GLM UNIVARIATE, ANOVA, AND ANCOVA

159 Pages·2013·4.19 MB·English

by  G. David Garson

GLM UNIVARIATE, ANOVA, AND ANCOVA 2013 Edition

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GLM UNIVARIATE, ANOVA, AND ANCOVA 2013 Edition

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GLM UNIVARIATE, ANOVA, AND ANCOVA 2013 Edition

Table of Contents

Overview ....................................................................................................................................... 11
Key Concepts ................................................................................................................................. 15
Why testing means is related to variance in analysis of variance ........................................... 15
One-way ANOVA ...................................................................................................................... 16
Simple one-way ANOVA in SPSS .......................................................................................... 16
Simple one-way ANOVA in SAS ........................................................................................... 20
Two-way ANOVA ...................................................................................................................... 23
Two-way ANOVA in SPSS ..................................................................................................... 24
Two-way ANOVA in SAS ....................................................................................................... 27
Multivariate or n-way ANOVA .................................................................................................. 29
Regression models ................................................................................................................... 29
Parameter estimates (b coefficients) for factor levels ............................................................ 31
Parameter estimates for dichotomies ................................................................................. 32
Significance of parameter estimates ................................................................................... 32
Research designs ........................................................................................................................... 32
Between-groups ANOVA design............................................................................................... 32
Completely randomized design ........................................................................................... 34
Full factorial ANOVA ............................................................................................................ 34
Balanced designs ................................................................................................................. 35
Latin square designs ............................................................................................................ 36
Graeco-Latin square designs ............................................................................................... 37
Randomized Complete Block Design (RCBD ANOVA) .......................................................... 37
Split plot designs .................................................................................................................. 39
Mixed design models ........................................................................................................... 39
Random v. fixed effects models ............................................................................................... 41
In SPSS .................................................................................................................................. 41
In SAS ................................................................................................................................... 42
Linear mixed models (LMM) vs. general linear models (GLM) ........................................... 43

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GLM UNIVARIATE, ANOVA, AND ANCOVA 2013 Edition

Effects .................................................................................................................................. 43
Treating a random factor as a fixed factor .......................................................................... 43
Mixed effects models .......................................................................................................... 44
Nested designs ..................................................................................................................... 44
Nested designs ......................................................................................................................... 45
In SPSS .................................................................................................................................. 46
In SAS ................................................................................................................................... 48
Treatment by replication design ......................................................................................... 48
Within-groups (repeated measures) ANOVA designs .............................................................. 48
Counterbalancing ................................................................................................................ 49
Reliability procedure ........................................................................................................... 50
Repeated measures GLM in SPSS ........................................................................................ 50
Repeated measures GLM in SAS .......................................................................................... 50
Interpreting repeated measures output ............................................................................. 51
Variables........................................................................................................................................ 51
Types of variables ..................................................................................................................... 52
Dependent variable ............................................................................................................. 52
Fixed and random factors .................................................................................................... 53
Covariates ............................................................................................................................ 53
WLS weights ......................................................................................................................... 53
Models and types of effects.......................................................................................................... 54
Full factorial models ................................................................................................................. 54
Effects ....................................................................................................................................... 55
Main effects ......................................................................................................................... 55
Interaction effects ............................................................................................................... 55
Residual effects .................................................................................................................... 58
Effect size measures...................................................................................................................... 59
Effect size coefficients based on percent of variance explained ............................................. 59
Partial eta-squared .............................................................................................................. 59

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GLM UNIVARIATE, ANOVA, AND ANCOVA 2013 Edition

