Abstract
Composite indicators of latent variables can be constructed by NonLinear Principal Components Analysis when data are collected by multiple-item scales. The aim of this paper is to establish the stability of the contribution made by each item to the composite indicator, by means of a resampling-based procedure able to take account of the hierarchical structure that often exists in the data, that is when individuals are nested in groups. The procedure modifies the standard nonparametric bootstrap technique and was applied to real data on job satisfaction from the most extensive survey on Italian social cooperatives.
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Notes
- 1.
In the literature (Michailidis and de Leeuw 2000), NL-PCA was also extended to hierarchical data structures to examine how variables are related across groups and how groups vary.
- 2.
All the computations were performed by R 2.13.1.
- 3.
Missing values were imputed according to Carpita and Manisera (2011). From the entire sample of the ICSI 2007 survey, the cooperatives with less than 5 workers were removed. The data used in this study result from a preliminary Rasch analysis (Carpita and Golia 2011), which identified the 11 selected job satisfaction items as related to a “global” job satisfaction latent trait, and suggested merging response categories to obtain a 5-point response scale for each item, ranging from 1 = “very dissatisfied” to 5 = “very satisfied”, with mid-point 3 = “neither dissatisfied nor satisfied”.
- 4.
Other papers investigated the drivers of job satisfaction by means of regression models with job satisfaction items as independent variables and the overall job satisfaction as a dependent variable (see, for example, Carpita and Zuccolotto 2007, Vezzoli and Zuccolotto 2010). Unlike the current paper, their aim was to identify which facets of job satisfaction drive, from a psychological point of view, the individual perception of the overall job satisfaction, since the latter is measured by a single item in the questionnaire.
References
Borzaga, C., & Depedri, S. (2005). Interpersonal relations and job satisfaction: some empirical results in social and community care services. In B. Gui, & R. Sugden (Eds.) Economics and social interaction: accounting for interpersonal relations (pp. 132–153). Cambridge: Cambridge University Press.
Carpita, M. (Ed.) (2009). La qualità del lavoro nelle cooperative sociali, misure e modelli statistici. Milano: Franco Angeli.
Carpita, M., & Golia, S. (2011). Measuring the quality of work: the case of the Italian social cooperatives. Quality and Quantity, 46, 1659–1685. On Line First, DOI 10.1007/s11135-011-9515-0.
Carpita, M., & Manisera, M. (2011). On the imputation of missing data in surveys with Likert-type scales. The Journal of Classification, 28, 93–112.
Carpita, M., & Zuccolotto, P. (2007). Mining the drivers of job satisfaction using algorithmic variable importance measures. In L. D’Ambra, P. Rostirolla, & M. Squillante (Eds.) Metodi, modelli, e tecnologie dell’informazione a supporto delle decisioni (vol. I, pp. 63–70). Milano: Franco Angeli.
Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. New York: Chapman and Hall.
Gifi, A. (1990). Nonlinear multivariate analysis. Chichester: Wiley.
Goldstein, H., & Healy, M. J. R. (1995). The graphical presentation of a collections of means. Journal of the Royal Statistical Society A, 158, 175–177.
Linting, M. (2007). Nonparametric inference in nonlinear principal components analysis: exploration and beyond. Leiden (NL): Leiden University.
Linting, M., van Os, B. J., & Meulman, J. (2011). Statistical significance of the contribution of variables to the PCA solution: an alternative permutation strategy. Psychometrika, 76, 440–460.
Manisera, M. (2011). A graphical tool to compare groups of subjects on categorical variables. Electronic Journal of Applied Statistical Analysis, 4, 1–22.
Meulman, J. J., Van der Kooij, A. J., & Heiser, W. J. (2004). Principal components analysis with nonlinear optimal scaling transformations for ordinal and nominal data. In Kaplan, D. (Ed.) Handbook of quantitative methodology for the social sciences (pp. 49–70). London: Sage.
Michailidis, G., & de Leeuw, J. (2000). Multilevel homogeneity analysis with differential weighting. Computational Statistics & Data Analysis, 32, 411–442.
Ren, S., Lai, H., Tong, W., Aminzadeh, M., Hou, X., & Lai, S. (2010). Nonparametric bootstrapping for hierarchical data. Journal of Applied Statistics, 37, 1487–1498.
Vezzoli, M., & Zuccolotto, P. (2010). CRAGGING measures of variable importance for data with hierarchical structure. In S. Ingrassia, R. Rocci, & M. Vichi (Eds.) New perspectives in statistical modeling and data analysis. Heidelberg: Springer.
Zani, S., & Cerioli, A. (2007). Analisi dei dati e data mining per le decisioni aziendali. Milano: Giuffrè.
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Manisera, M. (2013). Assessing Stability in NonLinear PCA with Hierarchical Data. In: Giudici, P., Ingrassia, S., Vichi, M. (eds) Statistical Models for Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00032-9_25
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DOI: https://doi.org/10.1007/978-3-319-00032-9_25
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