt / p; output out=out_ds r=resid_var; run; quit; data out_ds; set out_ds; time = _n_; run; proc gplot data=out_ds; plot resid_var * time; run; quit; data out_ds; set out_ds; time = _n_; run; proc gplot data=out_ds; plot resid_var * time; run; quit; First Order Autocorrelation 0.90931 Durbin-Watson D 0.12405 First Order Autocorrelation 0.00479029 Durbin-Watson D 1.96904290 Remedial Measures for Dependent Data • First defense against dependent data is proper study design and randomization – Designs could be implemented that takes correlation into account, e.g., crossover design • Look for environmental factors unaccounted for – Add covariates to the model if they are causing correlation, e.g., quantified learning curves • If no underlying factors can be found attributed to the autocorrelation – Use a different model, e.g., random effects model – Transform the independent variables using the correlation coefficient Homogeneity of Variances • Eisenhart (1947) describes the problem of unequal variances as follows – the ANOVA model is based on the proportion of the mean squares of the factors and the residual mean squares – The residual mean square is the unbiased estimator of 2, the variance of a single observation – The between treatment mean squares takes into account not only the differences between observations, 2, just like the residual mean squares, but also the variance between treatments – If there was non-constant variance among treatments, we can replace the residual mean square with some overall variance, a2, and a treatment variance, t2, which is some weighted version of a2 – The “neatness” of ANOVA is lost Homogeneity of Variances • The omnibus (overall) F-test is very robust against heterogeneity of variances, especially with fixed effects and equal sample sizes. • Tests for treatment differences like t-tests and contrasts are severely affected, resulting in inferences that may be too liberal or conservative. Tests for Homogeneity of Variances – Levene’s Test • computes a one-way-anova on the absolute value (or sometimes the square) of the residuals, |yij – ŷi| with t-1, N – t degrees of freedom • Considered robust to departures of normality, but too conservative – Brown-Forsythe Test • a slight modification of Levene’s test, where the median is substituted for the mean (Kuehl (2000) refers to it as the Levene (med) Test) – The Fmax Test • Proportion of the largest variance of the treatment groups to the smallest and compares it to a critical value table • Tabachnik and Fidell (2001) use the Fmax ratio more as a rule of thumb rather than using a table of critical values. – Fmax ratio is no greater than 10 – Sample sizes of groups are approximately equal (ratio of smallest to largest is no greater than 4) • No matter how the Fmax test is used, normality must be assumed. Tests for Homogeneity of Variances proc glm class model means run; quit; data=temp; trt; y = trt; trt / hovtest=levene hovtest=bf; Homogeneous Variances The GLM Procedure proc glm class model means run; quit; data=temp; trt; y = trt; trt / hovtest=levene hovtest=bf; Heterogenous Variances The GLM Procedure Levene's Test for Homogeneity of Y Variance ANOVA of Squared Deviations from Group Means Source DF Sum of Squares Mean Square TRT Error 1 98 10.2533 1663.5 10.2533 16.9747 Levene's Test for Homogeneity of y Variance ANOVA of Squared Deviations from Group Means F Value Pr > F Source DF Sum of Squares Mean Square 0.60 0.4389 trt Error 1 98 10459.1 27921.5 10459.1 284.9 Brown and Forsythe's Test for Homogeneity of Y Variance ANOVA of Absolute Deviations from Group Medians Source DF Sum of Squares Mean Square TRT Error 1 98 0.7087 124.6 0.7087 1.2710 F Value Pr > F 36.71 <.0001 Brown and Forsythe's Test for Homogeneity of y Variance ANOVA of Absolute Deviations from Group Medians F Value Pr > F Source DF Sum of Squares Mean Square 0.56 0.4570 trt Error 1 98 318.3 333.8 318.3 3.4065 F Value Pr > F 93.45 <.0001 Tests for Homogeneity of Variances • SAS (as far as I know) does not have a procedure to obtain Fmax (but easy to calculate) • More importantly: VARIANCE TESTS ARE ONLY FOR ONE-WAY ANOVA WARNING: Homogeneity of variance testing and Welch's ANOVA are only available for unweighted one-way models. Tests for Homogeneity of Variances (Randomized Complete Block Design and/or Factorial Design) • In a CRD, the variance of each treatment group is checked for homogeneity • In factorial/RCBD, each cell’s variance should be checked H0: σij2 = σi’j’2, For all i,j where i ≠ i’, j ≠ j’ Tests for Homogeneity of Variances (Randomized Complete Block Design and/or Factorial Design) • • Approach 1 – – Code each row/column to its