Omega-squared ................................................................................................................... 60
Herzberg's R2 ....................................................................................................................... 61
Intraclass correlation ........................................................................................................... 61
Effect size coefficients based on standardized mean differences ........................................... 61
Cohen's d ............................................................................................................................. 61
Glass's delta ......................................................................................................................... 63
Hedge's g.............................................................................................................................. 64
Significance tests ........................................................................................................................... 64
F-test ......................................................................................................................................... 64
Reading the F value ............................................................................................................. 64
Example 1 ............................................................................................................................ 65
Example 2 ............................................................................................................................ 65
Significance in two-way ANOVA .......................................................................................... 66
Computation of F ................................................................................................................. 66
F-test assumptions............................................................................................................... 66
Adjusted means ................................................................................................................... 67
Lack of fit test ...................................................................................................................... 67
Power level and noncentrality parameter .......................................................................... 68
Hotelling's T-Square ............................................................................................................. 69
Planned multiple comparison t-tests ....................................................................................... 69
Simple t-test difference of means ....................................................................................... 70
Bonferroni-adjusted t-test ................................................................................................... 71
Sidak test ............................................................................................................................. 72
Dunnett's test ...................................................................................................................... 73
HSU's multiple comparison with the best (MCB) test ......................................................... 73
Post-hoc multiple comparison tests......................................................................................... 73
The q-statistic ...................................................................................................................... 74
Output formats: pairwise vs. multiple range ...................................................................... 75
Tests assuming equal variances .......................................................................................... 75

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GLM UNIVARIATE, ANOVA, AND ANCOVA 2013 Edition

Least significant difference (LSD) test ................................................................................. 75
The Fisher-Hayter test ......................................................................................................... 76
Tukey's test, a.k.a. Tukey honestly significant difference (HSD) test .................................. 76
Tukey-b test, a.k.a. Tukey's wholly significant difference (WSD) test................................. 78
S-N-K or Student-Newman-Keuls test ................................................................................. 78
Duncan test .......................................................................................................................... 79
Ryan test (REGWQ) .............................................................................................................. 79
The Shaffer-Ryan test .......................................................................................................... 81
The Scheffé test ................................................................................................................... 81
Hochberg GT2 test ............................................................................................................... 83
Gabriel test .......................................................................................................................... 85
Waller-Duncan test .............................................................................................................. 85
Tests not assuming equal variances .................................................................................... 85
Tamhane's T2 test ................................................................................................................ 85
Games-Howell test .............................................................................................................. 86
Dunnett's T3 test and Dunnett's C test ............................................................................... 87
The Tukey-Kramer test ........................................................................................................ 87
The Miller-Winer test .......................................................................................................... 87
More than one multiple comparison/post hoc test ................................................................ 87
Example ............................................................................................................................... 87
Contrast tests ................................................................................................................................ 89
Overview .................................................................................................................................. 89
Types of contrasts .................................................................................................................... 90
Deviation contrasts .............................................................................................................. 90
Simple contrasts .................................................................................................................. 90
Difference contrasts ............................................................................................................ 90
Helmert contrasts ................................................................................................................ 90
Repeated contrasts .............................................................................................................. 90
Polynomial contrasts ........................................................................................................... 91

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GLM UNIVARIATE, ANOVA, AND ANCOVA 2013 Edition

Custom hypothesis tables ........................................................................................................ 91
Custom hypothesis tables index table ................................................................................. 91
Custom hypothesis tables.................................................................................................... 92
Estimated marginal means ........................................................................................................... 94
Overview .................................................................................................................................. 94
EMM Estimates table ............................................................................................................... 96
Other EMM output ................................................................................................................... 99
EMM Pairwise comparisons table ....................................................................................... 99
EMM Univariate tests table ................................................................................................. 99
Profile plots .......................................................................................................................... 99
GLM Repeated Measures ............................................................................................................ 100
Overview ................................................................................................................................ 100
Key Terms and Concepts ........................................................................................................ 101
Within-subjects factor ....................................................................................................... 101
Repeated measures dependent variables ......................................................................... 102
Between-subjects factors .................................................................................................. 103
Covariates .......................................................................................................................... 103
Models ............................................................................................................................... 104
Type of sum of squares ...................................................................................................... 105
Balanced vs. unbalanced models ...................................................................................... 105
Estimated marginal means ..................................................................................................... 106
Pairwise comparisons ............................................................................................................. 107
Statistics options in SPSS ........................................................................................................ 108
Descriptive statistics .......................................................................................................... 108
Hypothesis SSCP matrices.................................................................................................. 109
Partial eta-squared ............................................................................................................ 109
Within-subjects SSCP matrix and within-subjects contrast effects. .................................. 110
Multivariate tests. .................................................................................................................. 111
Univariate vs. multivariate models .................................................................................... 112

