Transcript Loglinear Models for Independence and Interaction in Three

Loglinear Models for
Independence and Interaction in
Three-way Tables
Veronica Estrada
Robert Lagier
Quick Review from Agresti, 4.3
• Poisson Loglinear Models are based on
Poisson distribution of Y counts and
employ log link function:
log μY = α + βx
μY = exp(α + βx)
Value of Loglinear Models?
• Used to model cell counts in contingency
tables where at least 2 variables are
response variables
• Specify how expected cell counts depend on
levels of categorical variables
• Allow for analysis of association and
interaction patterns among variables
Models for Two-way Tables
• Independence Model
–
–
–
–
μij = μαi βj
log μij = λ + λiX + λjY
where λiX is row effect, and λjY is column effect
odds for column response independent of row
• Saturated (Dependence) Model
– terms logμij = λ + λiX + λjY + λijXY
– where λijXY are association that represent
interactions between X and Y
– odds for column response depends on row
Loglinear Models for Three-way
(I x J x K) Tables
• Describe independence and association
patterns
• Assume a multinomial distribution of cell
counts with cell probabilities {πijk}
• Also apply to Poisson sampling with means
{µijk}
Types of Independence for Cell
Probabilities in I x J x K Tables
• Mutual Independence
• Joint Independence
• Conditional Independence
• Marginal Independence
Mutual Independence
• πijk = (πi++) (π+j+) (π++k)
for all i, j, k
• Loglinear Model for Expected Frequencies
– log μijk = λ + λiX + λjY + λkZ
• Interpretation:
– X independent of Y independent of Z
independent of X
– No association between variables
Joint Independence
• X jointly independent of Y and Z:
– πijk = (π+jk) (πi++)
for all i, j, k
• Loglinear Model for Expected Frequencies
– log μijk = λ + λiX + λjY + λkZ + λjkYZ
• Interpretation:
– X independent of Y and Z
– Partial association between variables Y and Z
• 3 Joint Independence Models
Conditional Independence
• X and Y conditionally independent of Z:
– πijk = (πi+k) (π+jk) / π++k
for all i, j, k
• Loglinear Model for Expected Frequencies
– log μijk = λ + λiX + λjY + λkZ + λikXZ + λjkYZ
• Interpretation:
– X and Y independent given Z
– Partial association between X,Z and Y,Z
• 3 Conditional Independence Models
Marginal Independence
• X and Y marginally independent of Z:
– πij+ = (πj++) (π+j+)
for all i, j, k
• Interpretation:
– X and Y independent in the two-way table that
has been collapsed over the levels of Z
– Variables may have different strength of
marginal association than conditional (partial)
association - Simpson’s Paradox
Partial v. Marginal Tables
Residence
Urban
Rural
Stress
Low
High
Total
Stress
Low
High
Total
Low
High
Total
Opinion
Favorable
Unfavorable
48
12
96
94
144
106
55
135
7
53
62
188
Opinion
Favorable
Unfavorable
103
147
103
147
206
294
Total
60
190
250
190
60
250
Total
250
250
500
Relationships Among Types of
XY Independence
Mutually
Independent
with Z
Jointly
Independent
of Z
Conditionally
Independent
given Z
Marginally
Independent
Homogenous Association Model
• Loglinear Model for Expected Frequencies
– log μijk = λ + λiX + λjY + λkZ + λijXY + λikXZ + λjkYZ
• Interpretation:
– Homogenous association:
• identical conditional odds ratios between any two
variables over the levels of the third variable
• θij(1) = θij(2) = … = θij(K)
for all i and j
