Using Factor Analysis andUsing Factor Analysis and Cronbach'sCronbach'sAlpha To Ascertain Relationships Alpha To Ascertain Relationships

Between Questions of a Dietary Between Questions of a Dietary Behavior Questionnaire Behavior Questionnaire

Eric GrauEric Grau

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OUTLINE

Study overviewStudy overview

Review of MethodsReview of Methods

Description of AnalysesDescription of Analyses

ResultsResults

SummarySummary

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Study Overview: Goal

Objective: Review and revise a draft questionnaire Objective: Review and revise a draft questionnaire on dietary behavior that was developed by another on dietary behavior that was developed by another contractorcontractor

–– Review content of questionsReview content of questions–– Review order of questionsReview order of questions–– Review whether questions should be droppedReview whether questions should be dropped

Goal: Develop a questionnaire (based on draft) that Goal: Develop a questionnaire (based on draft) that can assess respondents’ adherence to the Dietary can assess respondents’ adherence to the Dietary Guidelines for AmericansGuidelines for Americans

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Study Overview: Stages

Our focus: assessing results of field test Our focus: assessing results of field test with respect to relationships between itemswith respect to relationships between items

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Study Overview: Field test

Field TestField Test

–– Size of test: 453 white, AfricanSize of test: 453 white, African--American, and American, and Hispanic women food stamp recipientsHispanic women food stamp recipients

–– Time limit: core questions in the instrument Time limit: core questions in the instrument should take less than 15 minutes to administershould take less than 15 minutes to administer

–– Resolve: Level of redundancy within topical Resolve: Level of redundancy within topical modules modules

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Study Overview: Questionnaire Organization

Questionnaire organized into modulesQuestionnaire organized into modules

–– Dietary modules: recording weekly Dietary modules: recording weekly consumption of various food groupsconsumption of various food groups

–– Attitude and behavior modules: questions Attitude and behavior modules: questions about attitudes and behaviors related to about attitudes and behaviors related to food and nutritionfood and nutrition

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Review of Methods: Factor Analysis Definition

Describe a set of p random variables in terms Describe a set of p random variables in terms of a smaller number of unobserved random of a smaller number of unobserved random variables called FACTORSvariables called FACTORS

Factors are determined by interpreting Factors are determined by interpreting coefficients in factor model called LOADINGScoefficients in factor model called LOADINGS

Orthogonal transformation of factor loadings, Orthogonal transformation of factor loadings, called a ROTATION, allows for easier called a ROTATION, allows for easier interpretation of factors.interpretation of factors.

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Review of Methods: Factor Analysis

Variance of X can be decomposed into Variance of X can be decomposed into

common variance + specific variancecommon variance + specific variance

Data Assumed to be Multivariate NormalData Assumed to be Multivariate Normal

Methods of Estimation of Factor LoadingsMethods of Estimation of Factor Loadings–– Principle Component AnalysisPrinciple Component Analysis–– Principle Factor AnalysisPrinciple Factor Analysis–– Maximum Likelihood EstimationMaximum Likelihood Estimation

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Review of Methods: Factor Analysis Goodness-of-fit Measures

MSA (measure of sampling adequacy): MSA (measure of sampling adequacy): partial correlations between each pair of partial correlations between each pair of variables controlling for all other variablesvariables controlling for all other variables–– indicates how well variables fit in factor indicates how well variables fit in factor

modelmodel

Kaiser’s MSA is overall measure: Kaiser’s MSA is overall measure: –– Values below 0.5 are unacceptableValues below 0.5 are unacceptable–– Values above 0.8 indicate factor model fits Values above 0.8 indicate factor model fits

wellwell

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Review of Methods: Cronbach’s Alpha

A correlation coefficient that describes how A correlation coefficient that describes how well a group of items focuses on a single well a group of items focuses on a single idea or construct idea or construct

Establishes consistency of questions asked Establishes consistency of questions asked in different ways about a single attributein different ways about a single attribute

High levels indicate High levels indicate –– relative absence of item error variancerelative absence of item error variance–– Items contribute to a reliable scale for one Items contribute to a reliable scale for one

attributeattribute

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Review of Methods: Cronbach’sAlpha

Two calculations: Two calculations: –– Raw alpha (based on correlations)Raw alpha (based on correlations)–– Standardized alpha (based on Standardized alpha (based on covariancescovariances))

Rule of thumb: Rule of thumb:

>= 0.70 considered acceptable>= 0.70 considered acceptable

What does a lower level mean? What does a lower level mean?

