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,

and LN(154.3429) = 5.0392

3429.154)82)(47)(67)(301)(221(7876

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.

Interpreting the Interaction Parameters

• For (Happy, Married), = +.346There 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, Happye(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 Catmodoptions pageno=min nodate formdlim='-';data happy;input Happy Marital count;cards;1 1 7871 2 2211 3 3012 1 672 2 472 3 82proc 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.