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Loglinear Contingency Table AnalysisPowerPoint Presentation

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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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.

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