14-An-introduction-to-the-analysis-of-variance.ppt

14 An Introduction to the Analysis of VarianceXiaorong Cheng,Psychology,CCNUlAnalysis of Variance(ANOVA):A statistical procedure developed by R.A.Fisher that allows one to compare simultaneously the difference between two or more means.Xiaorong Cheng,Psychology,CCNUAdvantages of ANOVAlOmnibus statistical test:permits one to compare simultaneously several variables or levels of a variable.Xiaorong Cheng,Psychology,CCNUlOne-way ANOVA:allows one to compare the effects of different levels of a single IV.lTwo-way ANOVA:allows one to compare the effects of two IVs.Xiaorong Cheng,Psychology,CCNUControls experimentwise errorlOne t-tests significance level at.05(5%probability to have Type I error)l3 levels,3 t-tests.Type I error:1-0.95 X 0.95 X 0.95=.143C=number of potential pairsk=total number of groups!=factorial of a number,multiple all the whole numbers between 1 and the numberXiaorong Cheng,Psychology,CCNUlMultiple nonindependent t-ratios:greatly inflates the probability of committing a Type I error.lPairwise comparison:the probability of committing a Type I error when comparing any two means using a t-ratio.Experimentwise error rate:the probability of committing on or more type I errors when conducting multiple t-ratios from a single experiment.Xiaorong Cheng,Psychology,CCNUlFigure 14.2(p361)Xiaorong Cheng,Psychology,CCNUThe general linear modellPartition:a statistical procedure where the total variance is divided into separate components.lFactor:IVlLevels:the different values or conditions within an independent variable that are analyzed in an ANOVA.Xiaorong Cheng,Psychology,CCNUlFigure 14.3(p362)Xiaorong Cheng,Psychology,CCNUEquation 14.3(p363)General linear model:any individual score is the sum of the mean of the population,the effects of the treatment,and random error.Xiaorong Cheng,Psychology,CCNUlFigure 14.4(p364)Xiaorong Cheng,Psychology,CCNUlComponent of variancelTotal variance:the variance of all scores in the data set regardless of experimental group.lBetween-groups variance:estimate of variance between group meanslWithin-groups variance:estimate of the average variance within each group.Xiaorong Cheng,Psychology,CCNUlFigure 14.5(p364)Xiaorong Cheng,Psychology,CCNUlTotal variance=the grand mean,the mean of all the data.N=the total number of subjects in all groups.Xiaorong Cheng,Psychology,CCNUlWithin-groups variance:difference among the subjects caused by random error and factors not controlled by the researcher.An unbiased estimate of the population variance.Assumption:homogeneity of variance.nj=the number of subjects in the jth groupk=the number of groupsCalculate the variance of each group,add together,divided by the number of groups.Xiaorong Cheng,Psychology,CCNUlFigure 14.6(p366)Xiaorong Cheng,Psychology,CCNUlBetween-groups variance:the variance between the group means.=the mean of all observations(grand mean)=the mean of observations in one groupnj=the number of subjects in the jth groupk=the number of means(group means)Xiaorong Cheng,Psychology,CCNUlFigure 14.7(p367)Xiaorong Cheng,Psychology,CCNUlError variance:uncontrolled and unpredicted differences among individual scores.The within-group variance estimates the error variance.lTreatment variance:among group means that is due to the effects of the independent variable.