(geometry) a plane rectangle with four equal sides and four right angles; a four-sided regular polygon; "you can compute the area of a square if you know the length of its sides" (同)foursquare
the product of two equal terms; "nine is the second power of three"; "gravity is inversely proportional to the square of the distance" (同)second power
make square; "Square the circle"; "square the wood with a file" (同)square_up
something approximating the shape of a square
a hand tool consisting of two straight arms at right angles; used to construct or test right angles; "the carpenter who built this room must have lost his square"
any artifact having a shape similar to a plane geometric figure with four equal sides and four right angles; "a checkerboard has 64 squares"
someone who doesnt understand what is going on (同)lame
a formal and conservative person with old-fashioned views (同)square toes
without evasion or compromise; "a square contradiction"; "he is not being as straightforward as it appears" (同)straightforward, straight
be compatible with; "one idea squares with another"
cause to match, as of ideas or acts
having four equal sides and four right angles or forming a right angle; "a square peg in a round hole"; "a square corner"
leaving no balance; "my account with you is now all square"
pay someone and settle a debt; "I squared with him"
position so as to be square; "He squared his shoulders"
undergo a test; "She doesnt test well"
any standardized procedure for measuring sensitivity or memory or intelligence or aptitude or personality etc; "the test was standardized on a large sample of students" (同)mental test, mental testing, psychometric test
the act of undergoing testing; "he survived the great test of battle"; "candidates must compete in a trial of skill" (同)trial
the act of testing something; "in the experimental trials the amount of carbon was measured separately"; "he called each flip of the coin a new trial" (同)trial, run
a hard outer covering as of some amoebas and sea urchins
put to the test, as for its quality, or give experimental use to; "This approach has been tried with good results"; "Test this recipe" (同)prove, try, try out, examine, essay
achieve a certain score or rating on a test; "She tested high on the LSAT and was admitted to all the good law schools"
determine the presence or properties of (a substance)
show a certain characteristic when tested; "He tested positive for HIV"
an examination of the characteristics of something; "there are laboratories for commercial testing"; "it involved testing thousands of children for smallpox"
the act of subjecting to experimental test in order to determine how well something works; "they agreed to end the testing of atomic weapons"
tested and proved useful or correct; "a tested method" (同)tried, well-tried
tested and proved to be reliable (同)time-tested, tried, tried and true
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Chi-squared distribution, showing χ2 on the x-axis and p-value on the y-axis.
A chi-squared test, also written as χ2 test, is any statistical hypothesis test where the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true. Without other qualification, 'chi-squared test' often is used as short for Pearson's chi-squared test. The chi-squared test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.
In the standard applications of this test, the observations are classified into mutually exclusive classes, and there is some theory, or say null hypothesis, which gives the probability that any observation falls into the corresponding class. The purpose of the test is to evaluate how likely the observations that are made would be, assuming the null hypothesis is true.
Chi-squared tests are often constructed from a sum of squared errors, or through the sample variance. Test statistics that follow a chi-squared distribution arise from an assumption of independent normally distributed data, which is valid in many cases due to the central limit theorem. A chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent.
Also considered a chi-squared test is a test in which this is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be made to approximate a chi-squared distribution as closely as desired by making the sample size large enough.
Contents
1History
1.1Pearson's chi-squared test
2Other examples of chi-squared tests
2.1Fisher's exact test
2.2Binomial test
2.3Other chi-squared tests
3Yates's correction for continuity
4Chi-squared test for variance in a normal population
5Example chi-squared test for categorical data
6Applications
7See also
8References
9Further reading
History
In the 19th century, statistical analytical methods were mainly applied in biological data analysis and it was customary for researchers to assume that observations followed a normal distribution, such as Sir George Airy and Professor Merriman, whose works were criticized by Karl Pearson in his 1900 paper.[1]
Until the end of 19th century, Pearson noticed the existence of significant skewness within some biological observations. In order to model the observations regardless of being normal or skewed, Pearson, in a series of articles published from 1893 to 1916,[2][3][4][5] devised the Pearson distribution, a family of continuous probability distributions, which includes the normal distribution and many skewed distributions, and proposed a method of statistical analysis consisting of using the Pearson distribution to model the observation and performing the test of goodness of fit to determine how well the model and the observation really fit.
Pearson's chi-squared test
See also: Pearson's chi-squared test
In 1900, Pearson published a paper[1] on the χ2 test which is considered to be one of the foundations of modern statistics.[6] In this paper, Pearson investigated the test of goodness of fit.
