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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Derivation of model
  5. 3. Example
  6. 4. Discussion
  7. Acknowledgement
  8. References

This study aims to reparameterize ordinary factors into between- and within-person factor effects and utilize an array of the within-person factor loadings as a latent profile which encapsulates all score responses of individuals in a population. To illustrate, the Woodcock–Johnson III (WJ-III) tests of cognitive abilities were analysed and one between- and two within-person factors were identified. The scoring patterns of individuals in the WJ-III sample were interpreted according to the within-person factor patterns. Regression analyses were performed to examine how much the within-person factors accounted for the person scoring patterns and criterion variables. Finally, the importance and applications of the between- and within-person factors are discussed.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Derivation of model
  5. 3. Example
  6. 4. Discussion
  7. Acknowledgement
  8. References

Among multivariate data analysis techniques, factor analysis is most frequently used as a dimension-reducing or a factor structure-searching method. Factor analysis can be used for both exploratory and confirmatory purposes, depending on the research questions, but it is the exploratory aspect of factor analysis that will be the focus of this study. Within the exploratory approach, Spearman (1904) formulated the two-factor theory for studying the structure of human abilities, the two factors being a general factor (g), pervading all abilities, and an explicit factor, specific to an ability domain (e.g., long- or short-term memory, processing speed, spatial memory, etc.). Lenk, Wedel, and Bockenholt (2006) and Maydeu-Olivares and Coffman (2006) also differentiated between a general factor and specific factors, but they specified the general factor as an intercept in their factor model. We also include a general factor and specific (or group) factors in our model, but in addition we aim to extract person scoring pattern information from the factors. To do so, we consider a general factor as a basis of between-person effect which appears ubiquitously across variables but group factors as foundations of within-person effects which characterize overall individual persons' scoring patterns in a population.

Since Spearman's theory was based primarily on correlations of variables, factor analysis was viewed as variable-oriented until Cattell (1967) differentiated factor analysis into two approaches: one variable-oriented and the other person-oriented. The variable-oriented approach is known as R factor analysis and the person-oriented approach as Q factor analysis. R factor analysis usually involves factoring the correlations among variables to describe a factor structure of variables, whereas Q factor analysis typically involves factoring the correlations among people to describe a factor structure of persons. In both Q and R factor analysis, rotations are utilized to enhance factor interpretation. Like Q factor analysis, we analyse person information but, unlike Q factor analysis, this study explores latent profiles in a population, not latent factors. Moreover, no rotation is necessary for interpretation of latent profiles (e.g., Kim, Davison, & Frisby, 2007; Kim, Frisby, & Davison, 2004).

The current study aims to extract two layers of between- and within-person effects. The first layer is an individual person level and the second layer is a total people level:

  1. For an individual person, the between-person effect appears in the height or level of a person's observed score profile and the within-person effect appears in the pattern of the person score profile.
  2. For the group (sample) of people, the between-person effect is analogous to Spearman's general ability or g factor and often appears as the first unrotated factor in cognitive ability or personality tests. The within-person effect appears in arrays of the ipsatized factor loadings (around their mean loading) which portray latent profiles of individuals in a population.

To deduce summary information, inspecting every person's profile is not practically possible, and identification of latent profiles helps researchers understand individuals' scoring patterns (or person profiles) collectively first, and then it helps interpret individual profiles with this collective profile information. To make the connection between person profiles and latent profiles, we demonstrate decompositions for person-level and group-level profiles in between- and within-person effects. The person-level profiles are based on observed scores, but the group-level profiles are derived from latent scores. Then we illustrate an extension of the single factor decomposition to multiple orthogonal factors. The factor decomposition was first introduced by Davison, Kim, and Close (2009), but the Davison et al. study did not explicitly explain (1) how the ubiquitous between-person factor is defined by combining the between-person effects from multiple factors; (2) how the group-level (latent score) information is connected with the person-level (observed score) information; and (3) how person profiles are replicated by factor model parameters. The current study is mainly focused on how person profiles are interpreted in terms of between- and within-person factors.

