Confirmatory Factor Analysis

11.5 Confirmatory Factor Analysis

For confirmatory factor analysis specify the measurement model in advance of the analysis. One confirmatory

purpose of the analysis is to estimate the parameters of the measurement model, the values of the factor analysis: Analysis of a pattern coefficients, the λ ’s, and the factor correlations. From these estimates derive the imposed specified correlation matrix of the measured variables and compare to the actual item correlations. To measurement model. the extent that the imposed correlations match the actual correlations, the measurement model is empirically confirmed.

The general process of factor analysis based research for attitude surveys follows a standard pattern. First specify the measurement model and then analyze.

◦ Gather the responses to the items on the survey ◦ Calculate the correlations among the items ◦ Run the factor analysis program on the correlations to estimate the model parameters such

as the λ ’s ◦ Evaluate the fit of the estimated model by comparing the item correlations implied by the model to the actual item correlation

An explanation of the meaning of these imposed correlations follows.

11.5.1 The Covariance Structure

The confirmatory factor analysis estimates a specified measurement model. We apply our first confirmatory factor analysis to a correlation matrix in which the correct measurement model is known in advance of the analysis. We know the correct model because we define our own measurement model and then derive the item correlations computed not from data but from the

corresponding correlation pattern implied by this measurement model. We analyze the resulting covariance

correlations imposed by that model, the model’s covariance structure. 1 This model is devoid of structure: Covariances, such psychological content, but serves as a useful exercise to illustrate the conceptual basis of factor as correlations, of

analysis. the variables

implied by the

Our model includes six measured variables and two factors, with three variables linked to measurement

one factor and the other three variables linked to the other factor. This model contains multiple model. factors, but each item directly indicates only a single factor, an example of a multiple indicator

measurement model or MIMM. An MIMM is generally the most useful type of measurement MIMM: A model for scale development because the items on an attitude survey are usually partitioned measurement model with each into multi-item scales. Within this framework each item measures a single attitude, modeled as item directly linked

a factor. to only one factor. Define this MIMM in terms of six equations that link each of the measured items to one

of the two underlying factors. The correlation among the factors must also be specified. Here we specify our own population model, so use the symbol for the population correlation, ρ . Specify a moderate population factor correlation of 0.3, and a sequence of the population pattern coefficients, the λ ’s, that step from 0.8 to 0.6 to 0.4 for the three indicators of each factor.

X 1 =. 8 F 1 + u 1 X 4 =. 8 F 2 + u 4

X 2 =. 6 F 1 + u 2 X 5 =. 6 F 2 + u 5

X 3 =. 4 F 1 + u 3 X 6 =. 4 F 2 + u 6 ρ F 1, F 2 = 0 . 3

264 Factor/Item Analysis

path diagram: A

This model can also be presented as a path diagram, shown in Figure 11.4 . In this diagram the

diagram of causal relationships.

lines with arrowheads at one end represent a causal structure, which specifies that the response to each observed variable is due to the underlying shared factor, plus there is an implied uniqueness term. The curved line with double-headed arrows represents a correlation that has no underlying

specification of causality. That is, Factors F 1 and F 2 are correlated, but the model provides no explanation as to why they are correlated.

Figure 11.4 Specified population two-factor multiple indicator measurement model, three items per factor.

Now the critical question. How does a measurement model dictate or impose the correlations among the measured variables, here the correlation matrix for variables X 1 through X 6 ? Matrix algebra provides the general result. Fortunately, for the multiple indicator measurement model, which posits only one factor underlying each item, we can provide these equations without resorting to matrices.

External Consistency

external

The correlation of two items from different factors defined by a MIMM is provided by the

consistency: Covariance

product rule for external consistency (Spearman, 1914; Gerbing & Anderson, 1988). The extent of

structure of items

the correlation here depends not only on the strength of the relationship of each item with its

on a unidimensional

own factor, but also on the correlation of the factors. For example, if the factors have a low

scale with respect

correlation, then items from these two different factors also have a low correlation.

to other items.

The corresponding product rule provides the quantitative relationship between two items,

one from Factor F 1 and the other from Factor F 2 . product rule for external consistency: ρ X i X j =λ i ( ρ F 1, F 2 ) λ j

For example, the correlation of Items X 1 and X 5 consistent with the underlying measurement model in Figure 11.4 follows.