Suppose that n observations in a random sample from a population are classified into k mutually exclusive classes with respective observed numbers xi (for i = 1,2,…,k), and a null hypothesis gives the probability pi that an observation falls into the ith class. So we have the expected numbers mi = npi for all i, where
Pearson proposed that, under the circumstance of the null hypothesis being correct, as n → ∞ the limiting distribution of the quantity given below is the χ2 distribution.
Pearson dealt first with the case in which the expected numbers mi are large enough known numbers in all cells assuming every xi may be taken as normally distributed, and reached the result that, in the limit as n becomes large, X2 follows the χ2 distribution with k − 1 degrees of freedom.
However, Pearson next considered the case in which the expected numbers depended on the parameters that had to be estimated from the sample, and suggested that, with the notation of mi being the true expected numbers and m′i being the estimated expected numbers, the difference
will usually be positive and small enough to be omitted. In a conclusion, Pearson argued that if we regarded X′2 as also distributed as χ2 distribution with k − 1 degrees of freedom, the error in this approximation would not affect practical decisions. This conclusion caused some controversy in practical applications and was not settled for 20 years until Fisher's 1922 and 1924 papers.[7][8]
Other examples of chi-squared tests
One test statistic that follows a chi-squared distribution exactly is the test that the variance of a normally distributed population has a given value based on a sample variance. Such tests are uncommon in practice because the true variance of the population is usually unknown. However, there are several statistical tests where the chi-squared distribution is approximately valid:
Fisher's exact test
For an exact test used in place of the 2 x 2 chi-squared test for independence, see Fisher's exact test.
Binomial test
For an exact test used in place of the 2 x 1 chi-squared test for goodness of fit, see Binomial test.
Other chi-squared tests
Cochran–Mantel–Haenszel chi-squared test.
McNemar's test, used in certain 2 × 2 tables with pairing
Tukey's test of additivity
The portmanteau test in time-series analysis, testing for the presence of autocorrelation
Likelihood-ratio tests in general statistical modelling, for testing whether there is evidence of the need to move from a simple model to a more complicated one (where the simple model is nested within the complicated one).
Yates's correction for continuity
Main article: Yates's correction for continuity
Using the chi-squared distribution to interpret Pearson's chi-squared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chi-squared distribution. This assumption is not quite correct and introduces some error.
To reduce the error in approximation, Frank Yates suggested a correction for continuity that adjusts the formula for Pearson's chi-squared test by subtracting 0.5 from the absolute difference between each observed value and its expected value in a 2 × 2 contingency table.[9] This reduces the chi-squared value obtained and thus increases its p-value.
Chi-squared test for variance in a normal population
If a sample of size n is taken from a population having a normal distribution, then there is a result (see distribution of the sample variance) which allows a test to be made of whether the variance of the population has a pre-determined value. For example, a manufacturing process might have been in stable condition for a long period, allowing a value for the variance to be determined essentially without error. Suppose that a variant of the process is being tested, giving rise to a small sample of n product items whose variation is to be tested. The test statistic T in this instance could be set to be the sum of squares about the sample mean, divided by the nominal value for the variance (i.e. the value to be tested as holding). Then T has a chi-squared distribution with n − 1 degrees of freedom. For example, if the sample size is 21, the acceptance region for T with a significance level of 5% is between 9.59 and 34.17.
Example chi-squared test for categorical data
Suppose there is a city of 1,000,000 residents with four neighborhoods: A, B, C, and D. A random sample of 650 residents of the city is taken and their occupation is recorded as "white collar", "blue collar", or "no collar". The null hypothesis is that each person's neighborhood of residence is independent of the person's occupational classification. The data are tabulated as:
A
B
C
D
total
White collar
90
60
104
95
349
Blue collar
30
50
51
20
151
No collar
30
40
45
35
150
Total
150
150
200
150
650
Let us take the sample living in neighborhood A, 150, to estimate what proportion of the whole 1,000,000 live in neighborhood A. Similarly we take 349/650 to estimate what proportion of the 1,000,000 are white-collar workers. By the assumption of independence under the hypothesis we should "expect" the number of white-collar workers in neighborhood A to be
Then in that "cell" of the table, we have
The sum of these quantities over all of the cells is the test statistic. Under the null hypothesis, it has approximately a chi-squared distribution whose number of degrees of freedom are
If the test statistic is improbably large according to that chi-squared distribution, then one rejects the null hypothesis of independence.
A related issue is a test of homogeneity. Suppose that instead of giving every resident of each of the four neighborhoods an equal chance of inclusion in the sample, we decide in advance how many residents of each neighborhood to include. Then each resident has the same chance of being chosen as do all residents of the same neighborhood, but residents of different neighborhoods would have different probabilities of being chosen if the four sample sizes are not proportional to the populations of the four neighborhoods. In such a case, we would be testing "homogeneity" rather than "independence". The question is whether the proportions of blue-collar, white-collar, and no-collar workers in the four neighborhoods are the same. However, the test is done in the same way.
