Effect Size And Statistical Power
Effect Size and Statistical PowerJoseph Stevens, Ph.D., University of Oregon(541) 346-2445, stevensj@uoregon.edu Stevens, 20071
An Introductory Problem or Two:Which Study is Stronger?Answer: Study BStudy A: t (398) 2.30, p .022ω2 for Study A .01Study B: t (88) 2.30, p .024ω2 for Study B .05Examples inspired by Rosenthal & Gaito (1963)2
Study C Shows a Highly Significant ResultStudy C: F 63.62, p .0000001η2 for Study C .01, N 6,300Study D: F 5.40, p .049η2 for Study D .40, N 10Correct interpretation of statistical results requires consideration of statistical significance, effect size, and statistical power3
Three Fundamental Questions Asked inScienceIs there a relationship?Answered by Null Hypothesis Significance Tests(NHST; e.g., t tests, F tests, χ2, p-values, etc.)What kind of relationship?Answered by testing if relationshipis linear, curvilinear, etc.How strong is the relationship?Answered by effect size measures, notNHST’s (e.g., R2, r2, η2, ω2, Cohen’s d)4
The Logic of Inferential StatisticsThree Distributions Used in Inferential Statistics: Population: the entire universe of individuals we areinterested in studying (µ, σ, ) Sample: the selected subgroup that is actuallyobserved and measured ( X , ŝ , N)Sampling Distribution of the Statistic: A theoreticaldistribution that describes how a statistic behavesacross a large number of samples ( µ X , ŝX , )5
The Three Distributions Used inInferential StatisticsI. PopulationSelectionII. SampleInferenceIII. Sampling Distribution of theStatisticEvaluation6
The NHST Decision Model (based on thesampling distribution of the statistic)Statistical DecisionTrue StateFail toReject H0Reject H0H0 TrueH0 FalseCorrectDecision, (1 – α)Type II Error (β),Type I Error (α),Correct DecisionFalse Positive(1 – β), Statistical PowerFalse Negative
H0 True0.50.4Reject H0Reject H00.3Fail to reject H00.20.10-3-2-101α/2 .0251-α .9523α/2 .025Note: Sampling distributions are called Central Distributions when H0 is true
True StateStatistical DecisionThe value of α is set by convention which also determines 1 - αH0 TrueH0 FalseAnd,Failif Hto0 is reallytrue, thenβ 0CorrectDecisionReject H0(1 – α) .95Reject H0Type I Errorα .05But if H0 is false, what are the valuesof β and (1-β)?Type II Errorβ ?Statistical Power(1 – β) ?
What if H0 is False? If the null hypothesis is false, the sampling distributionand model just considered is incorrectIn that case, a different sampling distribution describesthe true state of affairs, the noncentral distribution In fact there is a family of sampling distributionswhen the null is false that depend on just how largean effect is present The size of the difference between the central andnoncentral distributions is described by anoncentrality parameter10
Central and Noncentral 0-3-2-10123The noncentrality parameter represents the lack of overlap ordisplacement of the two distributions that results from a trueNoncentraldifference between groups or nonzero relationship between variablesDistribution, H0 FalseCentral Distribution,H0 True
Assume an example using the tdistribution with Cohen’s d .4Note the disparity between the central andnoncentral sampling distributions
Reject H0β .67The portion of the noncentral distribution that is below therejection point represents the probability of a Type II error (β)
Reject H0β .23The portion of the noncentral distribution that is abovethe rejection point is statistical power (1 - β)
More overlap (smaller effect size)results in less statistical power
Less overlap (larger effect size)results in greater statistical powerESCI Software
