15.2 Chi-square tests

Chi-square tests are statistical tools used to analyze categorical data. They help us determine if there's a significant relationship between variables or if observed data fits an expected distribution. This is crucial for making informed decisions based on data.

In this section, we'll cover different types of chi-square tests, including goodness-of-fit and independence tests. We'll also learn how to interpret results, understand key components like degrees of freedom, and construct contingency tables for analysis.

Chi-square Tests and Hypotheses

Understanding Chi-square Statistics and Hypotheses

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• Chi-square statistic measures the difference between observed and expected frequencies in categorical data
• Null hypothesis assumes no significant difference between observed and expected frequencies
• Alternative hypothesis suggests a significant difference exists between observed and expected frequencies
• p-value indicates the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis is true
• Effect size quantifies the magnitude of the relationship or difference between variables (Cramer's V, Phi coefficient)
• Assumptions for chi-square tests include:
  • Independent observations
  • Mutually exclusive categories
  • Large enough sample size (expected frequencies > 5 in each cell)

Interpreting Chi-square Results

• Compare calculated chi-square statistic to critical value from chi-square distribution table
• Reject null hypothesis if calculated chi-square statistic exceeds critical value
• Use p-value to determine statistical significance (typically reject null hypothesis if p < 0.05)
• Consider effect size to assess practical significance of results
• Evaluate assumptions to ensure validity of test results

Types of Chi-square Tests

Goodness-of-fit and Independence Tests

• Goodness-of-fit test compares observed frequencies to expected frequencies based on a hypothesized distribution
  • Used to determine if sample data fits a specific probability distribution (uniform, normal)
  • Calculates chi-square statistic by comparing observed counts to expected counts in each category
• Test of independence examines relationship between two categorical variables in a contingency table
  • Determines if there is a significant association between row and column variables
  • Calculates expected frequencies assuming no relationship between variables
  • Compares observed frequencies to expected frequencies to compute chi-square statistic

Homogeneity Test and R Implementation

• Test of homogeneity assesses whether different populations have the same distribution of a categorical variable
  • Similar to test of independence but focuses on comparing multiple populations
  • Used when samples are drawn from different populations (comparing survey responses across different age groups)

• [chisq.test](https://www.fiveableKeyTerm:chisq.test)()

  function in R performs chi-square tests
  • Syntax:

    chisq.test(x, y = NULL, correct = TRUE, p = rep(1/length(x), length(x)), rescale.p = FALSE, simulate.p.value = FALSE, B = 2000)

  • x

    can be a vector, matrix, or data frame

  • y

    is optional for specifying a second variable in test of independence
  • Returns test statistic, degrees of freedom, and p-value

Key Components of Chi-square Tests

Understanding Degrees of Freedom and Frequencies

• Degrees of freedom represent the number of values that can vary freely in calculating the chi-square statistic
  • For goodness-of-fit test: df = (number of categories - 1)
  • For test of independence: df = (number of rows - 1) * (number of columns - 1)
• Expected frequencies are theoretical values calculated assuming the null hypothesis is true
  • For goodness-of-fit test: expected frequency = (total sample size * hypothesized proportion)
  • For test of independence: expected frequency = (row total * column total) / grand total
• Observed frequencies are actual counts obtained from the data collection process
  • Represent the real-world distribution of categorical data in the sample

Constructing and Analyzing Contingency Tables

• Contingency table organizes categorical data into rows and columns
  • Rows represent categories of one variable
  • Columns represent categories of another variable
  • Cell values show frequency or count of observations in each combination of categories
• Steps to create a contingency table:
  1. Identify two categorical variables of interest
  2. Determine categories for each variable
  3. Count observations in each combination of categories
  4. Arrange counts in a table format
• Analyze contingency tables by:
  • Calculating row and column totals
  • Computing expected frequencies for each cell
  • Identifying patterns or trends in the data
  • Applying chi-square test of independence to assess relationship between variables