Mixture Simultaneous Factor Analysis for Capturing Differences in Latent Variables Between Higher Level Units of Multilevel Data

Figures & data

TABLE 1 Base Loading Matrix and the Derived Cluster-Specific Loading Matrices for Clusters 1 and 2, in the Case of Two Factors (Top) and in the Case of Four Factors (Bottom)

	Base Loading Matrix				Cluster 1				Cluster 2
	Factor 1		Factor 2		Factor 1		Factor 2		Factor 1		Factor 2
Var. 1	1		0						1		0
Var. 2	1		0		1		0
Var. 3	1		0		1		0		1		0
Var. 4	1		0		1		0		1		0
Var. 5	1		0		1		0		1		0
Var. 6	1		0		1		0		1		0
Var. 7	0		1						0		1
Var. 8	0		1		0		1
Var. 9	0		1		0		1		0		1
Var. 10	0		1		0		1		0		1
Var. 11	0		1		0		1		0		1
Var. 12	0		1		0		1		0		1
	F1	F2	F3	F4	F1	F2	F3	F4	F1	F2	F3	F4
Var. 1	1	0	0	0			0	0	1	0	0	0
Var. 2	1	0	0	0	1	0	0	0			0	0
Var. 3	1	0	0	0	1	0	0	0	1	0	0	0
Var. 4	0	1	0	0			0	0	0	1	0	0
Var. 5	0	1	0	0	0	1	0	0			0	0
Var. 6	0	1	0	0	0	1	0	0	0	1	0	0
Var. 7	0	0	1	0	0	0	1	0	0	0	1	0
Var. 8	0	0	1	0	0	0	1	0	0	0	1	0
Var. 9	0	0	1	0	0	0	1	0	0	0	1	0
Var. 10	0	0	0	1	0	0	0	1	0	0	0	1
Var. 11	0	0	0	1	0	0	0	1	0	0	0	1
Var. 12	0	0	0	1	0	0	0	1	0	0	0	1

Note. In the case of medium similarity equals 0 and equals 1, whereas in the case of high similarity equals √(.6) and equals √(.4). When the number of clusters is four, the two additional loading matrices are constructed similarly; for example, in the four factor case, by shifting the primary loading or adding a cross-loading for Variables 3 and 6 for Cluster 3, and for Variables 4 and 7 for Cluster 4.

FIGURE 1 Scatter plot of CC_min versus ARI for the simulated data sets.

TABLE 2 Clustering of the Countries of the Mixture Simultaneous Factor Analysis Model With Two Clusters and Two Factors Per Cluster for the Emotion Norm Data From the 2001 ICS Study

Cluster 1	Bangladesh, Brazil, Bulgaria, Cameroon, Georgia, Ghana, India, Iran, Nepal, Nigeria, Slovakia (= .9980), South Africa (= .9965), Thailand, Turkey, Uganda, Zimbabwe
Cluster 2	Australia, Austria, Belgium, Canada, Chile, China, Colombia, Croatia, Cyprus, Egypt, Germany, Greece, Hong Kong, Hungary, Indonesia, Italy, Japan, Kuwait, Malaysia, Mexico, Netherlands, Philippines, Poland, Portugal, Russia, Singapore, Slovenia, South Korea, Spain, Switzerland, United States, Venezuela

Note. Except for Slovakia and South Africa, all countries are assigned to the clusters with a posterior probability of 1. The probabilities for Slovakia and South Africa are given in brackets.

FIGURE 2 Boxplots for (a) the Human Development Index (HDI) 1998 (United Nations Development Program, 2000) and (b) the level of conservatism (Schwartz, 1994) of the countries per cluster of the Mixture Simultaneous Factor Analysis model with two clusters and two factors per cluster for the International College Survey data set on emotion norms.

TABLE 3 Varimax (Top) and Echelon (Bottom) Rotated Loadings of the Mixture Simultaneous Factor Analysis Model With Two Clusters and Two Factors Per Cluster for the Emotion Norm Data From the 2001 ICS Study

	Cluster 1		Cluster 2
Varimax Rotation	Positive	Negative	Positive	Negative
Contentment	1.44	−0.25	1.21	−0.11
Happy	1.60	−0.26	1.42	−0.15
Love	1.39	−0.26	1.22	−0.06
Sad	−0.32	1.32	0.05	1.26
Jealousy (in romantic situations)	0.00	1.29	−0.02	1.27
Cheerful	1.18	−0.30	1.04	−0.05
Worry	−0.07	1.74	0.04	1.43
Stress	−0.25	2.01	−0.19	1.69
Anger	−0.37	1.97	−0.18	1.54
Pride	0.27	1.10	0.60	0.35
Guilt	0.05	1.24	0.11	1.10
Shame	0.18	1.03	0.08	1.07
Gratitude	0.95	−0.29	0.86	−0.12
	Cluster 1		Cluster 2
Echelon Rotation	Positive	Negative	Positive	Negative
Contentment	1.44**	−0.25	1.21**	−0.13
Happy	1.60**	−0.26	1.42**	−0.17
Love	1.39*	−0.26	1.22*	−0.08
Sad	−0.32**	1.32	0.07**	1.26
Jealousy (in romantic situations)	0	1.29	0	1.27
Cheerful	1.18	−0.30*	1.04	−0.06*
Worry	−0.07	1.74**	0.07	1.43**
Stress	−0.25	2.01**	−0.16	1.69**
Anger	−0.37	1.97**	−0.16	1.54**
Pride	0.27**	1.10**	0.61**	0.34**
Guilt	0.05	1.24	0.13	1.10
Shame	0.18	1.03	0.10	1.07
Gratitude	0.95	−0.29	0.86	−0.14

Note. For each emotion, the primary loading is shown in bold. Below, the restricted loadings are in italic and underlined and loadings that are significantly different across clusters (according to Wald tests and after Bonferroni correction) are indicated by **p < .01 and *p < .05.

TABLE 4 Unique Variances of the Mixture Simultaneous Factor Analysis Model With Two Clusters and Two Factors Per Cluster for the Emotion Norm Data From the 2001 ICS Study

	Cluster 1	Cluster 2
Contentment	1.47	3.48
Happy	0.63	1.39
Love	1.21	2.37
Sad	2.76	4.19
Jealousy (in romantic situations)	2.85	4.94
Cheerful	1.53	2.38
Worry	2.01	2.86
Stress	2.15	2.63
Anger	1.87	2.23
Pride	3.41	5.33
Guilt	2.80	4.42
Shame	3.01	4.85
Gratitude	2.88	3.95

FIGURE C.1  ‘Example.txt’ file communicating the clustering (‘Cluster’), the number of variables (‘V2’ to ‘V13’) and the data block structure (‘V1’ and ‘rows’) to the simulation syntax for LG 5.1. Note that, in general, the number of rows can differ across data blocks.

United Nations Development Program. (2000). Human development report 2000. New York, NY: Oxford University Press.

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