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Box's M test ....................................................................................................................... 113
Mauchly's test of sphericity ............................................................................................... 113
Univariate tests of within-subjects effects ............................................................................ 114
Parameter estimates .............................................................................................................. 116
Levene's test ........................................................................................................................... 117
Spread-versus-level plots ....................................................................................................... 118
Residual plots ......................................................................................................................... 118
Lack of fit test ......................................................................................................................... 120
General estimable function .................................................................................................... 121
Post hoc tests ......................................................................................................................... 121
Overview ............................................................................................................................ 121
Profile plots for repeated measures GLM .............................................................................. 124
Example ............................................................................................................................. 124
Contrast analysis for repeated measures GLM ...................................................................... 126
Types of contrasts for repeated measures ........................................................................ 127
Simple contrasts example .................................................................................................. 128
Saving variables in repeated measures GLM ......................................................................... 129
Cook's distance .................................................................................................................. 130
Leverage values ................................................................................................................. 130
Assumptions ................................................................................................................................ 131
Interval data ........................................................................................................................... 131
Homogeneity of variances ..................................................................................................... 131
Homogeneity of variance ....................................................................................................... 132
Appropriate sums of squares ................................................................................................. 136
Multivariate normality ........................................................................................................... 137
Adequate sample size ............................................................................................................ 138
Equal or similar sample sizes .................................................................................................. 138
Random sampling ................................................................................................................... 138
Orthogonal error .................................................................................................................... 139

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Data independence ................................................................................................................ 139
Recursive models ................................................................................................................... 139
Categorical independent variables ........................................................................................ 139
The independent variable is or variables are categorical. ..................................................... 139
Continuous dependent variables ........................................................................................... 139
Non-significant outliers .......................................................................................................... 139
Sphericity ................................................................................................................................ 140
Assumptions related to ANCOVA: .......................................................................................... 141
Limited number of covariates ........................................................................................... 141
Low measurement error of the covariate ......................................................................... 141
Covariates are linearly related or in a known relationship to the dependent .................. 141
Homogeneity of covariate regression coefficients ............................................................ 142
No covariate outliers ......................................................................................................... 142
No high multicollinearity of the covariates ....................................................................... 143
Additivity ............................................................................................................................ 143
Assumptions for repeated measures ..................................................................................... 143
Frequently Asked Questions ....................................................................................................... 144
How do you interpret an ANOVA table? ................................................................................ 145
Isn't ANOVA just for experimental research designs? ........................................................... 147
Should I standardize my data before using ANOVA or ANCOVA? ......................................... 147
Since orthogonality (uncorrelated independents) is an assumption, and since this is rare in
real life topics of interest to social scientists, shouldn't regression models be used instead of
ANOVA models? ..................................................................................................................... 147
Couldn't I just use several t-tests to compare means instead of ANOVA? ............................ 147
How does counterbalancing work in repeated measures designs? ...................................... 148
How is F computed in random effect designs? ...................................................................... 149
What designs are available in ANOVA for correlated independents? ................................... 149
If the assumption of homogeneity of variances is not met, should regression models be used
instead? .................................................................................................................................. 150

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Is ANOVA a linear procedure like regression? How is linearity related to the "Contrasts"
option? ................................................................................................................................... 150
What is hierarchical ANOVA or ANCOVA? ............................................................................. 150
Is there a limit on the number of independents which can be included in an analysis of
variance? ................................................................................................................................ 151
Which SPSS procedures compute ANOVA? ........................................................................... 151
I have several independent variables, which means there are a very large number of possible
interaction effects. Does SPSS have to compute them all? ................................................... 151
Do you use the same designs (between groups, repeated measures, etc.) with ANCOVA as
you do with ANOVA? .............................................................................................................. 151
How is GLM ANCOVA different from traditional ANCOVA? .................................................. 152
What are paired comparisons (planned or post hoc) in ANCOVA? ....................................... 152
Can ANCOVA be modeled using regression? ......................................................................... 152
How does blocking with ANOVA compare to ANCOVA? ....................................................... 152
What is the SPSS syntax for GLM repeated measures? ......................................................... 153
What is a "doubly repeated measures design"? .................................................................... 154
Bibliography ................................................................................................................................ 155

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