An Entropy-Based Approach for Measuring Factor Contributions in Factor Analysis Models.

In factor analysis, factor contributions of latent variables are assessed conventionally by the sums of the squared factor loadings related to the variables. First, the present paper considers issues in the conventional method. Second, an alternative entropy-based approach for measuring factor contributions is proposed. The method measures the contribution of the common factor vector to the manifest variable vector and decomposes it into contributions of factors. A numerical example is also provided to demonstrate the present approach.

1. Introduction

Factor analysis is a statistical method for extracting simple structures to explain inter-relations between manifest and latent variables. The origin dates back to the works of [1], and the single factor model was extended to the multiple factor model [2]. These days, factor analysis is widely applied in behavioral sciences [3]; hence, it is important to interpret the extracted factors and is critical to explain how such factors influence manifest variables, that is, measurement of factor contribution. Let be manifest variables; latent variables (common factors); unique factors related to ; and let be factor loadings that are weights of factors to explain . Then, the factor analysis model is given as follows:where For the simplicity of discussion, common factors are assumed to be mutually independent in this section, that is, we first consider an orthogonal factor analysis model. In the conventional approach, the contribution of factor to all manifest variables, , is defined as follows:The above definition of factor contributions is based on the following decomposition of the total of variances of the observed variables [4] (p. 59):What physical meaning does the above quantity have? Applying it to the manifest variables observed, however, such a decomposition leads to scale-variant results. For this reason, factor contribution is usually considered on the standardized versions of manifest variables. What does it mean to measure factor contributions by (2)? For standardized manifest variables, we have Then, (2) is the sum of the coefficients of determination for all standardized manifest variableswith respect to a single latent variable. The squared correlation coefficients (3), that is, , are the ratios of explained variances of a manifest variable, and in this sense, they can be interpreted as the contributions (effects) of factors to the manifest variable . Although, what does the sum of these with respect to all manifest variables , that is, (2), mean? The conventional method may be intuitively reasonable for measuring factor contributions; however, we think it is sensible to propose a method measuring factor contributions as the effects of factors on the manifest variable vector , which are interpretable and have a theoretical basis. There is no research on this topic as far as we have searched. The present paper provides an entropy-based solution to the problem. Entropy is a useful concept to measure the uncertainty in the systems of random variables and sample spaces [5] and it can be applied to measure multivariate dependences of random variables [6,7].

This paper proposes an entropy-based method for measuring factor contributions of to the manifest variable vector concerned, which can treat not only orthogonal factors, but also oblique cases. The present paper has five sections in addition to this section. In Section 2, the conventional method for measuring factor contributions is reviewed. Section 3 considers the factor analysis model in view of entropy and makes a preliminary discussion on measurement of factor contribution. In Section 4, an entropy-based path analysis is applied as a tool to measure factor contributions. Contributions of factors are defined by the total effects of the factors on the manifest variable vector, and the contributions are decomposed into those to manifest variables and subsets of manifest variables. Section 5 illustrates the present method using a numerical example. Finally, in Section 6, some conclusions are provided.

2. Relative Factor Contributions in the Conventional Method

In the conventional approach, for the orthogonal factor model (1), the contribution ratio of is defined by The above measure is referred to as the factor contribution ratio in the common factor space. Let be the multiple correlation coefficient of latent variable vector and manifest variable. Then, for standardized manifest variable, we have The above quantity can be interpreted as the effect (explanatory power) of latent variable vector on manifest variable; however, the denominator of (4) is the sum of those effects (5) and there is no theoretical basis to interpret it. Another contribution ratio ofis referred to as that in the whole space of , and is defined by If the manifest variables are standardized, we have Here, there is an issue similar to (4), because the denominator in (6) does not express the variation of the manifest variable vector . Indeed, it is the sum of the variances of manifest variables and does not include covariances between them. In the next section, the factor analysis model (1) is reconsidered in the framework of generalized linear models (GLMs), and the effects (contributions) of latent variables on the manifest variable vector , that is, factor contributions, are discussed through entropy [8].