Applications
In cryptanalysis, chi-squared test is used to compare the distribution of plaintext and (possibly) decrypted ciphertext. The lowest value of the test means that the decryption was successful with high probability.[10][11] This method can be generalized for solving modern cryptographic problems.[12]
In bioinformatics, chi-squared test is used to compare the distribution of certain properties of genes (e.g, genomic content, mutation rate, interaction network clustering, etc.) belonging to different categories (e.g., disease genes, essential genes, genes on a certain chromosome etc.).[13][14]
See also
Statistics portal
Contingency table
Chi-squared test nomogram
G-test
Minimum chi-square estimation
Nonparametric statistics
Wald test
Wilson score interval
References
^ ab
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: 157–175. doi:10.1080/14786440009463897.
^
Pearson, Karl (1893). "Contributions to the mathematical theory of evolution [abstract]". Proceedings of the Royal Society. 54: 329–333. doi:10.1098/rspl.1893.0079. JSTOR 115538.
^
Pearson, Karl (1895). "Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material". Philosophical Transactions of the Royal Society. 186: 343–414. Bibcode:1895RSPTA.186..343P. doi:10.1098/rsta.1895.0010. JSTOR 90649.
^
Pearson, Karl (1901). "Mathematical contributions to the theory of evolution, X: Supplement to a memoir on skew variation". Philosophical Transactions of the Royal Society A. 197: 443–459. Bibcode:1901RSPTA.197..443P. doi:10.1098/rsta.1901.0023. JSTOR 90841.
^
Pearson, Karl (1916). "Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation". Philosophical Transactions of the Royal Society A. 216: 429–457. Bibcode:1916RSPTA.216..429P. doi:10.1098/rsta.1916.0009. JSTOR 91092.
^
Cochran, William G. (1952). "The Chi-square Test of Goodness of Fit". The Annals of Mathematical Statistics. 23: 315–345. doi:10.1214/aoms/1177729380. JSTOR 2236678.
^
Fisher, Ronald A. (1922). "On the Interpretation of chi-squared from Contingency Tables, and the Calculation of P". Journal of the Royal Statistical Society. 85: 87–94. doi:10.2307/2340521. JSTOR 2340521.
^
Fisher, Ronald A. (1924). "The Conditions Under Which chi-squared Measures the Discrepancey Between Observation and Hypothesis". Journal of the Royal Statistical Society. 87: 442–450. JSTOR 2341149.
^Yates, Frank (1934). "Contingency table involving small numbers and the χ2 test". Supplement to the Journal of the Royal Statistical Society. 1 (2): 217–235. JSTOR 2983604.
^"Chi-squared Statistic". Practical Cryptography. Retrieved 18 February 2015.
^"Using Chi Squared to Crack Codes". IB Maths Resources. British International School Phuket.
^Ryabko, B. Ya.; Stognienko, V. S.; Shokin, Yu. I. (2004). "A new test for randomness and its application to some cryptographic problems" (PDF). Journal of Statistical Planning and Inference. 123: 365–376. doi:10.1016/s0378-3758(03)00149-6. Retrieved 18 February 2015.
^Feldman, I.; Rzhetsky, A.; Vitkup, D. (2008). "Network properties of genes harboring inherited disease mutations". PNAS. 105 (11): 4323–432. Bibcode:2008PNAS..105.4323F. doi:10.1073/pnas.0701722105. PMC 2393821. Retrieved 29 June 2018.
^"chi-square-tests" (PDF). Retrieved 29 June 2018.
Further reading
Weisstein, Eric W. "Chi-Squared Test". MathWorld.
Corder, G. W.; Foreman, D. I. (2014), Nonparametric Statistics: A Step-by-Step Approach, New York: Wiley, ISBN 978-1118840313
Greenwood, Cindy; Nikulin, M. S. (1996), A guide to chi-squared testing, New York: Wiley, ISBN 0-471-55779-X
Nikulin, M. S. (1973), "Chi-squared test for normality", Proceedings of the International Vilnius Conference on Probability Theory and Mathematical Statistics, 2, pp. 119–122
Bagdonavicius, V.; Nikulin, M. S. (2011), "Chi-squared goodness-of-fit test for right censored data", The International Journal of Applied Mathematics and Statistics, pp. 30–50[full citation needed]
Context: Behçet's disease (BD) is a multisystemic inflammatory disorder with unknown etiology. Many immunological changes were reported in BD previously and these changes may affect the frequency of contact sensitivity in these patients. Objective: We aimed to identify whether there is an interac
Chi-Square Test Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. For example, if, according to Mendel's laws, you expected 10 of 20 ...