The Relationship Between Effect Size andStatistical Significance It should be apparent that statistical significancedepends on the size of the effect (e.g., thenoncentrality parameter)And, statistical significance also depends on the sizeof the study (N)Statistical significance is the product of these twocomponents17
SignificanceTest Resultst Effect Sizer1 r2( X1 X 2 )t sˆXXXSize of Studydf11 n11n218
SignificanceTest Results Effect SizeXSize of Study2F r2 X df1 reta 2F 1 eta 2Xdf errordf means19
SignificanceTest Results Effect Size XSize of Study To make correct interpretations, additionalinformation beyond statistical significance is needed When results are statistically significant, it is veryimportant to estimate effect size to determine themagnitude of results20
Two Kinds of Metric for Measuring theMagnitude of Effects Standardized Difference Measures – Express thesize of group difference in standard deviation units(e.g., Cohen’s d ) Strength of Association Measures – Expressmagnitude of effect as a proportion or percentage(e.g., r2, η2, ω2 )21
Strength of Association Measures Pearson’s rMultiple RMultivariate Canonical rWilk’s Lambda (1 – Λ)Effect size can be interpreted in units of r (see BESDbelow) or after squaring and multiplying by 100 as PercentShared Variance (PSV)PSV r2 X 10022
Strength of Association MeasuresCorrelation ratio2 Omega squared (ω ) Eta squared (η2) Partial eta squared (η2p)23
Strength of Association Measures Cohen also uses f 2 as a metric of effect sizeThis is easily expressed as R2 or η 22R2f (1 R 2 )2ηf2 (1 η 2 )24
Strength of Association Measures: ω2Omega Squared for an independent t-test:ω2 ( t2 - 1 ) / ( t2 N1 N2 - 1)Example:Group 1Group 2Mean65.5069.00Variance20.6928.96N3030t 65.5 - 69 / 1.29 -2.71ω2 (2.71)2 - 1 / [(2.71)2 30 30 - 1] 0.096, about 10% shared variance25
Strength of Association Measures: ω2Omega Squared for a one-factor ANOVA:ω2 [SSBetween - (a-1)(MSResidual)](SSTotal MSResidual)26
Strength of Association Measures: ω2Omega Squared for a two-factor ANOVA:ω2 [SSA - (a-1)(MSResidual)] / (SSTotal MSResidual)ω2 [SSB - (b-1)(MSResidual)] / (SSTotal MSResidual)ω2 [SSAB - (a-1)(b-1)(MSResidual)] / (SSTotal MSResidual)27
Strength of Association Measures: 1Total757.008028
Strength of Association Measures: ω2ω2 [SSA - (a-1)(MSResidual)] / (SSTotal MSResidual) [3.61 – (1)(1.31)] / (757 1.31) .003ω2 [SSB - (b-1)(MSResidual)] / (SSTotal MSResidual) [13.94 – (3)(1.31)] / (757 1.31) .013ω2 [SSAB - (a-1)(b-1)(MSResidual)] / (SSTotal MSResidual) [12.34 – (3)(1)(1.31)] / (757 1.31) .01129
Strength of Association Measures: η2η2 SSEffect / SSTotalAn alternative measure is partialeta squared:η2p SSEffect / (SSEffect SSResidual)Note. Partial eta may sum to more than 100% in multifactor designs30
Strength of Association Measures: η2pAn alternative formula using only F and df:[( F )(dfeffect )]η [( F )(dfeffect ) dfresidual ]2pExample using the interaction effect from above:[( F )(dfeffect )](3.14)(3)η .116[( F )(dfeffect ) dfresidual ] [(3.14)(3) 72]2p31
Comparing Strength of AssociationMeasuresNote the problems with partials:22Effectωη Different denominator for each effectA may sum.003.005 in Partialsto more than 100%designsB multifactor.013.018AB.011.016η2p.037.129.116η2p SSEffect / (SSEffect SSResidual)Note that:ω2 η2 η2p32
Group Difference Indices There are a variety of indices that measure theextent of the difference between groupsCohen’s d is the most widely used index (twogroups only)Generalization of Cohen’s to multiple groups issometimes called δ, but there is great variation innotationHedges’ g (uses pooled sample standard deviations)For multivariate, Mahalanobis’ D233
The Standardized Mean Difference: Cohen’s d( X1 X 2 )d sˆ pooledsˆ pooled s12 (n1 1) s22 (n2 1)n1 n2 234
The Standardized Mean Difference: Cohen’s dExample:sˆ pooled Group 1Group 2Mean65.5069.00Variance20.6928.96N303020.69(29) 28.96(29)s12 (n1 1) s22 (n2 1) 4.98 30 30 2n1 n2 2( X 1 X 2 ) (65.5 69.0) 0.70d 4.98sˆ pooled35