2. Derivation of model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Derivation of model
  5. 3. Example
  6. 4. Discussion
  7. Acknowledgement
  8. References

2.1. Decomposition 1: Observed score profiles into between- and within-person effects

2.1.1. Between-person effect: Level information

The between-person effect is defined as a person-specific mean score which signifies the level or height of a person profile. The person-specific mean in person p's profile is represented by inline image, where v is a variable (e.g., scores of different cognitive ability domains) (= 1, …, V); xpv is standardized to have zero mean and unit variance for each variable v; and the squared standardized scores inline image represent the variances of the variables for a given person. Since both inline image and inline image remain constant across variables within person p, the total between-person variation for person p over all V variables equals inline image. The total between-person variation over all people is

  • display math(1)

which represents the variation due to individual differences in overall person profile heights.

2.1.2. Within-person effect: Pattern information

A deviation about the person-specific mean inline image will be called a within-person effect, because it varies across variables within a person, and the within-person variation for person p over variables is inline image. The total within-person variation over all people is defined as

  • display math(2)

For a given person, deviations about the person-specific mean comprise a vector of ipsative scores that describes the within-person effect which appears as the pattern of the person profile. If the variables were, for example, cognitive ability domains, the ipsative scores would describe a pattern of relative strengths and weaknesses: a positive value of inline image would correspond to a person's strength in that specific cognitive ability and a negative value would highlight a weakness.

2.2. Decomposition 2: Latent profiles into between- and within-person effects

This section demonstrates how ordinary factor loadings are reparameterized into modified factor loadings which include between- and within-person effects and shows how person profiles are related to these two effects. Davison et al. (2009) showed that for a given value of the factor score, where errors have expectation of zero and the factor scores are independent of errors, an expected person profile is proportional to the profile of factor loadings. Also, Davison et al.'s study indicates three ways to interpret the factor scores: (1) they represent information about both profile levels and patterns; (2) they only signify information about the overall profile levels; or (3) they only contain information about the profile patterns. If factor scores change, profile levels, profile patterns, or both will also change. The next section includes a hypothetical example to demonstrate engagements of a factor score in level and pattern, both together and separately.

2.2.1. Example: A single factor containing both a between- and a within-person effect

When an ordinary factor contains both effects, the factor can be decomposed into the level (between-person) effect and the pattern (within-person) effect in the following way:

  • display math(3)

where λv is a factor loading on variable v = 1, 2, 3, λ = [.7, .1, .7]T; fp is a factor score of person p on a single factor which is estimated by regressing person p's score vector onto a vector of factor loadings λ (hence, this factor score estimate is like a regression weight which indicates a positive or negative relationship with λ depending on its direction); inline image is the mean factor loading; and inline image is the ipsatized factor loading around the mean loading. In our example, if we have three variables where the factor loadings are λ = [.7, .1, .7]T and inline image, then λ can be decomposed as follows:

  • display math(4)

On the right-hand side of the equation, the elements in the first vector, [.5, .5, .5]T, are the mean factor loadings and determine the level of the profile (between-person effect). The elements in the second vector, [.2, −.4, .2]T, are the ipsatized factor loadings which differ from the ordinary factor loadings, [.7, .1, .7]T, only by an additive constant of .5. The elements of the ipsatized factor loadings determine the factor loading pattern (within-person effect) which is the same V shape as the ordinary factor loadings (if plotted), but they are not elevated since their mean equals 0 = inline image (.2 − .4 + .2), compared to the ordinary factor loadings V which are elevated .5 (mean) above 0.

Depending on the direction of factor scores, observed score profiles are expected to have the same pattern as the ipsatized factor loadings or will mirror them. Assuming factor scores are standardized with zero mean and unit variance, if fp = 1.0, person p's expected score profile would be the same V shape as the ipsatized factor loading shape and keep the same height of the mean of .5, whereas if fp = −1.0, person p's expected score profile would be the mirror image of V, ‘Λ’ shape, having the depressed height of −.5.

2.2.2. Example: A factor containing either a between- or a within-person effect

It is possible for a factor to contain only a between-person effect. For example, if there is no within-person effect in a factor, then the mean factor loading inline image equals an individual factor loading λv, resulting in a decomposition like equation (5). This type of a factor can be classified as a between-person factor,

  • display math(5)

Also, one might think of a factor that contains only a within-person effect. By definition (Davison et al., 2009), if the mean inline image, as in the equation

  • display math(6)

then this type of a factor is classified as a within-person factor. Usually group factors which represent specific domains of cognitive abilities can be categorized as within-person factors.

