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August+5th - Quantitative Methods in Psychology Psychology...

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Unformatted text preview: Quantitative Methods in Psychology Psychology August 4th, 2009: ANOVA week! Jeff Vietri corresponding to Chapter 14 in the corresponding text text Repeated Measures ANOVA Repeated Up until now we’ve just covered one-way Up ANOVA (the most basic type) ANOVA – This ANOVA merely determines if there are This significant differences between samples significant However, its possible that differences However, between samples could be attributed to the people in each group the Or, the differences are masked by Or, between subjects variability between Repeated-Measures ANOVA Repeated-Measures Just like repeated-samples t-test, Just Repeated-measures ANOVA uses the same people in different treatment conditions conditions – This reduces or may even eliminate extra This error error – This is an extremely powerful test – Often the same people are assessed at Often different points in time (or under different conditions) Repeated Measures ANOVA Repeated Recall: Independent measures ANOVA ANOVA In repeated measures, individual difference ONLY contributes to differences within treatments, because same individuals are used in all treatments. Variance-between Variance-within Independent measures Treatment effect + Experimental error + individual differences Experimental error + individual differences Repeated measures Treatment effect + Experimental error Experimental error + individual differences Now: Repeated-measures ANOVA Now: Repeated-measures ANOVA F= Variance between treatments Variance due to chance/error Now, we can eliminate individual differences in repeated measures ANOVA differences F= Variance between treatments (without individual differences) Variance due to error (without individual differences) = Treatment effect + error (excluding individual differences) Error (excluding individual differences) = Variance between treatments Variance within treatments – Variance between individuals Repeated Measures: SSTotal Repeated • SSTotal = sum of squares for the entire set of scores (without regard to group) of Same as with one-way ANOVA SSTotal = ΣX SSTotal 2 ( ΣX ) − N 2 G = ΣX − N 2 2 G = Grand Total G = ΣX dftotal = N - 1 Repeated Measures: SSBetween Repeated • SSBetween is the sum of squares of group mean differences differences Pretending that each mean of a group is an Pretending individual score you use the same process for the sum of squares the • dfbetween = k – 1 Again, same formula as with one-way ANOVA SS Between T12 T2 2 T3 2 G 2 − = + + n n2 n3 N 1 T = group total Repeated Measures: SSWithin Repeated Within Groups variance in One-way Within ANOVA is attributed to both individual differences and random error differences Repeated-measures ANOVA attempts to Repeated-measures separate out these two components separate • SSwithin = SSerror + SSsubjects • SSsubjects = individual differences – This analyses accounts for individual This differences which strengthens the test differences Repeated Measures: SSWithin Repeated • SSwithin is the sum of squares for each group added together group • SSwithin = SS1 + SS2 + SS3 + …+ SSk SS within • dfwithin = N – k Same as with regular ANOVA Repeated Measures: SSBetween Subjects Repeated Represents individual differences so they can be Represents removed removed 2 SS BetweenSubjects 2 2 Pn1 P P2 G2 =1+ + ... + − k k k N • dfbetween subjects = n1 – 1 P represents person totals represents For each person, square the person’s total and For divide by the number of samples Add all these up Add • Subtract [G2/N] Repeated Measures: SSError Repeated • SSError = SSwithin – SSbetween subjects • dferror = (N – k) – (n1 – 1) error – “N” is the number of people in study – “n1” is the number of people in each group For a repeated-measures ANOVA, which of the following is computed differently, compared to an independent-measures ANOVA? independent-measures A. total SS B. between treatment between SS SS C. within treatment SS D. the denominator of the the F ratio the Example Example Effect Size Effect Just like with other statistical tests, Just Repeated Measures ANOVA has a way to calculate effect size calculate Effect size gives a way to get a sense of Effect how important your results are how Just because they are significant does not Just mean that your IV has a huge impact on your DV your Post Hoc Tests Post Tukey’s HSD can be used for Repeated Tukey’s Measures Measures • Again, same formula as One-way ANOVA Again, except must use MSError instead of MSWithin except • Also, df for q-table is dferror not dfwithin for MS error HSD = q n1 Two-way ANOVA Two-way Two-way ANOVA Two-way Up until now we have only discussed One-way Up ANOVA ANOVA Where one discrete IV (“factor”) predicts one DV However, often times in research we not just However, concerned with one variable, researchers may want to study more than one at once to Remember confounding variables? We can include Remember them into our design them Two-way ANOVA Two-way One-way ANOVA design: Does my drug affect cholesterol levels (control, 5mg, Does 10mg)? 10mg)? Two-way ANOVA design: Does my drug affect cholesterol levels (control, 5mg, Does 10mg)? 