ANOVA II (Part 1) Class 15

Follow-up Points size of sample (n) and power of test. How are “inferential stats” inferential?

Logic of F Ratio F = Differences Among Treatment Means Differences Among Subjects Treated Alike F = Treatment Effects Experiment Error + Experiment Error F = Between Group Differences Within Group Differences

Variance CodeCalculationMeaning Mean Square Between Groups MS A SS A df A Between groups variance Mean Square Within Groups MS S/A SS S/A df S/A Within groups variance Mean Squares Calculations Degrees of Freedom (df) for between groups = (no. groups – 1) Degrees of Freedom (df) for within groups = (no. subjects – no. groups)

Decision Rule Regarding F Reject null hypothesis when F (x,y) observed >  (m,n) Reject null hypothesis when F (1,8) observed > F (1,8) = 8.88 > Decision : Reject null hypothesis Accept alternative hypothesis

Analysis of Variance Summary Table One Factor (One Way) ANOVA Source of Variation Sum of df Mean Square F Significance of Squares (MS) F Between Groups Within Groups Total25.889

TYPE I AND TYPE II ERRORS

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

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

Avoiding Type I and Type II Errors Avoiding Type I error: 1. ??? sample size 2. Reduce random error a. ??? b. ??? c. Pilot testing, etc. Avoiding Type II error 1. Reduce size of ????, BUT a. Not permitted by sci. community b. But, OK in some rare applied contexts

Avoiding Type I and Type II Errors Avoiding Type I error: 1. Increase sample size 2. Reduce random error a. Standardized instructions b. Train experimenters c. Pilot testing, etc. Avoiding Type II error 1. Reduce size of rejection region, BUT a. Not permitted by sci. community b. But, OK in some rare applied contexts

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

Non-Normal Distributions Distort 2%-3%

Effect of Different Variances on ANOVA

Effect of Different Variances on ANOVA

Additivity of Components AS ij =  + (  j -  ) + (AS ij -  j )

UNEQUAL SAMPLE SIZES Not a problem if: a. Differences in sample sizes is small b. Size of smallest sample is relatively large Is a problem if: a. Differences in sample sizes is large b. Samples differ in variances

Unplanned and Unequal Subject Loss 1. Subject loss due to "real world" circumstances 2. Subjects fail to meet inclusion criterion 3. Subjects fail to meet response-level criterion

Unequal Subject Loss and Compromised Randomness NTotal Subject Loss Motivated Subject Loss (n = 15) Unmotivated Subject Loss (n = 15) Peer Study Condition 305 (.17) 2 (.13) 3 (.20) Solo Study Condition 309 (.30) 2 (.13) 7 (.47)

Variance as a Descriptive Statistic How much do groups differ in their within groups variance? One-Way ANOVA??? Test for Homogeneity of Variance μ 1 = μ 2 = μ 3 = μ x σ 1 = σ 2 = σ 3 = σ x SPSS Conducts ??? Test for Homogeneity of Variance

Variance as a Descriptive Statistic How much do groups differ in their within groups variance? One-Way ANOVALevene Test for Homogeneity of Variance μ 1 = μ 2 = μ 3 = μ x σ 1 = σ 2 = σ 3 = σ x SPSS Conducts Levene Test for Homogeneity of Variance

Birth Order Means

Limitations of Main Effects Show “what” but not “???” Fail to account for the “what ifs” –Cannot show ???-eration –Cannot account for underlying ???

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

Verbal Definitions of Interaction Effects (Keppel, 178) 1. "Two variables interact when the effect of 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. ??? 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 ???. 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 ??? -eration.

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.

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

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

Birth Order Main Effect:NO Gender Main Effect:YES Interaction:N0

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

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

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

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

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