Abstract
Many questions in social sciences can only be addressed through individual perceptions, impressions, and judgments. A consumer’s willingness to pay for a product or service is a noisy signal, and the consumer has no obligation to follow through on a purchase intent, no matter how much the researcher might like to infer that “intention” is “action.” Such inherently unobservable constructs need to be modeled as a latent variable. Personal statements of intent, whether they are for purchases, good deeds, or other promises, can only be considered rough indicators; researchers like them because they are cheap and easy to collect by questioning the individual. But like confessions and New Year’s resolutions, intentions are pliable and yielding, and often mendacious. Psychologists have created improved polygraph protocols involving such questions over nearly a century; yet polygraph evidence is still not admissible in court. Obtaining truthful and accurate data from surveys and questionnaires is challenging and the quality of information is invariably lacking. Latent constructs that are of actual interest—ones that help us build theory—are often unobservable. The only way to understand them is through objective measurement of related constructs—the indicator variables.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
\( o\left({n}^{-1}\right) \) convergence implies that for the remaining terms v(n) go to zero faster than \( {n}^{-1} \); \( nv(n)\underset{n\to \infty }{\to }0 \).
References
Balakrishnan, N., & Rao, C. R. (1998). Order statistics: Applications (Handbook of statistics, Vol. 17). New York, NY: Elsevier.
Barclay, D. W., Higgins, C., & Thompson, R. (1995). The partial least squares (PLS) approach to causal modeling: Personal computer adaptation and use as an illustration. Technology Studies, 2(2), 285–309.
Bentler, P. M. (1989). EQS, structural equations, program manual, program version 3.0 (p. 6). Los Angeles, CA: BMDP Statistical Software.
Bollen, K. A. (1989). Structural equations with latent variables (p. 268). New York, NY: Wiley.
Boomsma, A. (1982a). Robustness of LISREL against small sample sizes in factor analysis models. In K. G. Joreskog & H. Wold (Eds.), Systems under indirect observations, causality, structure, prediction (Part 1) (pp. 149–173). Amsterdam, The Netherlands: North Holland.
Boomsma, A. (1982b). The robustness of LISREL against small sample sizes in factor analysis models. In K. G. Jöreskog & H. Wold (Eds.), Systems under indirect observation: Causality, structure, prediction (Vol. 149, pp. 149–173). Amsterdam, The Netherlands: North-Holland.
Brand, A., Bradley, M. T., Best, L. A., & Stoica, G. (2008). Accuracy of effect size estimates from published psychological research. Perceptual and Motor Skills, 106(2), 645–649.
Browne, M. W., & Cudeck, R. (1989). Single sample cross-validation indices for covariance structures. Multivariate Behavioral Research, 24, 445–455.
Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 136–162). Newbury Park, CA: Sage.
Cattell, R. B. (1966). Handbook of multivariate experimental psychology. Chicago, IL: Rand McNally.
Chin, W. W. (1998). The partial least squares approach to structural equation modeling. In G. A. Marcoulides (Ed.), Modern methods for business research (pp. 295–336). Mahwah, NJ: Lawrence Erlbaum Associates.
Chin, W. W., & Newsted, P. R. (1999). Structural equation modeling analysis with small samples using partial least squares. In R. Hoyle (Ed.), Statistical strategies for small sample research (pp. 307–341). Thousand Oaks, CA: Sage.
Cliff, N. (1988). The eigenvalues-greater-than-one rule and the reliability of components. Psychological Bulletin, 103(2), 276.
Cochran, W. G. (1977). Sampling techniques (3rd ed.). New York, NY: Wiley.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates.
Cohen, J. (1992). A power primer. Psychological Bulletin, 112, 155–159.
Copas, J. B., & Li, H. G. (1997). Inference for non-random samples. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 59(1), 55–95.
Cronbach, L. J., & Meehl, P. E. (1955). Construct validity in psychological tests. Psychological Bulletin, 52(4), 281.
Dhrymes, P. J. (1970). Econometrics—Statistical foundations and applications (p. 53). New York, NY: Evanston.
Dijkstra, T. (1983). Some comments on maximum likelihood and partial least squares methods. Journal of Econometrics, 22(1), 67–90.
Ding, L., Belicer, W. F., & Harlow, L. L. (1995). The effects of estimation methods, number of indicators per factor and improper solutions on structural equation modeling fit indices. Structural Equation Modeling, 2, 119–144.
Fava, J. L., & Velicer, W. F. (1992a). An empirical comparison of factor, image, component, and scale scores. Multivariate Behavioral Research, 27(3), 301–322.