ρ X 1 X 5 =λ 1 ( ρ F 1, F 2 ) λ 5 = (0 . 8)(0 . 3)(0 . 6) = 0 . 144

internal

consistency: Covariance

Internal Consistency

structure of items on a unidimensional

A special case of external consistency applies to the correlations of the items of the same factor,

scale.

that is, for a factor–factor correlation of 1.0. The product rule for internal consistency specifies

Factor/Item Analysis 265

that the correlation of two items that both measure the same factor depends only on their corresponding pattern coefficients or λ ’s. Here express the rule for Items X i and X j .

product rule for internal consistency: ρ X i X j =λ i (1 . 0) λ j =λ i λ j

One would expect that higher item correlations would occur for two items more highly related to their common factor. This pattern is shown quantitatively by applying the product

rule for internal consistency, here for Items X 1 and X 2 . ρ X 1 X 2 =λ 1 λ 2 = (0 . 8)(0 . 6) = 0 . 48

Similarly, apply the product rule to the second and third items.

ρ X 2 X 3 =λ 2 λ 3 = (0 . 6)(0 . 4) = 0 . 24

Apply the product rules to all six items to obtain the correlation matrix presented in Listing 11.5 . Each item correlates with itself perfectly, that is, the values of the diagonal elements are all 1’s.

> mycor

X1 X2 X3 X4 X5 X6

X1 1.000 0.480 0.320 0.192 0.144 0.096 X2 0.480 1.000 0.240 0.144 0.108 0.072 X3 0.320 0.240 1.000 0.096 0.072 0.048 X4 0.192 0.144 0.096 1.000 0.480 0.320 X5 0.144 0.108 0.072 0.480 1.000 0.240 X6 0.096 0.072 0.048 0.320 0.240 1.000

Listing 11.5 Population correlation matrix, the covariance structure, of the six observed variables imposed by their multiple indicator measurement model.

Correlation does not establish causation. A causal pattern, however, does imply correlation. If the items in the measurement model in Figure 11.4 are related to the two underlying factors as specified, then the population correlations of the six items in the model are constrained to those in Listing 11.5 .

Communalities

We have considered the correlations of two items from the same and different factors that are consistent with the underlying measurement model. What about the correlation of each item with itself, the diagonal elements in the correlation matrix such as in Listing 11.5 ? These diagonal elements, which all equal 1, are fundamentally different from, and are all larger than the off-diagonal elements.

The basis of this distinction is that the correlation of two different items directly depends only on the strength of their relationships with the underlying factor or factors. The correlation of an item with itself, however, depends not only on the influence of its underlying factor, but also on the influence of its uniqueness component. The relationship between the pattern coefficient λ and the variance of the item uniqueness attribute for the i th item follows.

ρ X i X i = 1 =λ 2 i +σ 2 u i

266 Factor/Item Analysis

communality:

An item’s communality is the proportion of the correlation of an item with itself that is

Proportion of an item’s variance due

due only to the shared factors. For a unidimensional model of the type specified here, this

only to the

communality, here λ 2 i , is due only to the one underlying factor.

common factors.

The expression of an item’s communality is based on a special case of the product rule for internal consistency, applied to the same item twice.

communality of i th item: λ i λ i =λ 2 i

To solve for this communality is to extract this common contribution to the full variation of the item, here a value of 1.0. This extraction of the common influence of the factors on item variability, the ability to distinguish between the communality and the uniqueness component of an item, is the solution for the factor pattern coefficient, λ i . The factor analysis, then, transforms the observed correlation matrix with 1’s in the diagonal to the correlation matrix with communalities in the diagonal.

The communalities are computed as an iterative process, from one iteration to the next.

iteration: A

Compute the communalities by starting with the 1’s in the diagonal of the input correlation

computed solution, the results

matrix. Then the corresponding pattern coefficients and factor–factor correlations are estimated

of which are

with one of the many available factor analysis estimation procedures. Next, the resulting

entered back into the model to

estimated pattern coefficients are squared according to the product rule for internal consistency,

compute another

and then those values inserted into the diagonal. This process is repeated until the solution

solution.

generally does not change much from iteration to iteration. The result is the final estimated parameters of the model.