=error variance+treatment variancetreatment varianceXiaorong Cheng,Psychology,CCNUWhats the function of ANOVA?To determine the proportion of between-groups variance that is due to error and the proportion due to treatment.Xiaorong Cheng,Psychology,CCNUThe F-RatioXiaorong Cheng,Psychology,CCNUlNo treatment effect(treatment variance=0)H0:Xiaorong Cheng,Psychology,CCNUlTreatment effect present(treatment variance0)H1:not H0F:does not specify which means are different from other means.H1:the difference between the means is great enough for us to reject the null hypothesis.Xiaorong Cheng,Psychology,CCNUlFratio sampling distributionK=the number of groupsnj=the number of observations in each groupXiaorong Cheng,Psychology,CCNUlTable 14.1(p372)Xiaorong Cheng,Psychology,CCNUlThe F-ratio and 2Calculate only if the statistic is significantXiaorong Cheng,Psychology,CCNUlt-ratio vs.F-ratiolt2=F when k=2,same conclusion with nondirectional hypothesislt-ratio can make directional comparisons between means while F-ratio cannotlt-distribution is based on the distribution of mean differences(can be negative),F-distribution is based on the distribution of variances between means.lFigure 14.10(p373)Xiaorong Cheng,Psychology,CCNUAssumptions of ANOVAl1.Normally distributed data within each groupl2.Homogeneity of variances,l3.Independent observations Robust when sample sizes are equal and large.Xiaorong Cheng,Psychology,CCNUObtaining variance estimateslTotal varianceMS:mean squares(estimated variance).The mean or average of the sum of squared deviation.Xiaorong Cheng,Psychology,CCNUlBetween-Groups Variance=the sum of all the observations in a groupK=the number of groupsFigure 14.11 hereXiaorong Cheng,Psychology,CCNUlWithin-groups varianceXiaorong Cheng,Psychology,CCNUWorked examplelExample table(p377)Xiaorong Cheng,Psychology,CCNUlH0:lH1:not H0 lStatistical test:F-ratiolSignificance level:=.05lSampling distribution:ldfbetween=4-1=3,dfwithin=(4-1)+(4-1)+(4-1)+(4-1)=12lCritical region for rejection of H0:Fcritical3.49Xiaorong Cheng,Psychology,CCNUlFmax test to examine homogeneity of variancesFmax=2/1.67=1.198Xiaorong Cheng,Psychology,CCNUlStep 1(p378)Xiaorong Cheng,Psychology,CCNUlStep 2(p378)Xiaorong Cheng,Psychology,CCNUlStep 3 Mean squareslp379Xiaorong Cheng,Psychology,CCNUlANOVA summary tableF(dfN,dfD)=Fo,p=_(p.05)F(3,12)=5.48,p=.013(p.05)dfN=degrees of freedom for numerator(between-groups),dfD=degrees of freedom for denominator(within-group)Xiaorong Cheng,Psychology,CCNUInterpreting the F-ratiolOmega squaredXiaorong Cheng,Psychology,CCNUlEffect size and powerlfCohen(1988):small effect size f=.10 medium effect size f=.25 large effect size f=.40Xiaorong Cheng,Psychology,CCNUlSmall effect size?lMany causeslInherent differences among humanslMeasurement with much errorXiaorong Cheng,Psychology,CCNUlTable M(table 14.2,p382)Xiaorong Cheng,Psychology,CCNUMultiple comparison of the meanslPlanned vs.unplanned comparisonslPriori(before the fact)comparison:the hypotheses for the multiple comparisons were stated before the start of the data analysis.lPosteriori(after the fact)comparison:a statistical test that is created after the data are collected and analyzed.Xiaorong Cheng,Psychology,CCNUlSimilarities:lDirectly compare meanslProtect against inflated experimentwise error.lDifferenceslWhen to make hypotheses:lPosteriori:made only after we have successfully rejected H0 with F-ratio.lPriori:made without having conducted an ANOVA.lHow to control experimentwise error:lPosteriori:use the potential number of comparisons regardless of the number of comparisons actually made.lPriori:use the number of comparisons made.-more powerfulXiaorong Cheng,Psychology,CCNUlA posteriori comparisons:Tukeys HSD(honestly significant difference)HSD=critical difference required to consider the means statistically differentqcritical=tabled value for a given level for number of means and dfwithin(Table L)MSwithin=mean square within groupsn=number of subjects in each group(equal number in each group)When ns are not equal:Xiaorong Cheng,Psychology,CCNUlTable L(p556)Xiaorong Cheng,Psychology,CCNUlFor the example:Table 14.3(p385)Xiaorong Cheng,Psychology,CCNUAn examplelData(p386)Xiaorong Cheng,Psychology,CCNUlFmaxlSSsldfslMSslFXiaorong Cheng,Psychology,CCNU