Pearson's chisquared test
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Pearson's chisquared test (χ^{2}) is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is the most widely used of many chisquared tests (e.g., Yates, likelihood ratio, portmanteau test in time series, etc.) – statistical procedures whose results are evaluated by reference to the chisquared distribution. Its properties were first investigated by Karl Pearson in 1900.^{[1]} In contexts where it is important to improve a distinction between the test statistic and its distribution, names similar to Pearson χsquared test or statistic are used.
It tests a null hypothesis stating that the frequency distribution of certain events observed in a sample is consistent with a particular theoretical distribution. The events considered must be mutually exclusive and have total probability 1. A common case for this is where the events each cover an outcome of a categorical variable. A simple example is the hypothesis that an ordinary sixsided die is "fair" (i. e., all six outcomes are equally likely to occur.)
Contents
Definition[edit]
Pearson's chisquared test is used to assess three types of comparison: goodness of fit, homogeneity, and independence.
 A test of goodness of fit establishes whether an observed frequency distribution differs from a theoretical distribution.
 A test of homogeneity compares the distribution of counts for two or more groups using the same categorical variable (e.g. choice of activity—college, military, employment, travel—of graduates of a high school reported a year after graduation, sorted by graduation year, to see if number of graduates choosing a given activity has changed from class to class, or from decade to decade).^{[2]}
 A test of independence assesses whether observations consisting of measures on two variables, expressed in a contingency table, are independent of each other (e.g. polling responses from people of different nationalities to see if one's nationality is related to the response).
For all three tests, the computational procedure includes the following steps:
 Calculate the chisquared test statistic, χ², which resembles a normalized sum of squared deviations between observed and theoretical frequencies (see below).
 Determine the degrees of freedom, df, of that statistic.
 For a test of goodnessoffit, df = Cats − Parms, where Cats is the number of observation categories recognized by the model, and Parms is the number of parameters in the model adjusted to make the model best fit the observations: The number of categories reduced by the number of fitted parameters in the distribution.
 For test of homogeneity, df = (Rows − 1)×(Cols − 1), where Rows corresponds to the number of categories (i.e. rows in the associated contingency table), and Cols corresponds the number of independent groups (i.e. columns in the associated contingency table).^{[2]}
 For test of independence, df = (Rows − 1)×(Cols − 1), where in this case, Rows corresponds to number of categories in one variable, and Cols corresponds to number of categories in the second variable.^{[2]}
 Select a desired level of confidence (significance level, pvalue or the corresponding alpha level) for the result of the test.
 Compare χ² to the critical value from the chisquared distribution with df degrees of freedom and the selected confidence level (onesided since the test is only one direction, i.e. is the test value greater than the critical value?), which in many cases gives a good approximation of the distribution of χ².
 Sustain or reject the null hypothesis that the observed frequency distribution is the same as the theoretical distribution based on whether the test statistic exceeds the critical value of χ². If the test statistic exceeds the critical value of χ², the null hypothesis ( = there is no difference between the distributions) can be rejected, and the alternative hypothesis ( = there is a difference between the distributions) can be accepted, both with the selected level of confidence. If the test statistic falls below the threshold χ² value, then no clear conclusion can be reached, and the null hypothesis is sustained (we failed to reject the null hypothesis), but not necessarily accepted.
Test for fit of a distribution[edit]
Discrete uniform distribution[edit]
In this case observations are divided among cells. A simple application is to test the hypothesis that, in the general population, values would occur in each cell with equal frequency. The "theoretical frequency" for any cell (under the null hypothesis of a discrete uniform distribution) is thus calculated as
and the reduction in the degrees of freedom is , notionally because the observed frequencies are constrained to sum to .
One specific example of its application would be its application for logrank test.
