Week_11_Lecture_Recording
Overview of Chi-Square and Log-Linear Analysis
Focus on tests of association using frequencies
Key Applications:
Analyze relationships between categorical variables
Utilize SPSS for computations and output
Explore effect sizes and odds ratios as interpretations of statistical significance
Categorical Data Analysis
Categorical data can be assessed using:
Chi-square tests
Log-linear models
Examples of categorical responses can include:
Voting frequency for politicians
Students passing or failing subjects
Patients diagnosed or free from diagnosis after treatment
Caution in analysis: Numeric values of categories are arbitrary; means are meaningless for categorical variables.
Chi-Square Test Basics
Pearson Chi-Square:
Used to assess the relationship between two categorical variables
Compares observed frequencies to expected frequencies to determine if deviations are due to chance
Example Scenario:
Analyzed training methods (food vs affection) on whether cats can learn to dance.
Utilizing a contingency table to visualize: training methods vs dance outcome (yes/no).
Equation for Chi-Square:
[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} ] where:
(O_{ij}) = observed data frequencies
(E_{ij}) = expected frequencies based on chance
Rows and columns represent categories in the contingency table
Key Terms:
Observed Frequencies: Real counts from the experiment
Expected Frequencies: Counts expected if there were no association
Use of standardized residuals to interpret results:
Standardized residuals are akin to z-scores, indicating significance at p < 0.05 if outside the range of +/- 1.96.
Log-Linear Models
Applied when assessing association among three or more categorical variables.
An example using dogs and cats, allowing for different predictor variables while analyzing training effects.
Key Concepts:
Models assess main effects and interaction effects (e.g., animal type, training method, and dance outcome).
Uses backward elimination to simplify models, assessing significance of interactions and removing non-significant terms.
Importance of assessing the k-way interactions and their individual main effects:
Statistical significance checks if removing interactions shortens the model significance.
Assumptions for Chi-Square Tests
Independence: Each data point contributes uniquely to contingency cells.
Expected Counts: No cell should have an expected frequency less than 5; overall, fewer than 20% of cells should meet this condition in larger tables.
Small samples may require Fishers Exact Test instead.
Conducting Chi-Square and Log-Linear Analysis in SPSS
SPSS Procedure for Chi-Square:
Enter categorical variables for training methods and outcomes under weight cases.
Utilize crosstab function to specify layout and options.
Output includes significant Pearson Chi-Square results and contingency tables with expected counts.
Reporting Findings:
Include chi-square statistic, degrees of freedom, p-value, and odds ratio in interpretation.
Odds Ratio Calculation
Odds of Different Outcomes:
Calculate probability ratios for cats based on reward types (food vs affection).
Compare odds for both methods and provide numerical results in analysis.
Interpretation may look like this:
"The odds of a cat dancing in response to food is 6.65 times higher than in response to affection."
Wrap-Up of Analysis Approach
For categorical data:
The main analytic tools are the Chi-square test (for two variables) and log-linear analysis (for three or more variables).
The process involves fitting models, evaluating deviations, assessing interactions, and using odds ratios as effect sizes to interpret results.
Students will practice these methods in upcoming labs, ensuring they can report results meaningfully with a focus on significance and effect size.