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Simple Group Comparisons Limits you to simple explanatory variables simple potential relationships

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Complex Group Difference Designs Types Multilevel Designs – single IV, 3 or more levels Single source of possible systematic variability, but more than a simple difference possible Multifactor Designs (Factorial) – multiple IVs Multiple possible sources of systematic variability Not all effects have single variable causes

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Multilevel Designs Situations requiring a multilevel approach Characteristics of the IV appropriate representation of IV variability Level of Sense of Humor and Perceived Personality No two levels of SoH are likely to represent important differences Characteristics of the IV DV relationship a nonlinear relationship anticipated Level of Test Anxiety and Test Performance Would not expect a linear relationship

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Evaluating the Results – searching for the systematic variability in the DV no single place (difference) to assess, variability could appear in multiple places - examples or in multiple forms - examples Multilevel Designs

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Two common approaches Overall test for systematic variability - with follow-up (post hoc) tests to see where differences might exist (Means vary systematically, as opposed to no more than unsystematically) Test for fit with predicted pattern across groups Using planned comparisons or contrasts to specify pattern (Means vary systematically in a predicted form or pattern) Multilevel Designs

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Overall test for systematic variability Ho: M 1 = M 2 = M 3 ………… variability among sample means will be no greater than variability expected due to unsystematic variability Possibly two steps in the analysis 1. Test to see if Ho can be rejected If yes, there is evidence of systematic variability 2. Examine differences between means to see where systematic variability occurs Multilevel Designs

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Test for fit to predicted pattern across groups Ho: M 1 = M 2 = M 3 ………… Specify pattern expected (contrast) and test to see if systematic variability does fit that pattern - allowed as many orthogonal contrasts as there are df overall - if doing nonorthogonal contrasts, may need to adjust Type 1 L = cM, where c is the weight assigned to each mean to represent the pattern predicted (and add up to 0) Multilevel Designs

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Test for fit to predicted pattern across groups Ho: M 1 = M 2 = M 3 ………… Specify pattern expected (contrast) and test to see if systematic variability does fit that pattern Contrast to test linear relationship between Sense of humor and Liking as SOH goes up, Liking goes up Contrast to test curvilinear relationship between Test Anxiety and outcome low and high anxiety lower the outcome, relative to moderate Multilevel Designs

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Situations requiring a Multifactor approach multiple possible Causes (each sufficient, not necessary) more efficient to assess in a single design laugh at your joke (or not) similar attitudes (or not) interactions among Causes needed to produce effect (necessary, not sufficient causal influence) high choice in behaving counter to ones attitude (vs low choice) high consequences for behavior (vs low consequences) need both to be High to get effect assess possible role of an EV in affecting DV psychological vs physical disorder (IV) gender or worker with disorder (EV) Multifactor Designs

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Main Effects vs. Interaction Effects Main effects – relationship of each IV with the DV -one possible main effect for each IV in the design Interaction effects – combinations of levels of different IVs -effect of changes in one IV (on DV) depends upon level of another IV present Multifactor Designs

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Understanding Multifactor Design notation How many main effects and interactions? 2 x 2 design - Chant (2) x Program (2) Kumbaya vs Chant 2 x 4 design – Program (2) x Chant (4) None, Kumbaya, Stats 1, Stats 2 2 x 2 x 3 design – Program (2) x Gender (2) x Chant (3) Kumbaya, Stats 1, Stats 2 Multifactor Designs

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What is being compared? Numbers that represent main effects and interactions main effects in margins interactions inside look at some examples with Means 2 x 2 design 2 x 4 design 2 x 2 x 3 design Multifactor Designs 7585 75 What if the groups have unequal ns?

