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Pearson's chi-squared test

From Wikipedia, the free encyclopedia

Pearson's chi-squared test or Pearson's 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 tests (e.g., Yates, likelihood ratio, portmanteau test in time series, etc.) – statistical procedures whose results are evaluated by reference to the chi-squared 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 χ-squared test or statistic are used.

It is a p-value test. The setup is as follows:[2][3]

  • Before the experiment, the experimenter fixes a certain number of samples to take.
  • The observed data is , the count number of samples from a finite set of given categories. They satisfy .
  • The null hypothesis is that the count numbers are sampled from a multinomial distribution . That is, the underlying data is sampled IID from a categorical distribution over the given categories.
  • The Pearson's chi-squared test statistic is defined as . 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 dice is "fair" (i. e., all six outcomes are equally likely to occur). In this case, the observed data is , the number of times that the dice has fallen on each number. The null hypothesis is , and . As detailed below, if , then the fairness of dice can be rejected at the level of .

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Transcription

I'm thinking about buying a restaurant, so I go and ask the current owner, what is the distribution of the number of customers you get each day? And he says, oh, I've already figure that out. And he gives me this distribution over here, which essentially says 10% of his customers come in on Monday, 10% on Tuesday, 15% on Wednesday, so forth, and so on. They're closed on Sunday. So this is 100% of the customers for a week. If you add that up, you get 100%. I obviously am a little bit suspicious, so I decide to see how good this distribution that he's describing actually fits observed data. So I actually observe the number of customers, when they come in during the week, and this is what I get from my observed data. So to figure out whether I want to accept or reject his hypothesis right here, I'm going to do a little bit of a hypothesis test. So I'll make the null hypothesis that the owner's distribution-- so that's this thing right here-- is correct. And then the alternative hypothesis is going to be that it is not correct, that it is not a correct distribution, that I should not feel reasonably OK relying on this. It's not the correct-- I should reject the owner's distribution. And I want to do this with a significance level of 5%. Or another way of thinking about it, I'm going to calculate a statistic based on this data right here. And it's going to be chi-square statistic. Or another way to view it is it that statistic that I'm going to calculate has approximately a chi-square distribution. And given that it does have a chi-square distribution with a certain number of degrees of freedom and we're going to calculate that, what I want to see is the probability of getting this result, or getting a result like this or a result more extreme less than 5%. If the probability of getting a result like this or something less likely than this is less than 5%, then I'm going to reject the null hypothesis, which is essentially just rejecting the owner's distribution. If I don't get that, if I say, hey, the probability of getting a chi-square statistic that is this extreme or more is greater than my alpha, than my significance level, then I'm not going to reject it. I'm going to say, well, I have no reason to really assume that he's lying. So let's do that. So to calculate the chi-square statistic, what I'm going to do is-- so here we're assuming the owner's distribution is correct. So assuming the owner's distribution was correct, what would have been the expected observed? So we have expected percentage here, but what would have been the expected observed? So let me write this right here. Expected. I'll add another row, Expected. So we would have expected 10% of the total customers in that week to come in on Monday, 10% of the total customers of that week to come in on Tuesday, 15% to come in on Wednesday. Now to figure out what the actual number is, we need to figure out the total number of customers. So let's add up these numbers right here. So we have-- I'll get the calculator out. So we have 30 plus 14 plus 34 plus 45 plus 57 plus 20. So there's a total of 200 customers who came into the restaurant that week. So let me write this down. So this is equal to-- so I wrote the total over here. Ignore this right here. I had 200 customers come in for the week. So what was the expected number on Monday? Well, on Monday, we would have expected 10% of the 200 to come in. So this would have been 20 customers, 10% times 200. On Tuesday, another 10%. So we would have expected 20 customers. Wednesday, 15% of 200, that's 30 customers. On Thursday, we would have expected 20% of 200 customers, so that would have been 40 customers. Then on Friday, 30%, that would have been 60 customers. And then on Friday 15% again. 15% of 200 would have been 30 customers. So if this distribution is correct, this is the actual number that I would have expected. Now to calculate chi-square statistic, we essentially just take-- let me just show it to you, and instead of writing chi, I'm going to write capital X squared. Sometimes someone will write the actual Greek letter chi here. But I'll write the x squared here. And let me write it this way. This is our chi-square statistic, but I'm going to write it with a capital X instead of a chi because this is going to have approximately a chi-squared distribution. I can't assume that it's exactly, so this is where we're dealing with approximations right here. But it's fairly straightforward to calculate. For each of the days, we take the difference between the observed and expected. So it's going to be 30 minus 20-- I'll do the first one color coded-- squared divided by the expected. So we're essentially taking the square of almost you could kind of do the error between what we observed and expected or the difference between what we observed and expect, and we're kind of normalizing it by the expected right over here. But we want to take the sum of all of these. So I'll just do all of those in yellow. So plus 14 minus 20 squared over 20 plus 34 minus 30 squared over 30 plus-- I'll continue over here-- 45 minus 40 squared over 40 plus 57 minus 60 squared over 60, and then finally, plus 20 minus 30 squared over 30. I just took the observed minus the expected squared over the expected. I took the sum of it, and this is what gives us our chi-square statistic. Now let's just calculate what this number is going to be. So this is going to be equal to-- I'll do it over here so you don't run out of space. So we'll do this a new color. We'll do it in orange. This is going to be equal to 30 minus 20 is 10 squared, which is 100 divided by 20, which is 5. I might not be able to do all of them in my head like this. Plus, actually, let me just write it this way just so you can see what I'm doing. This right here is 100 over 20 plus 14 minus 20 is negative 6 squared is positive 36. So plus 36 over 20. Plus 34 minus 30 is 4, squared is 16. So plus 16 over 30. Plus 45 minus 40 is 5 squared is 25. So plus 25 over 40. Plus the difference here is 3 squared is 9, so it's 9 over 60. Plus we have a difference of 10 squared is plus 100 over 30. And this is equal to-- and I'll just get the calculator out for this-- this is equal to, we have 100 divided by 20 plus 36 divided by 20 plus 16 divided by 30 plus 25 divided by 40 plus 9 divided by 60 plus 100 divided by 30 gives us 11.44. So let me write that down. So this right here is going to be 11.44. This is my chi-square statistic, or we could call it a big capital X squared. Sometimes you'll have it written as a chi-square, but this statistic is going to have approximately a chi-square distribution. Anyway, with that said, let's figure out, if we assume that it has roughly a chi-square distribution, what is the probability of getting a result this extreme or at least this extreme, I guess is another way of thinking about it. Or another way of saying, is this a more extreme result than the critical chi-square value that there's a 5% chance of getting a result that extreme? So let's do it that way. Let's figure out the critical chi-square value. And if this is more extreme than that, then we will reject our null hypothesis. So let's figure out our critical chi-square values. So we have an alpha of 5%. And actually the other thing we have to figure out is the degrees of freedom. The degrees of freedom, we're taking one, two, three, four, five, six sums, so you might be tempted to say the degrees of freedom are six. But one thing to realize is that if you had all of this information over here, you could actually figure out this last piece of information, so you actually have five degrees of freedom. When you have just kind of n data points like this, and you're measuring kind of the observed versus expected, your degrees of freedom are going to be n minus 1, because you could figure out that nth data point just based on everything else that you have, all of the other information. So our degrees of freedom here are going to be 5. It's n minus 1. So our significance level is 5%. And our degrees of freedom is also going to be equal to 5. So let's look at our chi-square distribution. We have a degree of freedom of 5. We have a significance level of 5%. And so the critical chi-square value is 11.07. So let's go with this chart. So we have a chi-squared distribution with a degree of freedom of 5. So that's this distribution over here in magenta. And we care about a critical value of 11.07. So this is right here. Oh, you actually even can't see it on this. So if I were to keep drawing this magenta thing all the way over here, if the magenta line just kept going, over here, you'd have 8. Over here you'd have 10. Over here, you'd have 12. 11.07 is maybe some place right over there. So what it's saying is the probability of getting a result at least as extreme as 11.07 is 5%. So we could write it even here. Our critical chi-square value is equal to-- we just saw-- 11.07. Let me look at the chart again. 11.07. The result we got for our statistic is even less likely than that. The probability is less than our significance level. So then we are going to reject. So the probability of getting that is-- let me put it this way-- 11.44 is more extreme than our critical chi-square level. So it's very unlikely that this distribution is true. So we will reject what he's telling us. We will reject this distribution. It's not a good fit based on this significance level.

