Lecture 12-14-2

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One-Way ANOVAChapter 11

Revised by Jeffrey B. Henriques, Ph.D.University of Wisconsin-Madison

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen1

t tests are great butHow often are we only interested in 2 groups?Male Vs. FemaleYoung Vs. OldSick Vs. HealthyAlcoholic Vs. Non-AlcoholicHow would you analyze more groups?

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenMake the point that often times we are far more interested in somewhat more nuanced questions than ones that can be answered using a simple 2 group dichotomization.

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Analysis of Variance (ANOVA)When to use an F distributionWorking with more than two samplesANOVAUsed with two or more nominal independent variables and an interval dependent variable


Why Not Use Multiple t Tests?The problem of too many t testsFishing for a findingProblem of Type I error

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The F Distribution

Analyzing variability to compare means

F = variance between groups variance within groups

That is, the difference among the sample means divided by the average of the sample variances

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenIt is similar to z and t tests.Variance between groups reflects differences in means.Variance within-groups is essentially an average of sample variances.

If you look at square root of F you will see that it is equal to t value, t statistic is computed on standard deviations, which are square roots of variances

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Types of VarianceBetween-groups varianceEstimate of the population variance based on differences among the meansWithin-groups varianceEstimate of population variance based on differences within (3 or more) sample distributions

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenIf both are similar, than the F will be approximately 1.00.We can think of the within-groups variance as reflecting the differences between means that we would expect to occur just by chance, while the between-groups variance reflects the groups differences in means that we have obtained from our data.6

Partitioning Variance in the Between-Groups ANOVA

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenMake the point Ive been making that this makes ANOVA just a slightly more complicated version of the same analyses weve been doing between group over within group variability7

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

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Types of ANOVAOne-way ANOVA: Hypothesis test including one nominal variable with more than two levels and a scale DVBetween-groups: more than two samples, with different participants in each sampleWithin-groups: more than two samples, with the same participants; also called repeated-measures


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Check Your LearningCan you come up with an example of a one-way between-groups ANOVA and an example of a one-way within-groups ANOVA?

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenAssumptions of ANOVAsRandom selection of samplesNormally distributed sampleHomoscedasticity: samples come from populations with the same varianceHomogeneity of variance

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenHomogeneity of variance is necessary because we are pooling the estimates of the variance in these different distributions to construct our sampling distribution.

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Everything About ANOVA but the CalculationsStep 1: Identify the populations, distribution, and assumptionsStep 2: State the null and research hypothesesH0: m1 = m2 = m3 ... = mnH1: m1 m2 m3 ... mn

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenAnother way to state the research hypothesis is at least one m is different from another m.

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Step 3: Characteristics of the comparison distribution

What are the degrees of freedom?If there are three levels of the independent variable?If there are a total of 20 participants in each of the three levels?

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzendfbetween = 3 1 = 2dfwiithin = (20-1) + (20-1) + (20-1) = 57dftotal = N 1 or dfbetween + dfwithin

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Step 4: Determine the critical value


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Determine Cutoffs for an F Distribution

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenBecause the F is a squared version of the z or t, we only have one cutoff for a two-tailed test (and all ANOVAs are two-tailed tests)15

Logic Behind the F Statistic

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenQuantifies overlap16

Logic Behind the F StatisticQuantifies overlapTwo ways to estimate population varianceBetween-groups variabilityWithin-groups variability


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The Source Table

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenThe Source Table presents important calculations and final results in a consistent, easy-to-read format.

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Formulae

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenWere presenting one group of formulae at a time, so students dont get overwhelmed.19

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenStep 5: Calculating the test statistic20


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Making a DecisionWhat is the ANOVA telling us to do about the null hypothesis?Do we reject or accept the null hypothesis?

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenIf calculated F is bigger than critical, we have a significant difference between means.

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Here the F statistic is 8.27 while the cutoff is 3.86. So we can reject the null hypothesis.An F Distribution


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Going Beyond Hypothesis TestingEffect sizePost hoc tests

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenWe will cover each separately30

Calculating Effect SizeR2 is a common measure of effect size for ANOVAs


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Post Hoc Tests to Determine Which Groups Are Different When you have three groups, and F is significant, how do you know where the difference(s) are?Tukey HSDBonferonniScheff

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenPost hoc tests are statistical procedures carried out after we have rejected the null hypothesis in ANOVA, they allow us to make multiple comparisons across several meansThere are a number of different post hoc procedures, but we will focus on Tukeys HSD test33

Tukey HSD TestWidely used post hoc test that uses means and standard error

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenWhen we have unequal Ns in our different groups we calculate a weighted sample size known as the harmonic mean.34


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One-Way Within-Groups ANOVASimilar to paired-samples t testsSame participants do something multiple (more than 2) timesAre used when we have one IV with at least 3 levels, a scale DV, and the same participants in each group

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenAlso known as repeated-measures ANOVABenefits:We reduce error due to differences between the groups. We know that the groups are identical for all of the relevant variables because each group includes exactly the same participants. We are able to reduce within-groups variability due to differences for the people in our study across groups.

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Partitioning Variance in the Within-Groups ANOVA


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Steps of Hypothesis TestingStep 1: Identify the populations, distribution, and assumptionsStep 2: State the null and research hypothesesStep 3: Determine the characteristics of the comparison distributionStep 4: Determine the critical value, or cutoffStep 5: Calculate the test statisticStep 6: Make a decision


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Step 3: Characteristics of the comparison distribution

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenF distribution is based on dfbegtween, dfwithin39

Step 4: Determine the critical values, or cutoffs

Critical F(2,8) = 4.46, for a = .05


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Formulae


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Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenStep 5: Calculate the test statisticCalculate SStotal42

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenCalculate SSbetween43

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenCalculate Sssubjects and then SSwithin

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Step 6: Make a decision

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenEffect SizeR2 can be calculated for this type of ANOVA, too


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Post hoc test to identify where you have differences if your F is significantYou need to calculate the standard error before using the Tukey HSD testTukey HSD

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenOnce we have standard error, we can calculate HSD for each pair of means. Cheap beer (34.4) versus mid-range beer (34.6): -0.074Cheap beer (34.4) versus high-end beer (52.6): -6.691Mid-range beer (34.6) versus high-end beer (52.6): -6.618

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Lecture 12-14-2

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