Department of Cognitive Science Michael Kalsher Adv. Experimental Methods & Statistics PSYC 4310 /...

Department of Cognitive Science

Michael Kalsher

Adv. Experimental Methods & Statistics

PSYC 4310 / COGS 6310

Factor Analysis

1PSYC 4310 Advanced Experimental Methods and Statistics © 2013, Michael Kalsher

PSYC 4310/6310 Advanced Experimental Methods and Statistics © 2009, Michael Kalsher and James Watt 2

• What Are Factors?

• Representing Factors– Graphs and Equations

• Extracting factors– Methods and Criteria

• Interpreting Factor Structures– Factor Rotation

• Reliability– Cronbach’s alpha

• Writing Results

Outline


When to use Factor Analysis?• Data ReductionData Reduction• Identification of underlying latent structuresIdentification of underlying latent structures

- Clusters of correlated variables are termed factors– Example:Example:

– Factor analysis could potentially be used to identify Factor analysis could potentially be used to identify the characteristics (out of a large number of the characteristics (out of a large number of characteristics) that make a person popular.characteristics) that make a person popular.

Candidate characteristics: Candidate characteristics: Level of social skills, selfishness, how Level of social skills, selfishness, how interesting a person is to others, the amount of time they spend interesting a person is to others, the amount of time they spend talking about themselves (Talk 2) versus the other person (Talk talking about themselves (Talk 2) versus the other person (Talk 1), their propensity to lie about themselves.1), their propensity to lie about themselves.


The R-Matrix

Meaningful clusters of large correlation coefficients between subsets of variables suggests these variables are measuring aspects of the same underlying dimension.

Factor 1: The better your social skills, the more interesting and talkative you tend to be.

Factor 2: Selfish people are likely to lie and talk about themselves.


What is a Factor?• Factors can be viewed as classification axes

along which the individual variables can be plotted.

• The greater the loading of variables on a factor, the more the factor explains relationships among those variables.

• Ideally, variables should be strongly related to (or load on) only one factor.


Graphical Representation of a factor plot

Note that each variable loads primarily on only one factor.

Factor loadings tell use about the relative contribution that a variable makes to a factor


Mathematical Representation of a factor plot

Yi = b1X1i +b2X2i + … bnXn + εi

Factori = b1Variable1i +b2Variable2i + … bnVariablen + εi

• The equation describing a linear model can be applied to the description of a factor.

• The b’s in the equation represent the factor loadings observed in the factor plot.

Note: there is no intercept in the equation since the lines intersection at zero and hence the intercept is also zero.



Sociabilityi = b1Talk 1i +b2Social Skillsi + b3interesti + b4Talk 2 + b5Selfishi + b6Liari + εi

There are two factors underlying the popularity construct: general sociability and consideration.

We can construct equations that describe each factor in terms of the variables that have been measured.

Considerationi = b1Talk 1i +b2Social Skillsi + b3interesti + b4Talk 2 + b5Selfishi + b6Liari + εi



Sociabilityi = 0.87Talk 1i +0.96Social Skillsi + 0.92Interesti + 0.00Talk 2 - 0.10Selfishi + 0.09Liari + εi

The values of the “b’s” in the two equations differ, depending on the relative importance of each variable to a particular factor.

Considerationi = 0.01Talk 1i - 0.03Social Skillsi + 0.04interesti + 0.82Talk 2 + 0.75Selfishi + 0.70Liari + εi

Ideally, variables should have very high b-values for one factor and very low b-values for all other factors.

Replace values of b with the co-ordinate of each variable on the graph.


Factor Loadings

• The b values represent the weights of a variable on a factor and are termed Factor Loadings.

• These values are stored in a Factor pattern matrix (A). • Columns display the factors (underlying constructs) and rows

display how each variable loads onto each factor.

VariablesFactors

Sociability Consideration

Talk 1 0.87 0.01

Social Skills 0.96 -0.03

Interest 0.92 0.04

Talk 2 0.00 0.82

Selfish -0.10 0.75

Liar 0.09 0.70

PSYC 4310/6310 Advanced Experimental Methods and Statistics © 2009, Michael Kalsher and James Watt

11

Factor Scores• Once factors are derived, we can estimate each

person’s Factor Scores (based on their scores for each factor’s constituent variables).