Interpreting Effect Size Results (How bigis big? There is no simple answer to “How large should aneffect size be?”The question begs another: “For what purpose?”The answer does not depend directly on statisticalconsiderations but on the utility, impact, and costsand benefits of the results36
Interpreting Effect Size ResultsCohen’s “Rules-of-Thumb” standardized mean difference effect size (Cohen’s d)small 0.20 medium 0.50 large 0.80 effect sizes (using fixed benchmarks) with the“If peopleinterpreted samerigidity that pcoefficient .05 has beenused inr)statistical testing, we correlation(Pearson’swould merelybe0.10being stupid in another metric”small (Thompson,2001; pp. 82–83). medium 0.30 large 0.5037
The Binomial Effect Size Display (BESD) Corresponding toVarious Values of r2 and rInterpreting Effect Size Results:Rosenthal.00& Rubin’sBESD.02.49.51Effect Sizesr2Success Rate IncreaserFromToSuccess .80.10.90.80.81.90.05.95.901.001.00.001.001.00.01 Small.08 Effects.46 Unimportant?.54.08Are
“Small” effects may be associated withimportant differences in outcomesSuccess Rate Increase Associated with an r2 of 466100Total100100200Note. Both tables from Rosenthal, R. (1984). Meta-analytic procedures for social research.Beverly Hills, CA: Sage.Also see Rosenthal, R., & Rubin, D. B. (1982). A simple, general purpose display ofmagnitude of experimental effect. Journal of Educational Psychology, 74, 166-169.
Confidence Intervals for Effect SizeX (original units)60626495% Confidenceinterval for Cohen’s d6668neg mean diffs7007274pos mean diffsCohen’s d - .70 (sameexample as slide 35)-1ESCI Software095% CI -1.2 to -0.2Note. See Cumming & Finch (2001) or http://www.latrobe.edu.au/psy/esci/40
Intermission
Statistical Power Statistical power, the probability of detecting aresult when it is presentOften the concern is “How many participants do Ineed?”While estimating N is important, a more productivefocus may be on effect size and design planningHow can I strengthen the research?42
Factors Affecting Statistical Power Sample Size Effect Size Alpha level Unexplained Variance Design Effects43
Effect of Sample Size on Statistical PowerAll things equal, sample size increases statistical powerat a geometric rate (in simple designs) This is accomplished primarily through reduction of thestandard error of the sampling distributionWith large samples, inferential statistics are very powerfulat detecting very small relationships or very smalldifferences between groups (even trivial ones)With small samples, larger relationships or differences areneeded to be detectable44
Effect of Sample Size on Statistical PowerσX σNsˆXsˆ NAs an example, if the estimated populationstandard deviation was 10 and sample size was4 then:But if sample was 16 (4 times larger) then thestandard error is 2.5 (smaller by half):sˆsˆXX10 5410 2.51645
0.4Ho true, Central tHa true, NonCentral tConsider the following example withN 10, note that power .21Probability .005.006.00
Versus a second example with N 30,notethat power .560.4Ho true, Central tHa true, NonCentral tProbability density0.30.20.10-3.00-2.00-1.000.001.002.00tESCI Software3.004.005.006.00
Impact of Sample Size on Statistical Power48
Impact of Effect Size on Statistical Power49
Impact of Sample and Effect Size onStatistical Power50
Effect of Alpha Level on Statistical Power One-tailed tests are more powerful than two-tailed tests Require clear a priori rationale Requires willingness to ignore results in the wrong direction Only possible with certain statistical tests (e.g., t but not F) Larger alpha values more powerful (e.g., p .10) May be difficult to convince reviewers Can be justified well in many program evaluation contexts (whenonly one direction of outcome is relevant) Justifiable with small sample size, small cluster size, or if, a priori,effect size is known to be small51
