An Overview Statistical Significance Only tells you sample results unlikely were the null true Null is usually that the effect size is absolutely zero If power is high the size of a significant effect could be trivial ID: 1001411
Download Presentation The PPT/PDF document "Effect Size Estimation Why and How" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
1. Effect Size EstimationWhy and HowAn Overview
2. Statistical SignificanceOnly tells you sample results unlikely were the null true.Null is usually that the effect size is absolutely zero.If power is high, the size of a significant effect could be trivial.If power is low, a big effect could fail to be detected
3. Nonsignificant ResultsEffect size estimates should be reported here too, especially when power was low.Will help you and others determine whether or not it is worth the effort to repeat the research under conditions providing more power.
4. Comparing MeansStudent’s T TestsEven with complex research, the most important questions can often be addressed by simple contrasts between means or sets of means.Reporting strength of effect estimates for such contrasts can be very helpful.
5. SymbolsDifferent folks use different symbols. Here are those I shall use – the parameter, Cohen’s.d – the sample statistic, There is much variation with respect to choice of symbols. Some use d to stand for the parameter, for example.
6. One SampleOn SAT-Q, is µ for my students same as national average?Point estimate does not indicate precision of estimation.We need a confidence interval.
7. Constructing the Confidence IntervalApproximate method – find unstandardized CI, divide endpoints by sample SD.OK with large sample sizes.With small sample sizes should use an exact method.Computer-intensive, iterative procedure, must estimate µ and σ.
8. Programs to Do ItSAS SPSSThe mean math SAT of my undergraduate statistics students (M = 535, SD = 93.4) was significantly greater than the national norm (516), t(113) = 2.147, p = .034, d = .20. A 95% confidence interval for the mean runs from 517 to 552. A 95% confidence interval for runs from .015 to .386.
9. Benchmarks for What would be a small effect in one context might be a large effect in another.Cohen reluctantly provided these benchmarks for behavioral research.2 = small, not trivial.5 = medium.8 = large
10. Reducing ErrorNot satisfied with the width of the CI, .015 to .386 (trivial to small/medium)?Get more data, orDo any of the other things that increase power.
11. Why Standardize?Statisticians argue about this.If the unit of measure is meaningful (cm, $, ml), do not need to standardize.Weight reduction intervention produced average loss of 17.3 pounds.Residents of Mississippi average 17.3 points higher than national norm on measure of neo-fascist attitudes.
12. Bias in Effect Size EstimationLab research may result in over-estimation of the size of the effect in the natural world.Sample HomogeneityExtraneous Variable ControlMean difference = 25Lab SD = 15, d = 1.67, whopper effectField SD = 100, d = .25, small effect
13. Two Independent Means
14. ProgramsWill do all this for you and give you a CI.Conf_Interval-d2.sas CI-d-SPSS.zip Confidence Intervals, Pooled and Separate Variances T
15. ExamplePooled t(86) = 3.267t = 3.267 ; df = 86 ; n1 = 33 ; n2 = 55 ;d = t/sqrt(n1*n2/(n1+n2));ncp_lower = TNONCT(t,df,.975);ncp_upper = TNONCT(t,df,.025);d_lower = ncp_lower*sqrt((n1+n2)/(n1*n2));d_upper = ncp_upper*sqrt((n1+n2)/(n1*n2));output; run; proc print; var d d_lower d_upper; run; Obs d d_lower d_upper 1 0.71937 0.27268 1.16212 Among Vermont school-children, girls’ GPA (M = 2.82, SD = .83, N = 33) was significantly higher than boys’ GPA (M = 2.24, SD = .81, N = 55), t(65.9) = 3.24, p = .002, d = .72. A 95% confidence interval for the difference between girls’ and boys’ mean GPA runs from .23 to .95 in raw score units and from .27 to 1.16 in standardized units. This is an almost large effect by Cohen’s guidelines.
16. Glass’ DeltaUse the control group SD rather than pooled SD as the standardizer.When the control group SD is a better estimate of SD in the population of interest.