Fava, J. L., & Velicer, W. F. (1992b). The effects of overextraction on factor and component analysis. Multivariate Behavioral Research, 27(3), 387–415.
Fèhèr, K. (1989). Comparison of LISREL and PLS estimation methods in latent variable models. Introducing latent variables into econometric models (Manuscript SFB, Vol. 303). Bonn, Germany: University of Bonn.
Fisher, R. A. (1921). On the ‘probable error’ of a coefficient of correlation deduced from a small sample. Metron, 1, 3–32.
Fisher, R. A. (1935). The design of experiments. Edinburgh, UK: Oliver & Boyd.
Fisher, R. A. (1990). Statistical methods, experimental design, and scientific inference. New York, NY: Oxford University Press.
Ford, E. D. (2000). Scientific method for ecological research. Cambridge, UK: Cambridge University Press.
Friedman, M. (1953). Essays in positive economics. Chicago, IL: University of Chicago Press.
Gerbing, D. W., & Anderson, J. C. (1985). The effects of sampling error and model characteristics on parameter estimation for maximum likelihood confirmatory factor analysis. Multivariate Behavioral Research, 20, 255–271.
Geweke, J. F., & Singleton, K. J. (1980). Interpreting the likelihood ratio statistic in factor models when sample size is small. Journal of the American Statistical Association, 7(369), 133–137.
Gibbons, J. D., & Chakraborti, S. (1992). Nonparametric statistical inference (3rd ed.). New York, NY: Marcel Dekker.
Goodhue, D., Lewis, W., & Thompson, R. (2006). PLS, small sample size, and statistical power in MIS Research, HICSS (Vol. 8, p. 202b). Proceedings of the 39th Annual Hawaii International Conference on System Sciences (HICSS‘06).
Goodhue, D., William, L., & Thompson, R. (2007). Statistical power in analyzing interaction effects: Questioning the advantage of PLS with product indicators (research note). Information Systems Research, 18(2), 211–227.
Hea, Q., & Nagarajab, H. N. (2009). Correlation estimation using concomitants of order statistics from bivariate normal samples. Communications in Statistics - Theory and Methods, 38(12), 2003–2015.
Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Annals of Mathematical Statistics, 19, 293–325.
Jöreskog, K. G. (1967). Some contributions to maximum likelihood factor analysis. Psychometrika, 32(4), 443–482.
Jöreskog, K. G. (1970). A general method for analysis of covariance structures. Biometrika, 57, 239–251.
Jöreskog, K. G., & Sörbom, D. (1996). LISREL 8 user’s reference guide. Chicago, IL: Scientific Software International.
Joreskog, K. G., Sorbom, D., & Magidson, J. (1979). Advances in factor analysis and structural equation models. Cambridge, MA: Abt Books.
Kahai, S. S., & Cooper, R. B. (2003). Exploring the core concepts of media richness theory: The impact of cue multiplicity and feedback immediacy on decision quality. Journal of Management Information Systems, 20(1), 263–299.
Kaiser, H. F. (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20, 141–151.
Kaiser, F. H. (1991). Coefficient alpha for a principal component and the Kaiser-Guttman rule. Psychological Reports, 68(3), 855–858.
Kaiser, H. F. (1992). On Cliff’s formula, the Kaiser-Guttman rule, and the number of factors. Perceptual and Motor Skills, 74(2), 595–598.
Kendall, M., & Gibbons, J. D. (1990). Rank correlation methods (5th ed.). New York, NY: Oxford University Press.
Kish, L. (1995). Survey sampling. New York, NY: Wiley.
Koopmans, T. C. (1951). Analysis of production as an efficient combination of activities. Activity Analysis of Production and Allocation, 13, 33–37.
Koopmans, T. C. (1957). Three essays on the state of economic science (Vol. 21). New York, NY: McGraw-Hill.
Koopmans, T. C. (1963). Appendix to ‘On the concept of optimal economic growth’. New Haven, CT: Cowles Foundation for Research in Economics, Yale University.
Lauro, C., & Vinzi, V. E. (2002). Some contributions to PLS path modeling and a system for the European customer satisfaction (Atti della XL1 riunione scientifica SIS). Milano, Italy: Universita di Milano Bicocca.
Leamer, E. E. (1978). Specification searches. New York, NY: Wiley.
Likert, R. (1932). A technique for the measurement of attitudes. Archives of Psychology, 22, 55.
Likert, R., Roslow, S., & Murphy, G. (1934). A simple and reliable method of scoring the Thurstone attitude scales. Journal of Social Psychology, 5(2), 228–238.
Lohr, S. L. (1999). Sampling: Design and analysis. Pacific Grove, CA: Duxbury.