α .05α .01Power .34Power .59
α .05Power .59α .10Power .70
Effect of Unexplained Variance onStatistical Power Terminology: “error” versus unexplained or residualResidual variance reduces power Anything that decreases residual variance, increasespower (e.g., more homogeneous participants, additionalexplanatory variables, etc.)Unreliability of measurement contributes to residualvarianceTreatment infidelity contributes to residual variance54
Effect of Design Features on StatisticalPower Stronger treatments!Blocking and matchingRepeated measuresFocused tests (df 1)Intraclass correlationStatistical control, use of covariatesRestriction of range (IV and DV)Measurement validity (IV and DV)55
Effect of Design Features on StatisticalPower Multicollinearity (and restriction of range)sb y1.2 s y212Σx12 (1 r122 )Statistical model misspecification Linearity, curvilinearity, Omission of relevant variablesInclusion of irrelevant variables56
Options for Estimating Statistical Power Cohen’s tablesStatistical Software like SAS and SPSS using syntaxfilesWeb calculatorsSpecialized software like G*Power, OptimalDesign, ESCI, nQuery57
Estimating Statistical Power Base parameters on best information availableDon’t overestimate effect size or underestimateresidual variance or ICCConsider alternative scenarios What kind of parameter values might occur in theresearch?Estimate for a variety of selected parametercombinationsConsider worst cases (easier to plan than recover)58
Recommendations for Study Planning Greater attention to study design featuresExplore the implications of research design features onpower Base power estimation on: Prior researchPilot studiesPlausible assumptionsThought experimentsCost/benefit analysis59
Power in Multisite and ClusterRandomized Studies More complex designs involving data that are arrangedin inherent hierarchies or levelsMuch educational and social science data is organizedin a multilevel or nested structure Students within schools Children within families Patients within physicians Treatments within sites Measurement occasions within individuals60
Power in Multisite and ClusterRandomized StudiesFactors affecting statistical power Intraclass Correlation (ICC) Number of participants per cluster (N) Number of clusters (J) Between vs. within cluster variance Treatment variability across clusters Other factors as discussed above61
Intraclass Correlation Coefficient (ρ)Total σ 2Y τ 2 σ 2population variance between unitsICC total variance τ 2 / (τ 2 σ 2 )As ICC approaches 0, multilevel modeling is not needed andpower is the same as a non-nested design, but even small values ofICC can impact power62
Intraclass Correlation (ρ) The Intraclass Correlation Coefficient (ICC)measures the correlation between a grouping factorand an outcome measureIn common notation there are 1 to J groupsIf participants do not differ from one group toanother, then the ICC 0As participants’ outcome scores differ due tomembership in a particular group, the ICC growslarge63
Intraclass Correlation (ρ) ICC becomes important in research design when: Random assignment is accomplished at the group levelMultistage sampling designs are usedGroup level predictors or covariates are usedIf there is little difference from one group toanother (ICC nears zero), power is similar to thetotal sample size ignoring the clustering of groupsThe more groups differ (ICC is nonzero), effectivesample size for power approaches the number ofgroups rather than the total number of participants64