Other distributions[edit]
When testing whether observations are random variables whose distribution belongs to a given family of distributions, the "theoretical frequencies" are calculated using a distribution from that family fitted in some standard way. The reduction in the degrees of freedom is calculated as , where is the number of covariates used in fitting the distribution. For instance, when checking a threecovariate Weibull distribution, , and when checking a normal distribution (where the parameters are mean and standard deviation), , and when checking a Poisson distribution (where the parameter is the expected value), . Thus, there will be degrees of freedom, where is the number of categories.
The degrees of freedom are not based on the number of observations as with a Student's t or Fdistribution. For example, if testing for a fair, sixsided die, there would be five degrees of freedom because there are six categories/parameters (each number). The number of times the die is rolled does not influence the number of degrees of freedom.
Calculating the teststatistic[edit]
Uppertail critical values of chisquare distribution ^{[3]}  

Degrees of freedom 
Probability less than the critical value  
0.90  0.95  0.975  0.99  0.999  
1  2.706  3.841  5.024  6.635  10.828 
2  4.605  5.991  7.378  9.210  13.816 
3  6.251  7.815  9.348  11.345  16.266 
4  7.779  9.488  11.143  13.277  18.467 
5  9.236  11.070  12.833  15.086  20.515 
6  10.645  12.592  14.449  16.812  22.458 
7  12.017  14.067  16.013  18.475  24.322 
8  13.362  15.507  17.535  20.090  26.125 
9  14.684  16.919  19.023  21.666  27.877 
10  15.987  18.307  20.483  23.209  29.588 
11  17.275  19.675  21.920  24.725  31.264 
12  18.549  21.026  23.337  26.217  32.910 
13  19.812  22.362  24.736  27.688  34.528 
14  21.064  23.685  26.119  29.141  36.123 
15  22.307  24.996  27.488  30.578  37.697 
16  23.542  26.296  28.845  32.000  39.252 
17  24.769  27.587  30.191  33.409  40.790 
18  25.989  28.869  31.526  34.805  42.312 
19  27.204  30.144  32.852  36.191  43.820 
20  28.412  31.410  34.170  37.566  45.315 
21  29.615  32.671  35.479  38.932  46.797 
22  30.813  33.924  36.781  40.289  48.268 
23  32.007  35.172  38.076  41.638  49.728 
24  33.196  36.415  39.364  42.980  51.179 
25  34.382  37.652  40.646  44.314  52.620 
26  35.563  38.885  41.923  45.642  54.052 
27  36.741  40.113  43.195  46.963  55.476 
28  37.916  41.337  44.461  48.278  56.892 
29  39.087  42.557  45.722  49.588  58.301 
30  40.256  43.773  46.979  50.892  59.703 
31  41.422  44.985  48.232  52.191  61.098 
32  42.585  46.194  49.480  53.486  62.487 
33  43.745  47.400  50.725  54.776  63.870 
34  44.903  48.602  51.966  56.061  65.247 
35  46.059  49.802  53.203  57.342  66.619 
36  47.212  50.998  54.437  58.619  67.985 
37  48.363  52.192  55.668  59.893  69.347 
38  49.513  53.384  56.896  61.162  70.703 
39  50.660  54.572  58.120  62.428  72.055 
40  51.805  55.758  59.342  63.691  73.402 
41  52.949  56.942  60.561  64.950  74.745 
42  54.090  58.124  61.777  66.206  76.084 
43  55.230  59.304  62.990  67.459  77.419 
44  56.369  60.481  64.201  68.710  78.750 
45  57.505  61.656  65.410  69.957  80.077 
46  58.641  62.830  66.617  71.201  81.400 
47  59.774  64.001  67.821  72.443  82.720 
48  60.907  65.171  69.023  73.683  84.037 
49  62.038  66.339  70.222  74.919  85.351 
50  63.167  67.505  71.420  76.154  86.661 
51  64.295  68.669  72.616  77.386  87.968 
52  65.422  69.832  73.810  78.616  89.272 
53  66.548  70.993  75.002  79.843  90.573 
54  67.673  72.153  76.192  81.069  91.872 
55  68.796  73.311  77.380  82.292  93.168 
56  69.919  74.468  78.567  83.513  94.461 
57  71.040  75.624  79.752  84.733  95.751 
58  72.160  76.778  80.936  85.950  97.039 
59  73.279  77.931  82.117  87.166  98.324 
60  74.397  79.082  83.298  88.379  99.607 
61  75.514  80.232  84.476  89.591  100.888 
62  76.630  81.381  85.654  90.802  102.166 
63  77.745  82.529  86.830  92.010  103.442 