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Description of Analyses

Separate analyses were done within dietary and Separate analyses were done within dietary and behavior/attitude topicsbehavior/attitude topics

Methods of estimation used for Factor Analysis:Methods of estimation used for Factor Analysis:–– Principle Factor AnalysisPrinciple Factor Analysis–– Maximum LikelihoodMaximum Likelihood

Factors rotated usingFactors rotated using VARIMAXVARIMAX rotationrotation

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Description of Analyses: Data Issues

Weekly consumption data positively skewedWeekly consumption data positively skewed–– Analyze data on square root scaleAnalyze data on square root scale

Some variables have only four categoriesSome variables have only four categories–– Other methods may be more appropriateOther methods may be more appropriate–– Ordinal response problematic: assumes Ordinal response problematic: assumes

equal distance between levelsequal distance between levels

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Results: Fruit and Vegetables (with Fries)

0.010.010.560.56Orange vegetablesOrange vegetables

0.410.410.590.59Dark green Dark green vegetablesvegetables

--0.290.29--0.050.05French friesFrench fries--0.260.260.390.39Potatoes (not fries)Potatoes (not fries)0.260.260.680.68All vegetableAll vegetable0.090.090.320.32Unsweetened JuiceUnsweetened Juice0.080.080.580.58FruitFruitFACTOR 2FACTOR 2FACTOR 1FACTOR 1VARIABLEVARIABLE

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Results: Fruit and Vegetables (with Fries)

First factor: true fruits and vegetablesFirst factor: true fruits and vegetablesSecond factor: distinguishes between green Second factor: distinguishes between green vegetables and potatoesvegetables and potatoes

89% of common variance explained by first 89% of common variance explained by first two two eigenvalueseigenvalues

Kaiser’s Overall MSA = 0.74Kaiser’s Overall MSA = 0.74

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Results: Fruit and Vegetables (with Fries)

Cronbach’s Cronbach’s alpha = 0.58alpha = 0.58

What does this mean?What does this mean?–– High level of error variance for items to be High level of error variance for items to be

considered reliable for single construct considered reliable for single construct scalescale

–– Not a single construct?Not a single construct?

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Results: Fruit and Vegetables (no Fries)

0.390.390.390.39Orange vegetablesOrange vegetables

0.130.130.730.73Dark green Dark green vegetablesvegetables

0.420.420.090.09Potatoes (not fries)Potatoes (not fries)0.340.340.650.65All vegetableAll vegetable0.170.170.290.29Unsweetened JuiceUnsweetened Juice0.510.510.350.35FruitFruitFACTOR 2FACTOR 2FACTOR 1FACTOR 1VARIABLEVARIABLE

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Results: Fruit and Vegetables (no Fries)

First factor: distinguishes type of vegetableFirst factor: distinguishes type of vegetableSecond factor: not clearSecond factor: not clear

88% of common variance explained by first 88% of common variance explained by first two two eigenvalues eigenvalues (78% by first (78% by first eigenvalueeigenvalue))

Kaiser’s Overall MSA = 0.74Kaiser’s Overall MSA = 0.74

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Results: Fruit and Vegetables (no Fries)

Cronbach’sCronbach’s alpha = 0.68alpha = 0.68

What does this mean?What does this mean?–– Removing single item that “didn’t fit” Removing single item that “didn’t fit”

improved alpha markedlyimproved alpha markedly

2020

Nine variables:Nine variables:–– Five refer to food consumptionFive refer to food consumption–– Four refer to behaviorsFour refer to behaviors

Problems:Problems:–– Some ordinal responsesSome ordinal responses–– Two behavioral variables are binaryTwo behavioral variables are binary–– Some levels needed collapsingSome levels needed collapsing–– Kaiser’s Overall MSA = 0.56Kaiser’s Overall MSA = 0.56

Results: Weight Consciousness

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Results: Weight Consciousness

0.000.000.090.090.710.71Attempted to lose weightAttempted to lose weight--0.110.110.350.350.050.05Snack or eat meals at TVSnack or eat meals at TV0.300.30--0.020.02--0.030.03Eat breakfast in morningEat breakfast in morning

0.060.060.110.110.690.69Switched to healthier dietSwitched to healthier diet0.080.080.450.450.020.02Fast foodFast food0.160.160.430.430.070.07SodaSoda0.150.150.260.260.110.11Sweetened fruit drinksSweetened fruit drinks0.460.460.120.120.100.10Fruit/vegetables as snacksFruit/vegetables as snacks0.480.480.090.090.030.03Fruit as dessertFruit as dessertFACFAC. 3. 3FACFAC. 2. 2FACFAC. 1. 1VARIABLEVARIABLE

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Results: Weight Consciousness

First factor: actions to improve healthFirst factor: actions to improve healthSecond factor: unhealthy eating habitsSecond factor: unhealthy eating habitsThird factor: healthy eating habitsThird factor: healthy eating habits

Two clusters across the three factors:Two clusters across the three factors:–– Switched to healthier diet/attempted to lose Switched to healthier diet/attempted to lose

weightweight–– Fruit as dessert/fruit or vegetable as snackFruit as dessert/fruit or vegetable as snack

All of common variance explained by first three All of common variance explained by first three eigenvalueseigenvalues

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Results: Weight Consciousness

Cronbach’sCronbach’s alpha = 0.48alpha = 0.48

–– High level of error variance for items to be High level of error variance for items to be considered reliable for single construct considered reliable for single construct scalescale

–– Not a single construct?Not a single construct?

