Schedule for Remainder of Semester    1. ANOVA: One way, Two way 2. Planned contrasts  3. Correlation and Regression 4. Moderated Multiple Regression 5. Power Analyses 6. Survey design  7. Non-experimental designs   IF TIME PERMITS 8. Writing up research   Quiz 2: Nov. 9 -- up to and including one-way ANOVA Quiz 3: Dec. 5 – What we’ve covered by Dec. 5 Class Assignment: Assigned Nov. 28, Due Dec. 7 Final: Dec. 19

HANDS ON STATS PRACTICE SPSS Demo in Computer Lab (Hill Hall Rm. 124) Thursday, Nov. 16 5:30 to 7:30 Hill Hall, Room 126 Homework: Extra Credit: 3 Pts full credit, 1 pt partial credit Homework corresponds to Computer Lab

F Distribution Notation   "F (1, 8)" means: The F distribution with: 1 df in the numerator (1 df associated with variance between treatment groups (= between-group variation)) and 8 df in the denominator (8 df associated with variance around group means (= within-group variation))

F Distribution for (2, 42) df Empirical Sampling (Monte Carlo Simulations) Theoretical Sampling

F is One Distribution; F‘ is Infinite Distributions Hello: I’m F. I contain Mean Group 1, Mean Group 2, Mean Group 3, Etc. Hello: I’m F ' I contain Mean Group 1, Mean Group 2, Mean Group 3, Etc. Hello: I’m F ' I contain Mean Group 1, Mean Group 2, Mean Group 3, Etc. Hello: I’m F ' I contain Mean Group 1, Mean Group 2, Mean Group 3, Etc.

Percent cases null hypothesis is true Criterion F and p Value For F (2, 42) = 3.48 Percent cases null hypothesis is true

F and F' Distributions (from Monte Carlo Experiments) Note how often we reject H1 when true! And the small proportion of H1 accepted when actually true.

Which Distribution Do Data Support: F or F′?   If F is correct, then Ho supported: u1 = u2 (First born = Last born) If F' is correct, then H1 supported: u1  u2 (First born ≠ Last born)

Critical Values for F (1, 8) [Result of Birth Order and Rating of Disclosers] What must our F be in order to reject null hypothesis? ≥ 5.32

Decision Rule Regarding F   Reject null hypothesis when F observed >  (m, n) Reject null hypothesis when F observed > 5.32 (1, 8). F (1,8) = 8.88 >  = 5.32 Decision: Reject or Accept null hypothesis? Reject or Accept alternative hypothesis? Have we proved alt. hypothesis? Format for reporting our result: F (1,8) = 8.88, p < .05 F (1,8) = 8.88, p < .02 also OK, based on our results. Conclusion: First Borns regard help-seekers as less "active" than do Last Borns. No, we supported it. There's a chance (p < .05), that we are wrong.

Summary of One Way ANOVA    1. Specify null and alt. hypotheses   2. Conduct experiment 3. Calculate F ratio = Between Group Diffs Within Group Diffs  4. Does F support the null hypothesis? i.e., is Observed F > Criterion F, at p < .05? p > .05, accept null hyp. p < .05, accept alt. hyp. BUT: OK to report p < .10 = “marginal difference” Sometimes OK to report p < .15 = “non significant trend”

TYPE I AND TYPE II ERRORS

Errors in Hypothesis Testing Reality Null Hyp. True Null Hyp. False Alt. Hyp. False Alt. Hyp. True Decision Reject Null Incorrect Correct Accept Alt. Type I Error Accept Null Correct Incorrect: Reject Alt Type II Error Type I Error Type II Error

Avoiding Type I and Type II Errors Avoiding Type I error: 1. Reduce the size of the Type I rejection region (i.e., go from p < .05 to p < .01 for example). Avoiding Type II error (= Expt’l Power) 1. Reduce size of Type II rejection region, BUT a. Not permitted by basic sci. community b. But, OK in some rare applied contexts 2. Reduce random error (within-groups variance) a. Increase sample size b. Improve experimental methods/design 1. Standardized instructions 2. Train experimenters c. Pilot testing , etc.

