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Linear statistical models 2009
Count data
Contingency tables and log-linear models
Poisson regression
Linear statistical models 2009
Contingency tables and log-linear models
Expected frequency:
Log-linear models are linear models of the log expected frequency
(log is used as link function)
SnoresHeart_problems Seldom Often TotalYes 59 51 110No 1958 416 2374
2017 467 2484
SnoresHeart_problems Seldom Often TotalYes p11 p12 p1.
No p21 p22 p2.
p.1 p.2 1
ijij pn
Linear statistical models 2009
A log-linear model for independence
The last parameter of each kind can be set to zero
jiijij ppnpn ..
jijiij ppn )log()log()log()log( ..
1
1
22
21
12
11
001
101
011
111
)log(
)log(
)log(
)log(
Linear statistical models 2009
The saturated log-linear model
Independence can be tested by relating the difference in deviance D2 – D1 to a 2 distribution with df2 – df1 degrees of freedom.
What is D1 and df1 for the saturated model?
ijij pn
ijjiij )()log(
Linear statistical models 2009
Analysis of example data (1)
proc genmod data=linear.snoring;
class snore heart;
model count = snore heart/link=log dist=Poisson;
run;
Can a Poisson distribution be justified?
Snore Hart CountOften Yes 51Often No 416Seldom Yes 59Seldom No 1958
Linear statistical models 2009
Analysis of example data (2)
• Analysis Of Parameter Estimates
• Standard Wald 95% Confidence Chi-• Parameter DF Estimate Error Limits Square Pr > ChiSq
• Intercept 1 4.4922 0.0958 4.3044 4.6801 2197.28 <.0001• Snore Often 1 -1.4630 0.0514 -1.5637 -1.3624 811.67 <.0001• Snore Seldom 0 0.0000 0.0000 0.0000 0.0000 . . • Heart No 1 3.0719 0.0975 2.8807 3.2630 992.02 <.0001• Heart Yes 0 0.0000 0.0000 0.0000 0.0000 . . • Scale 0 1.0000 0.0000 1.0000 1.0000•
Often SeldomYes 4.4922 3.0292No 7.5641 6.1009Estimates of log()
Linear statistical models 2009
Contingency table with one response variable
Consider the example data written in the following form
proc genmod data=linear.snoring2;
class snore;
model heart/total = snore/link=logit dist=binomial;
run;
Snore Heart TotalYes 51 467No 59 2017
Linear statistical models 2009
Analysis of example data (2)
• Analysis Of Parameter Estimates
• Standard Wald 95% Confidence Chi-• Parameter DF Estimate Error Limits Square Pr > ChiSq
• Intercept 1 -2.0989 0.1484 -2.3896 -1.8081 200.13 <.0001• Snore No 1 -1.4033 0.1987 -1.7927 -1.0139 49.89 <.0001• Snore Yes 0 0.0000 0.0000 0.0000 0.0000 . . • Scale 0 1.0000 0.0000 1.0000 1.0000
log(p/(1-p)) p
Yes -2.0989 0.109204No -3.5022 0.02925
log(p/(1-p)) pYes -2.0989 0.109204No -3.5022 0.02925
Snore Heart Total Rel. FrequencyYes 51 467 0.109208No 59 2017 0.029251
Linear statistical models 2009
The multinomial distribution
Consider a nominal random variable that takes k distinct values with probabilities p1, p2, …, pk
Assume that have made n independent observations of that variable
Then
where nj is the number of times the jth value is observed
Note that n is fixed in a multinomial distribution.
If the observations arrive randomly, a Poisson distribution is usually preferable.
knk
nn
kk ppp
nnn
nnnnP ...
!...!!
!)...,,,( 21
2121
21
Linear statistical models 2009
Higher order tables
Consider the following data on drug use
Model:
Alcohol Cigarette Marijuana Countyes yes yes 911yes yes no 538yes no yes 44yes no no 456no yes yes 3no yes no 43no no yes 2no no no 279
ijkjkikijkjiijk )()()()()log(
Linear statistical models 2009
Terminology
A = alcohol C = cigarette M = marijuana
Model A C M: mutual independence model
Model A C M A*C A*M C*M: homogeneous association model
Model A C M A*C A*M: Model in which C and M are mutually independent when controlling for A
Linear statistical models 2009
Poisson regression I
Poisson distribution
Log link
where x is a covariate
x10)log(
Linear statistical models 2009
Poisson regression II
Poisson distribution
Log link
where the parameters are row,
column and treatment effects
kji 0)log(
Row Column Treatment Count1 1 P 32 1 M 63 1 O 44 1 N 175 1 K 41 2 O 22 2 K 03 2 M 94 2 P 85 2 N 41 3 N 52 3 O 63 3 K 14 3 M 85 3 P 21 4 K 12 4 N 43 4 P 64 4 O 95 4 M 41 5 M 42 5 P 43 5 N 54 5 K 05 5 O 8