• Potential uses for Factor Scores.- Estimate a person’s score on one or more factors.- Answer questions of scientific or practical interest (e.g., Are females are

more sociable than males? using the factors scores for sociability).

• Methods of Determining Factor Scores- Weighted Average (simplest, but scale dependent)- Regression Method (easiest to understand; most typically used)- Bartlett Method (produces scores that are unbiased and correlate only with their

own factor).- Anderson-Rubin Method (produces scores that are uncorrelated and

standardized)


Approaches to Factor Analysis• Exploratory

– Reduce a number of measurements to a smaller number of indices or factors (e.g., Principal Components Analysis or PCA).

– Goal: Identify factors based on the data and to maximize the amount of variance explained.

• Confirmatory– Test hypothetical relationships between measures and more

abstract constructs.– Goal: The researcher must hypothesize, in advance, the number

of factors, whether or not these factors are correlated, and which items load onto and reflect particular factors. In contrast to EFA, where all loadings are free to vary, CFA allows for the explicit constraint of certain loadings to be zero.


Communality• Understanding variance in an R-matrix

– Total variance for a particular variable has two components:

• Common Variance – variance shared with other variables.• Unique Variance – variance specific to that variable (including

error or random variance).

• Communality– The proportion of common (or shared) variance present in a

variable is known as the communality.– A variable that has no unique variance has a communality of

1; one that shares none of its variance with any other variable has a communality of 0.


Factor Extraction: PCA vs. Factor Analysis

– Principal Component Analysis. A data reduction technique that represents a set of variables by a smaller number of variables called principal components. They are uncorrelated, and therefore, measure different, unrelated aspects or dimensions of the data.– Principal Components are chosen such that the first one accounts for as

much of the variation in the data as possible, the second one for as much of the remaining variance as possible, and so on.

– Useful for combining many variables into a smaller number of subsets.

– Factor Analysis. Derives a mathematical model from which factors are estimated.– Factors are linear combinations that maximize the shared portion of the

variance underlying latent constructs. – May be used to identify the structure underlying such variables and to

estimate scores to measure latent factors themselves.


Factor Extraction: Eigenvalues & Scree Plot

• Eigenvalues– Measure the amount of variation accounted for by each factor. – Number of principal components is less than or equal to the number

of original variables. The first principal component accounts for as much of the variability in the data as possible. Each succeeding component has the highest variance possible under the constraint that it be orthogonal to (i.e., uncorrelated with) the preceding components.

• Scree Plots– Plots a graph of each eigenvalue (Y-axis) against the factor

with which it is associated (X-axis).– By graphing the eigenvalues, the relative importance of each

factor becomes apparent.


Factor Retention Based on Scree Plots


Kaiser (1960) recommends retaining all factors with eigenvalues greater than 1.

- Based on the idea that eigenvalues represent the amount of variance explained by a factor and that an eigenvalue of 1 represents a substantial amount of variation.

- Kaiser’s criterion tends to overestimate the number of factors to be retained.

Factor Retention: Kaiser’s Criterion


• Students often become stressed about statistics (SAQ) and the use of computers and/or SPSS to analyze data.

• Suppose we develop a questionnaire to measure this propensity (see sample items on the following slides; the data can be found in SAQ.sav).

• Does the questionnaire measure a single construct? Or is it possible that there are multiple aspects comprising students’ anxiety toward SPSS?

Doing Factor Analysis: An Example


Doing Factor Analysis: Some Considerations

• Sample size is important! A sample of 300 or more will likely provide a stable factor solution, but depends on the number of variables and factors identified.

• Factors that have four or more loadings greater than 0.6 are likely to be reliable regardless of sample size.

• Correlations among the items should not be too low (less than .3) or too high (greater than .8), but the pattern is what is important.


