Principal Components Factor Analysis

8/6/2019 Principal Components Factor Analysis

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Principal Components Factor Analysis

The purpose of principal components factor analysis is to reduce the number of variables in the analysis

by using a surrogate variable or factor to represent a number of variables, while retaining the variance

that was present in the original variables. The data analysis indicates the relationship between the

original variables and the factors, so that we know how to make the substitutions.

Principal components are frequently used to simplify a data set prior to conducting a multiple regressionor discriminates analysis.

To demonstrate principal components analysis, we will use the sample problem in the text which begins

on page 120.

Stage 1: Define the Research Problem

The perceptions of HATCO on seven attributes (Delivery speed, Price level, Price flexibility, Manufacturer

image, Service, Salesforce image, and Product quality) are examined to (1) understand if these

perceptions can be "grouped" and (2) reduce the seven variables to a smaller number. (Text, page 120)

Stage 2: Designing a Factor Analysis

In this stage, we address issues of sample size and measurement issues. Since missing data has an

impact on sample size, we will analyze patterns of missing data in this stage.

Sample size issues

Missing Data Analysis

There is no missing data in the HATCO data set.

Sample size of 100 or more

There are 100 subjects in the sample. This requirement is met.

Ratio of subjects to variables should be 5 to 1

There are 100 subjects and 7 variables in the analysis for a ratio of 14 to 1. This requirement is met.

Variable selection and measurement issues

Dummy code non-metric variables

All variables in the analysis are metric so no dummy coding is required.

Parsimonious variable selection

The variables included in the analysis are core elements of the business. There do not appear to be any

extraneous variables.

Stage 3: Assumptions of Factor Analysis

In this stage, we do the tests necessary to meet the assumptions of the statistical analysis. For factor

analysis, we will examine the suitability of the data for a factor analysis.


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Metric or dummy-coded variables

All variables are metric in this analysis.

Departure from normality, homoscedasticity, and linearity diminish correlations

In the last class, we conducted an exploratory analysis of these variables. We know from this several

variables are not normally distributed. Non-normality will diminish the correlations among the

variables, such that the relationships between variables might be stronger than what we are able to

represent in this analysis.

Multivariate Normality required if a statistical criteria for factor loadings is used

We will use the criteria of 0.40 for identifying substantial loadings on factors, rather than a statistical

probability, so this criteria is not binding on this analysis.

Homogeneity of sample

There is nothing in the problem to indicate that subgroups within the sample have different patterns of

scores on the variables included in the analysis. In the absence of evidence to the contrary, we will

assume that this assumption is met.

Use of Factor Analysis is Justified

The determination that the use of factor analysis is justifiable is obtained from the statistical output that

SPSS provides in the request for a factor analysis.

Requesting a Principal Components Factor

Analysis

Specify the Variables to Include in the Analysis

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3/11

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4/11

Complete the Factor Analysis Request

Count the Number of

Correlations Greater than

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Measures of

Appropriateness of Factor

Analysis

Assessing the Sampling Adequacy Problem

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5/11

Removing X5 'Service' from the Analysis

The Revised

Measures of

Appropriateness

of Factor Analysis

T e

n t i - i m a g e C o r r e l a t i o n M a t r i x c o n t a i n s t e m e a s u r e s o f s a m p l i n g

a d e q u a c y fo r t e i n d iv id u a l v a r i a b l e s o n t e d i a g o n a l o f t e m a t r i x ,h i g h l ig h t e d i n c y a n . T h e m e a s u r e s fo r t h r e e v a r i a b l e s fa l l b e l o w t h ea c c e p t a b l e l e v e l o f 0 . 5 0 : X 1 ' e l i v e r y S p e e d ' ( . 3 4 4 ) , X 2 ' P r ic e L e v e l '

( . 3 3 0 ) , a n d X 5 'S e r v i c e ' ( . 2 8 8 ) . T h e c o r r e c t i v e a c t i o n i s t o d e l e t e t h e

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h e S ig . v a lu e fo r t h is a n a ly s i s le a d s u s t o r e j e c t t h e n u llh y p o t h e s is a n d c o n c lu d e t h a t t h e re a re c o r re la t io n s in t h e d a t a s e t t h a ta r e a p p r o p r ia t e fo r fa c t o r a n a ly s i s .


6/11

The Revised Anti-image CorrelationMatrix

Stage 4: Deriving Factors and Assessing Overall Fit 1

In the stage, several criteria are examined to determine the number of factors that represent the data.

If the analysis is designed to identify a factor structure that was obtained in or suggested by previous

research, we are employing an a priori criterion, i.e. we specify a specific number of factors. The three

other criteria are obtained in the data analysis: the latent root criterion, the percentage of variance

criterion, and the Scree test criterion. A final criterion influencing the number of factors is actually

deferred to the next stage, the interpretability criterion. The derived factor structure must make

plausible sense in terms of our research; if it does not we should seek a factor solution with a different

number of components.

