Confirmatory factor analysis (CFA) seeks to determine whether the number of factors and the loadings of measured (indicator) variables on them conform to the pre-established vision. Commonly displayed factor models are graphically displayed as path diagrams. Causal mod els can involve both manifest and latent variables (Cramer 2003, 28; Garson 2004). Such path diagrams can be created with computer programs like LISREL, which is an abbreviation for Linear Structural RELationships; or AMOS, which is an abbreviation for Analysis Of Moment Structures.
Manifest variables in Figure 1 are drawn as rectangles, while the latent factors are portrayed as ellipses. Lines with arrows that connect factors with rectangles are pathways. An arrow from the ellipse to the rectangle indicates a relationship between a variable
and a factor. For example, in Figure 1, there are arrows from the self-expression factor to self-improvement and employing abilities variables, showing that these variables load on or are related to that factor. There are no arrows from self-improvement and using abilities variables to the social recognition and social recognition factors, because these variables are not assumed to load on these factors.
The arrows point from the factor to the variables, rather than from variables to the factor, to indicate that the factor is expressed in terms of the variables. Factors are often described as latent variables, as they cannot be measured directly, while initial variables may be referred to as manifest variables. The values next to these arrows may be loosely thought of as the correlations or loadings between the variables and the factor in that they generally vary between –1 and +1. So in Figure 1, the loading between self-improvement and self-expression factor is .51; and between employing abilities and self-expression factor: .69.
Confirmatory factor analysis has certain advantages over exploratory techniques, as it enables us not only to analyse factor loadings but also to estimate possible variance error in every variable.
The arrows on the left of each variable and pointing to that variable indicate that each variable is not a perfect measure of the factor it is assumed to reflect. For example, the proportion of variance in the scores of the self-improvement variable that is thought to repre sent error is .74 and the error of the using abilities variable is expected to be .52. These errors may comprise both random errors and unique variance components.
We can observe the structure of the remaining two factors in a similar vein. Usefulness to other people (.90) and usefulness to society (.81) with the highest loadings confirming that social recog nition is the clearest factor in the observed model. Low errors, .18 and .34 respectively, confirm that assumption. The third factor, namely career, is built on the basis of five variables and presents loadings of these variables. High position (.68) and advancement (.72) are the highest loadings with respectively moderate errors (.53 and .48). Although other loadings in that factor are somewhat lower and errors respectively bigger, they characterise career factors in
addition to advancement values. Thus the loadings of easy work (.49), good salary (.48) and job security (.44) confirm that those school-leavers who are oriented on advancement are to a smaller extent interested in earning good salary at physically easy work in order to ensure peaceful and secure life.
It is not surprising to see that the factors in the Figure 1 are related to each other, as there were re markable correlations between the initial variables (see Appendix 1) that shape these factors. The curved lines with an arrowhead at both ends between factors signify these correlations between factors. Although it can be assumed that they should be related, the causal direction of this relation ship cannot be specified. The line and value on the right of each factor is often not shown in path diagrams and this indicates that the variance of these factors has been standardised as 1.00 (Cramer 2003, 31).
There are some indices that test statistical significance or goodness of fit of the suggested model. One of the most widely used indices is the chi-square test, that refers to the differences between the initial and the model correlation matrixes. A large chi-square value of 2227.33 indicates a level of difference that says this is not a good fit. There are some factors that affect chi-square. For example, the more observed variables and relations between them the estimated model contains, the bigger the result of chi-square test will be. The value of chi-square is a complex coefficient that takes into account all observed variables and relations between them through the degrees of freedom. The probability of chi-square being significant is less than the .001, which suggests that correlations in the presented model should differ remarkably from correlations between initial variables. However, the chi-square test for confirmatory analysis tends to be statistically more significant in case of lar ger samples.
Because of this problem, other measures of fit that depend less on sample size have been developed.
