CSD 5100CSD 5100Introduction to Research Introduction to Research

Methods in CSDMethods in CSD

Analysis of VarianceAnalysis of Variance

The Statistical Procedure for The Statistical Procedure for Factorial DesignFactorial Design

Factorial DesignFactorial Design

Experimental procedure for testing the Experimental procedure for testing the null hypothesis when two independent null hypothesis when two independent variables, or factors, are considered variables, or factors, are considered simultaneously in a research studysimultaneously in a research study

The ANOVA is the statistical procedure for The ANOVA is the statistical procedure for analyzing data from factorial designsanalyzing data from factorial designs

Advantages of Factorial Advantages of Factorial DesignDesign

• EfficiencyEfficiency

• ControlControl

• Allows the study of the interaction Allows the study of the interaction between two or more independent between two or more independent variablesvariables

Data TablesData Tables

How Does the ANOVA Work?How Does the ANOVA Work?

The ANOVA partitions the The ANOVA partitions the total variation of scores total variation of scores into four componentsinto four components

• Within cell varianceWithin cell variance• Variation among the Variation among the

row (age) meansrow (age) means• Variation among the Variation among the

column (gender) meanscolumn (gender) means• Variation due to the Variation due to the

interaction of age x interaction of age x gendergender

Within cell

Age

Age x gender

gender

What is Within Cell What is Within Cell Variance?Variance?

This is calculated as the variability of This is calculated as the variability of all individual cells of the data tableall individual cells of the data table

• Source of natural variabilitySource of natural variability

• Source of “error”Source of “error”

Mean Squares: Estimation of Mean Squares: Estimation of VarianceVariance

MS = SS / dfMS = SS / df

F-DistributionF-Distribution

The ANOVA uses the The ANOVA uses the F-statistic, which is F-statistic, which is based on an F-based on an F-distribution rather distribution rather than the normal than the normal distributiondistribution

The Three Null HypothesesThe Three Null Hypotheses

F F age age = MS = MS age age / MS / MS within within Age Main Effect Age Main Effect

F F gender gender = MS = MS gender gender / MS / MS within within Gender Main Gender Main

EffectEffect

F F AxG AxG = MS = MS AxG AxG / MS / MS within within Interaction Interaction

How Are the ANOVA Results How Are the ANOVA Results Reported?Reported?

The ANOVA The ANOVA SummarySummary

Here is the summary Here is the summary for voiced stops for voiced stops last timelast time

Illustrating Significant Illustrating Significant EffectsEffects

Here is an illustration Here is an illustration of the main effect of the main effect of age for the of age for the voiced phonemesvoiced phonemes

How Are the ANOVA Results How Are the ANOVA Results Reported?Reported?

The ANOVA The ANOVA SummarySummary

Here is the summary Here is the summary for voiceless stops for voiceless stops last timelast time

Illustrating Significant Illustrating Significant EffectsEffects

Here is an illustration Here is an illustration of the interaction of the interaction of age x gender for of age x gender for the voiceless the voiceless phonemesphonemes

Post-Hoc Tests for the Post-Hoc Tests for the ANOVAANOVA

Tests of the hypotheses in the two-way Tests of the hypotheses in the two-way ANOVA are only the first steps in the ANOVA are only the first steps in the analysis of a set of dataanalysis of a set of data

Multiple comparisons proceduresMultiple comparisons procedures

The Tukey MethodThe Tukey Method

Also known as the honestly significant Also known as the honestly significant difference testdifference test

Designed to make all pair-wise Designed to make all pair-wise comparisons of means while comparisons of means while maintaining the experiment-wise error maintaining the experiment-wise error rate at the pre-established probability rate at the pre-established probability levellevel

The test statistic is QThe test statistic is Q

Illustrating Significant Illustrating Significant EffectsEffects

Here is an illustration Here is an illustration of the main effect of the main effect of age for the of age for the voiced phonemesvoiced phonemes

Tukey Summary for the Tukey Summary for the Significant Age Effect (Voiced)Significant Age Effect (Voiced)

The pair-wise The pair-wise comparisons comparisons deemed deemed significantly significantly different by the different by the Tukey test are Tukey test are in bold.in bold.

Tukey Procedure for the Tukey Procedure for the Significant Age x Gender Significant Age x Gender Interaction (Voiceless)Interaction (Voiceless)