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Loglinear Contingency Table Analysis

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Loglinear Contingency Table Analysis. Karl L. Wuensch Dept of Psychology East Carolina University. The Data. Weight Cases by Freq. Crosstabs. Cell Statistics. LR Chi-Square. Model Selection Loglinear. HILOGLINEAR happy(1 2) marital(1 3) /CRITERIA ITERATION(20) DELTA(0)

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### Loglinear Contingency Table Analysis

Karl L. Wuensch

Dept of Psychology

East Carolina University

Model Selection Loglinear

HILOGLINEAR happy(1 2) marital(1 3)

/CRITERIA ITERATION(20) DELTA(0)

/PRINT=FREQ ASSOCIATION ESTIM

/DESIGN.

• No cells with count = 0, so no need to add .5 to each cell.
• Saturated model = happy, marital, Happy x Marital
The Model Fits the Data Perfectly, Chi-Square = 0
• The smaller the Chi-Square, the better the fit between model and data.
Both One- and Two-Way Effects Are Significant
• The LR Chi-Square for Happy x Marital has the same value we got with Crosstabs
The Model: Parameter Mu
• LN(cell freq)ij =  + i + j + ij
• We are predicting natural logs of the cell counts.
•  is the natural log of the geometric mean of the expected cell frequencies.
• For our data,

and LN(154.3429) = 5.0392

The Model: Lambda Parameters
• LN(cell freq)ij =  + i + j + ij
• i is the parameter associated with being at level i of the row variable.
• There will be (r-1) such parameters for r rows,
• And (c-1) lambda parameters, j, for c columns,
• And (r-1)(c-1) lambda parameters, for the interaction, ij.
Main Effect of Marital Status
• For Marital = 1 (married),  = +.397
• for Marital = 2 (single),  = ‑.415
• For each effect, the lambda coefficients must sum to zero, so
• For Marital = 3 (split), = 0 ‑ (.397 ‑ .415) = .018.
Main Effect of Happy
• For Happy = 1 (yes),  = +.885
• Accordingly, for Happy =2 (no),  is ‑.885.
Happy x Marital
• For cell 1,1 (Happy, Married),  = +.346
• So for [Unhappy, Married],  = -.346
• For cell 1,2 (Happy, Single),  = -.111
• So for [Unhappy, Single],  = +.111
• For cell 1,3 (Happy, Split),  = 0 ‑ (.346 ‑ .111) = ‑.235
• And for [Unhappy, Split], = 0 ‑ (‑.235) = +.235.
Interpreting the Interaction Parameters
• For (Happy, Married),  = +.346 There are more scores in that cell than would be expected from the marginal counts.
• For (Happy, Split),  = 0 ‑.235

There are fewer scores in that cell than would be expected from the marginal counts.

Predicting Cell Counts
• Married, Happy e(5.0392 + .397 +.885 +.346) = 786 (within rounding error of the actual frequency, 787)
• Split, Unhappy

e(5.0392 + .018 -.885 +.235) =82, the actual frequency.

Testing the Parameters
• The null is that lambda is zero.
• Divide by standard error to get a z score.
• Every one of our effects has at least one significant parameter.
• We really should not drop any of the effects from the model, but, for pedagogical purposes, ………
Drop Happy x Marital From the Model

HILOGLINEAR happy(1 2) marital(1 3)

/CRITERIA ITERATION(20) DELTA(0)

/PRINT=FREQ RESID ASSOCIATION ESTIM

/DESIGN happy marital.

• Notice that the design statement does not include the interaction term.
Uh-Oh, Big Residuals
• A main effects only model does a poor job of predicting the cell counts.
Big Chi-Square = Poor Fit
• Notice that the amount by which the Chi-Square increased = the value of Chi-Square we got earlier for the interaction term.
Pairwise Comparisons
• Break down the 3 x 2 table into three 2 x 2 tables.
• Married folks report being happy significantly more often than do single persons or divorced persons.
• The difference between single and divorced persons falls short of statistical significance.
SPSS Loglinear

LOGLINEAR Happy(1,2) Marital(1,3) /

CRITERIA=Delta(0) /

PRINT=DEFAULT ESTIM /

DESIGN=Happy Marital Happy by Marital.

• Replicates the analysis we just did using Hiloglinear.
• More later on the differences between Loglinear and Hiloglinear.
SAS Catmod

options pageno=min nodateformdlim='-';

data happy;

input Happy Marital count;

cards;

1 1 787

1 2 221

1 3 301

2 1 67

2 2 47

2 3 82

proccatmod;

weight count;

model Happy*Marital = _response_;

Loglin Happy|Marital;

run;

PASW GENLOG

GENLOG happy marital

/MODEL=POISSON

/PRINT=FREQ DESIGN ESTIM CORR COV

/PLOT=NONE

/CRITERIA=CIN(95) ITERATE(20) CONVERGE(0.001) DELTA(0)

/DESIGN.

GENLOG Coding
• Uses dummy coding, not effects coding.
• Dummy = One level versus reference level
• Effects = One level versus versus grand mean
• I don’t like it.
Catmod Output
• Parameter estimates same as those with Hilog and loglinear.
• For the tests of these paramaters, SAS’ Chi-Square = the square of the z from PASW.
• I don’t know how the entries in the ML ANOVA table were computed.