A statistical modelling of the visual acuity measurement and its multiple test procedure


To establish the computer assisted system of the visual acuity test, we propose a statistical modelling of the visual acuity measurement and its multiple test procedure. The psychometric functions for individual patients are produced by the logistic regression combined with the guessing rate. We adopt test statistics based on (i) psychometric functions (the Cochran-Mantel-Haenszel method) and (ii) psychophysical thresholds (the delta method). The multiple comparisons are performed by the step-down procedure with Ryan-Einot-Gabriel-Welsch (REGW) significance levels. To show the practical effectiveness of our system, we present a numerical example of four patient groups.

DOI Code: 10.1285/i20705948v13n1p1

Keywords: Cochran-Mantel-Haenszel test statistic; delta test statistic; psychometric functions; psychophysical thresholds; step-down procedure


Agresti, A. (2002). Categorical data analysis, Second Edition. Hoboken (NJ): Wiley.

Bach, M. (1996). The Freiburg visual acuity test – automatic measurement of visual acuity. Optometry and Vision Science, 73, 49–53.

Beck, R.W., Mokexi, P.S., Turpin, A.H., Ferris 3rd, F.L., SanGiovanni, J.P., Johnson, C.A., Birch, E.E., Chandler, D.L., Cox, T.A., Blair, R.C. and Kraker, R.T. (2003). A computerized method of visual acuity testing: Adaptation of the early treatment of diabetic retinopathy study testing protocol. American Journal of Ophthalmology, 135, 194–205.

Einot, I. and Gabriel, K.R. (1975). A study of the powers of several methods of multiple comparisons. Journal of the American Statistical Association, 70, 574–583.

Holland, P.W. and Thayer, D.T. (1988). Differential item performance and the Mantel-Haenszel procedure. In Wainer, H. and Braun, H.I., eds. Test Validity, pages 129–145. Hillsdale (NJ): Lawrence Erlbaum Associates.

Kingdom, F.A.A. and Prins, N. (2010). Psychophysics – A Practical Introduction. London: Academic Press.

Landis, J.R., Heyman, E.R. and Koch, G.G. (1978). Average partial association in three-way contingency tables: a review and discussion of alternative tests. International Statistical Review, 46, 237-254.

Mantel, N. and Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22, 719-748.

Mita, N., Hara, H., Kani, K. and Tabuchi, A. (2010). Use of statistical analysis for visual acuity measurement (in Japanese). Japanese Journal of Visual Science, 31, 19–25.

Mita, N., Sasaki, H., Kani, K., Tabuchi, A. and Hara, H. (2014). Multiple comparisons for a psychophysical test in bootstrap logistic regression. Journal of Algorithms and Computational Technology, 8, 339-355.

Mita, N., Sasaki, H., Kani, K., Tabuchi, A. and Hara, H. (2017). Multiple comparisons in two-factor testing of psychophysical thresholds by the delta test method. Communications in Statistcs – Simulation and Computation, 46, 669-678.

Nagai, T., Hoshino, T. and Uchikawa, K. (2006). Statistical significance testing of

thresholds estimated by constant stimuli method (in Japanese). Vision, 18, 113–123.

Pas, S.F. and Koenderink, J.J. (2004). Visual descrimination of spectral distributions. Perception, 33, 1483–1497.

Penfield, R.D. (2001). Assesing differential item functioning among multiple groups: A comparison of three Mantel-Haenszel procedures. Applied Measurement in Education, 14, 235–259.

Ryan, T.A. (1960). Significance tests for multiple comparison of proportions, variances and other statistics. Psychological Bulletin, 57, 318–328.

Schulze-Bonsel, K., Feltgen, N., Burau, H., Hansen, L. and Bach, M. (2006). Visual acuities “hand motion” and “counting fingers” can be quatified with the Freiburg visual acuity test. Investigative Ophthalmology and Visual Science, 47, 1236–1240.

Simpson, T.L. (1995). Vision thresholds from psychometric analyses: alternatives to probit analysis. Optometry and Visual Science, 72, 371–377.

Somes, G.W. (1986). The generalized Mantel-Haenszel Statistic. Amercan Statistician, 40, 106–108.

Tokutake, T., Yamauchi, T., Mita, N., Kawamoto, K., Tabuchi, H., Kani, K., Hara,

H., Tabuchi, A. and Sasaki, H. (2014). Repeatability of visual acuity testing using a psychometric function. Japanese Journal of Ophthalmology, 58, 381-387.

Watson, A.B. (1979). Probability summation over time. Vision Research, 19, 515–522.

Welsch, R.E. (1977). Stepwise multiple comparison procedures. Journal of the American Statistical Association, 72, 566–575.

Zhang, J. and Boos, D.D. (1996). Generalized Cochran-Mantel-Haenszel test statistics for correlated categorical data. Institute of Statistics Mimeo Series, No. 2284.

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