ANOVA II (Part 1) Class 17

Assumptions of Methods Class Kent is not inventing facts 2. Kent is actually a psychology professor 3. Readings are accurate 4. Statistics isn’t voodoo 5. This class is real, not a bad dream

Assumptions of ANOVA 2. Homogeneity of error variance   1. Normally distributed error variance 2. Homogeneity of error variance 3. Independence of error components 4. Additivity of components 5. Equal sample sizes

Opera Appreciation as a Function of Substance Sampling Opera Enjoyment Score Equal Variance Normal Variance Equal n’s Random assign’d Soda Group: n = 30, X = 2.5 σ = 2.6 Vodka Group: n = 30, X = 4.5 σ = 2.6 Ss randomly assigned to Soda Group or Vodka Group

Non-Normal Distributions Distort 2%-3% (Platykurtotic)

Effect of Different Variances on ANOVA: Violation of Homogeneity Assumption

Additivity of Components   ASij =  + (j - ) + (ASij - j) Individual Score Total Mean Group Mean minus Total Mean Individ. Score minus Total Mean

UNEQUAL SAMPLE SIZES Not a problem if sizes unequal and: a. Differences in sample sizes is small, or b. Size of smallest sample is relatively large (n > 30, e.g.) Is a problem if sizes unequal AND: a. Differences in sample sizes is large b. Samples differ in their variances

Unplanned and Unequal Subject Loss   1. Subject loss due to "real world" circumstances 2. Subjects fail to meet inclusion criterion Subjects fail to meet response-level criterion Problem occurs if unplanned loss is systematic, not random

Unequal Subject Loss and Compromised Randomness   N Total Subjects Quit Motivated Subjects Quit Unmotivated Subjects Quit Vodka Condition 30 5 (.17) 3 (.10) 2 (.07) Soda Condition 9 (.30) 8 (.27) 1 (.03)

ANOVA is Robust to Which Kinds of Violations?   1. Normally distributed error variance 2. Homogeneity of error variance 3. Independence of error components 4. Additivity of components 5. Equal sample sizes Yes Yes No No But not a worry Diffs are small, or smallest sample large AND Diffs not due to systematic attrition Yes, if:

Variance as a Descriptive Statistic   How much do groups differ in their within groups variance? One-Way ANOVA Levene Test for Homogeneity of Variance μ1 = μ2 = μ3 = μx σ1 = σ2 = σ3 = σx SPSS Conducts Levene Test for Homogeneity of Variance

Appreciate Friends if Suppress or Express Scary Movie Thoughts and Feelings Suppress Cond Express Cond Friend Appreciation Score How do conditions differ? Suppress Group: n = 10, σ = 0.69 Confide Group: n = 10, σ = 1.51 Does variability of suppress group differ from variability of confide group?

Levene's Test of Homogeneity of Variance Is a sig. Levene's Test a big problem? No. Recall ANOVA robust to violation of equal var. Levene's Test output How fix this problem? Data transformations. Run more subs. But, again ANOVA is robust!

Multiple Factor ANOVA

Multiple Factor Tests Birth Order Means

Anatomy of Factorial Design "FX" = "Effects"

Limitations of Main Effects Show “what” but not “why” Fail to account for the “what ifs” Cannot show moderation Cannot account for underlying causes

X X

Gender Main Effect X X

Birth Order Main Effect X X

Verbal Definitions of Interaction Effects (Keppel, 178) 1. "Two variables interact when the effect of (at least) one variable changes at different levels of the other variable".   2. "An interaction is present when the simple main effects of one variable are not the same at different levels of the second variable". 

Implications of Interaction   1. Main effects, alone, will not fully describe the results. 2. Each factor (or IV) must be interpreted in terms of the factor(s) with which it interacts. 3. Analysis of findings, when an interaction is present, will focus on individual treatment means rather than on overall factor (IV) means. 4. Interaction indicates moderation.

Interactions are Non-Additive Relationships Between Factors   1. Additive: When presence of one factor changes the expression of another factor consistently, across all levels. 2. Non-Additive: When the presence of one factor changes the expression of another factor differently, at different levels.

Ordinal and Disordinal Interactions

Eyeballing Interactions and Main Effects * Dem GOP * * X X X North South X Dem GOP * * * X X North South

Birth Order Main Effect: Gender Main Effect: Interaction: NO NO NO

Birth Order Main Effect: Gender Main Effect: Interaction: YES NO NO

Birth Order Main Effect: Gender Main Effect: Interaction: NO YES NO

Birth Order Main Effect: Gender Main Effect: Interaction: YES YES NO

Birth Order Main Effect: Gender Main Effect: Interaction: NO NO YES

Birth Order Main Effect: Gender Main Effect: Interaction: YES NO YES

Birth Order Main Effect: Gender Main Effect: Interaction: NO YES YES

Birth Order Main Effect: Gender Main Effect: Interaction: YES YES YES