own group Run HOVTESTS as before Approach 2 – – – data newgroup; set oldgroup; if block = 1 and treat = 1 then newgroup if block = 1 and treat = 2 then newgroup if block = 2 and treat = 1 then newgroup if block = 2 and treat = 2 then newgroup if block = 3 and treat = 1 then newgroup if block = 3 and treat = 2 then newgroup Recall Levene’s Test and BrownForsythe Test are ANOVAs based on residuals Find residual for each observation Run ANOVA proc sort data=oldgroup; by treat block; run; = 1; = 2; = 3; = 4; = 5; = 6; run; proc glm data=newgroup; class newgroup; model y = newgroup; means newgroup / hovtest=levene hovtest=bf; run; quit; proc means data=oldgroup noprint; by treat block; var y; output out=stats mean=mean median=median; run; data newgroup; merge oldgroup stats; by treat block; resid = abs(mean - y); if block = 1 and treat = 1 then newgroup = 1; ……… run; proc glm data=newgroup; class newgroup; model resid = newgroup; run; quit; Tests for Homogeneity of Variances (Repeated-Measures Design) • Recall the repeated-measures set-up: Treatment a1 a2 a3 s1 s1 s1 s2 s2 s2 s3 s3 s3 s4 s4 s4 Tests for Homogeneity of Variances (Repeated-Measures Design) • As there is only one score per cell, the variance of each cell cannot be computed. Instead, four assumptions need to be tested/satisfied – Compound Symmetry • Homogeneity of variance in each column – σa12 = σa22 = σa32 • Homogeneity of covariance between columns – σa1a2 = σa2a3 = σa3a1 – No A x S Interaction (Additivity) – Sphericity • Variance of difference scores between pairs are equal – σYa1-Ya2 = σYa1-Ya3 = σYa2-Ya3 Tests for Homogeneity of Variances (Repeated-Measures Design) • Usually, testing sphericity will suffice • Sphericity can be tested using the Mauchly test in SAS proc glm data=temp; class sub; model a1 a2 a3 = sub / nouni; repeated as 3 (1 2 3) polynomial / summary printe; run; quit; Sphericity Tests Variables Transformed Variates Orthogonal Components DF 2 2 Mauchly's Criterion Det = 0 Det = 0 Chi-Square 6.01 6.03 Pr > ChiSq .056 .062 Tests for Homogeneity of Variances (Latin-Squares/Split-Plot Design) • If there is only one score per cell, homogeneity of variances needs to be shown for the marginals of each column and each row – Each factor for a latin-square – Whole plots and subplots for split-plot • If there are repititions, homogeneity is to be shown within each cell like RCBD • If there are repeated-measures, follow guidelines for sphericity, compound symmetry and additivity as well Remedial Measures for Heterogeneous Variances • Studies that do not involve repeated measures – If normality is not violated, a weighted ANOVA is suggested (e.g., Welch’s ANOVA) – If normality is violated, the data transformation necessary to normalize data will usually stabilize variances as well – If variances are still not homogeneous, non-ANOVA tests might be your option • Studies with repeated measures – For violations of sphericity • modify the degrees of freedom have been suggested. – Greenhouse-Geisser – Huynh and Feldt • Only do specific comparisons (sphericity does not apply since only two groups – sphericity implies more than two) • MANOVA • Use an MLE procedure to specify variance-covariance matrix Other Concerns • Outliers and influential points – Data should always be checked for influential points that might bias statistical inference • Use scatterplots of residuals • Statistical tests using regression to detect outliers – DFBETAS – Cook’s D References • Casella, G. and Berger, R. (2002). Statistical Inference. United States: Duxbury. • Cochran, W. G. (1947). Some Consequences When the Assumptions for the Analysis of Variances are not Satisfied. Biometrics. Vol. 3, 22-38. • Eisenhart, C. (1947). The Assumptions Underlying the Analysis of Variance. Biometrics. Vol. 3, 1-21. • Ito, P. K. (1980). Robustness of ANOVA and MANOVA Test Procedures. Handbook of Statistics 1: Analysis of Variance (P. R. Krishnaiah, ed.), 199-236. Amsterdam: NorthHolland. • Kaskey, G., et al. (1980). Transformations to Normality. Handbook of Statistics 1: Analysis of Variance (P. R. Krishnaiah, ed.), 321-341. Amsterdam: North-Holland. • Kuehl, R. (2000). Design of Experiments: Statistical Principles of Research Design and Analysis, 2nd edition. United States: Duxbury. • Kutner, M. H., et al. (2005). Applied Linear Statistical Models, 5th edition. New York: McGraw-Hill. • Mardia, K. V. (1980). Tests of Univariate and Multivariate Normality. Handbook of Statistics 1: Analysis of Variance (P. R. Krishnaiah, ed.), 279-320. Amsterdam: North-Holland. • Tabachnik, B. and Fidell, L. (2001). Computer-Assisted Research Design and Analysis. Boston: Allyn & Bacon.