Jaymie Ticknor Quantitative Methods 2317 Sect 001 21 24 26 and 28 March 2014 Lecture 9 Review Descriptive Central Tendency Mean Median and Mode Variability spread from the mean and Inferential Variance and Standard Deviation Know everything use a Z test Not know variance over one group single sample t test Not know variance over one group before and after scores dependent t test Not know variance with two different groups independent sample t test Not know variance with three different groups ANOVA if using t test than Type I error probability increases Introduction to Analysis of Variance ANOVA Chapter 9 Works by comparing variability between what experimenter changes and within groups Compares variability between three groups Estimate 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 factors Even 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 groups Populations are exactly the same but the mean of each sample is somewhat different due to chance factors If 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 chance Different populations population means are different between group variability caused by actual differences between population means and variability within groups by chance If 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 tendency S2 Between S2 Between S2 Within S2 Within 1 However when there is a true difference between population research hypothesis the variability between groups will be based on actual differences between population means and variability due to chance When this occurs the ratio of variation between groups to variation within groups should greater than 1 research hypothesis sum of squares reject the null hypothesis S2 Within 1 F ratio this ratio variation between groups variation within groups is called the F Between S2 Between S2 Within S2 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 1 2 Within S2 X M 2 N 1 estimation of population Variance within groups S variability Within S2 S2 Variance between groups S 1 S2 2 S2 Last Ngroups 2 Between estimate the variance of the distribution of means M M GM 2 dfBetween calculate only once Grand Mean GM M Ngroups S2 dfBetween Ngroups 1 S2 F S2 Between S2 Between S2 Within M n n number of scores in each group Cutoff F value need to identify two degrees of freedom Numerator degrees of freedom between groups degrees of freedom dfBetween Ngroups 1 Denominator degrees of freedom within groups degrees of freedom dfWithin df1 df2 dfLast ANOVA does not specify where the differences are just that there will be a difference Assumptions of ANOVA mostly the same as t test for independent means sampling comparison distribution is normally distributed and populations have the same 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 comparison Procedure is similar to calculating the overall ANOVA We will calculate Between groups variance Within groups variance F statistic Cutoff F value Within groups estimate of variance will be the same as overall ANOVA In ANOVA we assume that groups come from populations with the same variance Between groups estimate of variance will be different Only comparing the two particular means F statistic is calculated in the same way Cutoff F value is found in same way dfBetween usually is exactly 1 because you are comparing two means dfBetween Ngroups 1 Step 1 Estimate the variance of the distribution of means Note only use the two means that are part of the contrast S2 M M GM 2 dfBetween Step 2 Estimate the variance for the population of individual scores dfWithin same as overall ANOVA Variance within groups S2 Within Same as overall ANOVA Variance between groups S2 Between Between S2 S2 Between S2 Within M n F statistic S2 Cutoff F value dfBetween Ngroups 1 dfWithin same as overall ANOVA Bonferroni 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 tests Bonferroni Procedure if you have multiple planned contrasts you use a more strict significance level Divide critical significance level by number of planned contrasts Post Hoc Comparisons Compare all the pairs of means more exploratory Different procedures available like Bonferroni to correct for increased chances of Type 1 error Effect Size for ANOVA R2 is a measure of the proportion of variance accounted for percentage of how much we did or changed to the groups For example of all the variability in people s happiness scores the character they favor accounts for 20 of that variability The other 80 is due to other factors Between dfBetween S2 Between dfBetween S2 Within dfWithin R2 S2 Another name for R2 is 2 eta squared R2 ranges from 0 to 1 Small Medium Large 01 06 14 Writing Up ANOVA Results F dfBetween dfWithin 800 p 001 Structural Model in ANOVA focus is on deviations how far a score deviates from the mean find means and then calculate grand mean Person s deviation from the grand mean is a combination of A person s deviation from the mean of their group A person s group s deviation from the Grand Mean Score grand mean group mean grand mean total deviation Next step is to sum the deviations in entire sample Need to square the deviations so that they don t cancel out because some are positive and some are negative Sum of the squared deviations from each person s score to the grand mean is