Total Variance Explained

7.290 31.696 31.696 7.290 31.696 31.696 3.730 16.219 16.219

1.739 7.560 39.256 1.739 7.560 39.256 3.340 14.523 30.7421.317 5.725 44.981 1.317 5.725 44.981 2.553 11.099 41.8421.227 5.336 50.317 1.227 5.336 50.317 1.949 8.475 50.317.988 4.295 54.612.895 3.893 58.504.806 3.502 62.007.783 3.404 65.410.751 3.265 68.676.717 3.117 71.793.684 2.972 74.765.670 2.911 77.676.612 2.661 80.337.578 2.512 82.849.549 2.388 85.236.523 2.275 87.511.508 2.210 89.721.456 1.982 91.704.424 1.843 93.546.408 1.773 95.319.379 1.650 96.969.364 1.583 98.552.333 1.448 100.000

Component

1234567891011121314151617181920212223

Total% of

VarianceCumulative

% Total% of

VarianceCumulative

% Total% of

VarianceCumulative

%

Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings

Extraction Method: Principal Component Analysis.

Factor Extraction


Scree Plot for the SAQ Data


Communalities

1.000 .4351.000 .4141.000 .5301.000 .4691.000 .3431.000 .6541.000 .5451.000 .7391.000 .4841.000 .3351.000 .6901.000 .5131.000 .5361.000 .4881.000 .3781.000 .4871.000 .6831.000 .5971.000 .3431.000 .4841.000 .5501.000 .4641.000 .412

Q01Q02Q03Q04Q05Q06Q07Q08Q09Q10Q11Q12Q13Q14Q15Q16Q17Q18Q19Q20Q21Q22Q23

Initial Extraction

Extraction Method: Principal Component

Table of Communalities Before and After Extraction

Component Matrixa

.701

.685

.679

.673

.669

.658

.656

.652 -.400

.643

.634 -.629 .593 .586 .556 .549 .401 -.417.437 .436 -.404

-.427 .627 .548 .465

.562 .571 .507

Q18Q07Q16Q13Q12Q21Q14Q11Q17Q04Q03Q15Q01Q05Q08Q10Q20Q19Q09Q02Q22Q06Q23

1 2 3 4Component

Extraction Method: Principal Component Analysis.4 components extracted.a.

Component Matrix Before Rotation (loadings of each variable onto each factor)

Note: Loadings less than 0.4 have been omitted.


Factor Rotation• To aid interpretation it is possible to maximize

the loading of a variable on one factor while minimizing its loading on all other factors.

• This is known as Factor Rotation.

• Two types:– OrthogonalOrthogonal (factors are uncorrelated)– ObliqueOblique (factors intercorrelate)


Orthogonal Rotation Oblique Rotation


Rotated Component Matrixa

.800

.684

.647

.638

.579

.550

.459 .677

.661

-.567

.473 .523

.516

.514 .496 .429 .833 .747 .747 .648 .645 .586

.543

.427

I have little experience of computersSPSS always crashes when I try to use itI worry that I will cause irreparable damage becauseof my incompetenece with computersAll computers hate meComputers have minds of their own and deliberatelygo wrong whenever I use themComputers are useful only for playing gamesComputers are out to get meI can't sleep for thoughts of eigen vectorsI wake up under my duvet thinking that I am trappedunder a normal distribtionStandard deviations excite mePeople try to tell you that SPSS makes statisticseasier to understand but it doesn'tI dream that Pearson is attacking me with correlationcoefficientsI weep openly at the mention of central tendencyStatiscs makes me cryI don't understand statisticsI have never been good at mathematicsI slip into a coma whenever I see an equationI did badly at mathematics at schoolMy friends are better at statistics than meMy friends are better at SPSS than I amIf I'm good at statistics my friends will think I'm a nerdMy friends will think I'm stupid for not being able tocope with SPSSEverybody looks at me when I use SPSS

1 2 3 4Component

Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

Rotation converged in 9 iterations.a.

