# Chapter 3: Multigroup Factor Analysis

### Example 1 on Multigroup factor analysis: A model under measurement invariance

Consider the data in our example for the questions D18-D23, using a 2-factor confirmatory factor analysis model where questions D18-D20 measure only the factor "obligation to obey the police" and questions D21-D23 only the factor "moral alignment with the police". Fit this as a multigroup model to data from all 27 countries, specifying the measurement model to have full invariance of measurement across the countries. Use the results of the model to compare the estimated means, variances and correlations of the factors between the countries.

// Example of multigroup analysis under full equivalence:
* Convert the country variable from string to numeric variabl
encode cntry, gen(country)
* Fit the model:
sem (Obey -> bplcdc@v1 doplcsy dpcstrb) ///
(MoralAlign -> plcrgwr@v2 plcipvl gsupplc), ///
var(1: Obey@1) var(1: MoralAlign@1) method(mlmv) ///
group(country) ginvariant(mcons mcoef merrvar)
** Standardized estimates, to get the estimated correlation
** (i.e. standardized covariance) between the factors:
sem, stand

# Example of multigroup analysis under full equivalence:
library(lavaan)
#
countries <- unique(ESS5Police\$cntry)
length(countries) # 27 countries
# The "c(1,rep(NA,26))*" constrains the factor variances to be 1 in the
# first country.
ModelSyntax <- '
Obey =~ NA*bplcdc + doplcsy + dpcstrb
MoralAlign =~ NA*plcrgwr + plcipvl + gsupplc
Obey ~~ c(1,rep(NA,26))* Obey
MoralAlign ~~ c(1,rep(NA,26))* MoralAlign
'
FittedModel.MG <- sem(model = ModelSyntax,
data = ESS5Police, group="cntry",
meanstructure = TRUE,missing="ml",
summary(FittedModel.MG)
# A more concise table of the estimates:
parameterEstimates(FittedModel.MG,fmi=F,stand=T)

Estimated means, variances and correlation of the two factors from the multigroup model are shown in Table 3.1 for each of the countries, and also in graphical form in Figures 3.1-3.4. Note that here the factor means are fixed at 0 and factor variances at 1 for the first country, Belgium. From these results, we may for example observe the following:

• There is a large amount of variation in the estimated means of the factors (see Figures 3.1 and 3.2). For both, the difference between the highest and the lowest means is around 2 units, in other words around two individual-level standard deviations of the factors. This means, for example, that almost all individuals in the countries where the average values of these factors are the highest have higher values than the average individual in the countries where the values are the lowest. The standard errors in the estimated means are fairly small, so that many of the differences between country means appear to be statistically significant.
• There are fairly clear geographic regularities in the levels of the means, although with exceptions. Levels of felt obligation to obey the police and moral alignment with the police tend to be higher in Northern and Western European countries than in Eastern and Southern Europe.
• The values of the two factors are positively correlated, both among individuals within the countries (Table 3.1) and between the country averages of the factors (Figure 3.3).
• The means and variances of the factors are negatively correlated (Figure 3.4). In other words, in countries where the average levels of felt obligation to obey the police or moral alignment with the police are highest, variation between individuals in these factors tends to be lowest.

Table 3.1: Estimated distributions of the factors for a 2-factor multigroup confirmatory factor analysis model for indicators of Obligation to obey the police and Moral alignment with the police, fitted to data from 27 countries in the ESS.

Means Variances
Country Obey MoralAl Obey MoralAl corr.
Belgium (BE) 0 0 1 1 0.35
Bulgaria (BG) -0.53 -0.37 2.56 1.81 0.46
Switzerland (CH) 0.36 0.26 1.25 0.83 0.12
Cyprus (CY) 0.45 -0.22 1.33 1.55 0.45
Czech Republic (CZ) 0.23 -0.50 1.54 1.47 0.31
Germany (DE) 0.31 0.36 1.17 0.84 0.35
Denmark (DK) 0.87 0.41 0.82 0.75 0.41
Estonia (EE) -0.23 0.12 1.74 0.82 0.24
Spain (ES) -0.10 0.10 0.94 1.06 0.39
Finland (FI) 0.75 0.57 0.60 0.62 0.50
France (FR) -0.15 -0.30 1.12 1.50 0.36
United Kingdom (GB) -0.01 0.07 1.10 1.03 0.44
Greece (GR) -0.23 -0.62 1.42 1.72 0.55
Croatia (HR) -0.36 -0.33 1.99 1.23 0.42
Hungary (HU) 0.38 -0.33 1.53 1.33 0.37
Ireland (IE) -0.18 0.17 1.30 1.44 0.50
Israel (IL) 0.60 -0.72 1.57 1.86 0.29
Lithuania (LT) -0.03 -0.21 1.64 0.96 0.40
Netherlands (NL) 0.26 0.10 0.87 0.78 0.35
Norway (NO) 0.39 0.41 0.94 0.67 0.52
Poland (PL) 0.04 0.11 1.37 0.87 0.27
Portugal (PT) 0.08 -0.11 1.26 0.94 0.43
Russia (RU) -0.89 -0.98 1.77 1.71 0.47
Sweden (SE) 0.54 0.30 0.98 0.57 0.44
Slovenia (SI) -0.71 -0.30 2.11 1.10 0.34
Slovakia (SK) -0.10 -0.11 1.76 1.14 0.23
Ukraine (UA) -0.67 -1.17 2.05 2.05 0.31

Figure 3.1: Estimated means of the factor Obligation to obey the police (with 95% confidence intervals) for each country, from the model shown in Table 3.1.

Figure 3.2: Estimated means of the factor Moral alignment with the police (with 95% confidence intervals) for each country, from the model shown in Table 3.1.

Figure 3.3: Estimated means of the two factors for each country, from the model shown in Table 3.1.

Figure 3.4: Estimated means against estimated standard deviations of the factor Moral alignment with the police for each country, from the model shown in Table 3.1.

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