The most common Chi-square significance tests are Pearson chi-square test and the likelihood ratio chi-square test.
Pearson chi-square test is by far the most common type of chi-square significance test. If simply “chi-square” is mentioned, it is probably Pearson’s chi-square. This statistic in Chi-square significance tests is used to test the hypothesis of no association of columns and rows in tabular data. This statistic in Chi-square significance tests can be used with nominal data. This statistic in Chi-square significance tests is more likely to establish significance to the extent that the relationship is strong, the sample size is large and the number of values of the two associated variables is also large. This statistic in Chi-square significance tests, with probability of .05 or less, is commonly interpreted by social scientists as the justification for rejecting the null hypothesis that the row variable is unrelated (that is, only randomly related) to the column variable. Its calculation in Chi-square significance tests is the sum of observed minus expected count squared and divided by the expected.
The goodness-of-fit test in Chi-square significance tests is simply a different usage of Pearsonian chi-square. It is used in Chi-square significance tests to test if an observed distribution conforms to any other distribution, such as one based on theory (eg., if the observed distribution is not significantly different from a normal distribution) or one based on some other known distribution (eg., if the observed distribution is not significantly different from a known national distribution based on Census data). The Kolmogorov-Smirnov goodness-of-fit test is preferred for interval data, for which it is more powerful than the chi-square goodness-of-fit in Chi-square significance tests.
Likelihood ratio chi-square test in Chi-square significance tests is also called the likelihood test or G test. It is an alternative procedure to test the hypothesis of no association of columns and rows in nominal-level tabular data. This test in Chi-square significance tests is based on maximum likelihood estimation. Though computed differently, likelihood ratio chi-square is interpreted the same way as goodness-of-fit test in Chi-square significance tests. For large samples, likelihood ratio chi-square will be similar to results to Pearson chi-square in Chi-square significance tests.
Mantel-Haenszel chi-square is also called the Mantel-Haenszel test in Chi-square significance tests. Unlike ordinary and likelihood ratio chi-square, it is an ordinal measure of significance. This test in Chi-square significance tests is preferred when testing the significance of the linear relationship between two ordinal variables because it is more powerful than Pearson chi-square (more likely to establish linear association). Mantel-Haenzel chi-square in Chi-square significance test is not appropriate for nominal variables. Like other chi-square statistics, M-H chi-square in Chi-square significance tests should not be used with tables with small cell counts.
Assumptions
In Chi-square significance tests, random sample data are assumed. If there is non-random sample data, Chi-square significance tests cannot be established.
A sufficiently large sample size is assumed in all Chi-square significance tests. Applying chi-square to small samples exposes the researcher to an unacceptable rate of Type II errors.
Adequate cell sizes are also assumed in Chi-square significance tests. Some require 5 or more, some require more than 5 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 count.
Observations must be independent in Chi-square significance tests. The same observation can only appear in one cell. This means chi-square cannot be used to test correlated data (eg., before-after, matched pairs, and panel data).
Observations must have the same underlying distribution in Chi-square significance tests. The hypothesized distribution is specified in advance so that the number of observations that are expected to appear in each cell of the table can be calculated without reference to the observed values in Chi-square significance tests.
Non-directional hypothesis is assumed in Chi-square significance tests. Chi-square tests the hypothesis that two variables are not related by chance. If a significant relationship is found, this is not equivalent to establish the researcher’s hypothesis, that A and B are related.
Observations must be grouped in categories in Chi-square significance tests.
Normal distribution of deviations (observed minus expected values) is assumed in Chi-square significance tests. It must be noted that chi-square is a nonparametric test in the sense that it does not assume the parameter of normal distribution for the data, it does so only for the deviations.
The most common Chi-square significance tests are Pearson chi-square test and the likelihood ratio chi-square test.
Pearson chi-square test is by far the most common type of chi-square significance test. If simply “chi-square” is mentioned, it is probably Pearson’s chi-square. This statistic in Chi-square significance tests is used to test the hypothesis of no association of columns and rows in tabular data. This statistic in Chi-square significance tests can be used even with nominal data. This statistic in Chi-square significance tests is more likely to establish significance to the extent that the relationship is strong, the sample size is large and the number of values of the two associated variables is also large. This statistic in Chi-square significance tests, with probability of .05 or less, is commonly interpreted by social scientists as the justification for rejecting the null hypothesis that the row variable is unrelated (that is, only randomly related) to the column variable. Its calculation in Chi-square significance tests is the sum of observed minus expected count squared and divided by the expected.
The goodness-of-fit test in Chi-square significance tests is simply a different usage of Pearsonian chi-square. It is used in Chi-square significance tests to test if an observed distribution conforms to any other distribution, such as one based on theory (ex., if the observed distribution is not significantly different from a normal distribution) or one based on some other known distribution (ex., if the observed distribution is not significantly different from a known national distribution based on Census data ). The Kolmogorov-Smirnov goodness-of-fit test is preferred for interval data, for which it is more powerful than the chi-square goodness-of-fit in Chi-square significance tests.
Likelihood ratio chi-square test in Chi-square significance tests is also called the likelihood test or G test. It is an alternative procedure to test the hypothesis of no association of columns and rows in nominal-level tabular data. This test in Chi-square significance tests is based on maximum likelihood estimation. Though computed differently, likelihood ratio chi-square is interpreted the same way as goodness-of-fit test in Chi-square significance tests. For large samples, likelihood ratio chi-square will be clone in results to Pearson chi-square in Chi-square significance tests. Even for smaller samples, it rarely leads to different substantive results.
Mantel-Haenszel chi-square is also called the Mantel-Haenszel test in Chi-square significance tests. Unlike ordinary and likelihood ratio chi-square, it is an ordinal measure of significance. This test in Chi-square significance tests is preferred when testing the significance of the linear relationship between two ordinal variables because it is more powerful than Pearson chi-square (more likely to establish linear association). Mantel-Haenzel chi-square in Chi-square significance test is not appropriate for nominal variables. Like other chi-square statistics, M-H chi-square in Chi-square significance tests should not be used with tables with small cell counts.
Assumptions
In Chi-square significance tests, random sample data are assumed. If there is non-random sample data, Chi-square significance tests cannot be established.
A sufficiently large sample size is assumed in all Chi-square significance tests. Applying chi-square to small samples exposes the researcher to an unacceptable rate of Type II errors.
Adequate cell sizes are also assumed in Chi-square significance tests. Some require 5 or more, some require more than 5 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 count.
Observations must be independent in Chi-square significance tests. The same observation can only appear in one cell. This means chi-square cannot be used to test correlated data (ex., before-after, matched pairs, and panel data).
Observations must have the same underlying distribution in Chi-square significance tests.
The hypothesized distribution is specified in advance so that the number of observations that are expected to appear in each cell of the table can be calculated without reference to the observed values in Chi-square significance tests.
Non-directional hypothesis is assumed in Chi-square significance tests. Chi-square tests the hypothesis that two variables are related only by chance. If a significant relationship is found, this is not equivalent to establish the researcher’s hypothesis, that A causes B, or that B causes A.
Observations must be grouped in categories in Chi-square significance tests.
Normal distribution of deviations (observed minus expected values) is assumed in Chi-square significance tests. It must be noted that chi-square is a nonparametric test in the sense that it does not assume the parameter of normal distribution for the data, it does so only for the deviations.
No assumption is made about level of the data in Chi-square significance tests. This means that nominal, ordinal, or interval data can be used with chi-square tests in Chi-square significance tests.
