# SAGE Journal Articles

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**Summary/Abstract**: The examination of cross-classified category data is common in evaluation and research, with Karl Pearson’s family of chi-square tests representing one of the most utilized statistical analyses for answering questions about the association or difference between categorical variables. Unfortunately, these tests are also among the more commonly misinterpreted statistical tests in the field. The problem is not that researchers and evaluators misapply the results of chi-square tests, but rather they tend to overinterpret or incorrectly interpret the results, leading to statements that may have limited or no statistical support based on the analyses preformed. This paper attempts to clarify any confusion about the uses and interpretations of the family of chi-square tests developed by Pearson, focusing primarily on the chi-square tests of independence and homogeneity of variance (identity of distributions). A brief survey of the recent evaluation literature is presented to illustrate the prevalence of the chi-square test and to offer examples of how these tests are misinterpreted. While the omnibus form of all three tests in the Karl Pearson family of chi-square tests—independence, homogeneity, and goodness-of-fit—uses essentially the same formula, each of these three tests is, in fact, distinct with specific hypotheses, sampling approaches, interpretations, and options following rejection of the null hypothesis. Finally, a little-known option, the use and interpretation of post hoc comparisons based on Goodman’s procedure (Goodman, 1963) following the rejection of the chi-square test of homogeneity, is described in detail.

#### Questions to Consider

1. One common misinterpretation of chi-square tests comes from not distinguishing between three different versions of the test. What are these three and how do they differ?

2. The chi-square ______ is used when a sample is compared on a variable of interest against a population with known parameters.

- test of independence
- goodness of fit test
- test of heterogeneity
- test of homogeneity

3. The review of the evaluation literature reveals that in ____ of the instances where a chi-square test was used, the wrong interpretation was presented.

- 28
- 49
- 57
- 63

**Summary/Abstract: **Statistical hypothesis testing is common in research, but a conventional understanding sometimes leads to mistaken application and misinterpretation. The logic of hypothesis testing presented in this article provides for a clearer understanding, application, and interpretation. Key conclusions are that (a) the magnitude of an estimate on its raw scale (i.e., not calibrated by the standard error) is irrelevant to statistical testing; (b) which statistical hypotheses are tested cannot generally be known a priori; (c) if an estimate falls in a hypothesized set of values, that hypothesis does not require testing; (d) if an estimate does not fall in a hypothesized set, that hypothesis requires testing; (e) the point in a hypothesized set that produces the largest *p *value is used for testing; and (f) statistically significant results constitute evidence, but insignificant results do not and must not be interpreted as evidence for or against the hypothesis being tested.

#### Questions to Consider

1. According to the author, what are some of the key issues related to hypothesis testing?

2. An ______ hypothesis is one for which empirical evidence can, in principle, bear on judgments of its truth or falsity. A _______ hypothesis is an empirical hypothesis about distribution parameters of random variables defined by a data-generating process.

- null; alternative
- alternative, null
- statistical; empirical
- empirical; statistical

3. Identifying predictors typically requires isolating a direction. Therefore, the researcher needs to first address three hypotheses. Which of the following is not one of them?

- The parameter is greater than zero.
- The parameter is less than zero.
- The parameter is equal to zero.
- The parameter is not equal to zero.