Glossary

11.4 Test for Homogeneity

3 min read•june 27, 2024

Tests for homogeneity compare distributions across multiple populations. They determine if proportions are consistent, like comparing smoker percentages between genders. This differs from goodness-of-fit tests, which examine a single population against an expected distribution.

The test uses contingency tables and chi-square statistics to analyze observed versus expected counts. Interpreting results involves comparing the test statistic to critical values, leading to conclusions about population distributions. Additional analyses can pinpoint specific differences between groups.

Test for Homogeneity

Homogeneity vs goodness-of-fit tests

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    • Compares the distribution of a categorical variable across two or more populations (gender, age groups)
    • Determines if the proportions of each category are the same across populations
    • Appropriate when you have two or more independent samples and want to know if they have the same distribution (comparing the proportion of smokers between men and women)
    • Compares the of categories in a single population to the based on a hypothesized distribution
    • Determines if the observed data fits the expected distribution (normal, uniform, )
    • Appropriate when you have one sample and want to know if it follows a specific distribution (testing if the outcomes of rolling a die follow a uniform distribution)

Test statistic for homogeneity

  • Organize data in a with rows representing populations and columns representing categories
  • Calculate expected counts for each cell using the formula Eij=(rowi total)(columnj total)overall totalE_{ij} = \frac{(row_i \text{ total})(column_j \text{ total})}{overall \text{ total}}
  • Calculate the using the formula χ2=∑(Oij−Eij)2Eij\chi^2 = \sum{\frac{(O_{ij} - E_{ij})^2}{E_{ij}}}, where OijO_{ij} is the observed count and EijE_{ij} is the expected count for each cell
  • calculated as df=(r−1)(c−1)df = (r - 1)(c - 1), where rr is the number of rows (populations) and cc is the number of columns (categories)
  • Compare the calculated test statistic to the from the with the given degrees of freedom and (0.05, 0.01)

Interpretation of homogeneity results

  • (H0H_0) states that the populations have the same distribution of the categorical variable
  • (HaH_a) states that the populations do not have the same distribution of the categorical variable
  • If the calculated test statistic is greater than the critical value (or if the is less than the significance level), reject the null hypothesis
    • Conclude that there is sufficient evidence to suggest that the populations do not have the same distribution (the proportion of smokers differs between men and women)
  • If the calculated test statistic is less than the critical value (or if the p-value is greater than the significance level), fail to reject the null hypothesis
    • Conclude that there is not enough evidence to suggest that the populations have different distributions (the proportion of favorite ice cream flavors is the same across age groups)

Additional Analysis

  • : Conduct further tests to identify which specific groups or categories contribute to the significant differences found in the homogeneity test
  • : Calculate measures such as Cramer's V to quantify the strength of the association between the categorical variables
  • : Compute to identify which specific cells in the contingency table contribute most to the overall chi-square statistic

Key Terms to Review (21)