64  78.860  83.675  88.004  93.217  104.716 
65  79.973  84.821  89.177  94.422  105.988 
66  81.085  85.965  90.349  95.626  107.258 
67  82.197  87.108  91.519  96.828  108.526 
68  83.308  88.250  92.689  98.028  109.791 
69  84.418  89.391  93.856  99.228  111.055 
70  85.527  90.531  95.023  100.425  112.317 
71  86.635  91.670  96.189  101.621  113.577 
72  87.743  92.808  97.353  102.816  114.835 
73  88.850  93.945  98.516  104.010  116.092 
74  89.956  95.081  99.678  105.202  117.346 
75  91.061  96.217  100.839  106.393  118.599 
76  92.166  97.351  101.999  107.583  119.850 
77  93.270  98.484  103.158  108.771  121.100 
78  94.374  99.617  104.316  109.958  122.348 
79  95.476  100.749  105.473  111.144  123.594 
80  96.578  101.879  106.629  112.329  124.839 
81  97.680  103.010  107.783  113.512  126.083 
82  98.780  104.139  108.937  114.695  127.324 
83  99.880  105.267  110.090  115.876  128.565 
84  100.980  106.395  111.242  117.057  129.804 
85  102.079  107.522  112.393  118.236  131.041 
86  103.177  108.648  113.544  119.414  132.277 
87  104.275  109.773  114.693  120.591  133.512 
88  105.372  110.898  115.841  121.767  134.746 
89  106.469  112.022  116.989  122.942  135.978 
90  107.565  113.145  118.136  124.116  137.208 
91  108.661  114.268  119.282  125.289  138.438 
92  109.756  115.390  120.427  126.462  139.666 
93  110.850  116.511  121.571  127.633  140.893 
94  111.944  117.632  122.715  128.803  142.119 
95  113.038  118.752  123.858  129.973  143.344 
96  114.131  119.871  125.000  131.141  144.567 
97  115.223  120.990  126.141  132.309  145.789 
98  116.315  122.108  127.282  133.476  147.010 
99  117.407  123.225  128.422  134.642  148.230 
100  118.498  124.342  129.561  135.807  149.449 
The value of the teststatistic is
where
 = Pearson's cumulative test statistic, which asymptotically approaches a distribution.
 = the number of observations of type i.
 = total number of observations
 = the expected (theoretical) count of type i, asserted by the null hypothesis that the fraction of type i in the population is
 = the number of cells in the table.
The chisquared statistic can then be used to calculate a pvalue by comparing the value of the statistic to a chisquared distribution. The number of degrees of freedom is equal to the number of cells , minus the reduction in degrees of freedom, .
The result about the numbers of degrees of freedom is valid when the original data are multinomial and hence the estimated parameters are efficient for minimizing the chisquared statistic. More generally however, when maximum likelihood estimation does not coincide with minimum chisquared estimation, the distribution will lie somewhere between a chisquared distribution with and degrees of freedom (See for instance Chernoff and Lehmann, 1954).
Bayesian method[edit]
In Bayesian statistics, one would instead use a Dirichlet distribution as conjugate prior. If one took a uniform prior, then the maximum likelihood estimate for the population probability is the observed probability, and one may compute a credible region around this or another estimate.
Testing for statistical independence[edit]
In this case, an "observation" consists of the values of two outcomes and the null hypothesis is that the occurrence of these outcomes is statistically independent. Each observation is allocated to one cell of a twodimensional array of cells (called a contingency table) according to the values of the two outcomes. If there are r rows and c columns in the table, the "theoretical frequency" for a cell, given the hypothesis of independence, is
where is the total sample size (the sum of all cells in the table), and
is the fraction of observations of type i ignoring the column attribute (fraction of row totals), and
is the fraction of observations of type j ignoring the row attribute (fraction of column totals). The term "frequencies" refers to absolute numbers rather than already normalised values.
