log-linear models & dependent samples feng ye, xiao guo, jing wang
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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.