3. Factor Analysis Model and Entropy

It is assumed that factors and are normally distributed, and the factor analysis model (1) is reconsidered in the GLM framework. Let be a factor loading matrix; let be an correlation matrix of common factor vector ; and let be the variance-covariance matrix of unique factor vector . The conditional density function of given, , is normal with mean and variance matrix, and is given as follows:where is the cofactor matrix of . Let and be the marginal density functions of and , respectively. Then, a basic predictive power measure for GLMs [9] is based on the Kullback–Leibler information [6], and applying it to the above model, we have The above measure was derived from a discussion on log odds ratios in GLMs [9], and is scale-invariant with respect to manifest variables . The numerator of (7) is the explained entropy of by , and the denominator is the dispersion of the unique factors in entropy, that is, the generalized variance of . Thus, (7) expresses the total effect (contribution) of factor vector on manifest variable vector in entropy, and is denoted by in the present paper. The entropy coefficient of determination (ECD) is calculated as follows [9]:The denominator of the above measure is interpreted as the variation of manifest variable vector in entropy and the numerator is the explained variation of random vector in entropy. In this sense, ECD (8) is the factor contribution ratio of for the whole entropy space of , and it expresses the standardized total effect of on the manifest variable vector , which is denoted by [8,10]. As for (6), in the present paper, the ECD is denoted by , that is, the relative contribution of factor vector for the whole space of manifest variable vector in entropy.

Let be the variance-covariance matrix of manifest variable vector and let be the correlation matrix of. Then, we have For assessing the goodness-of-fit of the models, the following overall coefficient of determination (OCD) is suggested ([11], p. 60) on the basis of (9):Determinant is the generalized variance of unique factor vector and is that of manifest variable vector . Then, OCD is interpreted as the ratio of the explained generalized variance of manifest variable vector by common factor vector in the p-dimensional Euclidian space. On the other hand, from (8), it follows that Hence, ECD is viewed as the ratio of the explained variation of the manifest variable vector in entropy.

Cofactor matrix is diagonal and the elements are . If common factors are statistically independent, it follows that Thus, (7) is decomposed as As detailed below, in the present paper, the contribution of factor to , , is defined by

The above contribution is different from the conventional definition of factor contribution (2); unless . In this sense, we may say that the standardization of manifest variables in entropy is obtained by setting all the unique factor variances to one.

In the next section, the contributions (effects) of factorsto manifest variable vector are discussed in a general framework through an entropy-based path analysis [8].

4. Measurement of Factor Contribution Based on Entropy

A path diagram for the factor analysis model is given in Figure 1, in which the single-headed arrows imply the directions of effects of factors and the double-headed curved arrows indicate the associations between the related variables. In this section, common factors are assumed to be correlated, that is, we consider an oblique case, and an entropy-based path analysis [8] is applied to make a general discussion in the measurement of factor contributions.

Figure 1

Path diagram for factor analysis model (1) .

In the factor analysis model (1),

Let be the conditional density functions of manifest variables , given factor vector ; let be the marginal density functions of ; let be the marginal density function of ; and let be the marginal density function of common factor vector. As the manifest variables are conditionally independent, given factor vector , the conditional density function of is From (7), we have  ☐

In model (1) with correlation matrix , we have The above quantity is referred to as the contribution of to , and is denoted as . Let be the multiple correlation coefficient ofand . Then, From Theorem 1, we then have Hence, Theorem 1 gives the following decomposition of the contribution of on into those on the single manifest variables (11):

Notice that in the denominator of (4), the total contribution of all factors is simply defined as the total sum assessed:On the other hand, in the present approach, the total effect (contribution) of factor vector on manifest variable vector is decomposed into those of manifest variables .

Let be any sub-vector of manifest variable vector . Then, the contribution of factor vector to is defined by From Theorem 1, we have the following corollary.

Let and be a decomposition of manifest variable vector , where

From a similar discussion to the proof of Theorem 1, we have Hence, the corollary follows.