2.3. Decomposition 3: Multiple orthogonal factors

The result from the one-factor model in equation (3) can be generalized to orthogonal multiple factors. If the expectations of factor scores and errors equal zero and factor scores are independent of each other and of errors, then the expected person profile given a factor k and its factor score fpk is proportional to the vector of factor loadings along factor k. Because of the orthogonal factor assumption, factor scores are uncorrelated with each other. The ordinary multiple factor model for person p is

  • display math(7)

where xpv is standardized to have zero mean and unit variance for each variable v = 1, …, V (e.g., cognitive subtest scores); k = 1, …, K is the number of factors; λvk is a factor loading of factor k; fpk is a score of person p on factor k; and epv is an error term including specificity and/or random error. Given the above assumptions, the expectation of the person profile or vector xpv is represented by a product of the factor k loading vector and the factor k score vector. Suppose, for example, a two-factor solution with three variables, as shown in the matrix form:

  • display math(8)

The expected score profile of person p can only be attributed to variation in the elements of the factor loadings, given factor scores f:

  • display math(9)

If the elements of factor loadings on factor 1 are approximately equal λ11 ≅ λ21 ≅ λ31, the factor loadings will be a flat line (there is no variation across the loadings). Then factor 1 is considered a between-person factor. In that case, depending on the sign of the factor 1 score, the height (level) of person p's profile will differ: if fp1 > 0 then person p's profile will be elevated from 0, but if fp1 < 0 then the profile will be depressed from 0. The contribution of the factor to the between-person effect can best be described by calculating a proportion of between-person variation to the total variation which includes both between- and within-person variation. The closer the proportion is to 1.00, the greater the between-person effect contribution to the total variation for a given factor.

On the other hand, if the factor loadings on factor 2, [λ12, λ22, λ32]T, is not flat (e.g., a ‘V’ pattern), the expected score profile will have the same pattern or the inverted form of it, depending on the sign of fp2. The contribution of the factor to the within-person effect can best be described by calculating the proportion of within-person variation to the factor variation. The closer the proportion is to 1.00, the greater is the within-person effect contribution to the total variation in a given factor.

2.3.1. For a single person: Reparameterization for between- and within-person effects

This section begins by describing a reparameterization of the orthogonal multiple factor model to decompose the total effect for an individual person p into between- and within-person effects for the person. The between-person element is represented by an intercept in the reparameterized model and the within-person element is specified by a deviation about the mean factor loading. Any ordinary factor solution can be reparameterized so as to include estimates of the intercepts for individuals, factor loadings, and factor scores.

In the multiple factor model described in equation (6), the ordinary (total) factor effect, λvkfpk, can be rewritten as between- and within-person effects separately: inline image, where inline image, the mean factor loading along factor k. Substituting these two terms into equation (7) yields an equation with an intercept:

  • display math(10)

where inline image. In this equation, cp is an intercept that describes the between-person effect of person p and inline image describes the within-person effect of person p. If the mean of a factor equals zero, the between-person effect, inline image, equals zero. In that case, the factor does not carry any between-person effect, and this factor can be seen as a pure within-person effect and categorized as a within-person factor. Note that in the modified model of equation (10), the original factor loadings (from equation (7)) are expressed as an intercept cp plus deviations about their means inline image, but the factor scores remain unchanged in both equations (7) and (10).

To make a connection between observed scores (or person profiles) and model parameters (or latent profiles), the decomposition is applied to observed scores: a score of person p on variable v, xpv, is decomposed into two additive effects, inline image, but they are also expressed with model parameters. Since within-person means, inline image, depend on the factor loading means inline image, person p's mean score can be expressed as inline image which approximates the between-person effect. The ipsatized score of person p can also be expressed as inline image, which approximates the within-person effect, considering the error turbulence (which is expected when raw scores are expressed with factor model parameters).

No matter how many factors underlie the data, the separate factor contributions to between-person means can always be aggregated into a single ubiquitous effect, and the between-person effect is supposed to appear as a single entity, such as inline image, although cp may not be identified as a single factor in ordinary exploratory factor analysis. However, cp can be specified as a single factor in a structural equation model in which all indicator variables have equal loadings. This combined between-person effect may help explain the ubiquity of a large general factor appearing in personality, cognitive, and school achievement data (e.g., Maydeu-Olivares & Coffman, 2006). Especially for confirmatory factor analysis, the use of constant loadings for the between-person effect is a mathematical representation to define a general or between-person factor, and does not overlap with other specific abilities.

