Pearson's chi-squared test

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The Pearson chi-squar test ( ² ) is a statistical test applied to a categorical data set to evaluate how likely differentiations are observed between sets appear coincidentally. It is suitable for unpaired data from large samples. This is the most widely used chi-square test (eg, Yates, likelihood ratio, portmanteau test in time series, etc.) - statistical procedures whose results are evaluated with reference to the chi-square distribution. Its properties were first investigated by Karl Pearson in 1900. In a context where it is important to increase the distinction between test statistics and their distribution, a name similar to Pearson? -quared test or statistics used.

It tests the null hypothesis that the frequency distribution of a given event observed in a sample is consistent with a particular theoretical distribution. Events that are deemed to be mutually exclusive and have a total probability 1. The general case for this is where each event includes the result of a category variable. A simple example is the hypothesis that the usual six-sided mortality is "fair" (i.e., all six results are equally likely to occur.)

Video Pearson's chi-squared test

Definisi

The Pearson chi-square test is used to assess three types of comparisons: goodness of fit, homogeneity, and independence.

• The goodness match test determines whether the observed frequency distribution is different from the theoretical distribution.
• The homogeneity test compares the distribution of numbers to two or more groups using the same category variables (eg choice of activities - lectures, military, occupations, travel - high school graduates report a year after graduation, sorted based on year of graduation, to see if the number of graduates who choose a particular activity has changed from class to class, or from decade to decade).
• The independency test assesses whether unpaired observations on two variables, expressed in the contingency table, are independent of each other (eg polls responses from people from different countries to see if one's nationality is related to response).

Dalam hal ini ${\ displaystyle N}  ÂÂ$ pengamatan dibagi di antara ${\ displaystyle n}  ÂÂ$ sel. Aplikasi sederhana adalah untuk menguji hipotesis bahwa, dalam populasi umum, nilai-nilai akan terjadi di setiap sel dengan frekuensi yang sama. "Frekuensi teoretis" untuk sel apa pun (di bawah hipotesis nol dari distribusi seragam diskrit) dihitung sebagai

${\ displaystyle E_ {i} = {\ frac {N} {n}} \ ,,}  ÂÂ$

dan pengurangan derajat kebebasan adalah ${\ displaystyle p = 1}  ÂÂ$ , secara notional karena frekuensi yang diamati ${\ displaystyle O_ {i}}  ÂÂ$ dibatasi untuk menjumlahkan ${\ displaystyle N}  ÂÂ$ .

One specific example of its application is its application for log-rank testing.

Other distributions

When testing whether observations are random variables whose distribution belongs to a particular distribution family, "theoretical frequencies" are calculated using the distribution of families that are installed in some standard way. The degrees of freedom count is calculated as ${\ displaystyle p = s 1}$ ${\ displaystyle s}$ is the number of co-variates used in adjusting the distribution. For example, when checking the three-co-variate Weibull distribution, ${\ displaystyle p = 4}$ , and when checking the normal distribution (where parameters are mean and standard deviation), ${\ displaystyle p = 3}$ , and when checking the Poisson distribution (where the parameter is the expected value), ${\ displaystyle p = 2}$ . Thus, there will be ${\ displaystyle n-p}$ degrees of freedom, where ${\ displaystyle n}$ is the number of categories.

It should be noted that the degrees of freedom are not based on the number of observations as in Student's or F's t distribution. For example, if testing to die, six sides are fair, there will be five degrees of freedom because there are six categories/parameters (each number). How many times the rolled die does not affect the number of degrees of freedom.

Calculate the

test statistics

Nilai statistik uji adalah

${\ displaystyle \ chi ^ {2} = \ jumlah _ {i = 1} ^ {n} {\ frac {(O_ {i} -E_ {i}) ^ {2}} {E_ {i}}} = N \ jumlah _ {i = 1} ^ {n} {\ frac {\ kiri (O_ {i}/N-p_ {i} \ kanan) ^ {2} } {p_ {i}}}}  ÂÂ$

dimana

${\ displaystyle \ chi ^ {2}}  ÂÂ$ = statistik uji kumulatif Pearson, yang secara asimtotik mendekati ${\ displaystyle \ chi ^ {2}}  ÂÂ$ distribusi.

${\ displaystyle O_ {i}}  ÂÂ$ = jumlah observasi tipe i .

