Sociology 690
Multivariate Analysis
Log Linear Models
The Analysis of Categories
IV
Category
Quantity
DV
Quantity
1) Analysis
of Variance
Models
(ANOVA)
2) Structural
Equation
Models
(SEM)
Linear Models
Category
3) Log
Linear
Models
(LLM)
4) Logistic
Regression
Models
(LRM)
Category Models
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:
Chi Square would be
determined by the
following formula:
2
(
f

f
)
2   e o
fe
Attend College
Sex
Yes
No
Total
Female
Male
40
10
50
35
65
100
75
75
150
Total
Where the expected frequencies are
determined by the formula (fc x fr) / ft
Chi Square Calculation:
Sex
Female
Male
Total
Attend College
Yes
No
40
35
10
65
50
100
Total
75
75
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.    2 / N  27 / 150  .42 )
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 crossclassification 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
Sex
Female
Male
Total
Attend College
Yes
No
40
35
10
65
50
100
Total
75
75
150
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.
SPSS Input
SPSS Output