It is generally recommended that each strategy be used in determining the number of factors in the data

set. It may be that multiple criteria suggest the same solution. If different criteria suggest different

conclusions, we might want to compare the parameters of our problem, i.e. sample size, correlations,

and communalities to determine which criterion should be given greater weight.


One of the most commonly used criteria for determining the number of factors or components to

include is the latent root criterion, also known as the eigenvalue-one criterion or the Kaiser criterion.

With this approach, you retain and interpret any component that has an eigenvalue greater than 1.0.

The rationale for this criterion is straightforward. Each observed variable contributes one unit of

variance to the total variance in the data set (the 1.0 on the diagonal of the correlation matrix). Any

component that displays an eigenvalue greater than 1.0 is accounting for a greater amount of variance

than was contributed by one variable. Such a component is therefore accounting for a meaningful

amount of variance and is worthy of being retained. On the other hand, a component with an

eigenvalue less than 1.0 is accounting for less variance than had been contributed by one variable.

The latent root criterion has been shown to produce the correct number of components when the

number of variables included in the analysis is small (10 to 15) or moderate (20 to 30) and the

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7/11

communalities are high (greater than 0.70). Low communalities are those below 0.40 (Stevens, page

366).


Proportion of Variance Accounted For

Another criterion in determining the number of factors to retain involves retaining a component if it

accounts for a specified proportion of variance in the data set, i.e. at least 5% or 10%. Alternatively, one

can retain enough components to explain some cumulative total percent of variance, usually 70% to

80%.

While this strategy has intuitive appeal (our goal is to explain the variance in the data set), there is not

agreement about what percentages are appropriate to use and the strategy is criticized for being

subjective and arbitrary. We will employ 70% as the target total percent of variance.


The Scree Test

With a Scree test, the eigenvalues associated with each component are plotted against their ordinal

numbers (i.e. first eigenvalue, second eigenvalue, etc.). Generally what happens is that the magnitude of

successive eigenvalues drops off sharply (steep descent) and then tends to level off.

The recommendation is to retain all the eigenvalues (and corresponding components) before the first

one on the line where they start to level off. (Hatcher and Stevens both indicate that the number to

accept is the number before the line levels off; Hair, et. al. say that the number of factors is the point

where the line levels off, but also state that this tends to indicate one or two more factors that are

indicated by the latent root criteria.)

The Scree test has been shown to be accurate in detecting the correct number of factors with a sample

size greater than 250 and communalities greater than 0.60.


The Interpretability Criteria

Perhaps the most important criterion in solving the number of components problem is the

interpretability criterion: interpreting the substantive meaning of the retained components and verifying

that this interpretation makes sense in terms of what is know about the constructs under investigation.

The interpretability of a component is improved if it is measured by at least three variables, when all of

the variables that load on the component have the same conceptual meaning, when the conceptual

meaning of other components appear to be measuring other constructs, and when the rotated factor

pattern shows simple structure, i.e. each variable loads significantly on only one component.


8/11

The Latent Root Criterion

Percentage of

Variance Criterion

Scree Test Criterion

Stage 5: Interpreting the Factors 1

All three of the criteria for determining the number of components indicate that two components

should be retained. If this number matches the number of components derived by SPSS, we can

continue with the analysis. If this number does not match the number of components that SPSS

derived, we need to request the factor analysis, specifying the number of components that we want

SPSS to extract.

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9/11

Once the extraction of factors has been completed satisfactorily, the resulting factor matrix, which

shows the relationship of the original variables to the factors, is rotated to make it easier to interpret.

The axes are rotated about the origin so that they are located as close to the clusters of related variables

as possible. The orthogonal VARIMAX rotation is the one found most commonly in the literature.

VARIMAX rotation keeps the axes at right angles to each other so that the factors are not correlated

with each other. Oblique rotation permits the factors to be correlated, and is cited as being a more

realistic method of analysis. In the analyses that we do for this class we will specify an orthogonal

rotation.

The first step in this stage is to determine if any variables should be eliminated from the factor solution.

A variable can be eliminated for two reasons. First, if the communality of a variable is low, i.e. less than

0.50, it means that the factors contain less than half of the variance in the original variable, so we might

want to exclude that variable from the factor analysis and use it in its original form in subsequent

analyses. Second, a variable may have loadings below the criteria level on all factors, i.e. it does not

have a strong enough relationship with any factor to be represented by the factor score, so that the

variables information is better represented by the original form of the variable.

Stage 5: Interpreting the Factors 2

Since factor analysis is based on a pattern of relationships among variables, elimination of one variable

will change the pattern of all of the others. If we have variables to eliminate, we should eliminate them

one at a time, starting with the variable that has the lowest communality or pattern of factor loadings.