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Following up on significant main effects and interactions Multilevel main effects Simple main effects in interactions Multifactor Designs

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Interpreting results when there are significant main effects and interaction effects Interactions take precedence in interpretations Multifactor Designs

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Independent Levels vs.Related Levels Independent Variables in Complex Group Designs Independent Levels Multilevel Related Levels Multilevel Independent Levels Multifactor Related Levels Multifactor Mixed Multifactor Complex Group Difference Designs

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Now – for the analyses Recall Variance = (x-M) 2 df Allows you to combine all the different deviations from the Mean to get a Sum (of squares), and then to find the typical deviation (squared) by dividing by df

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Need to be able to assess multiple possible differences that might reflect systematic variability along a single dimension - Multilevel Variables/Designs and/or could be the result of multiple independent sources of systematic variability – Multifactor Designs Evaluating Results in Complex Group Difference Designs – Interval/Ratio Data

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Building from the t-test single deviation (M – M) typical deviation (se) Numerator could be treated as two deviations Group 1 mean – Population mean (estimated) Group 2 mean – Population mean (estimated) Sum the deviations (after squaring), and find the average (what do you have?) t =

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Building from the t-test single deviation (treat as 2 deviations from Mean) typical deviation (square to convert to variance) Now you could compare that variance (numerator) (may be due to a situation where Ho is not true) to the typical variance (when Ho true) by squaring the denominator

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Building from the t-test (based on deviations) single deviation (systematic and unsystematic) typical deviation unsystematic F statistic – from Analysis of Variance Variance (systematic and unsystematic) Variance unsystematic t = F =

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When you have only two groups, F = t 2, but now, with F,you can include any number of means in the numerator of the F-ratio. What you have is: estimate of population variance One estimate is sensitive to systematic variability One estimate is unaffected by systematic variability F =

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Want to estimate variance in the population population Sample 2 IV level 2 Sample 3 IV level 3 Sample variance 1Sample variance 2 Sample variance 3 Most logical strategy is to use the variances from your samples to estimate the variance in the population. Sample 1 IV level 1

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Want to estimate variance in the population population Sample 1 IV level 1 Sample 2 IV level 2 Sample 3 IV level 3 Sample mean 1Sample mean 2 Sample mean 3 Can also estimate the population variance by using means from samples, but this estimate will only be accurate when the Ho is true

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estimate of population variance using means estimate of population variance using variances Means are sensitive to systematic variability Variances are unaffected by systematic variability systematic + unsystematic variability unsystematic variability F =

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Analysis of Variance involves partitioning the variability of the DV into the variability relevant to the parts of the F ratio. Recall that: Sum of Squares df So the sum of squares reflects the variability of DV (before it is averaged) Some of the variability (sum of squares) reflects only unsystematic variability Some of the variability (sum of squares) reflects systematic + unsystematic variability Variance =

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Partitioning the Sum of Squares (in a balanced design) Two groups (for simplicity) Non chanters (25) Chanters (25)

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x Sample Mean Group Mean SSTotal – sum of the squared deviations of individual scores from the mean of the entire sample – ignoring group membership SSWithin – sum of the squared deviations of individual scores from the mean of the individuals group, based on variability within each group Sample Mean Group Mean SSBetween – sum of the squared deviations of typical score from group from the mean of the entire sample – treats individuals as if they are typical for their group All 50 students as a sample Students divided by Level of IV Students as typical members of their group When equal n, SST = SSW + SSB

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Evaluating Results in Complex Group Difference Designs – Interval/Ratio Data SST = (x – M G ) 2 as if one group of participants SSW = (x g1 – M g1 ) 2 variability within each + (x g2 – M g2 ) 2 group – relative to + (x g3 – M g3 ) 2 each groups mean SSB = n(M g1 – M G ) 2 variability among group + n(M g2 – M G ) 2 means, relative to the + n(M g3 – M G ) 2 mean of the entire sample

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Evaluating Results in Complex Group Difference Designs – Interval/Ratio Data Analysis of Variance Assumptions 1. Interval/ratio data 2. Independent observations 3. Normal sampling distribution of the means (seldom a problem) 4. Homogeneity of variances (same guidelines as for t-test)

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Example – multilevel design IV is Sense of Humor of Target Person (3 levels: Below Ave - Average - Above Ave) DV is rated liking, from low (1) to (7) high 4 participants per group (power =.09) Levels of Sense of Humor of Target Below Ave Average Above Ave Means246 Variances.67.67.67

