Glossary

11.4 Test for Homogeneity

3 min read•june 25, 2024

The helps us compare proportions across different groups. It's like checking if political views are similar between men and women, or if blood types match up in different countries.

We use the to crunch the numbers and see if there's a real difference. This involves comparing what we see to what we'd expect if everything was the same. It's a powerful tool for spotting patterns in categorical data.

Test for Homogeneity

Homogeneity vs goodness-of-fit tests

Top images from around the web for Homogeneity vs goodness-of-fit tests

  • Goodness-of-fit calculations — IntroQG 2019 documentation View original

    Is this image relevant?

  • Goodness-of-Fit (2 of 2) | Concepts in Statistics View original

    Is this image relevant?

  • Goodness-of-fit calculations — IntroQG 2019 documentation View original

    Is this image relevant?

1 of 3

Top images from around the web for Homogeneity vs goodness-of-fit tests

  • Goodness-of-fit calculations — IntroQG 2019 documentation View original

    Is this image relevant?

  • Goodness-of-Fit (2 of 2) | Concepts in Statistics View original

    Is this image relevant?

  • Goodness-of-fit calculations — IntroQG 2019 documentation View original

    Is this image relevant?

1 of 3
  • Test for assesses whether the proportions of a categorical variable are consistent across multiple populations (comparing political party affiliation between men and women)
    • Null hypothesis: The proportions are the same across all populations
    • : The proportions differ in at least one population
  • evaluates if a sample's observed distribution matches an expected distribution based on a known population (testing if a sample's blood type distribution aligns with the general population)
    • Null hypothesis: The sample follows the expected distribution
    • Alternative hypothesis: The sample deviates from the expected distribution
  • Use homogeneity test when comparing distributions across populations and goodness-of-fit test when comparing a sample to an expected population distribution

Application of chi-square test statistic

  • Chi-square test statistic for homogeneity calculated using the formula χ2=∑i=1r∑j=1c(Oij−Eij)2Eij\chi^2 = \sum_{i=1}^{r} \sum_{j=1}^{c} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}
    • rr represents the number of categories (rows)
    • cc represents the number of populations (columns)
    • OijO_{ij} denotes the observed frequency in row ii and column jj
    • EijE_{ij} denotes the expected frequency in row ii and column jj
  • for the homogeneity test calculated as (r−1)(c−1)(r - 1)(c - 1)
  • EijE_{ij} calculated using the formula (row i total)×(column j total)grand total\frac{(\text{row } i \text{ total}) \times (\text{column } j \text{ total})}{\text{grand total}}
  • Compare the calculated chi-square value to the critical value at the desired
    • Reject the null hypothesis if the calculated value exceeds the critical value
    • Fail to reject the null hypothesis if the calculated value is less than or equal to the critical value
  • can be calculated to identify which specific cells contribute most to the

Interpretation of homogeneity test results

  • Null hypothesis (H0H_0): The categorical variable's distribution is consistent across all populations
  • Alternative hypothesis (HaH_a): The categorical variable's distribution differs in at least one population
  • Interpreting the results:
    • Rejecting the null hypothesis indicates a significant difference in the distribution across populations
    • Failing to reject the null hypothesis suggests insufficient evidence to conclude a difference in the distribution
  • Consider the test's assumptions (large sample size, expected frequencies ≥ 5) and use alternative tests () if assumptions are violated

Additional Analysis and Measures

  • can be calculated to determine the strength of the relationship between variables
  • may be conducted to identify specific differences between groups if the overall test is significant
  • can be used to measure the degree of association between in the homogeneity test

Key Terms to Review (23)

