Exploring How to Simply Approximate the P-value of a Chi-squared Statistic

  • Eric Beh University of Newcastle, Australia

Abstract

Calculating the p-value of any test statistic is of paramount importance to all statistically minded researchers across all areas of study. Many, these days, take for granted how the p-value is calculated and yet it is a pivotal quantity in all forms of statistical analysis. For the study of 2x2 tables where dichotomous variables are assessed for association, the chi-squared statistic, and its p-value, are fundamental quantities to all analysts, especially those in the health and allied disciplines. Examining the association between dichotomous variables is easily achieved through a very simple formula for the chi-squared statistic and yet the p-value of this statistic requires far more computational power.

This paper proposes and explores a very simple approximation of the p-value for a chi-squared statistic given its degrees of freedom. After providing a review a variety of common ways for determining the quantile of the chi-squared distribution given the level of significance and degrees of freedom, we shall derive an approximation based on the classic quantile formula given in 1977 by D. C. Hoaglin. We examine this approximation using a simple 2x2 contingency table example then show that it is extremely precise for all chi-squared values ranging from 0 to 50.

References

Aroian LA (1943). “A new approximation to the levels of significance of the chi-square distribution”. Annals of Mathematical Statistics 14, 93 – 95.

Beh EJ and Lombardo R (2014). Correspondence Analysis: Theory, Practice and New Strategies. Chichester: Wiley.

Beh EJ, Tran D and Hudson IL (2013). “A reformulation of the aggregate association index using the odds ratio”. Computational Statistics & Data Analysis 68, 52 – 65.

Chu L, Jacobs BL, Schwen Z and Schneck FX (2013). “Hydronephrosis in pediatric kidney transplant: Clinical relevance to graft outcome”. Journal of Pediatric Urology 9, 217 – 222, 2013.

Cressie N and Read TRC (1984). “Multinomial goodness-of-fit tests”. Journal of the Royal Statistical Society, Series B 46, 440 – 464.

Delucchi KL (1983). “The use and misuse of chi-square: Lewis and Burke revisited”. Psychological Bulletin 94, 166 – 176.

Dijkers MP (2005). “Misuse of the Pearson chi-square test of association”. Archives of Physical Medicine and Rehabilitation 86, 602.

Ding CG (1992). “Algorithm AS275: Computing the non-central chi-squared distribution function”. Applied Statistics 41, 478 – 482.

Edwards AL (1950). “On ‘The use and misuse of the chi-square test’ – the case of the 2x2 contingency table”. Psychological Bulletin 47, 341 – 346.

Elderton WP (1902). “Tables for testing the goodness of fit of theory to observation”. Biometrika 1, 155 – 163.

Fienberg SE (1979). “The use of chi-squared statistics for categorical data problems”. Journal of the Royal Statistical Society, Series B 41, 54 – 64.

Fisher RA (1928). Statistical Methods for Research Workers (2nd ed.). Oliver and Boyd.

Fleiss JL, Levin B and Paik MC (2003). Statistical Methods for Rates and Proportions (3rd edn). Wiley.

Goldberg H and Levine H (1946). “Approximate formulas for the percentage points and normalization of t and X2”. Annals of Mathematical Statistics 17, 216 – 225.

Hald A. and Sinkbaek SA (1950). “A table of percentage points of the X2 distribution”. Scandinavian Actuarial Journal 33, 168 – 175.

Haldane JBS (1940). “The mean and variance of 2 when used as a test of homogeneity, when expectations are small”. Biometrika 31, 346 – 355.

Harter HL (1964). New Tables of the Incomplete Gamma-Function Ratio and the Percentage Points of the Chi-square and Beta Distributions. US Government Printing Office, Washington, USA.

Heyworth MR (1976). “Approximation to chi-square”. The American Statistician 30, 204.

Hoaglin DC (1977). “Approximations for chi-squared percentage points”. Journal of the American Statistical Association 72, 508 – 515.

Khamis SH and Rudert W (1965). Tables of the Incomplete Gamma Function Ratio: Chi-square Integral, Poisson Distribution. Darmstad: Justus von Leibig.

Koehler KJ (1983). “A simple approximation for the percentiles of the t distribution”. Technometrics 25, 103 – 105.

Krauth J and Steinebach J (1976). “Extended tables of the percentage points of the chi-square distribution for at most ten degrees of freedom”. Biometrische Zeitschift 18, 13 – 22.

Lancaster HO (1969). The Chi-squared Distribution. Wiley.

Lewis D and Burke CJ (1949). “The use and misuse of the chi-square test”. Psychological Bulletin 46, 433 – 489.

Lin J–T (1988). “Approximating the cumulative chi-square distribution and its inverse”. The Statistician 37, 3 – 5.

Lin J–T (1990). “Alternative to Koehler’s approximation for the percentiles of the t distributions”. Probability in the Engineering and Informational Sciences 4, 535 – 537.

Merrington M (1941). “Numerical approximations to the percentage points of the χ2 distribution”. Biometrika 32, 200 – 202.

Pearson K (1904). “On the theory of contingency and its relation to association and normal correlation”. Drapers Memoirs, Biometric Series, Vol 1. London.

Pearson K (1922). Tables of the Incomplete -Function, Cambridge University Press.

Russell W and Lal M (1969). Tables of Chi-square Probability Function. Department of Mathematics, St. Johns: Memorial University of Newfoundland. (Reviewed in Mathematics of Computation 23, 211 – 212.)

Terrell GR (1987). “Chi-squared left-tail probabilities”. Journal of Statistical Computation and Simulation 28, 264 – 266.

Thompson CM (1941). Table of percentage points of the incomplete beta-function distribution. Biometrika 32, 151 – 181.

Vanderbeck JP and Cooke JR (1961). Extended Table of Percentage Points of the Chi-Square Distribution. Nauweps Report 7770, U.S. Naval Ordnance Test Station, China Lake, CA.

Wilson EB and Hilferty MM (1931). “The distribution of chi-square”. Proceedings of the National Academy of Sciences (USA) 17, 684 – 688.

Yates F (1934). “Contingency tables involving small numbers and the χ2 test”. Journal of the Royal Statistical Society Supplement 1, 217 – 235.

Published
2018-05-27
How to Cite
Beh, E. (2018). Exploring How to Simply Approximate the P-value of a Chi-squared Statistic. Austrian Journal of Statistics, 47(3), 63-75. https://doi.org/10.17713/ajs.v47i3.757