${\ displaystyle N}  ÂÂ$ = jumlah total pengamatan

${\ displaystyle E_ {i} = Np_ {i}}  ÂÂ$ = hitungan jenis teoritis yang diharapkan i , ditegaskan oleh hipotesis nol bahwa pecahan tipe i ${\ displaystyle p_ {i}}  ÂÂ$

${\ displaystyle n}  ÂÂ$ = jumlah sel dalam tabel.

Statistik chi-squared kemudian dapat digunakan untuk menghitung p-value dengan membandingkan nilai statistik dengan distribusi chi-kuadrat. Jumlah derajat kebebasan sama dengan jumlah sel ${\ displaystyle n}  ÂÂ$ , minus pengurangan derajat kebebasan, ${\ displaystyle p}  ÂÂ$ .

The result of the number of degrees of freedom applies when the original data is multinomial and hence the efficient estimation parameter to minimize the chi-square statistics. More generally, when the maximum probability estimate does not coincide with the minimum estimate of chi-square, the distribution will lie somewhere between the chi-square distribution with ${\ displaystyle n-1-p}$ and ${\ displaystyle n-1}$ degrees of freedom (See eg Chernoff and Lehmann, 1954).

Bayesian Bayesian Method

In Bayesian statistics, people would instead use Dirichlet's distribution as a previous conjugate. If a person takes the preceding uniform, then the maximum probability estimate for the probability of the population is the probability observed, and one can calculate the credible region around this or other estimates.

Maps Pearson's chi-squared test

Testing statistical independence

In this case, "observation" consists of the value of two results and the null hypothesis is that the occurrence of these results is statistically independent. Each observation is allocated to one cell from a cell two-dimensional cell (called a contingency table) according to the value of the two results. If there are r rows and columns c in the table, the "theoretical frequency" for cells, given the independence hypothesis, is

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          E                         me         Â mo moan,      Â                           =          N           Â ·                          me             ?                            Â ·                         ?      Â                           ,               {\ displaystyle E_ {i, j} = Np_ {i \ cdot} p _ {\ cdot j},}  Â}

di mana ${\ displaystyle N}  ÂÂ$ adalah ukuran total sampel (jumlah semua sel dalam tabel), dan

${\ displaystyle p_ {i \ cdot} = {\ frac {O_ {i \ cdot}} {N}} = \ jumlah _ {j = 1} ^ {c} {\ frac {O_ {i, j}} {N}},}  ÂÂ$

adalah pecahan dari pengamatan tipe i mengabaikan atribut kolom (pecahan dari total baris), dan

${\ displaystyle p _ {\ cdot j} = {\ frac {O_ {\ cdot j}} {N}} = \ jumlah _ {i = 1} ^ {r} {\ frac {O_ {i, j}} {N}}}  ÂÂ$

is a fraction of the observed type j ignores the line attribute (total column break). The term "frequency" refers to an absolute number rather than a normalized value.

Perhatikan bahwa ${\ displaystyle \ chi ^ {2}}  ÂÂ$ adalah 0 jika dan hanya jika ${\ displaystyle O_ {i, j} = E_ {i, j} \ forall i, j}  ÂÂ$ , yaitu hanya jika jumlah pengamatan yang diharapkan dan benar sama di semua sel.

The installation of the "independence" model reduces the number of degrees of freedom with p Ã, = Ã, r Ã, c Ã,-1. The number of degrees of freedom equal to the number of cells rc , reduced the degree of freedom, p , which reduces tol, ( c Ã, -Ã, 1).

For independence tests, also known as homogeneity tests, chi-square probabilities of less than or equal to 0.05 (or chi-square statistics at or greater than 0.05 critical points) are usually interpreted by workers who are applied as justifications for refusing the null hypothesis that the row variable is independent of the column variable. The alternative hypothesis corresponds to the variable having a relationship or relationship in which the structure of this relationship is not specified.

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Assumption

The chi-square test, when used with the standard approach that the chi-square distribution applies, has the following assumptions:

Simple random sample

The sample data is a random sampling of a fixed distribution or population in which each collection of population members of a given sample size has the same selection probability. Variant tests have been developed for complex samples, such as where the data is weighted. Other forms can be used such as purposive sampling.

Sample size (whole table)

Samples of large size are assumed. If a chi square test is performed on a sample of a smaller size, then a chi square test will result in inaccurate inference. Researchers, using a chi square test on a small sample, may end up with a Type II error.

Calculate expected cells

Approximate cell count is sufficient. Some require 5 or more, and others require 10 or more. The general rule is 5 or more in all cells of table 2-by-2, and 5 or more in 80% of cells in larger tables, but no cells with zero expected numbers. When this assumption is not met, Yates's correction is applied.