17. Point Biserial rSimply correlate group membership with the scores on the outcome variable.Or computeFor the regression Score = a + bGroup, b = difference in group means = .588.standardized slope = This is a medium-sized effect by Cohen’s benchmarks for r. Hmmmm. It was large when we used d.
18. Eta-SquaredFor two mean comparisons, this is simply the squared point biserial r.Can be interpreted as a proportion of variance.CI: Conf-Interval-R2-Regr.sas or CI-R2-SPSS.zip For our data, 2 = .11, CI.95 = .017, .240.Again, overestimation may result from EV control.2
19. Cohen’s Benchmarks for and 2.1 is small but not trivial (r2 = 1%).3 is medium (9%).5 is large (25%)2.01 (1%) is small but not trivial.06 is medium.14 is largeNote the inconsistency between these two sets of benchmarks.
20. Effect of n1/n2 on d and rpbn1/n2 = 1M1 = 5.5, SD1 = 2.306, n1 = 20, M2 = 7.8, SD2 = 2.306, n2 = 20t(38) = 3.155, p = .003M2-M1 = 2.30 d = 1.00 rpb = .456Large effect
21. Effect of n1/n2 on d and rpbn1/n2 = 25M1 = 5.500, SD1 = 2.259, n1 = 100, M2 = 7.775, SD2 = 2.241, n2 = 4t(102) = 1.976, p = .051M2-M1 = 2.30 d = 1.01 rpb = .192Large or (Small to Medium) Effect?
22. How does n1/n2 affect rpb?The point biserial r is the standardized slope for predicting the outcome variable from the grouping variable (coded 1,2). The unstandardized slope is the simple difference between group means. Standardize by multiplying by the SD of the grouping variable and dividing by the SD of the outcome variable. The SD of the grouping variable is a function of the sample sizes. For example, for N = 100, the SD of the grouping variable is.503 when n1, n2 = 50, 50 .473 when n1, n2 = 67, 33 .302 when n1, n2 = 90, 10
23. Common Language Effect Size StatisticFind the lower-tailed p forFor our data, p = .5,If you were to randomly select one boy & one girl. P(Girl GPA > Boy GPA) = .69.Odds = .69/(1-.69) = 2.23.
24. Two Related SamplesTreat the data as if they were from independent samples when calculating d.If you standardize with the SD of the difference scores, you will overestimate .There is not available software to get an exact CI, and approximation procedures are only good with large data sets.
25. Correlation/RegressionEven in complex research, many questions of great interest are addressed by zero-order correlation coefficients.Pearson r, are already standardized.Cohen’s Benchmarks:.1 = small, not trivial.3 = medium.5 = large
26. CI for , Correlation ModelAll variables random rather than fixed.Use R2 program to obtain CI for ρ2.
27. R2 Program (Correlation Model)
28. Oh my, p < .05, but the 95% CI includes zero.
29. That’s better. The 90% CI does NOT include zero. Do note that the “lower bound” from the 95% CI is identical to the “lower limit” of the 90% CI.
30. CI for , Regression ModelY random, X fixed.Tedious by-hand method: See handout.SPSS and SAS programs for comparing Pearson correlations and OLS regression coefficients.Web calculator at Vassar
31. Vassar Web App.
32. More Apps.R2 will not handle N > 5,000. Use this approximation instead:Conf-Interval-R2-Regr-LargeN.sas For Regression analysis (predictors are fixed, not random), use this:Conf-Interval-R2-Regr (SAS) orCI-R2-SPSS.zip (SPSS)
33. What Confidence Coefficient Should I Use?For R2, if you want the CI to be concordant with a test of the null that ρ2 = 0,Use a CC of (1 - 2α), not (1 - α).Suppose you obtain r = .26 from n = 62 pairs of scores. It is significant at .041.When you put a 95% confidence interval about r you obtain .01, .48. Zero is not included in the confidence interval.
