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Jaymie TicknorQuantitative Methods 2317 Sect. 00121, 24, 26, and 28 March 2014Lecture #9Review: Descriptive (Central Tendency: Mean, Median and Mode; Variability: spread from the mean) and Inferential (Variance and Standard Deviation)Know everything: use a Z testNot know variance over one group: single sample t testNot know variance over one group before and after scores: dependent t testNot know variance with two different groups: independent sample t testNot know variance with three different groups: ANOVA (if using t test, than Type I error probability increases)Chapter #9 : Introduction to Analysis of Variance (ANOVA) :Works by comparing variability between (what experimenter changes) and within groupsCompares variability between three groupsEstimate two types of variability: variability within each group (due to chance, not due to your experiment) and between each group (selection of people to certain groups)Same population, but the mean of each particular sample is somewhat different due to chance factorsEven if the null hypothesis is true and different populations have the exact same mean, variance, and shape, there will still be some variability; variability between groups is influenced by the same chance factors that influence the variability within groupsPopulations are exactly the same, but the mean of each sample is somewhat different due to chance factorsIf the null hypothesis is not true, and the populations have different means, variability between groups will be caused by actual differences between population means and variability within groups by chanceDifferent populations; population means are different; between group variability caused by actual differences between population means, and variability within groups by chanceIf there is no true difference between populations, the variability between groups and the variability within groups are based on the same thing (variation due to chance)When this occurs, the ratio of variation between groups to variation within groups should be about 1 = null hypothesis (fail to reject the null hypothesis); sum of squares measures of variability, not central tendencyS2Between = S2Within ; S2Between / S2Within = 1However, when there is a true difference between population (research hypothesis), the variability between groups will be based on actual differences between populationmeans and variability due to chanceWhen this occurs, the ratio of variation between groups to variation within groups shouldgreater than 1 = research hypothesis; sum of squares; reject the null hypothesisS2Between > S2Within ; S2Between / S2Within > 1F-ratio: this ratio (variation between groups / variation within groups) is called the F-ratio or F-statistic: just like the z-statistic or t-statistic but using different table; F-distribution skewed to the right (positive), critical F-value, and F table; M = 1Variance within groups (S 2 Within ): S2 = ∑ (X – M)2 / (N – 1) (estimation of population variability)S2Within = (S21 + S22 + … + S2Last) / NgroupsVariance between groups (S 2 Between ): estimate the variance of the distribution of means:Grand Mean (GM): ∑ M / NgroupsS2M = ∑ (M – GM)2 / dfBetween (calculate only once)(dfBetween = Ngroups - 1)S2Between = S2M* (n); n = number of scores in each groupF = S2Between / S2WithinCutoff F-value: need to identify two degrees of freedomNumerator degrees of freedom = between-groups degrees of freedomdfBetween = Ngroups - 1Denominator degrees of freedom = within-groups degrees of freedomdfWithin = df1 + df2 + … + dfLastANOVA: does not specify where the differences are, just that there will be a differenceAssumptions of ANOVA: mostly the same as t-test for independent means: sampling/comparison distribution is normally distributed and populations have thesame variance (homogeneity of variance)Planned Contrasts: ANOVA is an omnibus (overall) test; tells you that there is a difference, but it does not tell you where the difference is; researchers use planned contrasts or planned comparison to look at particular comparisonProcedure is similar to calculating the overall ANOVAWe will calculate:Between groups varianceWithin groups varianceF-statisticCutoff F-valueWithin groups estimate of variance will be the same as overall ANOVAIn ANOVA, we assume that groups come from populations with the same varianceBetween groups estimate of variance will be differentOnly comparing the two particular meansF-statistic is calculated in the same wayCutoff F-value is found in same waydfBetween = usually is exactly 1, because you are comparing two meansdfBetween = Ngroups – 1dfWithin = same as overall ANOVAVariance within groups (S2Within)Same as overall ANOVAVariance between groups (S2Between)Step 1: Estimate the variance of the distribution of means (Note: only use the two means that are part of the contrast)S2M = ∑ (M – GM)2 / dfBetweenStep 2: Estimate the variance for the population of individual scoresS2Between = S2M* (n)F-statistic = S2Between / S2WithinCutoff F-valuedfBetween = Ngroups – 1dfWithin = same as overall ANOVABonferroni Procedure: if you run many planned contrasts, your chances of Type 1 error (rejecting the null when shouldn’t) increase; same problem as running multiple t-testsBonferroni Procedure—if you have multiple planned contrasts, you use a morestrict significance levelDivide critical significance level by number of planned contrastsPost Hoc Comparisons:Compare all the pairs of means; more exploratoryDifferent procedures available (like Bonferroni) to correct for increased chances of Type 1 errorEffect Size for ANOVAR2 is a measure of the proportion of variance accounted for; percentage of how much we did or changed to the groupsFor example, of all the variability in people’s happiness scores, the character they favor accounts for 20% of that variabilityThe other 80% is due to other factorsR2 = (S2Between * dfBetween) / [(S2Between * dfBetween) + (S2Within * dfWithin)]Another name for R2 is η2 (eta squared)R2 ranges from 0 to 1Small .01Medium .06Large .14Writing Up ANOVA Results: F(dfBetween, dfWithin) = 800, p < .001Structural Model in ANOVA: focus is on deviations (how far a score deviates from the mean); find means and then calculate grand meanPerson’s deviation from the grand mean is a combination ofA person’s deviation from the mean of their groupA person’s group’s deviation from the Grand Mean(Score - grand mean) + (group mean - grand mean) = total

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