Alternative Hypothesis: The alternative hypothesis is a statement that suggests a potential outcome or relationship exists in a statistical test, opposing the null hypothesis. It indicates that there is a significant effect or difference that can be detected in the data, which researchers aim to support through evidence gathered during hypothesis testing.
Categorical variables: Categorical variables are variables that represent categories or groups and have a limited number of distinct values. These values are usually qualitative and describe characteristics or attributes.
Categorical Variables: Categorical variables are variables that represent a set of categories or groups, rather than numerical values. They are used to classify or group data based on qualitative characteristics or attributes, and are commonly used in statistical analysis and data visualization.
Chi-Square Distribution: The chi-square distribution is a continuous probability distribution that arises when independent standard normal random variables are squared and summed. It is widely used in statistical hypothesis testing, particularly in evaluating the goodness-of-fit of observed data to a theoretical distribution and in testing the independence of two categorical variables.
Chi-Square Statistic: The chi-square statistic is a statistical test used to determine if there is a significant difference between observed and expected frequencies in one or more categories. It is a fundamental concept in hypothesis testing and is widely applied in various fields, including the topics of test for homogeneity, comparison of chi-square tests, and chi-square goodness-of-fit.
Chi-Square Test: The chi-square test is a statistical hypothesis test that is used to determine if there is a significant difference between observed and expected frequencies in one or more categories. It is a versatile test that can be applied in various contexts, including contingency tables, goodness-of-fit, and tests for homogeneity.
Contingency Coefficient: The contingency coefficient is a statistical measure used to assess the degree of association between two categorical variables in a contingency table. It quantifies the strength of the relationship between these variables, providing insights into how the presence or absence of one variable influences the other. The coefficient ranges from 0 to 1, where 0 indicates no association and values closer to 1 suggest a stronger relationship.
Contingency table: A contingency table, also known as a cross-tabulation or crosstab, is a type of table in a matrix format that displays the frequency distribution of variables. It is commonly used to analyze the relationship between two categorical variables.
Contingency Table: A contingency table, also known as a cross-tabulation or a two-way table, is a type of table that displays the frequency distribution of two or more categorical variables. It is used to analyze the relationship between these variables and determine if they are independent or associated with each other.
Degrees of Freedom: Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation without breaking any constraints. It plays a crucial role in determining the appropriate statistical tests and distributions used for hypothesis testing, estimation, and data analysis across various contexts.
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 frequencies of observations in each category or cell of a contingency table, based on the assumption that the null hypothesis is true. They are a crucial component in the calculation and interpretation of the chi-square statistic, which is used to assess the goodness of fit between observed and expected frequencies.
Fisher's exact test: Fisher's exact test is a statistical significance test used to determine if there are nonrandom associations between two categorical variables in a contingency table. It is particularly useful when sample sizes are small, allowing researchers to evaluate the significance of the observed frequencies in relation to the expected frequencies under the null hypothesis, which states that there is no association between the variables. This test provides an exact p-value rather than an approximation, making it valuable in situations where traditional chi-square tests may not be applicable.
Goodness-of-Fit Test: A goodness-of-fit test is a statistical method used to determine how well a sample of observed data matches a theoretical probability distribution. This test assesses whether the differences between observed and expected frequencies are significant enough to reject the hypothesis that the observed data follow a specified distribution. It plays a critical role in evaluating models based on probability distributions, such as discrete random variables and exponential distributions.
Homogeneity: Homogeneity refers to the state of being uniform or consistent throughout, without variations or differences. In the context of statistical analysis, homogeneity is a crucial concept that is often examined when comparing groups or populations.
Independence: Independence is a fundamental concept in statistics that describes the relationship between events or variables. When events or variables are independent, the occurrence or value of one does not depend on or influence the occurrence or value of the other. This concept is crucial in understanding probability, statistical inference, and the analysis of relationships between different factors.
Nominal Data: Nominal data is a type of categorical data where the values represent labels or names rather than numerical quantities. It is the most basic level of measurement, where data is classified into distinct categories with no inherent order or numerical value associated with the categories.
Observed Frequencies: Observed frequencies refer to the actual or empirical counts of data points within each category or group in a statistical analysis. They represent the observed or measured values from a sample or experiment, as opposed to expected or theoretical frequencies.
P-value: The p-value is the probability of obtaining a test statistic at least as extreme as the one actually observed, assuming the null hypothesis is true. It is a crucial concept in hypothesis testing that helps determine the statistical significance of a result.
Post-Hoc Analysis: Post-hoc analysis, also known as an a posteriori analysis, is a statistical technique used to explore the relationships between variables after the initial hypothesis testing has been conducted. It allows researchers to identify specific differences or patterns that were not initially predicted or hypothesized.
Significance Level: The significance level, denoted as α (alpha), 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 rejecting the null hypothesis when it is actually true. The significance level is a crucial concept in hypothesis testing and statistical inference, as it helps determine the strength of evidence required to draw conclusions about a population parameter or the relationship between variables.
Standardized Residuals: Standardized residuals are the residuals (the difference between the observed and predicted values) divided by their standard error. They are used to assess the fit of a statistical model and identify potential outliers or influential observations.
Test for homogeneity: A test for homogeneity evaluates whether different populations have the same distribution of a categorical variable. It is a type of chi-square test used to compare the frequencies of categories across multiple groups.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide