Log-Linear Models by Ronald Christensen (auth.)

By Ronald Christensen (auth.)

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Log-Linear ModelsforTwo-Dimensional Tables where mij = u + u 1 (i) + u2(j) + ul2(ij) . Our goal is to examine the structure of the MLE's of the mij's which are Our statistical inferences are based dent with mij rv Oll mij 's. (2) To do that we use the the fact that the N(mij, a 2 47 mij 's are indepen- I K) and that the MSE is an estimate of a 2 , which is independent of the mij 's. It is of interest to note that although the MSE is not the MLE of a 2 , exactly the same tests and confidence intervals for the mij 's would be obtained if the MLE for a 2 was used in place of the MSE (and suitable adjustments in distributions were made).

N22 n2. n .. 2 PROBABILITIES Columns 1 2 Rows 1 Pu P12 2 P21 P22 Totals P·l P·2 Totals Pl· P2· p .. EXPECTED VALUES Columns 2 1 Rows Totals 1 mu m12 m1. 2 m21 m22 m2. 2 m .. Our interest is in finding estimates of the Pij 's, developing models for the Pij 's, and performing tests on the Pij 's. Equivalently, we can concern ourselves with estimates, models, and tests for the mij 's. , our interest is in whether sex is related to support for legalized abortion. Note that Pu +P12 = 1 and P21 +P22 = 1.

In other words, we wish to test the model Ho : Plj = P2j = ... = Pij for all j = 1, ... , J. (1) against the alternative HA : model (1) is not true. This is described as testing for homogeneity of proportions. We continue to use Pearson's chi-square test statistic to evaluate the appropriateness of the null hypothesis model. Pearson's chi-square requires estimates of the expected values mij. Each sample has a multinomial distribution so mij = ni·Pij. If Ho is true, Pij is the same for all values of i.

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