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Academic year: 2017



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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


s race


Race Death PenaltyYes No Percentage Yes

White White 19 132 12.6

Black 0 9 0


Marginal table


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



n VariablesP-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},


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, ifwhen 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



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


The variables are not independent

when we ignore major


Suppose Y is jointly independent of X and

Z, so


And summing both side over i we obtain




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


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



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


Loglinear Models for Three


Hierarchical Loglinear Models

Let {


} denote expected frequencies.

Suppose all


>0 and let


= 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, . . . ,


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


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



Chi-Squared Goodness-of-Fit Tests

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


values to observed counts

Where nijk = 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=


It is suggested by other criteria also, such as minimizing

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


Table 5.5 Joint Probability


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