Next, the standardized total effects of single factors on manifest variable vector, that is, , are calculated [8,10]. Let ; be the conditional density function of and given; be the conditional density function of given; be the conditional density function ofgiven; and be the marginal density function of . Then, we have where is a covariance matrix given , of which the elements are . The standardized total effect is given by The standardized total effect [8] is interpreted as the contribution ratio of factorin the whole entropy space of, and in the present paper, it is denoted by . The contribution of factor measured in entropy is defined by As for (6), the relative contribution of factor on is given by Concerning factor contributions of on the single manifest variables , that is, , the following theorem can be stated.

In the factor analysis model (1),

From Theorem 1, it follows that Then, we have and, From the above theorem, we have the following corollary.

Letandbedecomposition of manifest variable vector, where.

From a similar discussion in the proof of Theorem 2, the corollary follows. ☐

Let be any sub-vector of manifest variable vector . By substituting for in the above discussion, , , , and can be defined.

For orthogonal factor analysis models, the following theorem holds true.

In factor analysis model (1), if common factors

From model (1), we have This completes the theorem. ☐

From the above discussion, if common factors are statistically independent, (10) is derived. Moreover, we have This measure is the relative contribution ratio of for the variation of in entropy. The relative contributions of on in entropy are calculated as follows:

It is difficult to use OCD for assessing factor contributions, because cannot be decomposed as in the above discussion.

5. Numerical Example

In order to illustrate the present method, we use the data shown in Table 1 [12]. In this table, manifest variables are subjects in liberal arts and variables are those in sciences. First, orthogonal factor analysis (varimax method by S-PLUS ver. 8.2) is applied to the data and the results are illustrated in Table 2. From the estimated factor loadings, the first factor is interpreted as an ability relating to liberal arts, and the second factor as that for sciences. According to the factor contributions shown in Table 3, the contribution of factor is about twice as big than that of factor from a view point of entropy, and from the relative contributions , about 30% of variation of manifest variable vector in entropy is explained by factor and about 60% by factor . The relative contribution in Table 3 implies about 90% of the entropy of manifest variable vector is explained by the two factors. On the other hand, in the conventional method, the measured factor contributions of and , that is, , are almost equal (Table 4). As discussed in the present paper, the conventional method is intuitive and does not have any logical foundation for multidimensionally measuring contributions of factors to manifest variable vectors. Table 5 decomposes “the contribution of to ” into components . The contribution of to is prominent compared with the other contributions.

Table 1

Data for illustrating factor analysis.

Subject	Japanese X_1	English X_2	Social X_3	Mathematics X_4	Science X_5
1	64	65	83	69	70
2	54	56	53	40	32
3	80	68	75	74	84
4	71	65	40	41	68
5	63	61	60	56	80
6	47	62	33	57	87
7	42	53	50	38	23
8	54	17	46	58	58
9	57	48	59	26	17
10	54	72	58	55	30
11	67	82	52	50	44
12	71	82	54	67	28
13	53	67	74	75	53
14	90	96	63	87	100
15	71	69	74	76	42
16	61	100	92	53	58
17	61	69	48	63	71
18	87	84	64	65	53
19	77	75	78	37	44
20	57	27	41	54	30

Table 2

Factor loadings of orthogonal factor analysis (

	X_1	X_2	X_3	X_4	X_5
ξ_1	0.60	0.75	0.65	0.32	0.00
ξ_2	0.39	0.24	0.00	0.59	0.92
uniqueness	0.50	0.38	0.58	0.55	0.16

Table 3

Factor contributions based on entropy (orthogonal case).

	ξ_1	ξ_2	Total
C(ξ_j→X)	3.11	6.23	9.34=C(ξ→X)
RC˜(ξ_j→X)	0.30	0.60	0.90=RC˜(ξ→X)
RC(ξ_j→X)	0.33	0.67	1

Table 4

Factor contributions with the conventional method.