2.3.2. For a group of people: Reparameterization for between- and within-person effects

The between-person effect for a group of people is the sum of individuals' observed mean scores, inline image, which can be approximated by the latent scores, inline image. Similarly, the within-person effect for the group of people is a sum of individuals' ipsatized scores, inline image, and can be approximated by the ipsatized latent scores, inline image. The variation accounted for by between- or within-person effect appearing in the observed scores can be estimated with the reparameterized latent scores which are mean or ipsatized factor loadings.

Between-person variation in factor k

Assuming orthogonal factors, the variation accounted for by the effect of inline image in factor k will be approximated by inline image. This quantity inline image is the between-person variation for a single variable, but the variation is assumed the same for every variable. Aggregating across all of the variables, the total between-person variation of factor k will be inline image. In our hypothetical example, factor k loadings were .7, .1, and .7, and the mean factor loading was .5. The between-person variation accounted for in a single variable would be inline image. The total between-person variation accounted for by the factor would be inline image.

Within-person variation in factor k

Similarly, the variation accounted for by the within-person effect shown in observed scores inline image in factor k can be approximated by the squared ipsative latent scores inline image. The quantity inline image is the within-person variation for a single variable, but the total within-person variation across all variables will be inline image. In the previous example (equation (6)), the factor k ipsatized loadings were (.2, −.4, .2), so the within-person variation would be a sum of squared loadings, (.22 + (−.4)2 + .22) = .24.

Total variation in factor k

The variation accounted for by an ordinary factor can be represented as a linear combination of the between- and within-person variations. The total variation accounted for by factor k equals

  • display math(11)

Since inline image, inline image stands for the between-person variation in factor k and inline image characterizes the within-person variation in factor k. In the example above, the total variation accounted for by the factor equals .99, which is the sum of the between-person variation, inline image, and the within-person variation, inline image. The total variation is accounted for by the sum of the variation accounted for by each factor. The total between-person variation accounted for by the model equals

  • display math(12)

The total within-person variation accounted for by the model equals

  • display math(13)

2.4. Summary of between- and within-person effects

To further clarify the decomposition of total effect into between- and within-person effect expressed by observed and latent profiles, we include a summary table here:

Total effectBetween-person effectWithin-person effect
For a single variable:
xpv ≅ ∑kλvkfpk inline imagefpk Constant in person p (constant over variables) inline imagefpkVarying in person p (zero sum over variables)
For all variables (v = 1, …, V):
xp ≅ ∑kλkfpk inline imagefpk Varying over people inline imagefpk Varying over people

Note that inline image which are all V-vectors.

3. Example

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Derivation of model
  5. 3. Example
  6. 4. Discussion
  7. Acknowledgement
  8. References

We analyse the norm sample of the Woodcock-Johnson III (WJ-III) tests of cognitive abilities (Woodcock, McGrew, & Mather, 2001). The observations in the sample are assumed to be independent and identically distributed (i.i.d.). Each of the 14 cognitive subtests is shown in Table 1. After listwise deletion (of missing data) on the 14 cognitive subtest scores of 8,782 individuals, the sample size became 3,825. After excluding participants younger than 15 and older than 65, the sample was reduced to 1,767. We randomly split the sample into two parts: the first half of the sample (n = 855) was used to determine the number of factors by principal component analysis (PCA) and the other half (n = 912) was analysed using the maximum likelihood (ML) estimation procedure for identifying factors. Our total sample of participants was comprised of 56% females, 85% Whites, 10% Blacks, 2% Hispanics, and 3% other race. The mean age was 28 and its standard deviation was 14.

Table 1. WJ-III cognitive subtests and domains
SubtestsDomains
Note
  1. The numbers next to subtests are domain numbers. For example, VC1 and GI1 belong to the first domain, comprehension-knowledge.

Verbal comprehension (VC1)General information (GI1)1: Comprehension-knowledge (CKL)
Visual-auditory learning (VA2)Retrieval fluency (RF2)2: Long-term retrieval (LTR)
Spatial relations (SR3)Picture recognition (PR3)3: Visual-spatial thinking (VST)
Sound blending (SB4)Auditory attention (AA4)4: Auditory processing (ADP)
Concept formation (CF5)Analysis-synthesis (AS5)5: Fluid reasoning (FLR)
Visual matching (VM6)Decision speed (DS6)6: Processing speed (PRS)
Numbers reversed (NR7)Memory for words (MW7)7: Short-term memory (STM)

3.1. Determining number of factors using PCA

Principal component analysis was conducted with correlations among 14 WJ-III subtest scores. A 14 × 14 (cognitive subtests) correlation matrix from the first sample (n = 855) was analysed (Table 2). Because correlations were being analysed, the total variation was the same as the number of cognitive subtests (14.00). PCA not only provided the same number of components as the cognitive subtests, but also effectively separated between- and within-person factor variations because of the principal axis orientation exploited by PCA.

