Log-Linear Models & Dependent Samples

Feng Ye, Xiao Guo, Jing Wang

Outline

Symmetry, Quasi-independence & Quasi-symmetry.

Marginal Homogeneity & Quasi-symmetry. Ordinal Quasi-symmetry Model. Conclusion.

Quasi-independence Model

Model structure

Assumption:

Independent model holds for off diagonal cells.

Model fit

df = (I-1)^2-I

Symmetry

Model structure

Assumption:

The off diagonal cells have equal expected counts.

Model fit:

df = I^2-[I+I(I-1)/2]

Symmetry

  1 2 3 Total

1 20 5 8 17

2 5 20 10 20

3 8 10 20 23

Total 17 20 23  

Quasi-symmetry

Model structure

Model fit df = (I-2)(I-1)/2

Symmetry:

Independence:

Application

Each week Variety magazine summarizes reviews of new movies by critics in several cities. Each review is categorized as pro, con, or mixed, according to whether the overall evaluation is positive, negative, or a mixture of the two. April 1995 through September 1996 for Chicago film critics Gene Siskel and Roger Ebert.

Application

Reviews of new movies by critics.

    Ebert  

Siskel Con Mixed Pro

Con 24 8 13

Mixed 8 13 11

Pro 10 9 64

Output & Interpretation

Model df G2 p-value

Quasi-Independence 1 0.0061 0.938

Symmetry 3 0.5928 0.9

Quasi-symmetry 1 0.0061 0.938

Quasi-independence

Con-0.9603 Mixed-0.6239 pro-1.5069

e.g

exp(0.9603+0.6239)=4.41

Symmetry = Quasi-symmetry + Marginal homogeneity

Quasi-symmetry + Marginal homogeneity = Symmetry

Fit statistics for marginal homogeneity

Marginal Homogeneity & Quasi-symmetry

Symmetry Model Criteria For Assessing Goodness Of Fit

Criterion DF Value Value/DF

Deviance 3 0.5928 0.1976 Scaled Deviance 3 0.5928 0.1976 Pearson Chi-Square 3 0.5913 0.1971 Scaled Pearson X2 3 0.5913 0.1971 Log Likelihood 351.2829

Quasi-Symmetry Model Criteria For Assessing Goodness Of Fit

Criterion DF Value Value/DF

Deviance 1 0.0061 0.0061 Scaled Deviance 1 0.0061 0.0061 Pearson Chi-Square 1 0.0061 0.0061 Scaled Pearson X2 1 0.0061 0.0061 Log Likelihood 351.5763

G2(S/QS) = 0.5928 - 0.0061 = 0.5867 with df = 2, showing marginal homogeneity is plausible.

Marginal Homogeneity Testing

Marginal homogeneity is the special case βj=0.Specifying design matrix to produce expected frequency {µab}.Using G2 and X2 tests marginal homogeneity, with df=I-1

da = p+a – pa+ ; d’ =( d1,….dI-1) Covariance matrix V with elements: Vab = -(pab + pba) – (p+a – pa+)(p+b – pb+)

Vaa = p+a + pa+ -2paa – (p+a – pa+ )2

Under marginal homogeneity, E(d) = 0. W is asymptotically chi-squared with df = I-1.

Marginal Models

Criteria For Assessing Goodness Of Fit

Criterion DF Value Value/DF

Deviance 2 0.5868 0.2934 Scaled Deviance 2 0.5868 0.2934 Pearson Chi-Square 2 0.5855 0.2927 Scaled Pearson X2 2 0.5855 0.2927 Log Likelihood 351.2859

Analysis of Variance

Source DF Chi-Square Pr > ChiSq -------------------------------------------- Intercept 2 191.15 <.0001 review 2 0.59 0.7455

Residual 0 . .

Ordinal Quasi-symmetry Model

Quasi-independence, quasi-symmetry, symmetry models for square tables treat classifications as nominal.

Changing the constraints for log linear model to obtain reduced model for ordered response.

Model structure has a linear trend. Where is

the ordered scores

log ij a b b abu

ab ba Y Xb b bu

Ordinal Quasi-symmetry Model

Parameter estimation & interpretation

1. Fitted marginal counts ==(?) observed marginal counts

2. Dividing the first two equations by n indicates the same means.

Goodness of fit: Checking the distance between reduced model and saturated model.

, ,a a a a b b b ba a b b

ab ba ab ba

n n

n n

Logit Representation

Logit model

Interpretation1. difference between marginal

distribution

2. marginal homogeneity

3. Identify as binomial with trials, and

fit a logit model with no intercept and predictor

log( / ) ( )ab ba b au u

| | | |ab ba

0 ab ba ( , )ab ban n ab ban n

b ax u u

Marginal Homogeneity

Marginal model (cumulative logits)

marginal homogeneity :

Ordinal quasi-sym model1. At the condition of ordinal quasi-symmetry

marginal homogeneity is equivalent to symmetry

2. Fit statistic

2 2 2( | ) ( ) ( )G S QS G S G QS

log [ ( )]t j tit P Y j x

1 2( ) ( ) 0P Y a P Y a

Application Data 1

Reviews of new movies by critics.

    Ebert  

Siskel Con Mixed Pro

Con 24 8 13

Mixed 8 13 11

Pro 10 9 64

Output & Interpretation Output

Marginal homogeneity?1. No meaning to check if when ordinal quasi-symmetry fits poorly.2. Using marginal model is a good way.3. Check the symmetry under the condition of ordinal quasi-symmetry.

0

Model df G^2 p-valueOrdinal quasi_sym 2 0. 0917 0. 9555 Symmetry 3 0. 5928 0. 9Marginal homogeneity 1 0. 5011 0. 479

Conclusion

Summary statistics provide an overall picture of square tables.

Kappa & Percentage

Log-linear provides a valuable addition even an alternative to summary statistic.

1. Quasi-symmetry is the most general model for square table.

2. Adding or deleting variables from log-linear models provides different useful models.

3. Quasi-symmetry models proposes a good instrument for marginal homogeneity.