Sociology 690

Multivariate Analysis

Log Linear Models

The Analysis of Categories

Category

Quantity

QuantityCategory

IV

DV

Linear Models

Category Models

2) Structural Equation Models(SEM)

1) Analysis of Variance

Models(ANOVA)

4) Logistic Regression

Models(LRM)

3) LogLinear Models(LLM)

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, regression—in short the general linear model.

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.

Example of Chi Square

Suppose we have the following cross-classification of observed frequencies for two categorical variables:

756510Male

753540Female

NoYesSex Total

Attend College

Total 50 100 150

e

oe

f

ff 22 )(

Chi Square would be determined by the following formula:

Where the expected frequencies are determined by the formula (fc x fr) / ft

Chi Square Calculation:

N/2

756510Male753540Female

NoYesSex Total

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) Significance

And the measure of association would be derived from chi square (e.g. ) 42.150/27

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.

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.

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

Procedure for Log Linear Analysis

Choosing the model

Fitting the model

Estimating the Parameters

Testing the Goodness of Fit

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)

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.

756510Male753540Female

NoYesSex Total

Attend College

Total 50 100 150

SPSS Input

SPSS Output