1. Modeling Ordinal Associations Bin Hu
Linear-by-Linear Association in two-way table
Modeling Fitting
The “Sex Opinion” Example
Directed Ordinal Test of Independence

2. Linear-by-Linear Association
For two-way case, two ordinal variables assign the order score for
row score and the column scores are
and 
The model is
with constrains

3. L-by-L model: A model for ordinal variables uses association terms that permit trends. In this model, there are 1+(I-1)+(J-1)+1=I+J parameters and only one parameter describing the association. The residual DF: IJ-I-J;
beta>0, positive trend; beta<0, negative trend; otherwise, independent association.
Obviously, when the data display a positive or negative trend, the L*L model fits better than the independence model.

4. The result of MLE for L-by-L model is:
Since the marginal distributions and hence marginal means and variances are identical for fitted and observed distributions, the third equation implied the correlation between the scores for X and Y is the same for both distributions.
Since ui and vj are fixed, the L-by-L model has only one more parameter than independence model.

5. Parameter Estimates ‘Sex Opinion’ in LL model
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept
premar <.0001
premar
premar
premar
birth <.0001
birth <.0001
birth <.0001
birth
linlin <.0001
Exp( *1*1)=80.9
Exp( *2*2)=23.1

6. Birth Control Premarital Sex SDI DI AG SAI
Always Wrong (42.4) (51.2) (86.4) (67)
(7.6) (80.9) (3.1) (67.6) (-4.1) (69.4) (-4.8) (29.1)
Almost Wrong (16) (19.3) (32.5) (25.2)
(2.3) (20.8) (1.8) (23.1) (-.8) (31.5) (-2.8) (17.6)
Wrong Sometimes (30.1) (36.3) (61.2) (47.4)
(-2.7) (24.4) (1) (36.1) (2.2) (65.7) (-1) (48.8)
Not Wrong (70.6) (85.2) (143.8) (111.4)
(-6.1) (33) (-4.6) (65.1) (2.4) (157.4) (6.8) (155.5)
a(b)(c)(d): a: observed count; b: independence model fit; the Pearson residuals for independence model fit; Linear-by-linear association model fit.

7. Criteria For Assessing Goodness Of Fit For LL model
Criterion DF Value Value/DF
Deviance
Scaled Deviance
Pearson Chi-Square
Scaled Pearson X
Log Likelihood
For Indept. Model
Deviance
Scaled Deviance
Pearson Chi-Square
Scaled Pearson X
Log Likelihood

8. Independence Test The likelihood-ratio test statistic is
:   Independence, ie. Beta=0
The likelihood-ratio test statistic is
= = 116.1
With df=1, P<0.0001, there is strong evidence of association.
As for df, df of =1 is smaller than df of =(I-1)(J-1), the former is more powerful in testing the independence.