Intro to ANOVAIntro to ANOVA

What Is ANOVA?What Is ANOVA?

Analysis of Variance.Analysis of Variance. This is a hypothesis-testing procedure that This is a hypothesis-testing procedure that

is used to evaluate mean differences is used to evaluate mean differences between two of more treatments (or between two of more treatments (or populations).populations).

What is the difference between this and t What is the difference between this and t tests?tests? ANOVA allows us to look at more than two ANOVA allows us to look at more than two

groups at once.groups at once.

Some TerminologySome Terminology

In analysis of variance, the variable In analysis of variance, the variable (independent or quasi-independent) that (independent or quasi-independent) that designates the groups being compared is designates the groups being compared is called a factor.called a factor.

The individual conditions or values that The individual conditions or values that make up a factor are called the levels of make up a factor are called the levels of the factor.the factor.

FormulaeFormulae

SSw = SSw = (X (Xijij – X. – X.jj))22

SSbet = SSbet = nnjj (X. (X.jj – X..) – X..)22

SStot = SStot = (X (Xijij – X..) – X..)22

MSw = SSw/dfwMSw = SSw/dfw MSb = SSb/dfbMSb = SSb/dfb F = MSb/MSwF = MSb/MSw

Looking At ErrorLooking At Error

The F statistic is calculated by comparing The F statistic is calculated by comparing the error between groups (theoretically the error between groups (theoretically due to the treatment effect) to the error due to the treatment effect) to the error within groups (theoretically due to chance, within groups (theoretically due to chance, or error). We look to see if differences due or error). We look to see if differences due to our treatment effect is proportionally to our treatment effect is proportionally greater than differences due to chance greater than differences due to chance alone.alone.

Reporting ANOVAReporting ANOVASourceSource SSSS dfdf MSMS

BetweenBetween F = F =

WithinWithin

TotalTotal

Examples From ExcelExamples From Excel

Reading The TableReading The Table

Effect SizeEffect Size

For ANOVA we use eta squaredFor ANOVA we use eta squared 22

CalculatedCalculated 22 = SS = SSbetweenbetween/SS/SStotaltotal

Post Hoc TestsPost Hoc Tests

These are additional hypothesis tests that These are additional hypothesis tests that are done after an ANOVA to determine are done after an ANOVA to determine exactly which mean differences are exactly which mean differences are significant and which are not.significant and which are not.

If you reject the null, and there are three or If you reject the null, and there are three or more treatments, you may wish to explore more treatments, you may wish to explore which groups contain the mean which groups contain the mean differences.differences.

Accumulation of Type I ErrorAccumulation of Type I Error

Experimentwise alpha: This is the overall Experimentwise alpha: This is the overall probability of a Type I error that can probability of a Type I error that can accumulate over a series of separate accumulate over a series of separate hypothesis tests. Typically, the hypothesis tests. Typically, the experiment-wise alpha is substantially experiment-wise alpha is substantially greater than the value of alpha used for greater than the value of alpha used for any one of the individual tests.any one of the individual tests.

Planned ComparisonsPlanned Comparisons

Planned comparisons refer to specific Planned comparisons refer to specific mean differences that are relevant to mean differences that are relevant to specific hypotheses the researcher had in specific hypotheses the researcher had in mind before the study was conducted.mind before the study was conducted. For planned comparisons, we generally don’t For planned comparisons, we generally don’t

worry about accumulation of Type I error. worry about accumulation of Type I error. What we will do is use a smaller alpha level to What we will do is use a smaller alpha level to

test these hypotheses, often dividing alpha by test these hypotheses, often dividing alpha by the number of planned comparisons.the number of planned comparisons.

Unplanned ComparisonsUnplanned Comparisons

Unplanned comparisons involve sifting Unplanned comparisons involve sifting through the data to find significant results. through the data to find significant results. When doing this you have to worry about When doing this you have to worry about accumulation of Type I error in your accumulation of Type I error in your results. Two commonly used procedures results. Two commonly used procedures to protect against this accumulation are to protect against this accumulation are Tukey’s HSD and the Scheffe test.Tukey’s HSD and the Scheffe test.

Tukey’s HSDTukey’s HSD

Tukey’s Honestly Significant Difference (HSD) is Tukey’s Honestly Significant Difference (HSD) is used to compare two treatment conditions. If the used to compare two treatment conditions. If the mean difference between those treatment mean difference between those treatment conditions exceeds Tukey’s HSD, you conclude conditions exceeds Tukey’s HSD, you conclude that there is a significant difference between the that there is a significant difference between the treatmentstreatments

HSD = qHSD = q√√ MSw/n MSw/n Where the value of q is found in Table B.5 (p. 708)Where the value of q is found in Table B.5 (p. 708) Tukey’s requires that n be the same for all treatments.Tukey’s requires that n be the same for all treatments.

The Scheffe TestThe Scheffe Test

Scheffe is very conservative, therefore if Scheffe is very conservative, therefore if you use Scheffe and find a significant you use Scheffe and find a significant difference, you can feel safe you have difference, you can feel safe you have found a true difference.found a true difference.

For Scheffe’s test, you calculate a For Scheffe’s test, you calculate a serperate between groups MSb for the two serperate between groups MSb for the two groups you are looking at, and then groups you are looking at, and then compare this new MSb to the compare this new MSb to the experimentwise MSw.experimentwise MSw.

F as Compared to TF as Compared to T

For a two sample test, an ANOVA or a t-For a two sample test, an ANOVA or a t-test can be used. For these situations you test can be used. For these situations you would get F = twould get F = t22

AssumptionsAssumptions

The observations within each sample must be The observations within each sample must be independentindependent

The populations must be normalThe populations must be normal The populations must has equal variances The populations must has equal variances

(homogeneity of variance)(homogeneity of variance) This is important to test, and we due so by Hartley’s F-This is important to test, and we due so by Hartley’s F-

max test for homogeneity of variance (table on 704)max test for homogeneity of variance (table on 704) Compute the sample variance for each sampleCompute the sample variance for each sample Divide the largest by the smallestDivide the largest by the smallest K is the number of samplesK is the number of samples n is the sample size for each sample (assuming equal n is the sample size for each sample (assuming equal

n’s)n’s)