Intraclass Correlation (ρI) ICC varies with outcome and with type of group and participantsSmall groups that may be more homogenous (e.g., classrooms)are likely to have larger ICCs than large groups with moreheterogeneity (e.g., schools or districts)What size of ICCs are common? Concentrated between 0.01 and 0.05 for much social scienceresearch (Bloom, 2006) Between 0.05 and 0.15 for school achievement (Spybrook etal., 2006)The guideline of 0.05 to 0.15 is more consistent with the valuesof covariate adjusted intraclass correlations; unconditional ICCsmay be larger (roughly 0.15 to 0.25; Hedges & Hedberg, in press)“It is unusual for a GRT to have adequate power with fewerthan 8 to 10 groups per condition” (Murray et al., 2004)65
Relationship of ICC and power66
Relationship of ICC, Effect Size, Number ofClusters and Power
Relationship of ICC, Effect Size, Number ofClusters and Power When J is Small
Relationship of ICC, effect size, number of clustersand power1.0α 0.050n 500.9δ 0.20,ρ 0.05δ 0.20,ρ 0.10δ 0.40,ρ 0.05δ 0.40,ρ 0.100.80.7Power0.60.50.40.30.20.12342618099Number of clusters69
Effect of Cluster Size (n)1.0α 0.050J 200.9δ 0.20,ρ 0.05δ 0.20,ρ 0.10δ 0.40,ρ 0.05δ 0.40,ρ 0.100.80.7Power0.60.50.40.30.20.11324354657Number of subjects per cluster70
Effect of Number of Clusters (J)1.0α 0.050n 500.9δ 0.20,ρ 0.05δ 0.40,ρ 0.050.80.7Power0.60.50.40.30.20.12342618099Number of clusters71
The number of clustersNotehas thea strongerinfluenceon difference inpower for nj500asarranged50 per 10vs. nj 0 power than the cluster sizeICC asdepartsfromDifference500 arrangedas 25 perdue20 toclustersnumber of clustersDifference due tocluster size72
Ignoring Hierarchical Structure vs.Multilevel ModelingVariance of the treatment effect across clusters(τ 2 σ 2 / n)γ nJ
Effect of Effect Size Variability2σ( δ)
The number of clusters has a stronger influence onpower than the cluster size as ICC departs from 0 The standard error of the main effect of treatment is:4( ρ (1 ρ ) / n)SE (γˆ01 ) J As ρ increases, the effect of n decreasesIf clusters are variable (ρ is large), more power isgained by increasing the number of clusters sampledthan by increasing n75
Effect of a Covariate on Power1.0α 0.050n 500.9δ 0.20,ρ 0.102δ 0.20,ρ 0.10,RL2 0.560.80.7Power0.60.50.40.30.20.12342618099Number of clusters76
The Group Effect MultiplierRandomized group size (n)ICC 910.04Note: The group effect multiplier equals 1 (n 1) ρ ; table from Bloom (2006).
The Minimum Detectable Effect Expressed as a Multiple of the Standard ErrorNumber of groups (J)Two-tailed testOne-tailed te: The group effect multipliers shown here are for the difference between the mean program group outcome and the mean control groupoutcome, assuming equal variances for the groups, a significance level of .05, and a power level of .80; table from Bloom (2006).
The Minimum Detectable Effect SizeIntraclass correlation (ρI) 0.01Randomized group size (n)Number of groups Note: The minimum detectable effect sizes shown here are for a two-tailed hypothesis test, assuming a significancelevel of .05, a power level of .80, and randomization of half the groups to the program; table from Bloom (2006).
The Minimum Detectable Effect SizeIntraclass correlation (ρI) 0.05Randomized group size (n)Number of groups Note: The minimum detectable effect sizes shown here are for a two-tailed hypothesis test, assuming a significancelevel of .05, a power level of .80, and randomization of half the groups to the program; table from Bloom (2006).
The Minimum Detectable Effect SizeIntraclass correlation (ρI) 0.10Randomized group size (n)Number of groups Note: The minimum detectable effect sizes shown here are for a two-tailed hypothesis test, assuming a significancelevel of .05, a power level of .80, and randomization of half the groups to the program; table from Bloom (2006).
