Editing Pearson's chi-squared test (section)

{{Short description|Evaluates how likely it is that any difference between data sets arose by chance}}
{{broader|Chi-squared test}}
{{Use dmy dates|date=January 2020}}

'''Pearson's chi-squared test''' or '''Pearson's <math>\chi^2</math> test''' 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 [[chi-squared test]]s (e.g., [[Yates's correction for continuity|Yates]], [[Likelihood-ratio test|likelihood ratio]], [[Portmanteau test|portmanteau test in time series]], etc.) – [[Statistics|statistical]] procedures whose results are evaluated by reference to the [[chi-squared distribution]]. Its properties were first investigated by [[Karl Pearson]] in 1900.<ref>{{Cite journal | last = Pearson | first = Karl | author-link = Karl Pearson | title = 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 | doi = 10.1080/14786440009463897 | journal = Philosophical Magazine |series=Series 5 | volume = 50 | issue = 302 | pages = 157–175 | year = 1900 | url = https://zenodo.org/record/1430618 }}</ref> In contexts where it is important to improve a distinction between the [[test statistic]] and its distribution, names similar to ''Pearson χ-squared'' test or statistic are used.

It is a [[p-value]] test. The setup is as follows:<ref>{{cite arXiv|last1=Loukas|first1=Orestis|last2=Chung|first2=Ho Ryun|date=2022|title=Entropy-based Characterization of Modeling Constraints|eprint=2206.14105|class=stat.ME}}</ref><ref>{{cite arXiv|last1=Loukas|first1=Orestis|last2=Chung|first2=Ho Ryun|date=2023|title=Total Empiricism: Learning from Data|eprint=2311.08315|class=math.ST}}</ref>

* Before the experiment, the experimenter fixes a certain number <math>N</math> of samples to take.
* The '''observed data''' is <math>(O_1, O_2, ..., O_n)</math>, the count number of samples from a finite set of given categories. They satisfy <math display="inline">\sum_i O_i = N</math>.
* The '''null hypothesis''' is that the count numbers are sampled from a [[multinomial distribution]] <math>\mathrm{Multinomial}(N; p_1, ..., p_n)</math>. That is, the underlying data is sampled [[Independent and identically distributed random variables|IID]] from a [[categorical distribution]] <math>\mathrm{Categorical}(p_1, ..., p_n)</math> over the given categories.
* The Pearson's chi-squared '''test statistic''' is defined as <math display="inline">\chi^2 := \sum_i \frac{{\left(O_i - N p_i\right)}^2}{N p_i}</math>. The p-value of the test statistic is computed either numerically or by looking it up in a table.
* If the p-value is small enough (usually p < 0.05 by convention), then the null hypothesis is rejected, and we conclude that the observed data does not follow the multinomial distribution.

A simple example is testing the hypothesis that an ordinary six-sided die is "fair" (i. e., all six outcomes are equally likely to occur). In this case, the observed data is <math>(O_1, O_2, ..., O_6)</math>, the number of times that the dice has fallen on each number. The null hypothesis is <math>\mathrm{Multinomial}(N; 1/6, ..., 1/6)</math>, and <math display="inline">\chi^2 := \sum\limits_{i=1}^6 \frac{{\left(O_i - N/6\right)}^2}{N /6}</math>. As detailed below, if <math>\chi^2 > 11.07</math>, then the fairness of dice can be rejected at the level of <math>p < 0.05</math>.