10mg)? Does an exercise program affect cholesterol levels? Does drug affect cholesterol levels differently for people Does on exercise program vs. those not on exercise program? on Two-way ANOVA Two-way Two-way ANOVA is looking at two IVs and Two-way their effect on the DV their Two-way ANOVA has three null hypotheses: H0a: μA1 = μA2 = μA3 H0b: μB1 = μB2 = μB3 H0c: There is no interaction between A and B. There This last hypothesis tests what is called an This interaction interaction Two-way ANOVA Two-way Since we are looking at two variables at the Since same time, we need to know the effect of the first variable, the second variable, and the combined effects of both variables combined Example Example makes you feel good makes you feel good + = ? Example Example makes you feel good makes you feel good + makes you dead = Interaction Interaction An interaction means that the relationship An between one variable and the dependent variable is affected by another variable variable Q: What is the relationship of X and Y? If the answer is “it depends”, then you have If an interaction an What is the relationship between temperature and test scores? (A) Higher temperatures are associated with lower scores. (B) IT DEPENDS! If there is high humidity then higher temperatures are associated with lower scores. If there is low humidity, there is no relationship between temperature scores. Assumptions Assumptions Observations within each treatment condition Observations must be independent must – Independent observations Distribution of scores within each group must be Distribution normal normal – Normality Variances of scores within each group must be Variances normal normal – Homogeneity of variance – Test with Fmax test (just like t-test) but we won’t be doing that in this course doing Important stuff about ANOVAs What do you need to know to use ANOVAs? What do you need to know for the quiz/exam? Terms When to use Logic When to post-hoc How to interpret reports What patterns of results look like ANOVA terms Factor An independent (or quasi-independent) variable Statistics training Podcast listening sex Levels How many different levels of the factor do you have? None, undergraduate stats, graduate stats No listening, listening Female, male ANOVA terms Main effect In multifactorial ANOVA, this is if the factor affects the DV, ignoring the effects of the other factors Interaction In mulitfactorial ANOVA, this is when the effect of a factor depends on the level of another factor i.e., students do well in Jeff’s classes as long as it’s not stats Alcohol has a greater effect in women than men Types of ANOVA One-way (a.k.a. single-factor) One factor, multiple levels, btw-Ss designs Repeated measures One factor, multiple levels, within-Ss designs A one-way ANOVA with 2 levels of the factor is functionally equivalent to an independent-samples t-test A R-M ANOVA with 2 measurements is functionally equivalent to a paired-samples t-test Multifactorial ANOVA >1 factor, >1 level/factor, can have btw-Ss, within-Ss, or mixed designs ANOVA jargon # by # ANOVA i.e., 2 by 2 ANOVA This describes each factor by how many levels each has 2 by 2 means there are 2 factors, each with 2 levels 5 by 2 means there are 2 factors, one with 5 levels, one with 2 levels 5 by 2 by 2 means there are 3 factors, one with 5 and two factors with 2 levels each When should you use which ANOVA? One-way (single-factor) ANOVA: When you have a single factor with >=2 levels, and each level is a separate sample R-M ANOVA: Need post-hoc tests if levels > 2 When you’re trying to conduct a more sensitive test, single factor, multiple levels, each level applied to the same sample at different times Multifactorial ANOVA When you’re measuring the effect of >1 IV (or QIV) on the DV. Especially when you think the two might interact. ANOVA logic Not all that different from t-test logic Numerator: variability in scores due to treatments + error Denominator: variability in scores due to error If the numerator is much bigger than denominator, much of the variance in scores is due to our treatment ANOVA logic Numerator is variability between treatment conditions Denominator is either within-treatment differences or error, depending on test As your treatment means get more different, the numerator, and therefore the ration, gets bigger As the scores within a given treatment get more different, denominator gets bigger, F-ratio gets smaller, failure to reject null Bigger F-ratios = more the total variance is due to the treatment. More likely to reject null hypothesis Ratio logic The F-ratio is the ratio of one variance (mean squares) to another. Variances are always positive F >= 0 If there’s no treatment effect, Numerator = error Denominator = error F=1 Hypotheses Null is that there are no differences Therefore, reject the null means >= 1 mean is different from >= another mean Don’t know which unless there are only 2 means Post-hoc tests! Post-hoc tests Tukey’s HSD & Scheffe are pretty common Only do when: Reject null AND >2 groups How to interpret ANOVA in lit Reject null: F (df_btw, df_within/error) = F-value, p < alpha F (1, 28) = 7.23, p < .05 Fail to reject null F (df_btw, df_within/error) = F-value, p < alpha F (1, 28) = 2.10, p > .05 For multivariate Each effect will have it’s own F-ratio: Main effect for each factor There was no main effect of sex, F(1, 480) = 1.9, p > . 05 There was a main effect of studying, F (1, 480) = 6.2, p < .05 And for the interaction And studying had a stronger effect for women than for men, F (1, 480) = 5.1, p < .05 How to read reports This is important! Looking at the pattern of means can often be informative even before you perform calculations You can make good guesses at main effects & interactions group individual Main effect of studying technique? Is the mean for individual-study (averaged across sex) higher than the mean for group-study (averaged across sex)? Men Women Main effect of sex? Is the mean for men (averaged across studying technique) higher than the mean for women (averaged across studying technique)? Men Women Interaction between the two? Is the relationship between studying technique and quiz score the same for men and women? Are the lines not parallel? ...
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