Hi all,

I've been running loglinear models for three-way tables: one of the
variables having three levels, and the other two having two levels each.

An example looks like below:

> yes.no <- c("Yes","No")

> switch <- c("On","Off")

> att <- c("BB","AA","CC")

> L <- gl(2,1,12,yes.no)

> T <- gl(2,2,12,switch)

> A <- gl(3,4,12,att)

> n <- c(1136,4998,25,339,305,2752,31,692,251,1677,17,557)

> d.table <- data.frame(A,T,L,n)

> d.table

    A   T   L    n

1  BB  On Yes 1136

2  BB  On  No 4998

3  BB Off Yes   25

4  BB Off  No  339

5  AA  On Yes  305

6  AA  On  No 2752

7  AA Off Yes   31

8  AA Off  No  692

9  CC  On Yes  251

10 CC  On  No 1677

11 CC Off Yes   17

12 CC Off  No  557



First, I run the independence model and found a poor fit:

> library(MASS)
> loglm(n~A+T+L)
Call:
loglm(formula = n ~ A + T + L)

Statistics:
                      X^2 df P(> X^2)
Likelihood Ratio 1001.431  7        0
Pearson          1006.287  7        0



Thus, I went on and run the two-way association model and found a good fit:

> loglm(n~A:T+A:L+T:L)
Call:
loglm(formula = n ~ A:T + A:L + T:L)

Statistics:
                      X^2 df   P(> X^2)
Likelihood Ratio 4.827261  2 0.08948981
Pearson          4.680124  2 0.09632168


I compared the independence model (Model1), two-way association model (Model
2), and three-way interaction model (Saturated), and found that the two-way
association model was the most parsimonious one:

> ind <- loglm(n~A+T+L)
> twoway <- loglm(n~A:T+A:L+T:L)
> anova(ind,twoway)
LR tests for hierarchical log-linear models

Model 1:
 n ~ T + A + L
Model 2:
 n ~ A:L + A:T + T:L

             Deviance df Delta(Dev) Delta(df) P(> Delta(Dev)
Model 1   1001.430955  7
Model 2      4.827261  2 996.603694         5        0.00000
Saturated    0.000000  0   4.827261         2        0.08949


By running a Chi-square test, I found that all of the three two-way
associations are significant.
> drop1(twoway,test="Chisq")
Single term deletions

Model:
n ~ A:T + A:L + T:L
       Df    AIC    LRT   Pr(Chi)
<none>     24.83
A:T     2 645.91 625.08 < 2.2e-16 ***
A:L     2 152.93 132.10 < 2.2e-16 ***
T:L     1 143.60 120.77 < 2.2e-16 ***
---
Signif. codes:  0 ¡***¢ 0.001 ¡**¢ 0.01 ¡*¢ 0.05 ¡.¢ 0.1 ¡ ¢ 1


Then, I got the coefficients:
> coef(twoway)
$`(Intercept)`
[1] 5.866527

$A
         BB          AA          CC
 0.27277069 -0.01475892 -0.25801177

$T
       On       Off
 1.156143 -1.156143

$L
      Yes        No
-1.225228  1.225228

$A.T
    T
A            On        Off
  BB  0.4809533 -0.4809533
  AA -0.1783651  0.1783651
  CC -0.3025882  0.3025882

$A.L
    L
A            Yes          No
  BB  0.19166429 -0.19166429
  AA -0.15632604  0.15632604
  CC -0.03533825  0.03533825

$T.L
     L
T            Yes         No
  On   0.2933774 -0.2933774
  Off -0.2933774  0.2933774


I, then, hand-calculated odds ratio for A x T, A x L, and T x L.

T x L:
*èTL *=* e4(.293) *= 3.23

A x L:
*èAL(BB vs. AA) *= *e 2(.19166) + 2(.1563) = *2.01

*èAL(BB vs. CC) *= *e 2(.19166) + 2(.03533) = *1.57

A x T:

*èAT(BB vs. AA) *= *e 2(.48095) + 2(.17837) = 3.74*
*
*
*èAT(BB vs. CC) = e 2(.48095) + 2(.30259) = 4.79 *



Now, I'd like to know if BB and AA (or BB and CC) are significantly
different from each other (i.e., the odds of BB to be 2.01 times larger than
AA is significant) and the differences between BB and CC are significant
(i.e., the odds of BB to be 1.6 times bigger is significant) etc.


I'd really appreciate if someone can answer this question!

Thank you,
Sachi

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