11.5.2 Analysis of a Population Model

Now we have the information needed to proceed to the confirmatory factor analysis of the correlation matrix in Listing 11.5 . These correlations were generated according to the covariance structure rules imposed by a multiple-indicator measurement model presented in Figure 11.4 .

Scenario Recover the measurement model from the imposed correlations Recover the parameters of a measurement model used to construct the correlation matrix. The confirmatory factor analysis of those correlations should return the same parameter

values, the pattern coefficients and the factor correlations, originally used to construct the imposed correlations.

calculate correlation

The factor analysis begins from the correlations between the variables. One possibility is

matrix, Section 8.3 , p.194

to use the Correlation function to calculate the correlations from the original data table, the data frame, which contains the raw data. The other possibility is to read the already computed correlations from an external file, the only choice available in this example because the correlations were computed directly from the product rules imposed by the model. This is not a standard data analysis to evaluate a posited structure that may or may not fit the data. Instead this is a conceptual exercise, an attempt to recover a known structure from its implied correlations, its covariance structure.

Factor/Item Analysis 267

To read a correlation matrix into R , use the lessR function corRead , abbreviated rd.cor . The correlations are on the lessR web site.

> mycor <- corRead("http://lessRstats.com/data/MIMMperfect.cor")

correlation matrix

The correlations are stored in a data structure called mycor , which is the default name for in R, Section 8.2.1 , the input correlation matrix for the lessR factor analysis functions.

p. 182

The lessR confirmatory factor analysis program corCFA , abbreviated cfa , analyzes a cfa function: MIMM. The cor in the full name for the procedure indicates that the information entered Confirmatory factor analysis of a into the procedure for analysis is not the responses to the items, but their correlations. The multiple indicator

required arguments to measurement cfa are the lists of items that define each factor. Each factor is named

model.

with an F followed by the factor number, beginning with a 1 and continuing to the number of factors, of which 12 is the maximum. By default, cfa analyzes the correlations in the mycor matrix.

The most general technique to specify a list of items is to use the R combine function c , such as in c(X1,X2,X3) . If the variables in the list are sequential in the corresponding data

c function,

frame, then the colon notation, : , can be used instead.

Section 1.3.6 , p. 15

lessR Input Confirmatory factor analysis of a two factor MIMM > cfa(F1=X1:X3, F2=X4:X6)

The output of cfa is primarily text, but also includes one graphic, the heat map of the item-correlation matrix with communalities in the diagonal, as shown in Figure 11.5 . The heat heat map: A map displays a patch of color or gray scale for each correlation coefficient. The darker is the graph of item correlations. color, the stronger the correlation.

Item Correlations/Communalities

Figure 11.5 Heat map of six-variable correlation matrix with communalities in the diagonal. The pattern of the two three-item blocks in the heat map, in the top-left corner and bottom-

right corner, is the “hierarchical ordering” of the correlations of unidimensional items first noted

268 Factor/Item Analysis

by Spearman (1904) more than a century ago. The items that more strongly relate to their underlying factor correlate more highly with each other, as indicated by the darker shades of gray. Further, items within each factor correlate more highly than items in different factors, as shown by the lighter shades of gray in the bottom-left corner and top-right corner of Figure 11.5 .

The primary output of the confirmatory factor analysis, in Listing 11.6 , begins with the estimated pattern coefficients, the λ for each measured variable, as well as its corresponding uniqueness.

Indicator Analysis ------------------------------------------------------------------------- Fac Indi

Unique Factors with which an indicator correlates too tor cator tern

Pat

highly, and other indicator diagnostics. -------------------------------------------------------------------------

ness

F1 X1 0.800

F1 X2 0.600

F1 X3 0.400

F2 X4 0.800

F2 X5 0.600

F2 X6 0.400

Listing 11.6 Estimated factor pattern coefficients, uniquenesses and diagnostics.