Usage

Pearson's chi-squared 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).[4]
  • 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:

  1. Calculate the chi-squared test statistic, , which resembles a normalized sum of squared deviations between observed and theoretical frequencies (see below).
  2. Determine the degrees of freedom, df, of that statistic.
    1. For a test of goodness-of-fit, df = Cats − 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.
    2. For a 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 to the number of independent groups (i.e. columns in the associated contingency table).[4]
    3. For a test of independence, df = (Rows − 1)×(Cols − 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.[4]
  3. Select a desired level of confidence (significance level, p-value, or the corresponding alpha level) for the result of the test.
  4. Compare 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 .
  5. 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 . If the test statistic exceeds the critical value of , 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 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 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 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 , where is the number of parameters used in fitting the distribution. For instance, when checking a three-parameter Generalized gamma 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 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

Chi-squared distribution, showing X2 on the x-axis and P-value on the y-axis.

The value of the test-statistic 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 chi-squared statistic can then be used to calculate a p-value by comparing the value of the statistic to a chi-squared distribution. The number of degrees of freedom is equal to the number of cells , minus the reduction in degrees of freedom, .

The chi-squared statistic can be also calculated as

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 and 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

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

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 normalized values.

The value of the test-statistic 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 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.[6] The alternative hypothesis corresponds to the variables having an association or relationship where the structure of this relationship is not specified.

