LOGLINEAR MODELS FOR INDEPENDENCE ANDINTERACTION IN THREE-WAY TABLESBY ENI SUMARMININGSIH, SSI, MM

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 display the distribution of X-Y cell count at different level of Z using cross sections of the three way contingency table (called partial tables)

•The two way contingency table obtained by combining the partial table is called the X-Y marginal table (this table ignores Z)

Death Penalty ExampleDefendant’srace

Victim’s Race

Death Penalty Percentage YesYes No

White White 19 132 12.6

Black 0 9 0

Black White 11 52 17.5

Black 6 97 5.8

Marginal tableDefendant’sRace

Death Penalty Total

Yes No

White 19 141 160

Black 17 149 166

36 290 326

Partial and Marginal Odd Ratio

Partial Odd ratio describe the association when the third variable is controlledThe Marginal Odd ratio describe the association when the Third variable is ignored (i.e when we sum the counts over the levels of the third variable to obtain a marginal two-way table)

Association

Variables

P-D P-V D-V

Marginal 1.18 2.71 25.99

Partial Level 1 0.67 2.80 22.04

Level 2 0.79 3.29 25.90

Types of IndependenceA three-way IXJXK cross-classification of response variables X, Y, and Zhas several potential types of independenceWe assume a multinomial distribution with cell probabilities {i jk}, and

The models also apply to Poisson sampling with means }.

The three variables are mutually independent when

Similarly, X could be jointly independent of Y and Z, or Z could be jointlyindependent of X and Y. Mutual independence (8.5) implies joint independenceof any one variable from the others.X and Y are conditionally independent, given Z when independence holds for each partial table within which Z is fixed. That is, if

then

Homogeneous Association and Three-Factor Interaction

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 variables for new graduate of a university

Table 5.5 Joint ProbabilityMajor Gender Income

Low High

Liberal Art Female 0.18 0.12

Male 0.12 0.08

Science or Engineering

Female 0.02 0.08

Male 0.08 0.32

Total Female 0.20 0.20

Male 0.20 0.40

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 XGender 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 =2The 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 independentHowever, 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

Model 8.11. is called the loglinear model of homogeneous association or of no three-factor interaction.

Loglinear Models for Three Dimensions•Hierarchical Loglinear ModelsLet {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 modelsintroduced previously. Table 8.2 lists some of these models. To ease referringto models, Table 8.2 assigns to each model a symbol that lists the highest-order term(s) for each variable

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 todescribe 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 conditionalassociation.

Loglinear models have characterizations using constraints on conditional odds ratios. For instance, conditional independence of X and Yis equivalent to {i j(k)} = 1, i=1, . . . , I-1, j=1, . . . , J-1, k=1, . . . , K.substituting (8.11) for model (XY, XZ, YZ) into log i j(k) yields

Any model not having the three-factor interaction term has a homogeneousassociation for each pair of variables.

For 2x2x2 tables

Alcohol, Cigarette, and Marijuana Use Example

Table 8.3 refers to a 1992 survey by the Wright State University School ofMedicine and the United Health Services in Dayton, Ohio. The survey asked2276 students in their final year of high school in a nonurban area nearDayton, Ohio whether they had ever used alcohol, cigarettes, or marijuana.Denote the variables in this 222 table by A for alcohol use, C forcigarette use, and M for marijuana use.

Table 8.5 illustrates model association patterns by presenting estimatedconditional 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 common value of the AC fitted odds ratios at the two levels of M,

The entry 2.7 for the AC marginal association for this model is the odds ratiofor the marginal AC fitted tableTable 8.5 shows that estimated conditional odds ratios equal 1.0 for eachpairwise 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 maintainshomogeneous odds ratios between two variables at each level of the third.The AC fitted conditional odds ratios for this model equal 7.8.One can calculate this odds ratio using the model’s fitted values at either level of M, or from (8.14) using

INFERENCE FOR LOGLINEAR MODELS

Chi-Squared Goodness-of-Fit Tests

As usual, X 2 and G2 test whether a model holds by comparing cell fittedvalues 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 forseveral loglinear models.

𝑋 2=∑𝑖∑

𝑗∑𝑘

(𝑛𝑖𝑗𝑘−�̂�𝑖𝑗𝑘 )2

�̂�𝑖𝑗𝑘

=2

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)or equivalently, minimizing [G2- 2(df)].