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
22
OUTLINE
Study overviewStudy overview
Review of MethodsReview of Methods
Description of AnalysesDescription of Analyses
ResultsResults
SummarySummary
33
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
44
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
55
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
66
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
77
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.
88
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
99
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
1010
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
1111
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?
1212
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
1313
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
1414
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
1515
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
1616
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?
1717
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
1818
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
1919
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
2121
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
2222
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
2323
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?
2424
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
2525
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
2626
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
2727
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
2828
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)
2929
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
3030
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
3131
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?