Orthogonal Rotation (varimax) Fear of Computers

Fear of Statistics

Fear of Math

Peer Evaluation

Note: Varimax rotation is the most commonly used rotation. Its goal is to minimize the complexity of the components by making the large loadings larger and the small loadings smaller within each component. Quartimax rotation makes large loadings larger and small loadings smaller within each variable. Equamax rotation is a compromise that attempts to simplify both components and variables. These are all orthogonal rotations, that is, the axes remain perpendicular, so the components are not correlated.


Oblique Rotation:

Pattern Matrix

Pattern Matrixa

.706

.591

-.511

.405

.400 .643 .621 .615

.507

.885 .713 .653

.650

.588

.585

.412 .462

.411 -.902 -.774 -.774

I can't sleep for thoughts of eigen vectorsI wake up under my duvet thinking that I am trappedunder a normal distribtionStandard deviations excite meI dream that Pearson is attacking me with correlationcoefficientsI weep openly at the mention of central tendencyStatiscs makes me cryI don't understand statisticsMy friends are better at SPSS than I amMy friends are better at statistics than meIf I'm good at statistics my friends will think I'm a nerdMy friends will think I'm stupid for not being able tocope with SPSSEverybody looks at me when I use SPSSI have little experience of computersSPSS always crashes when I try to use itAll computers hate meI worry that I will cause irreparable damage becauseof my incompetenece with computersComputers have minds of their own and deliberatelygo wrong whenever I use themComputers are useful only for playing gamesPeople try to tell you that SPSS makes statisticseasier to understand but it doesn'tComputers are out to get meI have never been good at mathematicsI slip into a coma whenever I see an equationI did badly at mathematics at school

1 2 3 4Component

Extraction Method: Principal Component Analysis. Rotation Method: Oblimin with Kaiser Normalization.

Rotation converged in 29 iterations.a.

Fear of Statistics

Fear of Computers

Fear of Math

Peer Evaluation


Reliability: A measure should consistently reflect the construct it is measuring

• Test-Retest Method– What about practice effects/mood states?

• Alternate Form Method– Expensive and Impractical

• Split-Half Method– Splits the questionnaire into two random halves,

calculates scores and correlates them.• Cronbach’s Alpha

– Splits the questionnaire (or sub-scales of a questionnaire) into all possible halves, calculates the scores, correlates them and averages the correlation for all splits.

– Ranges from 0 (no reliability) to 1 (complete reliability)


Reliability: Fear of Computers Subscale


Reliability: Fear of Statistics Subscale


Reliability: Fear of Math Subscale


Reliability: Peer Evaluation Subscale


Reporting the ResultsA principal component analysis (PCA) was conducted on the 23 items with

orthogonal rotation (varimax). Bartlett’s test of sphericity, Χ2(253) = 19334.49, p< .001, indicated that correlations between items were sufficiently large for PCA. An initial analysis was run to obtain eigenvalues for each component in the data. Four components had eigenvalues over Kaiser’s criterion of 1 and in combination explained 50.32% of the variance. The scree plot was slightly ambiguous and showed inflexions that would justify retaining either 2 or 4 factors.

Given the large sample size, and the convergence of the scree plot and Kaiser’s criterion on four components, four components were retained in the final analysis. Component 1 represents a fear of computers, component 2 a fear of statistics, component 3 a fear of math, and component 4 peer evaluation concerns.

The fear of computers, fear of statistics, and fear of math subscales of the SAQ all had high reliabilities, all Chronbach’s α = .82. However, the fear of negative peer evaluation subscale had a relatively low reliability, Chronbach’s α= .57.


Step 1: Select Factor Analysis


Step 2: Add all variables to be included


Step 3: Get descriptive statistics & correlations


Step 4: Ask for Scree Plot and set extraction options


Step 5: Handle missing values and sort coefficients by size


Step 6: Select rotation type and set rotation iterations


Step 7: Save Factor Scores


Communalities


Variance Explained


Scree Plot


Rotated Component Matrix: Component 1


Rotated Component Matrix: Component 2


Component 1: Factor Score


Component (Factor): Score Values


Rename Components According to Interpretation

Department of Cognitive Science Michael Kalsher Adv. Experimental Methods & Statistics PSYC 4310 /...

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