==Usage==
Pearson's chi-squared test is used to assess three types of comparison: [[goodness of fit]], [[homogeneity (statistics)|homogeneity]], and [[Independence (probability theory)|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).<ref name="Bock">David E. Bock, Paul F. Velleman, Richard D. De Veaux (2007). "Stats, Modeling the World," pp.&nbsp;606-627, Pearson Addison Wesley, Boston, {{ISBN|0-13-187621-X}}</ref>
* 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 chi-squared test [[statistic]], <math>\chi^2</math>, which resembles a [[Normalization (statistics)|normalized]] sum of squared deviations between observed and theoretical [[Frequency (statistics)|frequencies]] (see below).
# Determine the [[degrees of freedom (statistics)|degrees of freedom]], '''df''', of that statistic.
## For a test of goodness-of-fit, {{nobreak|1= df = Cats &minus; Params}}, where ''Cats'' is the number of observation categories recognized by the model, and ''Params'' 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 a test of homogeneity, {{nobreak|1= df = (Rows &minus; 1)×(Cols &minus; 1)}}, where ''Rows'' corresponds to the number of categories (i.e. rows in the associated contingency table), and ''Cols'' corresponds to the number of independent groups (i.e. columns in the associated contingency table).<ref name="Bock" />
## For a test of independence, {{nobreak|1= df = (Rows &minus; 1)×(Cols &minus; 1)}}, where in this case, ''Rows'' corresponds to the number of categories in one variable, and ''Cols'' corresponds to the number of categories in the second variable.<ref name="Bock" />
# Select a desired level of confidence ([[significance level]], [[p-value|''p''-value]], or the corresponding [[alpha level]]) for the result of the test.
# Compare <math>\chi^2</math> to the critical value from the [[chi-squared distribution]] with ''df'' degrees of freedom and the selected confidence level (one-sided, since the test is only in 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 <math>\chi^2</math>.
# 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 <math>\chi^2</math>. If the test statistic exceeds the critical value of <math>\chi^2</math>, the null hypothesis (<math>H_0</math> = there is ''no'' difference between the distributions) can be rejected, and the alternative hypothesis (<math>H_1</math> = 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 <math>\chi^2</math> value, then no clear conclusion can be reached, and the null hypothesis is sustained (we fail to reject the null hypothesis), though not necessarily accepted.

==Test for fit of a distribution==

===Discrete uniform distribution===

In this case <math>N</math> observations are divided among <math>n</math> 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
<math display="block">E_i=\frac{N}{n}\, ,</math>
and the reduction in the degrees of freedom is <math>p=1</math>, notionally because the observed frequencies <math>O_i</math> are constrained to sum to <math>N</math>.

One specific example of its application would be its application for log-rank test.

===Other distributions===

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 <math>p=s+1</math>, where <math>s</math> is the number of parameters used in fitting the distribution. For instance, when checking a three-parameter [[Generalized gamma distribution]], <math>p=4</math>, and when checking a normal distribution (where the parameters are mean and standard deviation), <math>p=3</math>, and when checking a Poisson distribution (where the parameter is the expected value), <math>p=2</math>. Thus, there will be <math>n-p</math> degrees of freedom, where <math>n</math> is the number of categories.

The degrees of freedom are not based on the number of observations as with a [[Student's t]] or [[F-distribution]]. For example, if testing for a fair, six-sided die, there would be five degrees of freedom because there are six categories or parameters (each number); the number of times the die is rolled does not influence the number of degrees of freedom.

===Calculating the test-statistic===
[[File:Chi-square distributionCDF-English.png|thumb|right|300px|[[Chi-squared distribution]], showing ''X''<sup>2</sup> on the x-axis and P-value on the y-axis.]]
{| class="infobox wikitable collapsible collapsed" style="text-align:center;font-size:75%;line-height:0.9;"
! colspan="6" style="font-weight:normal;font-size:125%;"|Upper-tail critical values of chi-square distribution<ref>{{cite web|title=1.3.6.7.4. Critical Values of the Chi-Square Distribution|url=http://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm|access-date=14 October 2014}}</ref>
|-
! rowspan="2"|Degrees<br /> of<br />freedom
! colspan="5"|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 test-statistic is

<math display="block"> \chi^2 = \sum_{i=1}^{n} \frac{{\left(O_i - E_i\right)}^2}{E_i}
=  N \sum_{i=1}^n \frac{\left(O_i/N - p_i\right)^2}{p_i} </math>

where
*<math> \chi^2</math> = Pearson's cumulative test statistic, which asymptotically approaches a [[chi-squared distribution|<math>\chi^2</math> distribution]].
*<math>O_i</math> = the number of observations of type ''i''.
*<math>N</math> = total number of observations
*<math>E_i = N p_i</math> = the expected (theoretical) count of type ''i'', asserted by the null hypothesis that the fraction of type ''i'' in the population is <math> p_i</math>
*<math>n</math>  = the number of cells in the table.