Table 2. A correlation matrix of WJ-III's 14 cognitive subtests
 VC1GI1VA2RF2SR3PR3SB4AA4CF5AS5VM6DS6NR7MW7
Note
  1. Note that the lower triangle was used for the PCA (n = 855) and the upper triangle was use for the ML factor analysis (n = 912).

VC1 .794.513.332.373.236.477.331.556.468.289.249.371.377
GI1.786 .469.345.337.176.457.305.486.419.277.210.301.377
VA2.488.406 .241.359.321.376.310.535.462.328.259.413.307
RF2.403.379.226 .056.110.238.249.220.227.298.270.206.225
SR3.395.325.405.108 .199.253.229.375.351.278.260.288.238
PR3.259.214.351.179.234 .145.197.243.207.229.237.160.165
SB4.482.457.419.329.259.221 .443.366.273.291.230.326.399
AA4.340.310.287.280.199.177.453 .329.278.324.431.279.247
CF5.572.482.571.254.416.250.420.326 .533.337.288.400.326
AS5.507.432.484.248.383.251.346.228.568 .328.261.378.263
VM6.275.266.325.340.269.182.277.340.360.307 .527.362.291
DS6.303.285.276.348.246.203.319.443.318.211.520 .240.153
NR7.385.296.386.253.251.206.372.284.420.400.421.265 .460
MW7.371.330.307.209.209.188.411.242.343.289.252.172.464 

The first three components whose eigenvalues were larger than 1.00 were examined: the eigenvalues were 5.44, 1.29, and 1.03. According to the eigenvalue equal to or larger than 1.00 criterion, three factors could be obtained. However, to validate the three-factor solution, we conducted a parallel analysis. We fixed the number of variables at 14 and sample size n = 855 (which were the same as the original PCA variable and sample sizes) and generated 2,000 random data sets. The first three mean eigenvalues for the components (calculated from 2,000 random data sets) were 1.22, 1.17, and 1.13. The mean eigenvalue (1.13) for the third component estimated from the series of random data was larger than the third component's eigenvalue (1.03) estimated from our sample data, and the third component was disregarded, since it could have resulted from a random artefact. Therefore, we chose a two-factor solution for our further analyses (O'Connor, 2000).

3.2. Identifying factors using the ML method

After PCA was run as a preliminary method to decide the number of factors along with a parallel analysis, any factoring method could be used to identify factors, and we ran the maximum likelihood method factor analysis on the second sample (n = 912). Based on the parallel analysis results, the two-factor solution was chosen inline image which accounted for 39% of the total variance. The ML factor analysis results are summarized in Table 3.

Table 3. Factor loadings, total, between-, and within-person factors: ML factor analysis
 TPF1TPF2BPF1WPF1BPF2WPF2BPF*
Notes
  1. BPF* = a ubiquitous between-person factor combined with BPF1 and BPF2.

  2. Between-person effects are in bold and within-person effects underlined.

VC1.880−.233.546.335.160−.393.569
GI1.824−.261.546.279.160−.421.569
VA2.639.159.546.093.160−.001.569
RF2.402.102.546−.143.160−.058.569
SR3.466.158.546−.080.160−.002.569
PR3.307.205.546−.239.160.044.569
SB4.572.089.546.026.160−.071.569
AA4.461.315.546−.084.160.155.569
CF5.671.146.546.125.160−.015.569
AS5.583.168.546.037.160.008.569
VM6.460.499.546−.086.160.338.569
DS6.390.493.546−.155.160.333.569
NR7.500.277.546−.046.160.117.569
MW7.485.126.546−.061.160−.034.569
Mean.546.160.546.000.160.000.569
Eigen.4.514.9664.170 .344 .359 .607 4.529
BPF/WPF%  92% 37% 8% 63%  
TPV5.480100%     
BPV4.529 83%      
WPV .951 17%      

Specifically, factor 1 accounted for 32% of the total variance, and of that explained variance 92% was assigned to the between-person variation and 8% to the within-person variation. Factor 2 accounted for 7% of the total variance, and of that variance, 37% was assigned to the between-person variation and 63% to the within-person variation. Factors 1 and 2 contain both between- and within-person effects (as shown in equation (4)). To summarize, of the variance accounted for by two factors, 83% was assigned to the between-person variation and 17% assigned to the within-person variation (see Table 3).