The value of the teststatistic is
Note that is 0 if and only if , i.e. only if the expected and true number of observations are equal in all cells.
Fitting the model of "independence" reduces the number of degrees of freedom by p = r + c − 1. The number of degrees of freedom is equal to the number of cells rc, minus the reduction in degrees of freedom, p, which reduces to (r − 1)(c − 1).
For the test of independence, also known as the test of homogeneity, a chisquared probability of less than or equal to 0.05 (or the chisquared statistic being at or larger than the 0.05 critical point) is commonly interpreted by applied workers as justification for rejecting the null hypothesis that the row variable is independent of the column variable.^{[4]} The alternative hypothesis corresponds to the variables having an association or relationship where the structure of this relationship is not specified.
Assumptions[edit]
The chisquared test, when used with the standard approximation that a chisquared distribution is applicable, has the following assumptions:^{[citation needed]}
 Simple random sample
 The sample data is a random sampling from a fixed distribution or population where every collection of members of the population of the given sample size has an equal probability of selection. Variants of the test have been developed for complex samples, such as where the data is weighted. Other forms can be used such as purposive sampling.^{[5]}
 Sample size (whole table)
 A sample with a sufficiently large size is assumed. If a chi squared test is conducted on a sample with a smaller size, then the chi squared test will yield an inaccurate inference. The researcher, by using chi squared test on small samples, might end up committing a Type II error.
 Expected cell count
 Adequate expected cell counts. Some require 5 or more, and others require 10 or more. A common rule is 5 or more in all cells of a 2by2 table, and 5 or more in 80% of cells in larger tables, but no cells with zero expected count. When this assumption is not met, Yates's correction is applied.
 Independence
 The observations are always assumed to be independent of each other. This means chisquared cannot be used to test correlated data (like matched pairs or panel data). In those cases, McNemar's test may be more appropriate.
A test that relies on different assumptions is Fisher's exact test; if its assumption of fixed marginal distributions is met it is substantially more accurate in obtaining a significance level, especially with few observations. In the vast majority of applications this assumption will not be met, and Fisher's exact test will be over conservative and not have correct coverage.^{[6]}
Derivation[edit]
The null distribution of the Pearson statistic with j rows and k columns is approximated by the chisquared distribution with (k − 1)(j − 1) degrees of freedom.^{[7]}
This approximation arises as the true distribution, under the null hypothesis, if the expected value is given by a multinomial distribution. For large sample sizes, the central limit theorem says this distribution tends toward a certain multivariate normal distribution.
Two cells[edit]
In the special case where there are only two cells in the table, the expected values follow a binomial distribution,
where
 p = probability, under the null hypothesis,
 n = number of observations in the sample.
In the above example the hypothesised probability of a male observation is 0.5, with 100 samples. Thus we expect to observe 50 males.
If n is sufficiently large, the above binomial distribution may be approximated by a Gaussian (normal) distribution and thus the Pearson test statistic approximates a chisquared distribution,
Let O_{1} be the number of observations from the sample that are in the first cell. The Pearson test statistic can be expressed as
which can in turn be expressed as
By the normal approximation to a binomial this is the squared of one standard normal variate, and hence is distributed as chisquared with 1 degree of freedom. Note that the denominator is one standard deviation of the Gaussian approximation, so can be written
So as consistent with the meaning of the chisquared distribution, we are measuring how probable the observed number of standard deviations away from the mean is under the Gaussian approximation (which is a good approximation for large n).
The chisquared distribution is then integrated on the right of the statistic value to obtain the Pvalue, which is equal to the probability of getting a statistic equal or bigger than the observed one, assuming the null hypothesis.