Alternative Hypothesis: The alternative hypothesis, denoted as H1 or Ha, is a statement that contradicts the null hypothesis and suggests that the observed difference or relationship in a study is statistically significant and not due to chance. It represents the researcher's belief about the population parameter or the relationship between variables.
Categorical variable: A categorical variable is a type of variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category. This means that the values represent distinct categories or groups rather than numerical measurements, which makes them essential in organizing and analyzing data in various contexts.
Chi-Square Distribution: The chi-square distribution is a probability distribution that arises when independent standard normal random variables are squared and summed. It is a continuous probability distribution that is widely used in statistical hypothesis testing, particularly in assessing the goodness of fit of observed data to a theoretical distribution, testing the independence of two attributes, and testing the homogeneity of multiple populations.
Chi-square Test for Homogeneity: The chi-square test for homogeneity is a statistical test used to determine if there is a significant difference in the proportions or distributions of a categorical variable across two or more independent groups or populations. It evaluates whether the observed frequencies in each category are consistent with the expected frequencies under the null hypothesis of homogeneity.
Chi-Square Test Statistic: The chi-square test statistic is a statistical measure used to determine the goodness-of-fit between an observed set of data and an expected set of data. It is a fundamental concept in hypothesis testing that helps assess whether the differences between observed and expected frequencies are statistically significant.
Contingency Table: A contingency table, also known as a cross-tabulation or cross-tab, is a type of table that displays the frequency distribution of two or more categorical variables. It allows for the analysis of the relationship between these variables and is a fundamental tool in various statistical analyses.
Critical Value: The critical value is a threshold value in statistical analysis that determines whether to reject or fail to reject a null hypothesis. It is a key concept in hypothesis testing and is used to establish the boundaries for statistical significance in various statistical tests.
Degrees of Freedom: Degrees of freedom (df) is a fundamental statistical concept that represents the number of independent values or observations that can vary in a given situation. It is an essential parameter that determines the appropriate statistical test or distribution to use in various data analysis techniques.
Effect Size: Effect size is a quantitative measure that indicates the magnitude or strength of the relationship between two variables or the difference between two groups. It provides information about the practical significance of a statistical finding, beyond just the statistical significance.
Expected Frequencies: Expected frequencies refer to the anticipated or predicted values of frequencies in a statistical analysis, particularly in the context of hypothesis testing and the chi-square test. They represent the frequencies that would be expected to occur under the null hypothesis, assuming there is no significant difference or association between the variables being studied.
Goodness-of-Fit Test: The goodness-of-fit test is a statistical hypothesis test used to determine whether a sample of data fits a particular probability distribution. It evaluates how well the observed data matches the expected data under a specified distribution model.
Independence Assumption: The independence assumption is a fundamental statistical concept that underlies various hypothesis tests and statistical analyses. It states that the observations or data points in a sample are independent of one another, meaning that the value of one observation does not depend on or influence the value of another observation.
Multinomial Distribution: The multinomial distribution is a generalization of the binomial distribution, where the random variable can take on more than two possible outcomes. It is used to model the probabilities of obtaining different categories or outcomes from a single experiment with multiple possible results.
Null Hypothesis: The null hypothesis, denoted as H0, is a statistical hypothesis that states there is no significant difference or relationship between the variables being studied. It represents the default or initial position that a researcher takes before conducting an analysis or experiment.
Observed Frequencies: Observed frequencies refer to the actual or empirical counts of occurrences in a dataset, often displayed in a contingency table or frequency distribution. This term is central to understanding the application of chi-square tests in statistics, which compare observed frequencies to expected frequencies to determine statistical significance.
P-value: The p-value is a statistical measure that represents the probability of obtaining a test statistic that is at least as extreme as the observed value, given that the null hypothesis is true. It is a crucial component in hypothesis testing, as it helps determine the strength of evidence against the null hypothesis and guides the decision-making process in statistical analysis across a wide range of topics in statistics.
Post-Hoc Analysis: Post-hoc analysis refers to the statistical techniques used to explore relationships or make comparisons between groups after an initial hypothesis test has been conducted. It is often employed to gain deeper insights into the results of a study when significant findings are obtained.
Sample Size Requirement: The sample size requirement refers to the minimum number of observations or data points needed to conduct a valid statistical analysis, such as a test for homogeneity. It ensures that the sample size is large enough to provide reliable and accurate results that can be generalized to the larger population.
Significance Level: The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is true. It represents the maximum acceptable probability of making a Type I error, which is the error of concluding that an effect exists when it does not. The significance level is a critical component in hypothesis testing, as it sets the threshold for determining the statistical significance of the observed results.
Standardized Residuals: Standardized residuals are the residuals from a statistical model that have been standardized, or transformed, to have a mean of 0 and a standard deviation of 1. This standardization allows for easier interpretation and comparison of the magnitude of the residuals across different models or variables.
Test for Homogeneity: The test for homogeneity is a statistical method used to determine whether two or more samples come from populations with the same probability distribution. It is commonly used to assess the similarity or differences between groups or categories in a dataset.
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