Modeling Ordinal Associations Bin Hu

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Presentation transcript:

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

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

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.

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.

Parameter Estimates ‘Sex Opinion’ in LL model Standard Wald 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 0.4735 0.4339 -0.3769 1.3239 1.19 0.2751 premar 1 1 1.7537 0.2343 1.2944 2.2129 56.01 <.0001 premar 2 1 0.1077 0.1988 -0.2820 0.4974 0.29 0.5880 premar 3 1 -0.0163 0.1264 -0.2641 0.2314 0.02 0.8972 premar 4 0 0.0000 0.0000 0.0000 0.0000 . . birth 1 1 1.8797 0.2491 1.3914 2.3679 56.94 <.0001 birth 2 1 1.4156 0.1996 1.0243 1.8068 50.29 <.0001 birth 3 1 1.1551 0.1291 0.9021 1.4082 80.07 <.0001 birth 4 0 0.0000 0.0000 0.0000 0.0000 . . linlin 1 0.2858 0.0282 0.2305 0.3412 102.46 <.0001 Exp(0.4735+1.7537+1.8797+0.2858*1*1)=80.9 Exp(0.4735+0.1077+1.4156+0.2858*2*2)=23.1

Birth Control Premarital Sex SDI DI AG SAI Always Wrong 81 (42.4) 68 (51.2) 60 (86.4) 38 (67) (7.6) (80.9) (3.1) (67.6) (-4.1) (69.4) (-4.8) (29.1) Almost Wrong 24 (16) 26 (19.3) 29 (32.5) 14 (25.2) (2.3) (20.8) (1.8) (23.1) (-.8) (31.5) (-2.8) (17.6) Wrong Sometimes 18 (30.1) 41(36.3) 74 (61.2) 42 (47.4) (-2.7) (24.4) (1) (36.1) (2.2) (65.7) (-1) (48.8) Not Wrong 36(70.6) 57(85.2) 161(143.8) 157(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.

Criteria For Assessing Goodness Of Fit For LL model Criterion DF Value Value/DF Deviance 8 11.5337 1.4417 Scaled Deviance 8 11.5337 1.4417 Pearson Chi-Square 8 11.5085 1.4386 Scaled Pearson X2 8 11.5085 1.4386 Log Likelihood 3041.7446 For Indept. Model Deviance 9 127.6529 14.1837 Scaled Deviance 9 127.6529 14.1837 Pearson Chi-Square 9 128.6836 14.2982 Scaled Pearson X2 9 128.6836 14.2982 Log Likelihood 2983.6850

Independence Test The likelihood-ratio test statistic is : Independence, ie. Beta=0 The likelihood-ratio test statistic is = 127.6-11.5 = 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.