Once we have eliminated variables that do not belong in the factor analysis, we complete the analysis by

naming the factors which we obtained. Naming is an important piece of the analysis, because it assures

us that the factor solution is conceptually valid.

Analysis of the Communalities

Once the extraction of factors has been completed, we examine the table of 'Communalities' which tells

us how much of the variance in each of the original variables is explained by the extracted factors. For

example, in the table shown below, 65.8% of the variance in the original X1 'Delivery Speed' variable is

explained by the two extracted components. Higher communalities are desirable. If the communality

for a variable is less than 50%, it is a candidate for exclusion from the analysis because the factor

solution contains less that half of the variance in the original variable, and the explanatory power of that

variable might be better represented by the individual variable.

The table of Communalities for this analysis shows communalities for all variables above 0.50, so we

would not exclude any variables on the basis of low communalities. If we did exclude a variable for a

low communality, we should re-run the factor analysis

without that variable before proceeding.

The table of Communalities for this analysis shows

communalities for all variables above 0.50, so we would not

exclude any variables on the basis of low communalities. If

we did exclude a variable for a low communality, we should re-run the factor analysis without that

variable before proceeding.


10/11

Analysis of the Factor Loadings 1

When we are satisfied that the factor solution explains sufficient variance for all of the variables in the

analysis, we examine the 'Rotated Factor Matrix ' to see if each variable has a substantial loading on

one, and only one, factor. The size of the loading termed substantial is a subject about which there are

a lot of divergent opinions. We will use a time-honored rule of thumb that a substantial loading is 0.40

or higher. Whichever method is employed to define substantial, the process of analyzing factor loadings

is the same.

The methodology for analyzing factor loading is to underline or mark all of the loadings in the rotated

factor matrix that are higher than 0.40. For the rotated factor matrix for this problem, the substantial

loadings are highlighted in green.

Analysis of the Factor Loadings 2

We examine the pattern of loadings for

what is called 'simple structure' which means that each variable has a substantial loading on one andonly one factor.

In this component matrix, each variable does have one substantial loading on a component. If one or

more variables did not have a substantial loading on a factor, we would re-run the factor analysis

excluding those variables one at a time, until we have a solution in which all of the variables in the

analysis load on at least one factor.

In this component matrix, each of the original variables also has a substantial loading on only one

factor. If a variable had a substantial loading on more than one variable, we refer to that variable as

"complex" meaning that it has a relationship to two or more of the derived factors. There are a variety

of prescriptions for handling complex variables. The simple prescription is to ignore the complexity and

treat the variable as belonging to the factor on which it has the highest loading. A second simple

solution to complexity is to eliminate the complex variable from the factor analysis. I have seen other

instances where authors chose to include it as a variable in multiple factors, or to arbitrarily assign it to a

factor for conceptual reasons. Other prescriptions are to try different methods of factor extraction and

rotation to see if a more interpretable solution can be found.

Naming the Factors

Once we have an interpretable pattern of loadings, we name the factors or components according to

their substantive content or core. The factors should have conceptually distinct names and content.

Variables with higher loadings on a factor should play a more important role in naming the factor.

If the factors are not conceptually distinct and cannot be named satisfactorily, the factor solution may

be a mathematical contrivance that has not useful application.

The naming should take into account the signs of the factor loadings, i.e. negative signs imply an inverse

relationship to the factor. For example, X1 'Delivery Speed', X2 'Price Level', X3 'Price Flexibility', and

X7 'Product Quality' load on the first factor. Two of these variables: X1 'Delivery Speed' and X3 'Price

Flexibility' have a negative sign meaning that they vary inversely to the two variables which have a


11/11

positive loading: 'Price Level' and X7 'Product Quality'. The name for this factor, which the authors term

'basic value' on page 126 of the text, attempts to take into account the direction of the relationships

among all of these variables.

The two variables loading on the second factor X4 'Manufacturer Image' and X6 'Salesforce Image' both

have positive signs and are named 'HATCO image' by the authors on page 127 of the text.

Stage 6: Validation of Factor Analysis

In the validation stage of the factor analysis, we are concerned with the issue generalizability of the

factor model we have derived. We examine two issues: first, is the factor model stable and

generalizable, and second, is the factor solution impacted by outliers.

The only method for examining the generalizability of the factor model in SPSS is a split-half validation.

To identify outliers, we will employ a strategy proposed in the SPSS manual.

Split Half Validation

As in all multivariate methods, the findings in factor analysis are impacted by sample size. The larger the

sample size, the greater the opportunity to obtain significant findings that are present only because of

the large sample. The strategy for examining the stability of the model is to do a split-half validation to

see if the factor structure and the communalities remain the same.

Principal Components Factor Analysis

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