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Deviations from Overall Sample Mean Deviations within each group from group mean Deviations of Group Means from Sample Mean

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7 1 2 4 5 3 6 Rating Subjects 1 2 3 4 5 6 7 8 9 10 11 12 Below AverageAverageAbove Average Overall sample mean Group mean

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Mean Square Between = Sum of Squares Between/df between (estimate of population variance based upon Means) Mean Square Within = Sum of Squares Within/df within (estimate of population variance based upon group Variances) MSB -- biased estimate MSW -- unbiased estimate ?? Does the variability among the means (relative to the estimated population mean) appear to be greater than the variability expected when the Null Hypothesis is true (error variability only)? F (df b, df w ) =

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The variance in each group is.8165 2 or.667 – the same as the MSW in the Table below Since this is the best estimate of the population variance If we assume each persons rating in a group was equal to the mean (typical) rating for that group, the estimated population variance is the MSB (ivsoh) Why are these the dfs NOT Partial eta 2 = SSB/SST SSB SSW SST Power to detect a moderate effect (eta 2 of 6%) =.09

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Go through the steps in SPSS Analyze General Linear Model Univariate Choose DV Choose IV (Fixed Factor) Options Estimated Marginal Means only needed in Multifactor Designs Must use for Main Effects if unequal n in cells Can use Pairwise Comparisons to get CIs for Main Effects Descriptive Statistics Estimates of Effect Size Homogeneity Tests Post hoc Choose as desired Can reset the alpha level in Options

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Examples in Handouts Multilevel ANOVA Multilevel Chi square

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Situations requiring a Multilevel approach - single IV in the design – 3 or more levels SST = SSW + SSB iv MSB MSW Multilevel Designs F (df b, df w ) =

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Situations requiring a Multifactor approach - including multiple IVs in the design so for the 2 x 2 multifactor design SST = SSW + SSB iv becomes SST = SSW + SSB iv1 + SSB iv2 + SSB interaction To answer 3 questions Multifactor Designs

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Independent Variable 1 – Music Level 1 = Rock Level 2 = Classical Independent Variable 2 = Volume Level 1 = Low Level 2 = High Dependent Variable = Number of words recalled from a list of 20. Independent Variable 1 MUSIC Rock Classical_ _ _ 13 8 10 5 sum = 90 Level 1 9 M = 10 11 M = 8 Low 7 8 M = 9 11 8 Independent Variable 2 ___________________________ ____ VOLUME 19 3 16 6 sum = 110 Level 2 18 M = 16 9 M = 6 High 13 6 M = 11 14 6 _________________ _______ _______ sum = 130 sum = 70 sum = 200 M = 13 M = 7 M s = 10 Overall Sample Mean Power =.18 For eta 2 =.06 (moderate effect)

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(1) (2) SSW (4) SST (6) SSB || (1) (2) SSW (4) SST (6) SSB Xi (Xi-Mg) (Xi-Mg)2 (Xi-MS) (Xi-MS)2 (Mg-MS) (Mg-MS)2 13 13-10 9 13-10 9 10-10 0 || 8 8-8 0 8-10 48-10 4 10 10-10 0 10-10 0 10-10 0 || 5 5-8 9 5-10 258-10 4 9 9-10 1 9-10 1 10-10 0 || 11 11-8 9 11-10 18-10 4 7 7-10 9 7-10 9 10-10 0 || 8 8-8 0 8-10 48-10 4 11 11-10 1 11-10 1 10-10 0 || 8 8-8 0 8-10 48-10 4 50 0 20 0 20 0 0 || 40 0 18 -10 38 -10 20 |G1 G2|__________________________________________________ ---------------------------------------------------|G3 G4|--------------------------------------------------- 19 19-16 9 19-10 81 16-10 36 || 3 3-6 9 3-10 496-1016 16 16-16 0 16-10 36 16-10 36 || 6 6-6 0 6-10 166-1016 18 18-16 4 18-10 64 16-10 36 || 9 9-6 9 9-10 16-1016 13 13-16 9 13-10 9 16-10 36 || 6 6-6 0 6-10 166-1016 14 14-16 4 14-10 16 16-10 36 || 6 6-6 0 6-10 166-1016 80 0 26 30 206 30 180 || 30 0 18 -20 98-2080 SST = (X i – M S ) 2 SST = 20 + 38 + 206 + 98 = 362 SSW = (X ig1 – M g1 ) 2 + (X ig 2 – M g2 ) 2 + (X ig3 – M g3 ) 2 + (X ig4 – M g4 ) 2 SSW = 20 + 18 + 26 + 18 = 82 SSBIV1 = n(M gl+g3 – M S ) 2 + n(M g2+g4 - M S ) 2 SSBIV1 = 10(13-10) 2 + 10(7-10) 2 = 180 SSBIV2 = n(M g1+g2 – M S ) 2 + n(M g3+g4 – M S ) 2 SSBIV2 = 10(9-10) 2 + 10(11-10) 2 = 20 SSBX = SSB – SSBIV1 – SSBIV2 SSBX = 280 – 180 – 20 = 80