Independence

Observations are always considered independent of each other. This means that chi-squared can not be used to test correlated data (such as matching pairs or panel data). In such cases, the McNemar test may be more appropriate.

Tests that depend on different assumptions are Fisher's exact test; if the assumption of a marginal distribution remains satisfied then it is substantially more accurate in gaining significance levels, especially with little observation. In most applications, this assumption will not be met, and Fisher's exact test will be more conservative and lack the correct coverage.

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Derivation

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Example

Justice dice

Dice 6 sides thrown 60 times. The number of times it landed with 1, 2, 3, 4, 5 and 6 facing upwards is 5, 8, 9, 8, 10 and 20, respectively. Is the dice biased, according to Pearson's chi-square test at a 95% and/or 99% significance level?

n = 6 because there are 6 possible results, 1 through 6. The null hypothesis is that the dice is not biased, then each number is expected to occur in the same amount, in this case 60 / n = 10. The results can be tabulated as follows:

The number of degrees of freedom is n - 1 = 5. Table The upper critical critical value of the chi-square distribution gives a critical value of 11,070 at the 95% level of significance:

Since the chi-square statistic of 13.4 exceeds this critical value, we reject the null hypothesis and conclude that the dice bias at a 95% significance level.

At the 99% significance level, the critical value is 15,086. Since the chi-squared statistics do not go beyond it, we fail to reject the null hypothesis and conclude that there is insufficient evidence to show that the dice bias is at a 99% significance level.

Appropriate goodness

In this context, the theoretical and empirical distribution frequencies are abnormal quantities, and for the total chi-square test the sample size ${\ displaystyle N}$ of these distributions (the sum of all cells from the corresponding contingency table) should be the same.

For example, to test the hypothesis that a random sample of 100 people has been taken from a population where men and women have the same frequency, the number of men and women observed will be compared with the theoretical frequencies of 50 men and 50 women.. If there are 44 men in the sample and 56 women, then

${\ displaystyle \ chi ^ {2} = {{44-50} ^ {2} \ over 50} {(56-50) ^ {2} \ more than 50} = 1.44.}$

If the null hypothesis is true (ie, men and women are chosen with the same probability), the test statistic will be taken from the chi-square distribution with one degree of freedom (because if the male frequency is known, the female frequency is determined).

Consulting the chi-square distribution for 1 degree of freedom suggests 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 about 0.23. This probability is higher than the conventional criterion for statistical significance (0.01 or 0.05), so normally we will not reject the null hypothesis that the number of men in the population equals the number of women (ie we will consider our sample in what range which we expect for a 50/50 male/female ratio.)

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Problem

The approximation to the chi-square distribution is damaged if the expected frequency is too low. This will usually be acceptable for not more than 20% of the events have expected frequencies below 5. Where there is only 1 degree of freedom, the approach is unreliable if the expected frequency is below 10. In this case, better can be obtained by reducing the absolute value of any difference between the observed and expected frequency by 0.5 before squaring; this is called Yates's correction for continuity.

In the case where the expected value, E, is found to be small (indicates the probability of a small underlying population, and/or a small number of observations), the normal estimate of multinomial distribution may fail, and in such cases it is found more appropriately to use G-test, based on likelihood ratio. When the total sample size is small, it is necessary to use appropriate precise tests, usually either a binomial test or (for contingency tables) the exact test of Fisher. This test uses the conditional distribution of the test statistic given the total marginal; However, it does not assume that the data generated from the experiment in which the marginal amount is fixed and apply whether or not that is the case.

Hal ini dapat ditunjukkan bahwa ${\ displaystyle \ chi ^ {2}}  ÂÂ$ test adalah pendekatan urutan rendah dari ${\ displaystyle \ Psi}  ÂÂ$ test. Alasan di atas untuk masalah di atas menjadi jelas ketika istilah order yang lebih tinggi diselidiki.

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Lihat juga

• Uji-G, uji di mana tes chi-kuadrat adalah perkiraan
• Derajat kebebasan (statistik)
• Tes pasti Fisher
• Tes median
• Rasio Lexis, statistik sebelumnya, digantikan oleh chi-squared
• nomogram Chi-kuadrat
• Deviance (statistik), ukuran lain dari kualitas fit
• Tes Mann-Whitney U
• CramÃÆ' © r's V - ukuran korelasi untuk tes chi-squared
• Estimasi chi-kuadrat minimum

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Catatan

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Referensi

Source of the article : Wikipedia