34. Now let us put a 95% confidence interval about the r2 (.0676) using Steiger & Fouladi’s R2 Oh my, the CI includes zero. We should use a 90% CI.
35. Bias in Sample R2Sample R2 overestimates population ρ2.With large dfnumerator this can result in the CI excluding the point estimate.This should not happen if you use the shrunken R2 as your point estimate.
36. Common Language StatisticSample two cases (A & B) from paired X,Y.CL = P(YA > YB | XA > XB)For one case, CL = P(Y > My | X > Mx)
37. r to CLr.00.10.30.50.70.90.99CL50%53%60%67%75%86%96%Odds11.131.5236.124
38. Multiple R2Cohen:.02 = small (2% of variance).15 = medium (13% of variance).35 = large (26% of variance)
39. Partial and Semipartial
40. ExampleGrad GPA = GRE-Q, GRE-V, MAT, ARR2 = .6405For GRE-Q, pr2=.16023, sr2=.06860
41. One-Way ANOVAsdsdfsdsd
42. CI for 2 Conf-Interval-R2-Regr.sas CI-R2-SPSS at my SPSS Programs Page CI.95 = .84, .96If you want the CI to be consistent with the F-test, obtain a CI with (1-2) confidence.
43. 2Sample 2 overestimates population 2 2 is less biasedFor our data, 2 = .93.
44. Cohen’s Benchmarks for 2 .01 = small.06 = medium.14 = large.942 = somebody made up these data.
45. Misinterpretation of Estimates of Proportion of Variance Explained6% (Cohen’s benchmark for medium 2 sounds small.Aspirin study: Outcome = Heart Attack?Preliminary results so dramatic study was stopped, placebo group told to take aspirinOdds ratio = 1.83r2 = .0011Report r instead of r2? r = .033
46. Extraneous Variable ControlMay artificially inflate strength of effect estimates (including d, r, , , etc.).Effect estimate from lab research >> that from field research.A variable that explains a large % of variance in highly controlled lab research may explain little out in the natural world.
47. Standardized Differences Between Means When k > 2Plan focused contrasts between means or sets of means.Chose contrasts that best address the research questions posed.Do not need to do ANOVA.Report d for each contrast.
48. Standardized Differences Among Means in ANOVAFind an average value of d across pairs of means.Or the average standardized difference between group mean and grand mean.Steiger has proposed the RMSSE as the estimator.
49. Root Mean Square Standardized Effectk is the number of groups, Mj is group mean, GM is grand mean.Standardizer is pooled SD, SQRT(MSE)For our data, RMSSE = 4.16. Godzilla.The population parameter is .
50. Place a CI on RMSSEhttp://www.statpower.net/Content/NDC/NDC.exe
51. Click ComputeGet CI for lambda, the noncentrality parameter.
52. Transform CI to RMSSEThe CI for lambda = 102.646, 480.288CI for = 2.616, 5.659.
53. CI → Hypothesis TestH0: = 0. cannot be less than 0, so a one-tailed p would be appropriate.Accordingly we find a 100(1-2α)% CI.For the usual .05 test, that is a 90% CI.If the CI excludes 0, then the ANOVA is significant.
54. Factorial Analysis of Variance For each effect, 2 = SSeffect/SStotal2: as before, use SSeffect in place of SSAmongGroupsNow suppose that one of the factors is experimental (present in the lab but not in the natural world).And the other is variable in both lab and the natural world.
55. Modify the Denominator of 2Sex x Experimental Therapy ANOVASex is variable in lab and natural worldExperimental Therapy only in labEstimate effect of sex with variance due to Therapy and Interaction excluded from denominator.The resulting statistic is called partial eta-squared.
56. Partial 2When estimating 2 for Therapy and theInteraction, one should not remove the effect of Sex from the denominator.
57. Explaining More Than 100% of the VariancePierce, Block, and Aguinis (2004)Many articles, in good journals, where partial 2 was wrongly identified as 2 Even when total variance explained exceeded 100%.In one case, 204%.Why don’t authors, reviewers, and editors notice such foolishness?