Using G*Power Free software for power estimation available n/aap/gpower3/download-and-register Estimates power for a variety of situations includingt-tests, F-tests, and χ2 G*Power82
Examples using G*PowerLuft & Vidoni (2002) examined preservice teachers’ knowledge about schoolto career transitions before and after a teacher internship. Some of theobtained results were:BeforeKnowledge about:WritingUse of Hands-on activitiesClass assignmentssd2.92 1.444.58 .673.67 .49XAftersdX3.92 .794.75 .454.08 .79tp-2.25 .05-1.00 .34-1.82 .10r.59.71.56Twelve students participated in the study and completed the pre andpost testing.83
Example 1. Using G*Power, estimate the power of the repeatedmeasures t-test for knowledge of hands-on activities. Use thesupplied information in the table.Choose t-testsChoose matched pairsChoose post hoc: Computeachieved power84
Next calculate an effect size based on the suppliedtable information:Add required informationClick DetermineClick Calculate and transfer85
Click calculate86
Example 2. Using the same information as example 1, determinethe necessary sample size to achieve a power of .80Graphing in G*Power87
Example 3. Continue with the same information and determine theminimum detectable effect size if power is .8088
Using the Optimal Design Software The Optimal Design Software can also be used toestimate power in a variety of situationsThe particular strength of this software is itsapplication to multilevel situations involving clusterrandomization or multisite designsAvailable at:http://sitemaker.umich.edu/group-based/optimal design software Optimal Design89
Using Optimal Design (OD), estimate the power for a grouprandomized study under several conditions. Start by choosing“File/Mode” on the toolbar and then “Optimal Design for GroupRandomized Trials”Next choose Power vs.number of clusters90
Now enter values to produce power estimates.Use α .05, n 10, δ .5, and ρ .05
Range and legend for axes canalso be modifiedNote that if you mouse over the powercurve, exact values are displayed
Now explore the use of OD for examiningpower as a function of n, ρ, δ, and R2
The OD software can also be used to determine the bestcombination of design features under cost constraintsChoose Optimal sample allocationEnter values of 10,000 Total budget, 400 per cluster, 20 per member, ρ .03, and δ .4; then compute94
Optimal DesignWhat if the ICC was lower, .01?What if the ICC was higher, .08?For the given budget, n is 21, J is12 and power is .62Note the loss of power with higher ICCWhat if the budget was increased?Note the increase in both n and powerNote the ratio of n to J giventhe higher ICC95
One Last Example: Multisite CRT The primary rationale in this approach is to extend the ideaof blocking to the multilevel situationClusters are assigned to blocks with other similar clustersand then randomly assigned to treatmentBlocking creates greater homogeneity and less residualvariance, thereby increasing powerFor example, schools are collected into blocks based onwhether school composition is low, medium, or high SESSchools are within each block are randomly assigned totreatmentBetween school SES variability is controlled by the blocking96
Multisite CRTTwo additional parameters are used in estimation: Number of sites or blocks, K2 The effect size variability, σδ σ 2 represents the variability of effect size fromδone cluster to another within a site This variability represents within site replicationsof the study97
Multisite CRTExample:2 5 cities, 12 schools per city, d .4, ICC .12, σ δ .01,blocking accounts for 50% of the variation in the outcome98
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Applications For the remainder of the workshop you may complete exercises on power estimationcalculate power estimates for your own researchExercises can be downloaded from:http://www.uoregon.edu/ stevensj/workshops/exercises.pdf When you finish the exercises, you can obtainanswers at:http://www.uoregon.edu/ stevensj/workshops/answers.pdf Discussion as time permits100
