LOGLINEAR MODELS FOR INDEPENDENCE AND

LOGLINEAR MODELS FOR

INDEPENDENCE AND

INTERACTION IN THREE-WAY

TABLES

Table Structure For Three Dimensions

•

When all variables are categorical, a

multidimensional contingency table

displays the data

•

We illustrate ideas using thr

three-variables case.

•

Denote the variables by X, Y, and Z. We

Death Penalty Example

Defendant’

s race

Victim’s

Race Death Penalty_{Yes No} Percentage Yes

White White 19 132 12.6

Black 0 9 0

Marginal table

Defendant’s

Race Death PenaltyYes No Total

White 19 141 160

Black 17 149 166

Partial and Marginal Odd Ratio

Partial Odd ratio describe the association

when the third variable is controlled

The Marginal Odd ratio describe the

association when the Third variable is

Associatio

n Variables_{P-D P-V D-V}

Types of Independence

A three-way IXJXK cross-classification of response variables X, Y, and Z

has several potential types of independence

We assume a multinomial distribution with cell probabilities {i jk},

and  

The models also apply to Poisson sampling with means }.  

Similarly, X could be jointly independent of Y and Z, or Z could be jointly

independent of X and Y. Mutual independence (8.5) implies joint independence

of any one variable from the others.X _{holds for each partial table within which} and Y are conditionally independent, given Z _{Z is fixed. That is, if}when independence

Marginal vs Conditional Independence

•

Partial association can be quite different

from marginal association

•

For further illustration, we now see that

conditional independence of X and Y,

given Z, does not imply marginal

independence of X and Y

•

The joint probability in Table 5.5 show

hypothetical relationship among three

Table 5.5 Joint Probability

Major Gender Income

Low High Liberal Art Female 0.18 0.12

Male 0.12 0.08 Science or

Engineering

The association between Y=income at first

job(high, low) and X=gender(female, male)

at two level of Z=major discipline (liberal

art, science or engineering) is described by

the odd ratios

Income and gender are conditionally

independent, given major

Marginal Probability of Y and X

Gender Income

low high

Female 0.18+0.02=0.20 0.12+0.08=0.20 Male 0.12+0.08=0.20 0.08+0.32=0.40 Total 0.40 0.60

The odd ratio for the (income,

gender) from marginal table

=2

The variables are not independent

when we ignore major

•

Suppose Y is jointly independent of X and

Z, so

Then

And summing both side over i we obtain

=

Therefore

So X and Y are also conditionally independent.

In summary, mutual indepedence of the variables

implies that Y is jointly independent of X and Z,

which itself implies that X and Y are conditionaaly

independent.



Suppose Y is jointly independent of X and Z, that

is .

Summing over k on both side, we obtain

Thus, X and Y also exhibit marginal independence

So, joint independence of Y from X and Z (or X

from Y and Z) implies X and Y are both

marginally and condotionally independent.

Since mutual independence of X, Y and Z implies

that Y is jointly independent of X and Z, mutual

independence also implies that X and Y are

both marginally and conditionally independent

However, when we know only that X and Y are

conditionally independent,

Summing over k on both sides, we obtain

•

All three terms in the summation involve

k, and this does not simplify to marginal

independence

A model that permits all three pairs to be conditionally dependent is

Loglinear Models for Three

Dimensions

•

Hierarchical Loglinear Models

Let {



ijk

} denote expected frequencies.

Suppose all



ijk

>0 and let



ijk

= log



ijk .

A dot in a subscript denotes the average

with respect to that index; for instance,

We set

, ,

The sum of parameters for any index

equals zero. That is

The general loglinear model for a three-way table is

This model has as many parameters as observations and describes all possible positive _{i jk}

Setting certain parameters equal to zero in 8.12. yields the models introduced previously. Table 8.2 lists some of these models. To ease referring

Interpreting Model Parameters

Interpretations of loglinear model parameters use their highest-order terms.

For instance, interpretations for model (8.11). use the two-factor terms to

describe conditional odds ratios

At a fixed level k of Z, the conditional association between X and Y

uses (I- 1)(J – 1). odds ratios, such as the local odds ratios

Similarly, ( I – 1)(K – 1) odds ratios {i (j)k} describe XZ conditional

association, and (J – 1)(K – 1) odds ratios {_(i)jk} describe YZ

Loglinear models have characterizations using constraints on

conditional odds ratios. For instance, conditional independence of

X and Y

is equivalent to {_ij(k)} = 1, i=1, . . . , I-1, j=1, . . . , J-1, k=1, . . . ,

K.

substituting (8.11) for model (XY, XZ, YZ) into log _ij(k) yields

Any model not having the three-factor interaction term has a homogeneous

Alcohol, Cigarette, and Marijuana Use Example

Table 8.3 refers to a 1992 survey by the Wright State University School of

Medicine and the United Health Services in Dayton, Ohio. The survey asked

2276 students in their final year of high school in a nonurban area near

Dayton, Ohio whether they had ever used alcohol, cigarettes, or marijuana.

Table 8.5 illustrates model association patterns by presenting estimated

conditional and marginal odds ratios

For example, the entry 1.0 for the AC conditional association for the model (AM, CM) of AC conditional independence is the

The entry 2.7 for the AC marginal association for this model is the odds ratio

for the marginal AC fitted table

Table 8.5 shows that estimated conditional odds ratios equal 1.0 for each

pairwise term not appearing in a model, such as the AC

association in model ( AM, CM).

For that model, the estimated marginal AC odds ratio differs from 1.0, since conditional independence does not imply marginal

independence.

Model (AC, AM, CM) permits all pairwise associations but maintains

homogeneous odds ratios between two variables at each level of the third.

The AC fitted conditional odds ratios for this model

INFERENCE FOR LOGLINEAR MODELS

Chi-Squared Goodness-of-Fit Tests

As usual, X 2 and G2 test whether a model holds by comparing cell

fitted

values to observed counts

Where n_ijk = observed frequency and =expected frequency Here df equals the number of cell counts minus the number of model parameters.

 

For the student survey (Table 8.3), Table 8.6 shows results of testing fit for

several loglinear models.  

Models that lack any association term fit poorly

The model ( AC, AM, CM) that has all pairwise associations fits well (P=

0.54)

It is suggested by other criteria also, such as minimizing

AIC= - 2(maximized log likelihood - number of parameters in model)