58. CI for 2 or Partial 2 Use Conf-Interval-R2-Regr.sas Use the ANOVA F to get CI for partial 2To get CI for 2 will need compute a modified F.See Two-Way Independent Samples ANOVA on SAS
59. Contingency Table Analysis2 x 2 table: Phi = Pearson r between dichotomous variables.Cramér’s φ = similar, for a x b tables where a and/or b > 2.Odds ratio: (odds of A|B)/(odds of A| not B)
60. Small EffectPhi = .1Odds ratio = (55/45)(45/55) = 1.49
61. Medium EffectPhi = .3Odds ratio = (65/35)(35/65) = 3.45
62. Large EffectPhi = .5Odds ratio = (75/25)(25/75) = 9.00
63. Phi and Odds RatiosThe marginals were uniform in the contingency tables above.For a fixed odds ratio, phi decreases as the marginals deviate from uniform.See http://core.ecu.edu/psyc/wuenschk/StatHelp/Phi-OddsRatio.docx
64. CI for Odds RatioConduct a binary logistic regression and ask for confidence intervals for the odds ratios.
65. Multivariate AnalysisMost provide statistics similar to r2 and 2Canonical correlation/regressionFor each root get a canonical r Is the corr between a weighted combination of the Xs and a weighted combination of the YsOther analyses are just simplifications or special cases of canonical corr/regr.
66. MANOVA and DFA: Canonical rFor each root get a squared canonical r.There will be one root for each treatment df.If you were to use ANOVA to compare the groups on that root, this canonical r2 would be
67. MANOVA and DFA: 1 - For each effect, Wilks is, basically, Accordingly, you can compute a multivariate 2 as 1 - .If k = 2, 1 - is the canonical r2.
68. Binary Logistic RegressionCox & Snell R2 Has an upper boundary less than 1.Nagelkerke R2 Has an upper boundary of 1.Classification results speak to magnitude of omnibus effect.Odds ratios speak to magnitude of partial effects.
69. Comparing Predictors’ ContributionsIt may help to standardize continuous predictors prior to computing odds ratiosConsider these results
70. Relative Contributions of the PredictorsThe event being predicted is retention in ECU’s engineering program.Each one point increase in HS GPA multiplies the odds of retention by 3.656.A one point inrease in Quantitative SAT increases the odds by only 1.006But a one point increase in GPA is a helluva lot larger than a one point increase in SAT.
71. Standardized PredictorsHere we see that the relative contributions of the three predictors do not differ much. BS.E.WalddfSig.Exp(B)ZHSGPA.510.1996.5691.0101.665ZSATQ.440.2014.7911.0291.553ZOpenness.435.1984.8321.0281.545Constant-.121.188.4141.520.886
72. Why Confidence Intervals?They are not often reported.So why do I preach their usefulness?IMHO, they give one everything given by a hypothesis test p AND MORE.Let me illustrate, using confidence intervals for
73. Significant Results, CI = .01, .03 We can be confident of the direction of the effect.We can also be confident that the size of the effect is so small that it might as well be zero.“Significant” in this case is a very poor descriptor of the effect.
74. Significant Results, CI = .02, .84 We can be confident of the direction of the effect& it is probably not trivial in magnitude,But it is estimated with little precision.Could be trivial, could be humongous.Need more data to get more precise estimation of size of effect.
75. Significant Results, CI = .51, .55 We can be confident of the direction of the effect& that it is large in magnitude (in most contexts).We have great precision.
76. Not Significant, CI = -.46, +.43 Effect could be anywhere from large in one direction to large in the other direction.This tells us we need more data (or other power-enhancing characteristics).
77. Not Significant, CI = -.74, +.02 Cannot be very confident about the direction of the effect, butIt is likely that is negative.Need more data/power.
78. Not Significant, CI = -.02, +.01 A very impressive result.Tells us that the effect is of trivial magnitude.Suppose X = generic vs. brand-name drugY = response to drug.We have established bioequivalence.
79. The Effect Size Bible