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ANALYSIS OF VARIANCE TABLE Source SS df MS F partial eta 2 Total 362 19 Between Music 180 1 180.00 35.12.687 Volume 20 1 20.00 3.90.196 Interaction 80 1 80.00 15.61.494 Within 82 16 5.125 Critical F(1,16) = 4.49, p<.05, two-tailed test. Partial eta 2 SSB/(SSB+SSW) eta 2 SSB/SST Music 180/(180 + 82) =.687 180/362 =.497 Volume 20/(20 + 82) =.196 20/362 =.055 Interaction 80/(80+82) =.494 80/362 =.221 Total 1.377.773

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Follow up analyses that might be needed with Multifactor Analyses If Multilevel IV main effect is significant need Multiple Comparison Tests Note – if unbalanced design, use tests available in Options to insure unweighted means are used If Interaction is significant need Simple Main effects tests which could include Multiple Comparisons Go to handouts for examples

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Assumptions interval/ratio data independent observations (relatedness removed) normality of sampling distribution homogeneity of variances of difference scores (sphericity) violations of homogeneity can be serious, and inflate the F values in post hoc tests, may want to use conservative test Repeated Measures Multilevel Design

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Research – testing 3 separate recipes for Chocolate Chip Cookies IV Cookie Recipe – 3 levels DV Rated taste 1 2 3456789 Awful Delicious 5 participants, each tastes and rates each cookie (design issues?) cookie-acookie-bcookie-c Means for Ps p1579 7 p2357 5 p3138 3 p4378 6 p5579 7 M cookie 3.45.88.2 5.8 Main effect for Participants – reflects differences across the participants Main effect for Cookie

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Examples of SPSS in Handouts Show how to define RM variables in SPSS

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Combines aspects of both types (independent levels and related levels) but the error (MSW) used depends on whether the effect is: truly independent levels, or related levels that have been adjusted Assumptions interval ratio data normality of the sampling distribution independent observations for independent levels effects – homogeneity of variances for related levels effects – homogeneity of variances of difference scores (sphericity) Mixed Designs

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Mixed example Cookie Recipe (2) x Temperature (2) [ x Participants (n per group)] DV Rated taste Awful 1 2 3 4 5 6 7 8 9 Delicious Two cookies tasted by each person - within subjects effect some get warm cookies (n = 5), - between subjects effect some get room temperature (cold) cookies (n = 5), WarmCold Cookie 1 Cookie 2MCookie 1 Cookie 2M 1576 354 2687 475.5 3877.5 354 4687 354 5687 465 6.27.66.9 3.45.64.5 SSB – uncontaminated by relatedness SSB (Temperature – 2 levels) SSW (Participants within groups – variability among means of participants) SSW – systematic, but contaminated by relatedness SSB (cookies – 2 repeated measures levels) SSB interaction (2 x 2) SSW variability among participants 2 x 2 Mixed Multifactor (treated as if 2 x 2 x 5) – but ignore Participants effects Partition SS into

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