With the retained factors, utilizing model parameters, the between-person effect for person p, inline image, can be estimated. Inclusion of factor 1 had a dominant effect on the estimation of levels, while factor 2 had certain effects on the construction of the levels as well, since 37% of the variance explained by factor 2 was between-person variance. We examined correlations between factor scores and level estimates and mean factor loadings to understand how much each factor contributed to between-person variance. The correlation between scores on factor 1 and the levels was Cor(fp1, cp) = .96, whereas the correlations of factor 2 scores with the levels were small, Cor(fp2, cp) = .28. This correlation analysis indicates that the between-person effect for factor 1 was much larger than that for factor 2. Also, the mean factor 1 loading was much larger than the mean factor 2 loading: inline image and inline image. The mean factor 1 loading contributes more than three times the mean factor 2 loading for the level estimation. Although the model parameters (mean factor loadings and factor scores) are used to determine the quantity of person p's level cp, it can also be estimated by the total scores (Totalp) of the cognitive subtests. To verify this, the correlations were computed between cp and Totalp: Cor(cp, Totalp) = .99, confirming that cp ≅ Totalp.

Figure 1 displays the initial (or total) factor 1 loadings and the between- and within-person factor 1 loadings. The initial factor loadings were decomposed into between- and within-person factor loadings. The initial factor 1 displays some fluctuation in its pattern (when compared to the between-person factor pattern in the same figure), which is caused by the within-person effect (8% of factor 1 variation). However, there was a noticeable difference between the levels of the initial and the within-person factor loadings, but little difference was found between the levels of the initial and the between-person factor. Inspection of the figure informs us that most of the factor 1 variation accounted for by the level represents an eigenvalue (the ordinate) in the figure. The manifest level difference shown in Figure 1 confirms that most of factor 1 variation (92%) was accounted for mainly by the between-person effect.

image

Figure 1. Maximum likelihood total person factor 1 (TPF1), between-person factor (BPF1), and within-person factor 1 (WPF1) patterns. Note. Although the cognitive batteries measured by subtests are categorical, for the purpose of considering the linear relationships among them, they are connected by dotted lines.

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Consistent with the summary in Table 3, factor 2 was substantially accounted for by within-person variation, 63%, and there was a small difference found in the profile level between total and within-person factor loading pattern, as shown in Figure 2, suggesting that factor 2 accounts for more within-person variation than between-person variation.

image

Figure 2. Maximum likelihood total person factor 2 (TPF2), between-person factor 2 (BPF2), and within-person factor 2 (WPF2) patterns. Note. Although the cognitive batteries measured by subtests are categorical, for the purpose of considering the linear relationships among them, they are connected by dotted lines.

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3.2.1. Identifying a ubiquitous between-person factor

A between-person factor was identified by combining 92% of factor 1 variation with 37% of factor 2 variation inline image. The between-person factor loadings were calculated by inline image. Fourteen between-person factor loadings were assigned the same value, .569 (see Table 3). Although observed scores are not the same for every variable, the latent general factor effect is assumed to be the same for every variable in a between-person factor. To assess person p's overall performance with the between-person factor information, one can utilize either cp or a within-person mean, Totalp. Recall that cp and Totalp shared 98% of variance. When cp is standardized using the z-score metric, one can compare how much the average performance of person p is above or below the grand mean 0. If person p is above zero in cp, his/her overall performance across the 14 cognitive subtests is above the average performance. The between-person factor here was not defined by a single factor but by a combination of two between-person effects: one from factor 1 and the other from factor 2. Since the newly constructed between-person factor was not directly estimated from the factor analysis paradigm, factor scores are not available. However, person parameter estimates cp (or Totalp) can be used as substitutes for between-person factor scores which locate individuals on the continuum of general cognitive ability, rather than assess relationships between those individuals and the between-person factor.