Twobytwo contingency tables[edit]
When the test is applied to a contingency table containing two rows and two columns, the test is equivalent to a Ztest of proportions.^{[citation needed]}
Many cells[edit]
Similar arguments as above lead to the desired result.^{[citation needed]} Each cell (except the final one, whose value is completely determined by the others) is treated as an independent binomial variable, and their contributions are summed and each contributes one degree of freedom.
Let us now prove that the distribution indeed approaches asymptotically the distribution as the number of observations approaches infinity.
Let be the number of observations, the number of cells and the probability of an observation to fall in the ith cell, for . We denote by the configuration where for each i there are observations in the ith cell. Note that
Let be Pearson's cumulative test statistic for such a configuration, and let be the distribution of this statistic. We will show that the latter probability approaches the distribution with degrees of freedom, as
For any arbitrary value T:
We will use a procedure similar to the approximation in de Moivre–Laplace theorem. Contributions from small are of subleading order in and thus for large we may use Stirling's formula for both and to get the following:
By substituting for
we may approximate for large the sum over the by an integral over the . Noting that:
we arrive at
By expanding the logarithm and taking the leading terms in , we get
Now, it should be noted that Pearson's chi, , is precisely the argument of the exponent (except for the 1/2; note that the final term in the exponent's argument is equal to ).
This argument can be written as:
is a regular symmetric matrix, and hence diagonalizable. It is therefore possible to make a linear change of variables in so as to get new variables so that:
This linear change of variables merely multiplies the integral by a constant Jacobian, so we get:
Where C is a constant.
This is the probability that squared sum of independent normally distributed variables of zero mean and unit variance will be greater than T, namely that with degrees of freedom is larger than T.
We have thus shown that at the limit where the distribution of Pearson's chi approaches the chi distribution with degrees of freedom.
Examples[edit]
Fairness of dice[edit]
A 6sided dice is thrown 60 times. The number of times it lands with 1, 2, 3, 4, 5 and 6 face up is 5, 8, 9, 8, 10 and 20, respectively. Is the die biased, according to the Pearson's chisquared test at a significance level of 95% and/or 99%?
n = 6 as there are 6 possible outcomes, 1 to 6. The null hypothesis is that the die is unbiased, hence each number is expected to occur the same number of times, in this case, 60/n = 10. The outcomes can be tabulated as follows:
i  O_{i}  E_{i}  O_{i} −E_{i}  (O_{i} −E_{i} )^{2}  (O_{i} −E_{i} )^{2}/E_{i} 

1  5  10  −5  25  2.5 
2  8  10  −2  4  0.4 
3  9  10  −1  1  0.1 
4  8  10  −2  4  0.4 
5  10  10  0  0  0 
6  20  10  10  100  10 
Sum  13.4 
The number of degrees of freedom is n − 1 = 5. The Uppertail critical values of chisquare distribution table gives a critical value of 11.070 at 95% significance level:
Degrees of freedom 
Probability less than the critical value  

0.90  0.95  0.975  0.99  0.999  
5  9.236  11.070  12.833  15.086  20.515 
As the chisquared statistic of 13.4 exceeds this critical value, we reject the null hypothesis and conclude that the die is biased at 95% significance level.
At 99% significance level, the critical value is 15.086. As the chisquared statistic does not exceed it, we fail to reject the null hypothesis and thus conclude that there is insufficient evidence to show that the die is biased at 99% significance level.
Goodness of fit[edit]
In this context, the frequencies of both theoretical and empirical distributions are unnormalised counts, and for a chisquared test the total sample sizes of both these distributions (sums of all cells of the corresponding contingency tables) have to be the same.
For example, to test the hypothesis that a random sample of 100 people has been drawn from a population in which men and women are equal in frequency, the observed number of men and women would be compared to the theoretical frequencies of 50 men and 50 women. If there were 44 men in the sample and 56 women, then
If the null hypothesis is true (i.e., men and women are chosen with equal probability), the test statistic will be drawn from a chisquared distribution with one degree of freedom (because if the male frequency is known, then the female frequency is determined).
