Loglinear Contingency Table Analysis

1
Loglinear Contingency Table Analysis

• Karl L. Wuensch
• Dept of Psychology
• East Carolina University

2
The Data
3
Weight Cases by Freq
4
Crosstabs
5
Cell Statistics
6
LR Chi-Square
7
Model Selection Loglinear

• HILOGLINEAR happy(1 2) marital(1 3)
• /CRITERIA ITERATION(20) DELTA(0)
• /PRINTFREQ ASSOCIATION ESTIM
• /DESIGN.
• No cells with count 0, so no need to add .5 to
  each cell.
• Saturated model happy, marital, Happy x Marital

8
In Each Cell, OE, Residual 0
9
The Model Fits the Data Perfectly, Chi-Square 0

• The smaller the Chi-Square, the better the fit
  between model and data.

10
Both One- and Two-Way Effects Are Significant

• The LR Chi-Square for Happy x Marital has the
  same value we got with Crosstabs

11
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

12
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.

13
Lambda Parameter Estimates
14
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.

15
Main Effect of Happy

• For Happy 1 (yes), ? .885
• Accordingly, for Happy 2 (no), ? is -.885.

16
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
  .

17
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.

18
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.

19
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,

20
Drop Happy x Marital From the Model

• HILOGLINEAR happy(1 2) marital(1 3)
• /CRITERIA ITERATION(20) DELTA(0)
• /PRINTFREQ RESID ASSOCIATION ESTIM
• /DESIGN happy marital.
• Notice that the design statement does not include
  the interaction term.

21
Uh-Oh, Big Residuals

• A main effects only model does a poor job of
  predicting the cell counts.

22
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.

23
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.

24
SPSS Loglinear

• LOGLINEAR Happy(1,2) Marital(1,3) /
• CRITERIADelta(0) /
• PRINTDEFAULT ESTIM /
• DESIGNHappy Marital Happy by Marital.
• Replicates the analysis we just did using
  Hiloglinear.
• More later on the differences between Loglinear
  and Hiloglinear.

25
SAS Catmod

• options pagenomin 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 HappyMarital _response_
• Loglin HappyMarital
• run

26
PASW GENLOG

• GENLOG happy marital
• /MODELPOISSON
• /PRINTFREQ DESIGN ESTIM CORR COV
• /PLOTNONE
• /CRITERIACIN(95) ITERATE(20) CONVERGE(0.001)
  DELTA(0)
• /DESIGN.

27
GENLOG Coding

• Uses dummy coding, not effects coding.
• Dummy One level versus reference level
• Effects One level versus versus grand mean
• I dont like it.

28
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 dont know how the entries in the ML ANOVA
  table were computed.