Assumptions

The chi-squared test, when used with the standard approximation that a chi-squared distribution is applicable, has the following assumptions:[7]

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.[8]
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. For small sample sizes the Cash test is preferred.[9][10]
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 2-by-2 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 chi-squared 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.[11]

Derivation

Derivation using Central Limit Theorem

The null distribution of the Pearson statistic with j rows and k columns is approximated by the chi-squared distribution with (k − 1)(j − 1) degrees of freedom.[12]

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

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 chi-squared distribution,

Let O1 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 chi-squared 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 chi-squared 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 chi-squared distribution is then integrated on the right of the statistic value to obtain the P-value, which is equal to the probability of getting a statistic equal or bigger than the observed one, assuming the null hypothesis.

Two-by-two contingency tables

When the test is applied to a contingency table containing two rows and two columns, the test is equivalent to a Z-test of proportions.[citation needed]

Many cells

Broadly similar arguments as above lead to the desired result, though the details are more involved. One may apply an orthogonal change of variables to turn the limiting summands in the test statistic into one fewer squares of i.i.d. standard normal random variables.[13]

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 i-th cell, for . We denote by the configuration where for each i there are observations in the i-th 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

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.

An alternative derivation is on the multinomial distribution page.

Examples

Fairness of dice

A 6-sided die 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 chi-squared test at a significance level of 95% and/or 99%?

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:

1 5 10 −5 25
2 8 10 −2 4
3 9 10 −1 1
4 8 10 −2 4
5 10 10 0 0
6 20 10 10 100
Sum 134

We then consult an Upper-tail critical values of chi-square distribution table, the tabular value refers to the sum of the squared variables each divided by the expected outcomes. For the present example, this means

This is the experimental result whose unlikeliness (with a fair die) we wish to estimate.

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

The experimental sum of 13.4 is between the critical values of 97.5% and 99% significance or confidence (p-value). Specifically, getting 20 rolls of 6, when the expectation is only 10 such values, is unlikely with a fair die.

Chi-squared goodness of fit test

In this context, the frequencies of both theoretical and empirical distributions are unnormalised counts, and for a chi-squared 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 chi-squared distribution with one degree of freedom (because if the male frequency is known, then the female frequency is determined).

Consultation of the chi-squared 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

The approximation to the chi-squared 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 G-test, a likelihood ratio-based 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, and thus assumes that the margins were determined before the study; alternatives such as Boschloo's test which do not make this assumption are uniformly more powerful.

It can be shown that the test is a low order approximation of the test.[14] The above reasons for the above issues become apparent when the higher order terms are investigated.

See also

Notes

  1. ^ 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". Philosophical Magazine. Series 5. 50 (302): 157–175. doi:10.1080/14786440009463897.
  2. ^ Loukas, Orestis; Chung, Ho Ryun (2022). "Entropy-based Characterization of Modeling Constraints". arXiv:2206.14105 [stat.ME].
  3. ^ Loukas, Orestis; Chung, Ho Ryun (2023). "Total Empiricism: Learning from Data". arXiv:2311.08315 [math.ST].
  4. ^ a b c David E. Bock, Paul F. Velleman, Richard D. De Veaux (2007). "Stats, Modeling the World," pp. 606-627, Pearson Addison Wesley, Boston, ISBN 0-13-187621-X
  5. ^ "1.3.6.7.4. Critical Values of the Chi-Square Distribution". Retrieved 14 October 2014.
  6. ^ "Critical Values of the Chi-Squared Distribution". NIST/SEMATECH e-Handbook of Statistical Methods. National Institute of Standards and Technology.
  7. ^ McHugh, Mary (15 June 2013). "The chi-square test of independence". Biochemia Medica. 23 (2): 143–149. doi:10.11613/BM.2013.018. PMC 3900058. PMID 23894860.
  8. ^ See Field, Andy. Discovering Statistics Using SPSS. for assumptions on Chi Square.
  9. ^ Cash, W. (1979). "Parameter estimation in astronomy through application of the likelihood ratio". The Astrophysical Journal. 228: 939. Bibcode:1979ApJ...228..939C. doi:10.1086/156922. ISSN 0004-637X.
  10. ^ "The Cash Statistic and Forward Fitting". hesperia.gsfc.nasa.gov. Retrieved 19 October 2021.
  11. ^ "A Bayesian Formulation for Exploratory Data Analysis and Goodness-of-Fit Testing" (PDF). International Statistical Review. p. 375.
  12. ^ Statistics for Applications. MIT OpenCourseWare. Lecture 23. Pearson's Theorem. Retrieved 21 March 2007.
  13. ^ Benhamou, Eric; Melot, Valentin (2018). "Seven Proofs of the Pearson Chi-Squared Independence Test and its Graphical Interpretation". SSRN (preprint): 5-6. arXiv:1808.09171. doi:10.2139/ssrn.3239829. S2CID 88524653. SSRN 3239829. {{cite journal}}: Cite journal requires |journal= (help)
  14. ^ Jaynes, E.T. (2003). Probability Theory: The Logic of Science. C. University Press. p. 298. ISBN 978-0-521-59271-0. (Link is to a fragmentary edition of March 1996.)

References

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