Consultation of the chisquared distribution for 1 degree of freedom shows that the probability of observing this difference (or a more extreme difference than this) if men and women are equally numerous in the population is approximately 0.23. This probability is higher than conventional criteria for statistical significance (0.01 or 0.05), so normally we would not reject the null hypothesis that the number of men in the population is the same as the number of women (i.e., we would consider our sample within the range of what we would expect for a 50/50 male/female ratio.)
Problems[edit]
The approximation to the chisquared distribution breaks down if expected frequencies are too low. It will normally be acceptable so long as no more than 20% of the events have expected frequencies below 5. Where there is only 1 degree of freedom, the approximation is not reliable if expected frequencies are below 10. In this case, a better approximation can be obtained by reducing the absolute value of each difference between observed and expected frequencies by 0.5 before squaring; this is called Yates's correction for continuity.
In cases where the expected value, E, is found to be small (indicating a small underlying population probability, and/or a small number of observations), the normal approximation of the multinomial distribution can fail, and in such cases it is found to be more appropriate to use the Gtest, a likelihood ratiobased test statistic. When the total sample size is small, it is necessary to use an appropriate exact test, typically either the binomial test or (for contingency tables) Fisher's exact test. This test uses the conditional distribution of the test statistic given the marginal totals; however, it does not assume that the data were generated from an experiment in which the marginal totals are fixed^{[dubious – discuss]} and is valid whether or not that is the case.^{[dubious – discuss]}^{[citation needed]}
It can be shown that the test is a low order approximation of the test.^{[8]} The above reasons for the above issues become apparent when the higher order terms are investigated.
See also[edit]
 Gtest, test to which chisquared test is an approximation
 Degrees of freedom (statistics)
 Fisher's exact test
 Median test
 Lexis ratio, earlier statistic, replaced by chisquared
 Chisquared nomogram
 Deviance (statistics), another measure of the quality of fit
 Mann–Whitney U test
 Cramér's V – a measure of correlation for the chisquared test
 Minimum chisquare estimation
Notes[edit]
 ^ Pearson, Karl (1900). "On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling" (PDF). Philosophical Magazine. Series 5. 50 (302): 157–175. doi:10.1080/14786440009463897.
 ^ ^{a} ^{b} ^{c} David E. Bock, Paul F. Velleman, Richard D. De Veaux (2007). "Stats, Modeling the World," pp. 606627, Pearson Addison Wesley, Boston, ISBN 013187621X
 ^ "1.3.6.7.4. Critical Values of the ChiSquare Distribution". Retrieved 14 October 2014.
 ^ "Critical Values of the ChiSquared Distribution". NIST/SEMATECH eHandbook of Statistical Methods. National Institute of Standards and Technology.
 ^ See Field, Andy. Discovering Statistics Using SPSS. for assumptions on Chi Square.
 ^ "A Bayesian Formulation for Exploratory Data Analysis and GoodnessofFit Testing" (PDF). International Statistical Review. p. 375.
 ^ Statistics for Applications. MIT OpenCourseWare. Lecture 23. Pearson's Theorem. Retrieved 21 March 2007.
 ^ Jaynes, E.T. (2003). Probability Theory: The Logic of Science. C. University Press. p. 298. ISBN 9780521592710. (Link is to a fragmentary edition of March 1996.)
References[edit]
 Chernoff, H.; Lehmann, E. L. (1954). "The Use of Maximum Likelihood Estimates in Tests for Goodness of Fit". The Annals of Mathematical Statistics. 25 (3): 579–586. doi:10.1214/aoms/1177728726.
 Plackett, R. L. (1983). "Karl Pearson and the ChiSquared Test". International Statistical Review. International Statistical Institute (ISI). 51 (1): 59–72. doi:10.2307/1402731. JSTOR 1402731.
 Greenwood, P.E.; Nikulin, M.S. (1996). A guide to chisquared testing. New York: Wiley. ISBN 047155779X.