The Pearson chi-square test (χ^{2}) compares differences between groups on variables measured at the nominal level. The χ^{2} compares the frequencies that are observed with the frequencies that are expected. When a study requires that researchers compare proportions (percentages) in one category versus another category, the χ^{2} is a statistic that will reveal if the difference in proportion is statistically improbable.

A one-way χ^{2} is a statistic that compares different levels of one variable only. For example, a researcher may collect information on gender and compare the proportions of males to females. If the one-way χ^{2} is statistically significant, it would indicate that proportions of one gender are significantly higher than proportions of the other gender than what would be expected by chance (Daniel, 2000). If more than two groups are being examined, the χ^{2} does not determine where the differences lie; it only determines that a significant difference exists. Further testing on pairs of groups with the χ^{2} would then be warranted to identify the significant differences.

A two-way χ^{2} is a statistic that tests whether proportions in levels of one nominal variable are significantly different from proportions of the second nominal variable. For example, the presence of advanced colon polyps was studied in three groups of patients: those having a normal body mass index (BMI), those who were overweight, and those who were obese (Siddiqui, Mahgoub, Pandove, Cipher, & Spechler, 2009). The research question tested was: “Is there a difference between the three groups (normal weight, overweight, and obese) on the presence of advanced colon polyps?” The results of the χ^{2} test indicated that a larger proportion of obese patients fell into the category of having advanced colon polyps compared to normal weight and overweight patients, suggesting that obesity may be a risk factor for developing advanced colon polyps. Further examples of two-way χ^{2} tests are reviewed in Exercise 19.

Research designs that may utilize the Pearson χ^{2} include the randomized experimental, quasi-experimental, and comparative designs (Gliner, Morgan, & Leech, 2009). The variables may be active, attributional, or a combination of both. An active variable refers to an intervention, treatment, or program. An attributional variable refers to a characteristic of the participant, such as gender, diagnosis, or ethnicity. Regardless of the whether the variables are active or attributional, all variables submitted to χ^{2} calculations must be measured at the nominal level.

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Use of the Pearson χ^{2} involves the following assumptions (Daniel, 2000):

1. Only one datum entry is made for each subject in the sample. Therefore, if repeated measures from the same subject are being used for analysis, such as pretests and posttests, χ^{2} is not an appropriate test.

2. The variables must be categorical (nominal), either inherently or transformed to categorical from quantitative values.

3. For each variable, the categories are mutually exclusive and exhaustive. No cells may have an expected frequency of zero. In the actual data, the observed cell frequency may be zero. However, the Pearson χ^{2} test is sensitive to small sample sizes, and other tests, such as the Fisher’s exact test, are more appropriate when testing very small samples (Daniel, 2000; Yates, 1934).

The test is distribution-free, or nonparametric, which means that no assumption has been made for a normal distribution of values in the population from which the sample was taken (Daniel, 2000).

The formula for a two-way χ^{2} is:

χ 2 =n[(A)(D)−(B)(C)] 2 (A+B)(C+D)(A+C)(B+D)

The contingency table is labeled as follows. A contingency table is a table that displays the relationship between two or more categorical variables (Daniel, 2000):

A | B |

C | D |

With any χ^{2} analysis, the degrees of freedom (df) must be calculated to determine the significance of the value of the statistic. The following formula is used for this calculation:

df=(R−1)(C−1)

where

R=Number of rows

C=Number of columns

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