The confirmatory factor analysis perfectly recovered the underlying true measurement model. The λ ’s computed by the factor analysis, listed under the heading of Pattern , are the exact values used to generate the covariance structure, in this case the item correlation matrix.

misspecification:

There is also room for diagnostic information that helps identify aspects of the model that do

A specified model that does not

not fit the data, what are called misspecifications. In this example this area is blank because the

match reality.

covariance structure input in the analysis fits the estimated model perfectly, as designed. The next output, in Figure 11.6 , is the complete item–factor correlation matrix with

item–factor

communalities in the diagonal of the item correlations. Here we observe all of the item–factor

correlation: A correlation of an

correlations, the correlations of the items, the measured variables, with the factors, the latent

item with a factor.

variables. Thus we see, for example, that the first item, X 1 , correlates 0.8 with its own factor,

F 1 , and 0.24 with the second factor, F 2 . Note that the factor correlation of an item on its own factor for an MIMM is the corresponding pattern coefficient, λ .

X1 X2 X3 X4 X5 X6 F1 F2 X1 0.64 0.48 0.32 0.19 0.14 0.10 0.80 0.24 0.64 0.48 0.32 0.64 0.48 0.32 X2 0.48 0.36 0.24 0.14 0.11 0.07 0.60 0.18 0.48 0.36 0.24 0.48 0.36 0.24 X3 0.32 0.24 0.16 0.10 0.07 0.05 0.40 0.12 0.32 0.24 0.16 0.32 0.24 0.16 X4 0.19 0.14 0.10 0.64 0.48 0.32 0.24 0.80 0.64 0.48 0.32 X5 0.14 0.11 0.07 0.48 0.36 0.24 0.18 0.60 0.48 0.36 0.24 X6 0.10 0.07 0.05 0.32 0.24 0.16 0.12 0.40 0.32 0.24 0.16 F1 0.80 0.60 0.40 0.24 0.18 0.12 1.00 0.30 1.00 0.30 F2 0.24 0.18 0.12 0.80 0.60 0.40 0.30 1.00 0.30 1.00

Figure 11.6 Annotated output correlation matrix with communalities in the diagonal of all measured and latent variables in the analysis.

The factor correlation matrix appears in the bottom right of the complete matrix in Figure 11.6 . The correlation of the two factors, F 1 and F 2 , is 0.3. This value for the factor

Factor/Item Analysis 269

correlation specified in the constructed model has also been recovered by the confirmatory factor analysis of the implied population covariance structure, validating the analysis for the estimation and evaluation of measurement models for data analysis.

For an analysis of many variables it may be convenient to list only the item–factor item.cor option: and factor–factor correlations. To display just those correlations on the cfa output, add the If FALSE, then the item correlations item.cor=FALSE option to the function call.

are not displayed.

How well does the model fit? Examine Listing 11.7 , where each residual is the difference residual: between the corresponding item correlation and its value imposed by the estimated multiple Difference between the actual indicator measurement model.

correlation and the imposed by the model.

Residuals --------------------

X1 X2 X3 X4 X5 X6

X1 0 0 0 0 0 0 X2 0 0 0 0 0 0 X3 0 0 0 0 0 0 X4 0 0 0 0 0 0 X5 0 0 0 0 0 0 X6 0 0 0 0 0 0

Listing 11.7 Residuals, the difference between observed correlations and those specified by the estimated model.

The analysis perfectly recovered the structure of the underlying model. The fit is perfect. All residuals are zero, as it should be because the correlation matrix was generated to be perfectly consistent with the underlying model.

The preceding analysis was accomplished with the default 25 iterations to obtain the communality estimates. The number of iterations can be specified by setting the iter option.

For example, to restrict the analysis to the observed scores, specify 0 iterations, which is the iter option:

Specify the number

equivalent of specifying the scales function. The result is an analysis of the item and scale of iterations for the (total) correlations. It is as if the scale scores were computed by summing the relevant item estimates.

scores and then the correlations computed between the items and the scale scores.

scales function, Section 11.4 , p. 260

11.5.3 Proportionality

An implication of the product rule for external consistency is that for any two items of the external same factor, their correlations of all other items, including items from other factors and for consistency, Section 11.5.1 , the communalities, are proportional. To illustrate, return to Figure 11.6 , which lists the item p. 264

correlations with communalities in the diagonal. Consider the first two items, X 1 and X 2 , their correlations with all other items, and the corresponding ratios, shown in Table 11.1 . How can these proportionality constants help diagnose and suggest measurement models consistent with the data?

