“chi-squared test”的概念、定义、翻译、参考文献-科学参考

chi-squared test

Mathematics

A test, normally written as the χ‎²-test, to determine how well a set of observations fits a particular discrete distribution or some other given null hypothesis (see hypothesis testing). The observed frequencies in different groups are denoted by O_i, and the expected frequencies from the statistical model are denoted by E_i. For each i, the value (O_i-E_i)²/E_i is calculated, and these are summed. The result is compared with a chi-squared distribution with an appropriate number of degrees of freedom. The number of degrees of freedom depends on the number of groups and the number of parameters being estimated. The test requires that the observations are independent and that the sample size and expected frequencies exceed minimum numbers depending on the number of groups.

Statistics

A goodness-of-fit test, introduced by Karl Pearson in 1900, that is popular because of its simplicity. Let O denote the observed frequency of an outcome in a sample and let E denote the corresponding expected frequency under some model. The test statistic is X², defined by $chi-squared test$ where the summation is over all the outcomes whose frequencies are being compared, and r, the Pearson residual, is defined by $chi-squared test$ It is important to note that the comparison involves frequencies and not proportions.
If there are m observed frequencies, and p parameters have been estimated using these frequencies, then, if the model is correct, the observed value of X² will approximate an observation from a chi-squared distribution with (m−p−1) degrees of freedom. In the case of a continuous random variable, the frequencies will refer to ranges of values. For other random variables it is usual to combine together neighbouring rare values into a single category, since the chi-squared approximation fails if there are too many small expected frequencies.
As an example, suppose it is hypothesized that a type of sweet pea occurs in shades of white, red, pink, and blue, with proportions ¼, p, (¾−3p), and 2p, respectively. A random sample of 120 seeds is sown. All germinate with 20 having white flowers, 10 having red flowers, 40 pink, and 50 blue. The question is whether these results are consistent with the hypothesis. In this case the maximum likelihood estimate of p is 0.15, so the expected frequencies are 30, 18, 36, and 36. Thus $chi-squared test$ There are 4−1−1=2 degrees of freedom. Since 12.8 is a very large value (compared with the percentage points (Appendix VIII) of a -distribution), the hypothesis can confidently be rejected.
One situation in which the use of the chi-squared test is frequently encountered is as a test for independence in a J×K contingency table that cross-classifies the variables A and B. Let the observed frequency of data belonging to category j of variable A and to category k of variable B be f_jk. Write $chi-squared test$ Then, according to the null hypothesis of independence, the expected frequency e_jk is given by $chi-squared test$ and the test statistic X² is given by $chi-squared test$ If the null hypothesis of independence is correct, then the distribution of X² can be approximated by a chi-squared distribution with (J−1)(K−1) degrees of freedom. In the special case where J=K=2 (a two-by-two table), the chi-squared approximation is improved by using the Yates-corrected chi-squared test (see two-by-two table).
In 1946 Cramér suggested that a measure of association could be based on the value of X². This is Cramér's V, given by $chi-squared test$ where M is the smaller of J−1 and K−1.

Geology and Earth Sciences

In statistics, a hypothesis test used to determine the goodness of fit of a particular data set with that expected from a theoretical distribution. The test statistic is a function of the difference between observed and expected values, which is compared to the chi-squared distribution. The chi-squared distribution is a distribution of sample variance based on a single parameter, the degrees of freedom.

单词	chi-squared test
释义	chi-squared test Mathematics A test, normally written as the χ‎²-test, to determine how well a set of observations fits a particular discrete distribution or some other given null hypothesis (see hypothesis testing). The observed frequencies in different groups are denoted by O_i, and the expected frequencies from the statistical model are denoted by E_i. For each i, the value (O_i-E_i)²/E_i is calculated, and these are summed. The result is compared with a chi-squared distribution with an appropriate number of degrees of freedom. The number of degrees of freedom depends on the number of groups and the number of parameters being estimated. The test requires that the observations are independent and that the sample size and expected frequencies exceed minimum numbers depending on the number of groups. Statistics A goodness-of-fit test, introduced by Karl Pearson in 1900, that is popular because of its simplicity. Let O denote the observed frequency of an outcome in a sample and let E denote the corresponding expected frequency under some model. The test statistic is X², defined by $chi-squared test$ where the summation is over all the outcomes whose frequencies are being compared, and r, the Pearson residual, is defined by $chi-squared test$ It is important to note that the comparison involves frequencies and not proportions. If there are m observed frequencies, and p parameters have been estimated using these frequencies, then, if the model is correct, the observed value of X² will approximate an observation from a chi-squared distribution with (m−p−1) degrees of freedom. In the case of a continuous random variable, the frequencies will refer to ranges of values. For other random variables it is usual to combine together neighbouring rare values into a single category, since the chi-squared approximation fails if there are too many small expected frequencies. As an example, suppose it is hypothesized that a type of sweet pea occurs in shades of white, red, pink, and blue, with proportions ¼, p, (¾−3p), and 2p, respectively. A random sample of 120 seeds is sown. All germinate with 20 having white flowers, 10 having red flowers, 40 pink, and 50 blue. The question is whether these results are consistent with the hypothesis. In this case the maximum likelihood estimate of p is 0.15, so the expected frequencies are 30, 18, 36, and 36. Thus $chi-squared test$ There are 4−1−1=2 degrees of freedom. Since 12.8 is a very large value (compared with the percentage points (Appendix VIII) of a -distribution), the hypothesis can confidently be rejected. One situation in which the use of the chi-squared test is frequently encountered is as a test for independence in a J×K contingency table that cross-classifies the variables A and B. Let the observed frequency of data belonging to category j of variable A and to category k of variable B be f_jk. Write $chi-squared test$ Then, according to the null hypothesis of independence, the expected frequency e_jk is given by $chi-squared test$ and the test statistic X² is given by $chi-squared test$ If the null hypothesis of independence is correct, then the distribution of X² can be approximated by a chi-squared distribution with (J−1)(K−1) degrees of freedom. In the special case where J=K=2 (a two-by-two table), the chi-squared approximation is improved by using the Yates-corrected chi-squared test (see two-by-two table). In 1946 Cramér suggested that a measure of association could be based on the value of X². This is Cramér's V, given by $chi-squared test$ where M is the smaller of J−1 and K−1. Geology and Earth Sciences In statistics, a hypothesis test used to determine the goodness of fit of a particular data set with that expected from a theoretical distribution. The test statistic is a function of the difference between observed and expected values, which is compared to the chi-squared distribution. The chi-squared distribution is a distribution of sample variance based on a single parameter, the degrees of freedom.
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