1. Sociology 690 Multivariate Analysis Log Linear Models

2. The Analysis of Categories Category Quantity Category IV DV Linear Models Category Models 2) Structural Equation Models (SEM) 1) Analysis of Variance Models (ANOVA) 4) Logistic Regression Models (LRM) 3) Log Linear Models (LLM)

3. Cross-classification Ironically, while categorical data are among the most prevalent form of information collected in sociology, until recently the most dominant types of statistical analysis have been based on continuous data: e.g. t-tests, ANOVA, correlation, regressionin short the general linear model.

4. Typical Goodness of Fit Model The analysis of effects among categorical variables has been traditionally accomplished through cross-tabulation tables, utilizing a goodness of fit method such as chi square. To the extent the observed frequencies deviate from expected cell frequencies, we would reject the assumption that the variables are independent and accept the alternative that they are related.

5. Example of Chi Square Suppose we have the following cross-classification of observed frequencies for two categorical variables: 756510Male 753540Female NoYesSexTotal Attend College Total 50 100150 Chi Square would be determined by the following formula: Where the expected frequencies are determined by the formula (fc x fr) / ft

6. Chi Square Calculation: 756510Male 753540Female NoYesSexTotal Attend College Total 50 100 150 Here chi square would be calculated as follows: (25-40) 2 /25 + (50-35) 2 /50 + (25-10) 2 /25 + (50-65) 2 /50 = 9+4.5+9+4.5 = 27. With 1 d.f. (r-1 x c-1) SignificanceSignificance And the measure of association would be derived from chi square (e.g. )

7. What chi square does not cover But what if we wanted to examine more than two categorical variables (as in a 2 x 2 x 2 cross- classification table). This kind of multi-way frequency analysis (sometimes called MFA) could be done by calculating chi-squares on all the possible two-way tables. However, that would (among other things), prevent us from calculations of any interactions between the variables.

8. Purpose of Log Linear Analysis Log-linear models are typically used with multi-way dichotomous or categorical variables. They focus on a procedure for accounting for the distribution of cases in a cross-tabulation of categorical variables. Based on the association of categorical data (rather than the causal sequencing of independent and dependent variables), LLA looks at all levels of possible interaction effects. In this sense, Log-linear analysis is a type of multi-way frequency analysis (MFA) and sometimes log-linear analysis is labeled MFA.

9. Definitions in Log linear Analysis Ln(Fij) = + iA + jB + ijAB, where: Ln(Fij) = is the log of the expected cell frequency of the cases for cell ij in the contingency table. = is the overall mean of the natural log of the expected frequencies = terms each represent effects which the variables have on the cell frequencies A and B = the variables i and j = refer to the categories within the variables

10. Procedure for Log Linear Analysis Choosing the model Fitting the model Estimating the Parameters Testing the Goodness of Fit

11. Choosing the Model Saturated vs. Unsaturated If all possible effects are included in the model, is it considered saturated. Unsaturated models are useful when the number of effects equals the number of cell (as would be the case in a 2 x 2 table). Hierarchical vs. Non-Hierarchical The former implies that if we have a higher interaction effect in our model (e.g. AxBXC), we must include a lower interaction effect (e.g. AxB)

12. Estimating Parameters Odds and Odds Ratios: In our original cross-tabulation table, the odds of being female is 75/75 or 1.0. The odds of being in college is 40/10 or 4.0 and the odds of no being in college are 35/65 or.54. An odds ratio is the conditional odds of one category divided by the conditional odds of the of the other category. Hence the odds ratio for women being in college is 4.0/.54 or 7.55. Odds ratios greater than one = a relationship. 756510Male 753540Female NoYesSexTotal Attend College Total 50 100 150

13. SPSS Input

14. SPSS Output