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The Chi-square test corrections for small tables

These tests are based on data collected in the form of a contingency table of 2 features ($X$, $Y$), each of which has possible $2$ categories $X_1, X_2$ and $Y_1, Y_2$ (look at the table(\ref{tab_kontyngencji_obser})).

The Chi-square test with the Yate's correction for continuity

The $\chi^2$test with the Yate's correction (Frank Yates (1934)1)) is a more conservative test than the Chi-square test (it rejects a null hypothesis more rarely than the $\chi^2$ test). The correction for continuity guarantees the possibility of taking in all the values of real numbers by a test statistic, according to the $\chi^2$ distribution assumption.

The test statistic is defined by:


The Fisher test for (2×2) tables

The Fisher test for $2\times 2$ tables is also called the Fisher exact test (R. A. Fisher (1934)2), (1935)3)). This test enables you to calculate the exact probability of the occurrence of the particular number distribution in a table (knowing $n$ and defined marginal sums.

P=\frac{{O_{11}+O_{21} \choose O_{11}}{O_{12}+O_{22} \choose O_{12}}}{{O_{11}+O_{12}+O_{21}+O_{22} \choose O_{11}+O_{12}}}.

If you know each marginal sum, you can calculate the $P$ probability for various configurations of observed frequencies. The exact $p$ significance level is the sum of probabilities which are less or equal to the analysed probability.

The mid-p test

The mid-p is the Fisher exact test correction. This modified p-value is recommended by many statisticians (Lancaster 19614), Anscombe 19815), Pratt and Gibbons 19816), Plackett 19847), Miettinen 19858) and Barnard 19899), Rothman 200810)) as a method used in decreasing the Fisher exact test conservatism. As a result, using the mid-p the null hypothesis is rejected much more qucikly than by using the Fisher exact test. For large samples a p-value is calculated by using the $\chi^2$ test with the Yate's correction and the Fisher test gives quite similar results. But a p-value of the $\chi^2$ test without any correction corresponds with the mid-p.

The p-value of the mid-p is calculated by the transformation of the probability value for the Fisher exact test. The one-sided p-value is calculated by using the following formula:

p_{I(mid-p)}=p_{I(Fisher)}-0.5\cdot P_{punktu(tabeli\quad zadanej)},


$p_{I(mid-p)}$ – one-sided p-value of mid-p,

$p_{I(Fisher)}$ – one-sided p-value of Fisher exact test,

and the two-sided p-value is defined as a doubled value of the smaller one-sided probability: \begin{displaymath}


$p_{II(mid-p)}$ – two-sided p-value of mid-p.

The settings window with the chi-square test and its corrections can be opened in Statistics menu → NonParametric testsChi-square, Fisher, OR/RR or in ''Wizard''.

Yates F. (1934), Contingency tables involving small numbers and the chi-square test. Journal of the Royal Statistical Society, 1,2 17-235
Fisher R.A. (1934), Statistical methods for research workers (5th ed.). Edinburgh: Oliver and Boyd.
Fisher R.A. (1935), The logic of inductive inference. Journal of the Royal Statistical Society, Series A, 98,39-54
Lancaster H.O. (1961), Significance tests in discrete distributions. Journal of the American Statistical Association 56:223-234
Anscombe F.J. (1981), Computing in Statistical Science through APL. Springer-Verlag, New York
Pratt J.W. and Gibbons J.D. (1981), Concepts of Nonparametric Theory. Springer-Verlag, New York
Plackett R.L. (1984), Discussion of Yates' „Tests of significance for 2×2 contingency tables”. Journal of Royal Statistical Society Series A 147:426-463
Miettinen O.S. (1985), Theoretical Epidemiology: Principles of Occurrence Research in Medicine. John Wiley and Sons, New York
Barnard G.A. (1989), On alleged gains in power from lower p-values. Statistics in Medicine 8:1469-1477
Rothman K.J., Greenland S., Lash T.L. (2008), Modern Epidemiology, 3rd ed. (Lippincott Williams and Wilkins) 221-225
en/statpqpl/porown2grpl/nparpl/fisher2x2pl.txt · ostatnio zmienione: 2022/02/12 12:49 przez admin

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