3.2.2. Identifying within-person factors

We identified two within-person factors: one from factor 1 and the other from factor 2. Although only 8% of factor 1 variance was accounted for by its within-person factor, a clear contrast appeared in the lowest and highest loadings of subtests as shown in Figure 1. The first within-person factor (WPF1) has the highest loading on VC1 (where ‘1’ represents the cognitive domain number) and the lowest loading on PR3 (where ‘3’ represents the cognitive domain number). This within-person factor may be labelled as ‘high verbal comprehension versus low picture recognition profile’. Those who have similar patterns of WPF1 would expect to perform better in tasks related to verbal comprehension than in tasks related to picture recognition. The second within-person factor (WPF2) from factor 2 has the highest loadings on VM6 and DS6 (Domain 6: processing speed) and the lowest loadings on VC1 and GI1 (Domain 1: comprehension-knowledge). This within-person factor may be labelled as ‘high processing speed versus low comprehension-knowledge profile’. Participants who have similar patterns to WPF2 would expect to perform better on processing speed tasks than on comprehension-knowledge tasks.

3.3. Person score profiles versus latent profiles

In order to assess matching between person profiles and the two latent profiles, we estimated factor scores by regressing person profiles onto the latent profiles. These factor score estimates are similar to regression weights and assess linear relationships between person profiles and the latent profiles. Positive factor scores imply positive relationships of person profiles with latent profiles, but negative factor scores indicate negative relationships with them. The first factor score fp(1) has mean −.04 and standard deviation 2.34 and the second factor score fp(2) has mean −.02 and standard deviation 1.85. However, to interpret factor scores in terms of standard deviation units, they have been standardized with mean zero and standard deviation one.

3.3.1. Replicating person profiles

Considering the magnitudes and directions of factor scores, three persons were selected to compare their person profiles with profiles replicated by factor scores and within-person factors according to equation (10). The first person (no. 37) had a substantial (negative) score on factor 1 (f37(1) = −1.28), that is, 1.28 standard deviation units below the mean of zero, and a positive a score on factor 2 (f37(2) = 0.50), that is, half a standard deviation above the mean. To replicate the profile of person 37, the person's factor scores were multiplied by each relevant within-person factor (or latent profile): −1.28 × WPF1 + .50 × WPF2. Because of the person level information, c37 = −2.74, this person was 2.74 standard deviations below the average performance on cognitive subtests. Note that levels of latent profiles are set to zero, and the level of a replicated profile is also zero. So, factor scores and within-person factors are used to replicate person profile patterns, not levels. As shown in Figure 3, the two profile patterns are similar, but there is a distinctive difference in their levels. The correlation between the two profiles was .89. Since person 37 had a substantial negative factor score on WPF1, compared to the score on WPF2, this person profile pattern would be more explained as a mirror image of WPF1 than the WPF2 pattern. As seen in Figure 3, VC1 and GI1 (which are the highest points in WPF1) are at the lowest points, whereas PR3 (which is the lowest point in WPF1) are at the highest point. Thus, person 37 is expected to do well on PR3 type tasks, but do poorly on VC1 and GI1 type tasks. The correlation between the profile and WPF1 for person 37 was −.87.

image

Figure 3. Profile for person 37 (c37 = −2.74); profile replicated by two within-person factors (−1.28WPF1 + .50WPF2); and correlation (r = .89) between profile and replicated profile. Note. Although the cognitive batteries measured by subtests are categorical, for the purpose of considering the linear relationships among them, they are connected by dotted lines.

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The second person (no. 63) had inline image. This person's average performance in the cognitive subtests was about one standard deviation above the average performance. The profile pattern was replicated by a linear combination of two within-person factors weighted by factor scores .60 × WPF1 − .77 × WPF2. As seen in Figure 4, the two profile patterns are somewhat similar and the correlation between them was .70, although the person profile was superimposed on the replicated profile because of c63 = 1.03. This person's profile pattern would be rather similar to a mirror image of WPF2, since the factor score on WPF2 was more substantial (but negative) than the score on WPF1. Thus, we see VC1/GI1 at the highest points and DS6 at a somewhat lower point in the profile of this person as a mirror image of WPF2. Notice that, as shown in Figure 4, the replicated profile of this person reproduced the mirror image of WPF2 perfectly, at least in the highest and the lowest points (see the closed square profile in Figure 4). Those who have similar profiles to this person would expect to do well on VC1 and GI1 type tasks but poorly on VM6 and DS6 type tasks.

image

Figure 4. Profile for person 63 (c63 = 1.03); profile replicated by two within-person factors (.60WPF1 − .77WPF2); and correlation (r = .70) between profile and replicated profile. Note. Although the cognitive batteries measured by subtests are categorical, for the purpose of considering the linear relationships among them, they are connected by dotted lines.