	ξ_1	ξ_2	Total
C_j	1.44	1.39	2.83
RC˜_j	0.29	0.28	0.57
RC_j	0.51	0.49	1

Table 5

Decomposition of factor contribution into

	X_1	X_2	X_3	X_4	X_5	Total =C(ξ_j→X )
ξ_1	0.72	1.49	0.72	0.19	0.00	3.11
ξ_2	0.30	0.15	0	0.63	5.14	6.23
total =C(ξ→X_i)	1.01	1.64	0.72	0.82	5.14	9.34

From the discussion in the previous section, the contributions of factors are flexibly calculated. For example, it is reasonable to divide the manifest variable vector into and , because the first sub-vector is related to the liberal arts and the second one to the sciences. First, the contributions of and to are calculated according to the present method, and the details are given as follows: From (14), 77% of the entropy of manifest variable sub-vector are explained by the two factors, in which 67% of that are explained by factor (15) and 10% by factor . From the relative contributions (17) and (18), 87% of the total contribution of the two factors are made by factor and 13% by factor .

On the other hand, the contributions of and on are calculated as follows: From (19), 86% of entropy of manifest variable sub-vector is explained by the two factors, in which 3% of the entropy are explained by factor (20) and 83% by factor (21). The contribution ratios of the factors to sub-vector are calculated in (22) and (23). Ninety-seven percent of the entropy was made by factor .

Second, factor contributions in an oblique case are calculated. The estimated factor loadings and the correlation matrix of factors based on the covarimin method are shown in Table 6 and Table 7, respectively. Based on factor loadings in Table 6, factor is interpreted as an ability for subjects in the liberal arts and factor as an ability for subjects in sciences. The results are similar to those in the orthogonal case mentioned above, because the correlation between the factors is not strong. Table 8 shows the decomposition of based on Theorems 1 and 2. In this case, it is noted that ; however, . According to the table, the contributions of and to sub-vectors of manifest variable vector can also be calculated as in the above orthogonal factor analysis. Table 9 illustrates the contributions of factors on manifest variable vector. Factor explains 42% of the entropy of and factor explains 71%.

Table 6

Factor loadings of oblique factor analysis (

	X_1	X_2	X_3	X_4	X_5
ξ_1	0.59	0.77	0.68	0.29	0
ξ_2	0.24	0.00	−0.12	0.52	0.92
uniqueness	0.50	0.41	0.58	0.55	0.16

Table 7

Correlation matrix of factors.

	ξ_1	ξ_2
ξ_1	1	0.315
ξ_2	0.315	1

Table 8

Decomposition of factor contribution into (oblique case).

	X_1	X_2	X_3	X_4	X_5	Total=C(ξ_j→X )
ξ_1	0.90	1.44	0.70	0.37	0.54	3.95
ξ_2	0.37	0.14	0.01	0.68	5.43	6.65
C(ξ→X_i)	1.01	1.44	0.73	0.82	5.43	C(ξ→X)=9.43

Table 9

Factor contributions based on entropy (oblique case).

	ξ_1	ξ_2	Effect of ξ on X
C(ξ_j→X)	3.95	6.65	C(ξ→X)=9.43
RC˜(ξ_j→X)	0.38	0.64	RC˜(ξ→X)=0.90
RC(ξ_j→X)	0.42	0.71

6. Discussion

For orthogonal factor analysis models, the conventional method measures factor contributions (effects) by the sums (totals) of squared factor loadings related to the factors (2); however, there is no logical foundation for how they can be interpreted. It is reasonable to measure factor contributions as the effects of factors on the manifest variable vector concerned. The present paper has proposed a method of measuring factor contributions through entropy, that is, applying an entropy-based path analysis approach. The method measures the contribution of factor vector to manifest variable vector and decomposes it into those of factors to manifest variables and/or those to sub-vectors of . Comparing (2) and (10), for standardization of unique factor variances , the present method equals to the conventional method. As discussed in this paper, the present method can be employed in oblique factor analysis as well, and it has been illustrated in a numerical example. The present method has a theoretical basis for measuring factor contributions in a framework of entropy, and it is a novel approach for factor analysis. The present paper confines itself to the usual factor analysis model. A more complicated model with a mixture of normal factor analysis models [13] is excluded, and a further study is needed to apply the entropy-based method to the model.