Saturated Model
• Loglinear Model for Expected Frequencies
– log μijk = λ + λiX + λjY + λkZ + λijXY + λikXZ + λjkYZ +
λijkXYZ
• Interpretation:
– Each pair of variables may be conditionally
dependent
– Odds ratios for any pair of variables may vary over
levels of the third variable
– perfect fit to observed data
Inference for Loglinear Models
• Interpretation of Loglinear model
parameters is at the level of the highestorder terms
• χ2 or G2 Goodness of Fit Tests can be used
to select best fitting model
• Parameter estimates are log odds ratios for
associations
Example:
Alcohol, Cigarette, and Marijuana
Data
Alcohol Use
Yes
No
Cigarette
Use
Yes
No
Yes
No
Marijuana
Use: Yes
911
44
3
2
Source: Data courtesy of Harry Khamis, Wright State University
Marijuana
Use: NO
538
456
43
279
SAS Code
•
•
•
•
•
•
data drugs; input a c m count;
cards;
1 1 1 911 1 1 2 538 1 2 1 44 1 2 2 456
2 1 1 3 2 1 2 43 2 2 1 2 2 2 2 279 ;
proc genmod; class a c m; model count = a c m / dist=poi link=log obstats;
run;
•
•
•
•
•
•
•
•
•
•
•
•
•
proc genmod; class a c m; model count = a c m c*m / dist=poi link=log obstats;
run;
proc genmod; class a c m; model count = a c m a*m / dist=poi link=log obstats;
run;
proc genmod; class a c m; model count = a c m a*c / dist=poi link=log obstats;
run;
proc genmod; class a c m; model count = a c m a*c a*m / dist=poi link=log obstats;
run;
proc genmod; class a c m; model count = a c m a*c c*m / dist=poi link=log obstats;
run;
proc genmod; class a c m; model count = a c m a*c a*m c*m / dist=poi link=log obstats;
run;
proc genmod; class a c m; model count = a c m a*c a*m c*m a*c*m/ dist=poi link=log
obstats;
run;
•
Fitted Values for Loglinear Models
Alcohol
Use
Yes
No
a
Cigarette Marijuan
Use
a Use
(A, C,
M)
Loglinear Model
(AC, M)
(AM,
CM)
(AC,
AM,
CM)
(ACM)
Yes
Yes
No
540.0
740.2
611.2
837.8
909.24
438.84
910.4
538.6
911
538
No
Yes
No
282.1
386.7
210.9
289.1
45.76
555.16
44.6
455.4
44
456
Yes
Yes
No
90.6
124.2
19.4
26.6
4.76
142.16
3.6
42.4
3
43
No
Yes
No
47.3
64.9
118.5
162.5
0.24
179.84
1.4
279.6
2
279
A, alcohol use; C, cigarette use; M, marijuana use.
Estimated Odds Ratios for Loglinear
Models
Model
Conditional Association Marginal Association
AC AM CM
AC AM CM
1.0 1.0
1.0
(AC,M)
17.7 1.0 1.0
(AM,CM)
1.0 61.9 25.1
(AC,AM,CM) 7.8 19.8 17.3
(ACM)
13.8 24.3 17.5
(A,C,M)
1.0
1.0
1.0
17.7 1.0 1.0
2.7 61.9 25.1
17.7 61.9 25.1
17.7 61.9 25.1
Computation of the Odds Ratio
909.24  0.24
438.84  179.84

 1.0 Thisis the entry for AC
45.76  4.76
55.16  142.16
conditional association for the
mod el ( AM , CM )
6112
.  118.5 837.8  162.5

 17.7 Thisis the Conditional
210.9  19.4
289.1  26.6
Association for AC for the mod el AC , M
b
g
• Model (AC, AM, CM) permits all pairwise
associations but maintains homogeneous
odds rations between two variables at each
level of the third.
• The previous table shows that estimated
odds ratios are very dependent on the
model, and from this we can only say that
the model fits well.
Conditional independence has implications regarding
marginal (in) dependence; however, marginal (in)
dependence does not have implications regarding conditional
(in) dependence.
Conditional independence->marginal independence
Conditional independence->marginal dependence
Marginal independence does not ->conditional
independence
Marginal dependence does not ->conditional dependence.