The Chi-Square Test

How to do a Pearson's Product Moment Correlational Analysis (Research Method Lab) + (links)

Chi-Square Test

Contingency table
(see also this)
Contigency tables

Chi Square test in Excel using the QI Macros

Chi-Square Test (

KvantiMOTV (in Finnish)
The Chi-Square Test (A very good page!: Mainiot sivut. Perusteellinen opastus.)
The chi square statistics


SPSS -pikaohjeita (in Finnish)


Validity and Reliability Compared

So what is the relationship between validity and reliability? The two do not necessarily go hand-in-hand.

Graphic of Target

At best, we have a measure that has both high validity and high reliability. It yields consistent results in repeated application and it accurately reflects what we hope to represent.

It is possible to have a measure that has high reliability but low validity - one that is consistent in getting bad information or consistent in missing the mark. *It is also possible to have one that has low reliability and low validity - inconsistent and not on target.

Finally, it is not possible to have a measure that has low reliability and high validity - you can't really get at what you want or what you're interested in if your measure fluctuates wildly.

Two-Way Tables and the Chi-Square Test

P-value calculator

Statistics Calculator

Vassar Stats (calculator)


Bentler, P.M. (1995).EQS structural equations program manual. Encino, CA: Multivariate Software.

Bentler, P.M., & Dudgeon, P. (1996). Covariance structure analysis: Statistical practice, theory, and directions.Annual Review of Psychology, 47, 541–570.

Bentler, P.M., & Yuan, K.-H. (1999). Structural equation modeling with small samples: Test statistics.Multivariate Behavioral Research, 34, 183–199.

Bollen, K.A. (1989).Structural equations with latent variables. New York, NY: Wiley.

Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures.British Journal of Mathematical and Statistical Psychology, 37, 62–83.

Byrne, B.M., & Campbell, T.L. (1999). Cross-cultural comparisons and the presumption of equivalent measurement and theoretical structure: A look beneath the surface.Journal of Cross-Cultural Psychology, 30, 557–576.

Chou, C.-P., Bentler, P.M., & Satorra, A. (1991). Scaled test statistics and robust standard errors for nonnormal data in covariance structure analysis: A Monte Carlo study.British Journal of Mathematical and Statistical Psychology, 44, 347–357.

Curran, P.J., West, S.G., & Finch, J.F. (1996). The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis.Psychological Methods, 1, 16–29.

Fuller, W.A. (1987).Measurement error models. New York, NY: Wiley.

Hu, L., Bentler, P.M., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted?Psychological Bulletin, 112, 351–362.
CrossRef PubMed

Jöreskog, K., & Sörbom, D. (1994).LISREL 8 user's reference guide. Mooresville, IN: Scientific Software.

Kano, Y. (1992). Robust statistics for test-of-independence and related structural models.Statistics and Probability Letters, 15, 21–26.

Magnus, J., & Neudecker, H. (1999).Matrix differential calculus with applications in statistics and econometrics. (rev. ed.) New York, NY: Wiley.

H. B. Mann and A. Wald (1942) On the Choice of the Number of Class Intervals in the Application of the Chi Square Test The Annals of Mathematical Statistics, Vol. 13, No. 3 (Sep., 1942), pp. 306-317 (link, PDF)

Muthén, B. (1993). Goodness of fit test with categorical and other nonnormal variables. In K.A. Bollen & J.S. Long (Eds.),Testing structural equation models (pp. 205–234). Newbury Park, CA: Sage Publications.

Rao, C.R., (1973).Linear statistical inference and its applications (2nd. ed.). New York, NY: Wiley.

Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach.Psychometrika, 54, 131–151.

Satorra, A. (1992). Asymptotic robust inferences in the analysis of mean and covariance structures.Sociological Methodology, 22, 249–278.

Satorra, A. (2000). Scaled and adjusted restricted tests in multisample analysis of moment structures. In D.D.H. Heijmans, D.S.G. Pollock, & A. Satorra (Eds.),Innovations in multivariate statistical analysis: A Festschrift for Heinz Neudecker (pp. 233–247). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Satorra, A., & Bentler, P.M. (1986). Some robustness properties of goodness of fit statistics in covariance structure analysis.1986 ASA Proceedings of the Business and Economic Statistics Section (549–554). Alexandria, VA: American Statistical Association.

Satorra, A., & Bentler, P.M. (1988a). Scaling corrections for chi-square statistics in covariance structure analysis.ASA 1988 Proceedings of the Business and Economic Statistics Section (308–313). Alexandria, VA: American Statistical Association.

Satorra, A., & Bentler, P.M. (1988b).Scaling corrections for statistics in covariance structure analysis (UCLA Statistics Series #2). Los Angeles, CA: University of California.

Satorra, A., & Bentler, P.M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C.C. Clogg (Eds.),Latent variables analysis: Applications for developmental research (pp. 399–419). Thousand Oaks, CA: Sage.

V. G. Voinov and M. S. Nikulin (1990). CHI-SQUARE GOODNESS-OF-FIT TEST FOR ONE- AND

Yuan, K.-H., & Bentler, P.M. (1997). Mean and covariance structure analysis: Theoretical and practical improvements.Journal of The American Statistical Association, 92, 767–774.

Yuan, K.-H., & Bentler, P.M. (1998). Normal theory based test statistics in structural equation modelling.British Journal of Mathematical and Statistical Psychology, 51, 289–309.

1. Smith, Kenneth J., Advances in Accounting Education 2009

2. Diamantopoulou, Sofia (2009) Testing Developmental Pathways to Antisocial Personality Problems. Journal of Abnormal Child Psychology

3. Morhart, Felicitas M (2009) Brand-Specific Leadership: Turning Employees into Brand Champions. Journal of Marketing 73(5)

4. Soest, Tilmann (2009) A Validation Study of the Resilience Scale for Adolescents (READ). Journal of Psychopathology and Behavioral Assessment

5. Ólafsson, Ragnar P. (2009) Yale-Brown Obsessive Compulsive Scale: Psychometric Properties of the Self-Report Version in a Student Sample. Journal of Psychopathology and Behavioral Assessment

6. Williams, Larry J, Research Methodology in Strategy and Management 2004

7. Borsci, Simone (2009) On the dimensionality of the System Usability Scale: a test of alternative measurement models. Cognitive Processing

8. Satorra, Albert (2009) Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic. Psychometrika

9. Klüppelberg, Claudia (2009) Copula structure analysis. Journal of the Royal Statistical Society Series B (Statistical Methodology) 71(3)

10. Wu, Pei-Chen (2009) Differential Functioning of the Chinese Version of Beck Depression Inventory-II in Adolescent Gender Groups: Use of a Multiple-Group Mean and Covariance Structure Model. Social Indicators Research