Transcript Loglinear Contingency Table Analysis Karl L. Wuensch Dept of Psychology

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)
/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
In Each Cell, O=E, Residual = 0
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,
  6 787(221)(301)(67)( 47)(82)  154.3429
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.
Lambda Parameter Estimates
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.
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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 ChiSquare increased = the value of ChiSquare 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 nodate formdlim='-';
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
proc catmod;
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.