T038A
Manifest variables: T038A - Self-improvement; T038B - Income; T038C - Job security;
T038D - Position in society; T038E - Useful for society; T038F - Opportunity for advancement;
T038G - Clean and physically easy work; T038H - Useful for people; T038I - Using one’s abilities
Latent variables: self – self-realisation; social – social recognition; career – career Figure 1. Path diagram for the three-factor model.
One of these is the root mean square error of approximation (RMSEA). Cramer (2003, 34) suggests that the values below .100 are indicators of good fit. As the value of this statistic is .106, and thus slightly above that level, we can assume that our model does not offer a very good fit to the data. As neither chi-square nor the root mean square error of approximation shows that this model provides a perfect fit, it can be assumed that our model contains correlations not only with common factors, but also with unmeasured variables.
However, after removing the job security item, RMSEA re ported good fit with the data. Relevant literature (Tomarken & Waller 2005) suggests that when model does not fit perfectly, there is no need to change theoretical expectations. The problem that some items remain unmeasured by the questionnaire is very common in the study of beliefs, values and orientations.
Conclusion
The three different techniques discussed here study value orientations from somewhat different an gles. None of them is thoroughly superior;
they all have certain advantages and limitations. The choice of suitable method depends on the researcher’s intentions. If the aim is to reduce the infor mation of many measured variables into smaller set of components, principal components analysis should be used.
If the aim is to obtain parameters’ reflecting latent orientations or factors, (explor ative) factor analysis should be preferred. Rotation of the axes causes the factor loadings of every variable to be more clearly differentiated by factor, in order to make the solution more easily inter pretable. The advantage of the rotation is that it does not affect the goodness of fit of a factor solu tion. As the study of youth values is concerned with the detection of latent structures, factor analysis must be preferred to principal component analysis.
Whereas it was difficult to identify meaningful factors in the un-rotated matrix, the un-rotated solution proved to be much more easily interpretable. Explorative factor analysis is a fruitful method for the study of new types of data, as it is the extrac tion of latent structures that can be interpreted as orientations.
However, researchers frequently have a particular ad hoc vision of the relations between variables. This is especially true in the case of analysis of value orientations that were done in Estonia for nearly 30 years. As it was possible to outline a vision of the structure of values on the basis of pre vious studies, it became possible to create a relevant model that suggests, besides initial values, also latent orientations. Initial variables had different impact on these orientations that are manifested through factor loadings and errors in every single variable. Path analysis drew a detailed picture of school-leavers’ work orientations that reflects their understanding of future challenges, which is in a large sense idealistic, because they do not have any real working experience before graduation.
The clearest orientation is towards social recognition, which reflects social and altruistic rewards that can be achieved through work. Self-expression that converges with intrinsic rewards did not emerge as
powerfully as in the previous studies of school-leavers during Soviet times. That discrepancy can be assumed to be related to the shift in the dominant ideology – away from the instrumental aspects of work stressed by the Soviet ideology, towards more personal ambitions, central to a more liberal ideology. Career orientation on may seem questionable here, but we should not forget that adoles cents do not have any working experience which would affect their world view, including value ori entations. It consists not only of advancement values but also of aspirations toward good salary, easy work and job security.
A common problem in the study of values is the capturing and analysis of these latent and vague constructs. Different researchers have used different methods on different samples at different times, but some aspects still remain unclear. These aspects in the presented study were described by errors and goodness of fit indices on the basis of the presented confirmatory factor analysis model.
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Appendix 1. Correlation matrix of values items.
T038A T038B T038C T038D T038E T038F T038G T038H T038I T038A
T038B .04**
T038C .00 .29**
T038D .12** .23** .21**
T038E .19** .01 .11** .26**
T038F .19** .24** .17** .45** .17**
T038G .03** .17** .23** .26** .07** .31**
T038H .20** -.00 .09** .15** .64** .15** .14**
T038I .27** .07** .08** .09** .23** .16** .07** .29**
T038A - Self-improvement; T038B - Income; T038C - Job security; T038D - Position in society; T038E - Useful for society; T038F - Opportunity for advancement; T038G - Clean and physically easy work; T038H - Useful for people; T038I - Using one’s abilities