Mari, D. D., & Kotz, S. (2001). Correlation and dependence. London: Imperial College Press.
Marsh, H. W., & Bailey, M. (1991). Confirmatory factor analyses of multitrait-multimethod data: A comparison of alternative models. Applied Psychological Measurement, 15(1), 47–70.
Marsh, H. W., Balla, J. R., & Hau, K.-T. (1996). An evaluation of incremental fit indices: A clarification of mathematical and empirical properties. In G. A. Marcoulides & R. E. Schumacker (Eds.), Advanced structural equation modeling: Issues and techniques (pp. 315–353). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Marsh, H. W., Balla, J. R., & McDonald, R. P. (1988). Goodness of fit indexes in confirmatory factor analysis: The effect of sample size. Psychological Bulletin, 103, 391–410.
Marsh, H. W., Hau, K.-T., Balla, J. R., & Grayson, D. (1998). Is more ever too much? The number of indicators per factor in confirmatory factor analysis. Multivariate Behavioral Research, 33, 181–220.
Merton, R. K. (1988). The Matthew effect in science, II: Cumulative advantage and the symbolism of intellectual property. Isis, 79, 606–623.
Nakagawa, S., & Cuthill, I. C. (2007). Effect size, confidence interval and statistical significance: A practical guide for biologists. Biological Reviews of the Cambridge Philosophical Society, 82, 591–605.
Nunnally, J. C. (1967). Psychometric theory (p. 355). New York, NY: McGraw-Hill.
Podsakoff, P. M., & Organ, D. W. (1986). Self-reports in organization research: Problems and prospects. Journal of Management, 12, 531–544.
Raspe, R. E. (2004). The surprising adventures of Baron Munchausen. Whitefish, MT: Kessinger Pub.
Schechtman, E., & Yitzhaki, S. (1987). A measure of association based on Gini’s mean difference. Communications in Statistics - Theory and Methods, 16, 207–231.
Schneeweiß, H. (1990). Models with latent variables: LISREL versus PLS. Contemporary Mathematics, 112, 33–40.
Schneeweiß, H. (1991). Models with latent variables: LISREL versus PLS. Statistica Neerlandica, 45, 145–157.
Schneeweiß, H. (1993). Consistency at large in models with latent variables. In K. Hagen, D. J. Barthdomew, & M. Deistler (Eds.), Statistical modelling and latent variables (pp. 299–320). Amsterdam, The Netherlands: Elsevier.
Shevlyakov, G. L., & Vilchevski, N. O. (2002). Robustness in data analysis: Criteria and methods (Modern probability and statistics). Utrecht, The Netherlands: VSP.
Snedecor, G. W., & Cochran, W. G. (1989). Statistical methods (8th ed.). Ames, IA: Iowa University Press.
Soper, D. (2012). A priori sample size calculator for structural equation models. http://www.danielsoper.com/statcalc3/calc.aspx?id=89
Stuart, A., & Keith Ord, J. (1987). Kendall’s advanced theory of statistics. New York, NY: Oxford University Press.
Tabachnick, B. G., & Fidell, L. S. (1989). Using multivariate statistics. New York, NY: Harper and Row.
Tanaka, J. S. (1987). “How big is big enough?”: Sample size and goodness of fit in structural equation models with latent variables. Child Development, 58, 134–146.
Thomas, D. R., Lu, I. R. R., & Cedzynski, M. (2005). Partial least squares: A critical review and a potential alternative. Proceedings of the Annual Conference of Administrative Sciences Association of Canada, Management Science Division, Toronto.
Velicer, W. F., & Fava, J. L. (1998). Affects of variable and subject sampling on factor pattern recovery. Psychological Methods, 3(2), 231.
Westland, J. C. (2010). Lower bounds on sample size in structural equation modeling. Electronic Commerce Research and Applications, 9(6), 476–487.
Westland, J. C., & See-To, W. K. (2007). The short-run price-performance dynamics of microcomputer technologies. Research Policy, 36(5), 591–604.
Wilkinson, L., & APA Task Force on Statistical Inference. (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594–604.
Wolfram, S. (2002). A new kind of science (Vol. 5). Champaign, IL: Wolfram media.
Xu, W., Hung, Y. S., Niranjan, M., & Shen, M. (2010). Asymptotic mean and variance of Gini correlation for bivariate normal samples. IEEE Transactions on Signal Processing, 58(2), 522–534.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Westland, J.C. (2015). Data Collection, Control, and Sample Size. In: Structural Equation Models. Studies in Systems, Decision and Control, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-16507-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-16507-3_6
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16506-6
Online ISBN: 978-3-319-16507-3
eBook Packages: EngineeringEngineering (R0)