BibliographyBloom, H. S. (2006). Learning More from Social Experiments: Evolving Analytic Approaches. New York,NY: Russell Sage Foundation Publications.Boling, N. C., & Robinson, D. H. (1999). Individual study, interactive multimedia, or cooperativelearning: Which activity best supplements lecture-based distance education? Journal ofEducational Psychology, 91, 169-174.Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.Cohen, J. (1992). A Power Primer, Psychological Bulletin, 112, 155-159.Cohen, J. (1994). The earth is round (p .05). American Psychologist, 49, 997– 1003.Cooper, H., & Hedges, L. (1994). The Handbook of Research Synthesis. New York, NY: RusselSage Foundation.Cumming, G., & Finch, S. (2001). A primer on the understanding, use and calculation of confidenceintervals that are based on central and noncentral distributions. Educational and PsychologicalMeasurement, 61, 532–575.Elashoff, J. D. (2002). NQuery Advisor Version 5.0 User’s Guide. Los Angeles, CA: Statistical SolutionsLimited.Elmore, P., & Rotou, O. (2001, April). A primer on basic effect size concepts. Paper presented at theannual meeting of the American Educational Research Association, Seattle, WA.101
Hallahan & Rosenthal (1996). Statistical Power: Concepts, Procedures and Applications, BehaviorResearch and Therapy, 34, 489-99.Harlow, L. L. Mulaik, S. A. , & Steiger, J. H. (1997). What if there were no significance tests? Hillsdale, NJ:Erlbaum.Hays, W. L. (1963). Statistics for psychologists. New York: Holt, Rinehart & Winston.Hedges, L. V., & Hedburg, E. C. (in press). Intraclass correlation values for planning grouprandomized trials in education. Educational Evaluation and Policy Analysis.Huberty, C. (2002). A History of Effect Size Indices, Educational and Psychological Measurement, 62, 227240.Luft, V. D., & Vidoni, K. (2002). Results of a school-to-careers preservice teacher internshipprogram, Education, 122, 706-714.Murray, D. M., Varnell, S. P., & Blitstein, J. L. (2004). Design and analysis of group-randomizedtrials: A review of recent methodological developments, American Journal of Public Health,94, 423-432.Olejnik, S., & Algina, J. (2000). Measures of effect size for comparative studies: Applications,interpretations, and limitations. Contemporary Educational Psychology, 25, 241–286.Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials,Psychological Methods, 2(2), 173-185.Rosenthal & Gaito (1963). The interpretation of levels of significance by psychological researchers.Journal of Psychology, 55, 33-38.102
Rosenthal, R. & Rosnow, R. L. (1991). Essentials of behavioral research (2nd Ed.). New York: McGrawHill, Inc.Rosenthal, R., & Rubin, D. B. (1982). A simple, general purpose display of magnitude ofexperimental effect. Journal of Educational Psychology, 74, 166-169.Spybrook, J., Raudenbush, S., & Liu, X.-f. (2006). Optimal design for longitudinal and multilevel research:Documentation for the Optimal Design Software. New York: William T. Grant Foundation.Thompson, B. (1996). AERA editorial policies regarding statistical significance testing: Threesuggested reforms. Educational Researcher, 25 (2), 26– 30.Thompson (2002). What Future Quantitative Social Science Research Could Look Like: ConfidenceIntervals for Effect Sizes, Educational Researcher, 31, 25-32.Wilkinson, L. & Task Force on Statistical Inference (1999). Statistical Methods in PsychologyJournals: Guidelines and Explanations, American Psychologist, 54 (8), 594–604. [Retrievedfrom: http://www.apa.org/journals/amp/amp548594.html#c1 ].103
3 Study C Shows a Highly Significant Result Study C: F 63.62, p .0000001 Study D: F 5.40, p .049 η2 for Study C .01, N 6,300 η2 for Study D .40, N 10 Correct interpretation of statistical results requires consider-ation of statistical significance, effect size, and statistical power
about statistical power? Why should researchers perform power analysis to plan sample size? Statistical power depends on three parameters: significance level (α level), effect size, and sample size. Given an effect size value and a fixed α level, recruiting more participants in a study increases statistical power and the accuracy of the result.
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