Scenario Obtain the matrix of proportionalities

A heuristic aid to specify a multiple indicator measurement model consistent with the data is a matrix of proportionality coefficients, which provides, for each pair of items, the average proportionality of their correlations with all other items.

270 Factor/Item Analysis

Table 11.1 Correlations of Items X 1 and X 2 to three decimal digits across all six items in the model and their corresponding ratios, that is, proportions.

Item

X1 X2 Ratio

The lessR function corProp , abbreviated prop , provides the matrix of average proportion- alities (Hunter, 1973). Again, the default input correlation matrix is mycor . The output of prop is in this example not written to a data structure, so the output instead is directed to the console as text output.

lessR Input Proportionality coefficients > prop()

The output is shown in Figure 11.7 .

X1 X2 X3 X4 X5 X6 X1 1.00 1.00 1.00 0.55 0.56 0.57 1.00 1.00 1.00 X2 1.00 1.00 1.00 0.56 0.55 0.55 1.00 1.00 1.00 X3 1.00 1.00 1.00 0.57 0.55 0.55 1.00 1.00 1.00 X4 0.55 0.56 0.57 1.00 1.00 1.00 1.00 1.00 1.00 X5 0.56 0.55 0.55 1.00 1.00 1.00 1.00 1.00 1.00 X6 0.57 0.55 0.55 1.00 1.00 1.00 1.00 1.00 1.00

Figure 11.7 Annotated proportionality matrix.

For these perfect correlations according to their fit with the underlying model, the proportionalities of items that measure the same factor are all 1.00.

perfect model,

Recovering the postulated model from the implied covariance structure is a conceptual

Section 11.5.2 , p. 266

exercise. In this situation there is no model specification error because the model is perfectly specified. There is also no sampling error because the implied population correlations are used as the input to the analysis. In the real world of data analysis, even for a well specified model, sampling error ensures that the model parameters are not perfectly recovered. If multiple samples were taken, each sample would generate different estimates because each sample of data is different from every other sample, and no sample perfectly represents the population.

The process of a confirmatory factor analysis with actual data for the analysis of Machiavel- lianism follows.

Factor/Item Analysis 271

11.5.4 Confirmatory Analysis of the Mach IV Scale

One tradition in social science research is to use exploratory factor analysis to define and/or revise a set of multi-item scales. The procedure is straightforward. Define a scale for each extracted factor and then place each item on the scale for which it has the highest factor loading, perhaps only retaining those items that load on the factor at some minimum value such as 0.4. Another part of this tradition, at least as practiced by some researchers, is to stop at this point and declare the scales unidimensional.

The problem here is that exploratory factor analysis defined the factors as dimensions that set up a coordinate system in some multi-dimensional space. A set of coordinate axes is not a set of scales. The construction of the scales from these factors is an ad-hoc procedure without any formal statistical evaluation of dimensionality. Fortunately, exploratory factor analysis can

be useful for suggesting multiple indicator measurement models (Gerbing & Hamilton, 1996), but a suggestion is not an evaluation of fit. The re-emergence of confirmatory factor analysis during the last several decades of social science research has once again provided access to tools for formally testing the dimensionality of the resulting scales.

Scenario Evaluate the implied Mach IV measurement model

A set of four Mach IV subscales was constructed based on an exploratory factor analysis of all 20 Mach IV items. Now formally evaluate the dimensionality of these scales with

a confirmatory factor analysis of the resulting multiple indicator measurement model.

The confirmatory factor analysis specification provided by the four-factor exploratory analysis from the efa function follows. To run this analysis, copy the instructions for the cfa function call at the end of the efa output in Listing 11.8 , and then paste back into R .

> corCFA( F1=c(m03,m06,m07,m09,m10), F2=c(m01,m05,m08,m12,m13,m18), F3=c(m04,m14,m17,m20), F4=c(m02,m11,m15,m16)

Listing 11.8 Confirmatory factor analysis code from the exploratory factor analysis.