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The last example (no. 1) is shown in Figure 5. Person no. 1 had inline image. This person performed .93 standard deviations below average in cognitive subtests. The profile pattern for this person was replicated by a linear combination of two within-person factors weighted by factor scores .31 × WPF1 − .23 × WPF2. However, as shown in Figure 5, there was little resemblance in their profile patterns and the correlation between them was .24. In short, these three examples demonstrate how critical the magnitudes of factor scores are to replicate person profile patterns: The larger, the better in replication!

image

Figure 5. Profile for person 1 (c1 = −.93); profile replicated by two within-person factors (.31WPF1 −.23WPF2); and correlation (r = .25) between profile and replicated profile. Note. Although the cognitive batteries measured by subtests are categorical, for the purpose of considering the linear relationships among them, they are connected by dotted lines.

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3.4. Utility of within-person factors

3.4.1. Within-person variation in person profiles

We conducted regression analyses to see the extent to which the within-person factors accounted for the within-person variation in 912 person profiles (from the sample used in the ML factor analysis). The two within-person factors accounted for 22% of the within-person variation in person profile patterns.

3.4.2. Relationships with criterion variables

In addition, we conducted multivariate regression with two ML factors as predictor variables for broad reading (BR), broad maths skill (BM), and broad written language (BW) from the WJ-III achievement test battery. The two factors accounted for 59% of BR, 50% of BM, and 52% of BW. The total variance was composed of 83% of between-person variance and 17% by within-person variance (see Table 3). Two within-person factors accounted for 10% of BR, 8% of BM, and 9% of BW.

4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Derivation of model
  5. 3. Example
  6. 4. Discussion
  7. Acknowledgement
  8. References

Ordinary factor analysis has been used mostly to seek latent factors from variable correlations, utilizing a rotated solution in pursuit of a simple structure. Such a rotated factor structure does not provide any information about person score profiles. We reparameterized the ordinary unrotated factors to get the between- or within-person factors because they provide information about person score profiles (level and pattern). To determine the number of factors, we recommend using PCA along with parallel analysis.

Interestingly, in our example, two factors, when viewed as ordinary (ML) factors, accounted for only 39% of total variance, but the level information inline image that was estimated by the factor model parameters accounted for 98% (the squared correlation between the level estimates and total subscale scores) of total subscale scores. Moreover, two within-person factors which accounted for only 7% of total variance in the ordinary factor solution, when all person score profiles (n = 912) were regressed onto them, accounted for 22% of total person scoring pattern variance.

The assumption of orthogonality of factors is required in the current study as in Davison et al. (2009) and in the random intercept model in Maydeu-Olivares and Coffman (2006). Thus, one can replicate person score profiles with within-person factors weighted by factor scores: dominant factor scores determine person profile patterns according to within-person factor patterns as shown in Figures 3 and 4.

Ordinary factors tend to confound between- and within-person factor variation, and it is important to differentiate factors according to their sources of variation. When a factor is primarily explained by either between- or within-person variation, it should be classified accordingly. However, in reality it is not possible for a single factor to be completely explained by either between- or within-person variation as shown in our example. For factor 1, 92% of the factor variance was accounted for by between-person variation and 8% by within-person variation; for factor 2, 37% of the factor variation was from between-person variation and 63% from within-person variation. The two between-person variations (from factors 1 and 2) were added to constitute total between-person variation with which a between-person factor was defined. This between-person factor (or level) information is useful to summarize all of a person's subtest scores as a single number which can be easily compared to other persons' level information. Ipsatized loadings were utilized to identify WPF1 and WPF2. These within-person factors can be used as a tool to understand the patterning of person profiles, and to examine their validity by assessing their relationship with criterion variables of interest.

Acknowledgement

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Derivation of model
  5. 3. Example
  6. 4. Discussion
  7. Acknowledgement
  8. References

Thanks are due to Kevin McGrew at Institute for Applied Psychometrics for the data analysed in this paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Derivation of model
  5. 3. Example
  6. 4. Discussion
  7. Acknowledgement
  8. References
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