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Summary

RedundanciesRedundancies–– Eating fish and eating dry beansEating fish and eating dry beans–– Behaviors: switched to healthier diet and Behaviors: switched to healthier diet and

attempted to lose weightattempted to lose weight–– Eating fruit as dessert and eating fruit or Eating fruit as dessert and eating fruit or

vegetables as snacksvegetables as snacks

In many cases, alpha may not be an appropriate In many cases, alpha may not be an appropriate measure, given the number of underlying factors is measure, given the number of underlying factors is greater than one.greater than one.

Final recommendation for ERS: Remove one of the Final recommendation for ERS: Remove one of the fish and dry bean questionsfish and dry bean questions

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Acknowledgements

This work was done as part of a contract that This work was done as part of a contract that MPR had with the U.S. Department of MPR had with the U.S. Department of Agriculture, Economic Research ServiceAgriculture, Economic Research Service

Project: Project: Development of a Questionnaire on Development of a Questionnaire on Dietary Behavior for Use in LowDietary Behavior for Use in Low--Income Income PopulationsPopulations (MPR project number 6191)(MPR project number 6191)

–– Project Officer: David Smallwood, ERSProject Officer: David Smallwood, ERS–– MPR Project Director: Rhoda Cohen MPR Project Director: Rhoda Cohen

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Review of Methods: Factor Model

Factor Model:Factor Model:

XXii = a= ai1i1FF11 + a+ ai2i2FF22 + … + + … + aaimimFFmm + + εεii; i = 1,2,… p; i = 1,2,… p

XXii = = ith ith variable, centered with mean 0 variance 1variable, centered with mean 0 variance 1εεii = = ith ith error (specific factor)error (specific factor)aaijij = = jth jth factor loading for Xfactor loading for XiiFFjj = uncorrelated common factors with unit = uncorrelated common factors with unit

variancevariance

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Review of Methods: Factor Analysis Rotations

Factor ModelFactor Model–– Orthogonal transformation of factor Orthogonal transformation of factor

loadings corresponds to a rotation of the loadings corresponds to a rotation of the coordinate axescoordinate axes

–– Communalities and specific variances Communalities and specific variances remain unchangedremain unchanged

–– Original loadings may not be readily Original loadings may not be readily interpretableinterpretable——rotate until simple structure rotate until simple structure is achievedis achieved

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Review of Methods: Factor Analysis Rotations

Factor Model Factor Model VARIMAXVARIMAX Rotation:Rotation:Maximize V, which is proportional toMaximize V, which is proportional to

ΣΣjj VarVar((aaijij22))

Spreads out the squares of the loadings on Spreads out the squares of the loadings on each factoreach factorForces large and negligible coefficients in Forces large and negligible coefficients in any column of the rotated loadings matrix any column of the rotated loadings matrix (I.e., associated with each factor)(I.e., associated with each factor)

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Results: High Protein Foods

0.280.280.410.410.100.10Dry beansDry beans0.460.460.040.040.030.03Peanut butterPeanut butter0.290.290.120.120.240.24EggsEggs0.240.240.470.470.090.09FishFish0.070.070.220.220.620.62Deli meatsDeli meats0.090.090.020.020.650.65Red meat/porkRed meat/pork--0.070.070.510.510.090.09PoultryPoultryFACTOR 3FACTOR 3FACTOR 2FACTOR 2FACTOR 1FACTOR 1VARIABLEVARIABLE

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Results: High Protein Foods

First factor: measure of less healthy proteinsFirst factor: measure of less healthy proteinsSecond factor: healthier proteinsSecond factor: healthier proteinsThird factor: peanut butterThird factor: peanut butter

Beans and fish cluster together across all Beans and fish cluster together across all three factorsthree factors

All of common variance explained by first All of common variance explained by first three three eigenvalueseigenvalues

Kaiser’s Overall MSA = 0.63Kaiser’s Overall MSA = 0.63

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Results: High Protein Foods

Cronbach’sCronbach’s alpha = 0.56alpha = 0.56

–– High level of error variance for items to be High level of error variance for items to be considered reliable for single construct considered reliable for single construct scalescale

–– Not a single construct?Not a single construct?