The chi-squared statistic can then be used to calculate a [[p-value]] by [[Chi-squared distribution#Table of χ2 values vs p-values|comparing the value of the statistic]] to a [[chi-squared distribution]]. The number of [[degrees of freedom (statistics)|degrees of freedom]] is equal to the number of cells <math>n</math>, minus the reduction in degrees of freedom, <math>p</math>.

The chi-squared statistic can be also calculated as

<math display="block"> \chi^2 = \sum_{i=1}^{n} \frac{O_i^2}{E_i} - N.   </math>

This result is the consequence of the Binomial theorem.

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 chi-squared statistic. More generally however, when maximum likelihood estimation does not coincide with minimum chi-squared estimation, the distribution will lie somewhere between a chi-squared distribution with <math>n-1-p</math> and <math>n-1</math> degrees of freedom (See for instance Chernoff and Lehmann, 1954).

The chi-squared test indicates a statistically significant association between the level of education completed and routine check-up attendance (chi2(3) = 14.6090, p = 0.002). The proportions suggest that as the level of education increases, so does the proportion of individuals attending routine check-ups. Specifically, individuals who have graduated from college or university attend routine check-ups at a higher proportion (31.52%) compared to those who have not graduated high school (8.44%). This finding may suggest that higher educational attainment is associated with a greater likelihood of engaging in health-promoting behaviors such as routine check-ups.

===Bayesian method===

{{details|Categorical distribution#Bayesian inference using conjugate prior}}
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==
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 two-dimensional 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

<math display="block">E_{i,j}= N  p_{i\cdot} p_{\cdot j} ,</math>

where <math>N</math> is the total sample size (the sum of all cells in the table), and

<math display="block"> p_{i\cdot} = \frac{O_{i\cdot}}{N} = \sum_{j=1}^c \frac{O_{i,j}}{N},</math>

is the fraction of observations of type ''i'' ignoring the column attribute (fraction of row totals), and
<math display="block"> p_{\cdot j} = \frac{O_{\cdot j}}{N}  = \sum_{i = 1}^r \frac{O_{i,j}}{N} </math>

is the fraction of observations of type ''j'' ignoring the row attribute (fraction of column totals).  The term "[[frequency distribution|frequencies]]" refers to absolute numbers rather than already normalized values.

The value of the test-statistic is

<math display="block">\begin{align}
 \chi^2 &= \sum_{i=1}^r \sum_{j=1}^c \frac{{\left(O_{i,j} - E_{i,j}\right)}^2}{E_{i,j}} \\[1ex]
 &= N \sum_{i,j} p_{i\cdot} p_{\cdot j} {\left(\frac{\left(O_{i,j}/N\right) - p_{i\cdot} p_{\cdot j}}{p_{i\cdot} p_{\cdot j}}\right)}^2
\end{align}</math>

Note that <math> \chi^2 </math> is 0 if and only if <math> O_{i,j} = E_{i,j} \forall i,j </math>, 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 {{math|1=''p'' = ''r'' + ''c'' − 1}}.  The number of [[degrees of freedom (statistics)|degrees of freedom]] is equal to the number of cells ''rc'', minus the reduction in degrees of freedom, ''p'', which reduces to&nbsp;{{math|(''r'' − 1)(''c'' − 1)}}.

For the test of independence, also known as the test of homogeneity, a chi-squared probability of less than or equal to 0.05 (or the chi-squared 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.<ref>{{cite web|title=Critical Values of the Chi-Squared Distribution |url=http://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm |work=NIST/SEMATECH e-Handbook of Statistical Methods |publisher=National Institute of Standards and Technology}}</ref>
The [[alternative hypothesis]] corresponds to the variables having an association or relationship where the structure of this relationship is not specified.