The formal analysis of the corresponding MIMM indicates that the model does not fit well. For example, 10 of the 19 items have loadings on their own factor at 0.4 or less, and 7 of the

19 items in the model correlate more with another factor than they do their own. Clearly the model direct from the exploratory analysis needs refinement. The size of the average absolute value of all the residuals is also large, 0.051.

How to respecify the corresponding measurement model to improve fit? One strategy is delete from the model some of the poor fitting items, such as those with low λ values. Another strategy is to recognize that two items of the same underlying factor have, in the population,

the same constant of proportionality across all other variables, both items that also measure the item proportionality

of correlations,

same factor, or items or any other variable from anywhere else.

Figure 11.7 , p. 270

272 Factor/Item Analysis

Scenario Obtain and re-order proportionality coefficients As an aid to constructing a multiple indicator measurement model, an MIMM, calculate

the proportionality coefficient for each pair of Mach IV items. Then re-order the matrix of proportionality coefficients so that similar items tend to appear next to each other in the matrix. The goal is identify a MIMM that complements the guidance provided by the exploratory factor analysis.

To obtain these re-ordered proportionality coefficients, first invoke the prop function as

reorder function:

before. But now save the new matrix into another R object, here called myprop , so that the

Re-order the variables that

proportionalities can be re-ordered with the lessR function corReorder , or just reord . The

define a correlation

output from reord here is not directed to a data structure, so R directs its output to the console

matrix.

for viewing as text output.

lessR Input Re-ordered proportionalities > myprop <- prop() > reord(myprop)

The reord function identifies the variable in the matrix that is most strongly related to all the other variables, the variable with the highest sum of squared coefficients. This variable is placed first in the re-ordered matrix. Then the second variable is the variable that has the highest coefficient with the first variable. The third variable has the highest coefficient with the second, and so on. The algorithm is very simple, but is designed to list variables with parallel profiles of correlation coefficients next to each other.

Figure 11.8 lists the proportionality coefficients for the first eight items listed in the output. The gray regions in the figure highlight how this technique isolated two groups of items such that the proportionality coefficients within each group are relatively larger than those coefficients between the groups.

m09 m10 m07 m06 m11 m16 m04 m09 1.00 0.95 0.93 0.91 0.82 0.80 0.84 m10 0.95 1.00 0.96 0.90 0.82 0.82 0.81 m07 0.93 0.96 1.00 0.97 0.85 0.85 0.77 m06 0.91 0.90 0.97 1.00 0.83 0.82 0.76 m11 0.82 0.82 0.85 0.83 1.00 0.92 0.85 1.00 0.92 0.85 m16 0.80 0.82 0.85 0.82 0.92 1.00 0.85 0.92 1.00 0.85 m04 0.84 0.81 0.77 0.76 0.85 0.85 1.00 0.85 0.85 1.00

Figure 11.8 Annotated proportionality coefficients for the items that define the first two factors in the confirmatory factor analysis of Mach IV.

Integrating the information from the exploratory analysis, the re-ordered proportionality coefficients and the content and meaning of the items, and then formulating several more models lead to the final measurement model advocated by Hunter et al. (1982). The two groups

Factor/Item Analysis 273

of items delineated in Figure 11.8 are a subset of the final model, of which the corresponding call to the cfa function is shown in Listing 11.9 .

> cfa(F1=c(m06,m07,m09,m10), F2=c(m15,m02), F3=c(m04,m11,m16), F4=c(m01,m05,m12,m13))

Listing 11.9 Specification of the final four-factor Machiavellianism model.

The average residual of the confirmatory factor analysis of this four-factor model is just 0.036. So the model has reasonable fit, but what does it mean?

Scenario List the content of the items for each derived unidimensional scale Factor and item analysis are statistical techniques. Ultimately, however, the results need to make conceptual sense, they need to be interpretable. An understanding of content should guide the entire process of scale construction. Display the content for each of the four scales that correspond to the four-factor multiple indicator measurement model.

Variable labels can be read into R with the labels option of the lessR function Read . Here read variable labels, the variable label for each item is its content. The cfa function lists the items in each group, Section 2.4 , p. 45 and, if the variable labels are present, automatically displays their content.

