Statistics for Social Studies. Chi-Square Assumptions Coursework

Abstract

The use of the chi-square test in research is needed to compare the distributions of two or more groups whose variables fall into two or more different classes or categories. With the precondition, there are five or more observations expected in each cell of a contingency table. The aim of this essay is to provide the assumptions for using a chi-square test and give a practical example from research to support these assumptions.

Introduction

Binomial data distribution is a method to summarize nonordinal (data showing no order or relative position) categorical (outcome) data. Data can be in the form of proportion (yes or no) or incidence (+ or -). Binomial data is found when the research focuses on the occurrence of events rather than the magnitude of events. In other words, a binomial distribution occurs whenever the possession or lack of a particular attribute is under investigation. Examples are clinical trials, animal or plant studies, and public opinion surveys (Neher, 2003).

The use of Chi-square statistic (X²) is to assess whether an observed binomial (or other) distribution accords with that expected either on the basis of knowledge of true population parameters or on theoretical backgrounds (Pepkin, 1986).

The Chi-square (X²) statistic test

The use of the X² test aims to compare the distributions of two or more groups. The variables fall into two or more different classes or categories provided that five or more observations are expected in each cell of a 2 x 2 contingency table. A contingency table is a table of the number of individuals in each group that fall in each category. The different characteristics or categories are the columns of the table, and the groups are the rows of the table (or the opposite). Each cell in the table lists the number of individuals for that combination of category and group. The power, or sensitivity, of a Chi-Square test, is the probability that the test will detect a difference among the groups if there really is a difference. As the power is close to 1, the more sensitive the test. Chi-Square power is affected by the sample size and the observed proportions of the samples (Pepkin, 1986).

Types of Chi-square test

1. Pearson X²: The commonest X² test used, it aims at testing the hypothesis of no association (correlation) of rows and columns in a contingency table (Garson, 2007).
2. Yates correction: It aims at adjusting the X² and therefore the P-value for 2 x 2 tables to more accurately reflect the true distribution of X² (Garson, 2007).
3. X² test for goodness of fit: It is to test a hypothesis of no association (correlation) of columns and rows in a nominal contingency table (Garson, 2007).

Assumptions of Chi-square test

According to Garson (2007), the assumptions of the X² test are:

1. Contained random (arbitrary) sample data. With non-random sample data, recognition of significance cannot be made.
2. Good sample size to avoid type II statistical error (the probability of accepting a null hypothesis when it is invalid).
3. Sufficient cell sizes: Some advise 5 or more (common), others require 10 or more. If this assumption is not fulfilled, application of Yates correction is in place.
4. Observations should be independent (not related, separate) and grouped into categories.
5. About observations, they should be of similar distribution.
6. The hypothesized distribution should be specified (known) beforehand. This means the number of expected observations to appear in each cell in the 2 x 2 table can be assessed without the values observed being the source of information.
7. The variables relate only by chance (non-directional hypothesis).
8. Being a non-parametric test, X² assumes the deviations are normally distributed (not the data).

An example from research

Dhand and others (2005) conducted a random survey to study the epidemiology of brucellosis in the Punjab region in India. They conducted their study in two stages, the first selection of the villages, and the second was to select the animals. They used the X² test (goodness of fit) to compare the actually observed frequencies in their sample assuming no relationship between the variables, and based on the assumption that sample frequencies are distributed normally around the expected value. They had no cell value of Zero. These assumptions necessitated the grouping of some variables as villages in close proximity, and the month of abortion to make the X² test valid. They designed comparisons within different categories based on the cell contribution to the X² test. Their results showed the prevalence of serological brucellosis is significantly higher (chi-square = 24.50, p < 0.001) in animals with a history of abortion (33.87%) than in those without such a history (11.63%).

Conclusion

The Chi-square test is a non-parametric statistical test used to compare the distributions of two or more groups. The variables fall into two or more different classes or categories provided that five or more observations are expected in each cell of a 2 x 2 contingency table. There are two main types of X² tests (Pearson and goodness of fit). Yates correction is applied if the assumption of sufficient cell sizes is not fulfilled (less than 5). A practical example from research literature summarizes the application of X² goodness of fit.

References

Dhand, N. K., Gumber, S., Singh, B. B., et al (2005). A study on the epidemiology of brucellosis in Punjab (India) using Survey Toolbox. Rev. sci. tech. Off. int. Epiz., 24(3), 879-885.

Garson, D. (2007). Chi-Square Significance Tests. Web.

Neher, D, A. (2003). Binomial Distributions (2003). Web.

Pipkin, F B. (1986). Medical Statistics Made Easy. 2nd. Edition. Philadelphia: W.B. Saunders.