The call to cfa in Listing 11.9 displays the item content that identifies the four specified distinct content domains of Machiavellianism. The content for the final Mach IV measurement model, as output by the cfa function, appears in Listing 11.10 for Deceit, 11.11 for Flattery,

11.12 for Immorality, and 11.13 for Cynicism.

F1: m06 m07 m09 m10 ------------------------------ m06: Honesty is the best policy in all cases. m07: There is no excuse for lying to someone else. m09: All in all, it is better to be humble and honest than to be important

and dishonest. m10: When you ask someone to do something for you, it is best to give the real reasons for wanting it rather than giving reasons which carry more weight.

Listing 11.10 Deceit Machiavellian subdomain.

F2: m15 m02 ------------------------------ m15: It is wise to flatter important people. m02: The best way to handle people is to tell them what they want to hear.

Listing 11.11 Flattery Machiavellian subdomain.

The four different subscales are conceptually distinct in terms of the covariance pattern, the structure of the item correlation matrix. The subscales are also distinct in terms of their content.

274 Factor/Item Analysis

F3:

m04 m11 m16 ------------------------------ m04: Most people are basically good and kind. m11: Most people who get ahead in the world lead clean, moral lives. m16: It is possible to be good in all respects.

Listing 11.12 Immorality Machiavellian subdomain.

F4:

m01 m05 m12 m13 ------------------------------

m01: Never tell anyone the real reason you did something unless it is useful to do so. m05: It is safest to assume that all people have a vicious streak and it

will come out when they are given a chance. m12: Anyone who completely trusts anyone else is asking for trouble. m13: The biggest difference between most criminals and other people is that

the criminals are stupid enough to get caught.

Listing 11.13 Cynicism Machiavellian subdomain.

Further, the underlying factors correlate moderately, but not excessively. The factor correlation matrix from the cfa output is reproduced in Listing 11.14 .

F1 F2 F3 F4

F1 1.00 0.50 0.64 0.44 F2 0.50 1.00 0.16 0.51 F3 0.64 0.16 1.00 0.61 F4 0.44 0.51 0.61 1.00

Listing 11.14 Factor correlations between the four distinct Machiavellian content domains: Deceit(F 1 ),

Flattery(F 2 ), Immorality(F 3 ), and Cynicism(F 4 ).

These factor correlations are the corresponding observed scale correlations corrected for attenuation due to measurement error. That is, they are the estimated correlations among latent

scales abbreviation

variables, the factors, which are larger than the observed scale–score correlations. The later are

for cfa, Section 11.4 ,

obtained by setting iter=0 , or, equivalently, running the abbreviation scales in place of cfa .

p. 260

Another issue aside from the dimensionality of the subscales is the reliability of their respective scale scores. This assessment is provided by several possible reliability coefficients,

coefficient alpha,

of which two of the most widely used are Coefficient Alpha and Coefficient Omega. Alpha

Section 11.4.2 , p. 262

is historically prior but Omega is more appropriate to the analysis of MIMM’s because it incorporates the communality estimates into its underlying calculations, which Alpha ignores.

The reliability of the scales tends to be too low to be of practical use for identifying individuals in terms of the scores on the various subscales. Still, the reliabilities are somewhat higher than for the initial model derived from the exploratory factor analysis, in which only Coefficient Alpha was reported. The factor analysis is accomplished with communalities in the diagonal to partition out the unique variance of each item, which then permits the calculation of Coefficient Omega. The more appropriate Omegas are somewhat larger than the corresponding Alpha coefficients.

Factor/Item Analysis 275

Scale Alpha Omega -------------------

F1 0.691 0.701 F2 0.400 0.400 F3 0.402 0.421 F4 0.515 0.517

Listing 11.15 Reliability of the four Mach IV scales.

This delineation of unidimensional Machiavellian subscales advances our knowledge of the meaning of this social construct. By itself, however, the uncovering of these subscales does not invalidate Machiavellianism as a generic concept, but does demonstrate that Machiavellianism should not be conceived simply as a unitary construct. To further explore that question could include more advanced analyses than a simple confirmatory factor analysis, methodologies briefly discussed next, but which are beyond the scope of this text in terms of their detail.