Multilevel Modeling-Logistic
Raul Cruz-Cano, HLTH653
Spring 2013
Schedule
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3/18/2013 = Spring Break
3/25/2013 = Longitudinal Analysis
4/1/2013 = Midterm (Exercises 1-5,
not Longitudinal)
Raul Cruz-Cano, HLTH653
Spring 2013
Introduction
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Just as with linear regression, logistic regression allows you to
look at the effect of multiple predictors on an outcome.
Consider the following example: 15- and 16-year-old adolescents
were asked if they have ever had sexual intercourse.
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The outcome of interest is intercourse.
The predictors are race (white and black) and gender (male and female).
Example from Agresti, A. Categorical Data Analysis, 2nd ed. 2002.
Raul Cruz-Cano, HLTH653
Spring 2013
Here is a table of the data:
Intercourse
Race
Gender
Yes
No
White
Male
43
134
Female
26
149
Male
29
23
Female
22
36
Black
Raul Cruz-Cano, HLTH653
Spring 2013
Data Set Intercourse
DATA intercourse;
INPUT white male intercourse count;
DATALINES;
1 1 1 43
1 1 0 134
1 0 1 26
1 0 0 149
0 1 1 29
0 1 0 23
0 0 1 22
0 0 0 36
;
RUN;
Raul Cruz-Cano, HLTH653
Spring 2013
SAS:
PROC LOGISTIC DATA = intercourse descending;
weight count;
MODEL intercourse = white male/rsquare lackfit;
RUN;
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“descending” models the probability that intercourse
= 1 (yes) rather than = 0 (no).
“rsquare” requests the R2 value from SAS; it is
interpreted the same way as the R2 from linear
regression.
“lackfit” requests the Hosmer and Lemeshow
Goodness-of-Fit Test. This tells you if the model
you have created is a good fit for the data.
Raul Cruz-Cano, HLTH653
Spring 2013
SAS Output: R2
Raul Cruz-Cano, HLTH653
Spring 2013
Interpreting the R2 value
The R2 value is 0.9907. This means
that 99.07% of the variability in our
outcome (intercourse) is explained
by including gender and race in our
model.
Raul Cruz-Cano, HLTH653
Spring 2013
PROC LOGISTIC Output
The odds of having intercourse is 1.911
times greater for males versus females.
Hosmer and Lemeshow GOF Test
H-L GOF Test
The Hosmer and Lemeshow Goodness-of-Fit Test tests the hypotheses:
Ho: the model is a good fit, vs. Ha: the model is NOT a good fit
With this test, we want to FAIL to reject the null hypothesis, because that
means our model is a good fit (this is different from most of the
hypothesis testing you have seen).
Look for a p-value > 0.10 in the H-L GOF test. This indicates the model
is a good fit.
In this case, the pvalue = 0.2419, so we do NOT reject the null
hypothesis, and we conclude the model is a good fit.
Raul Cruz-Cano, HLTH653
Spring 2013
Model Selection in SAS
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Often, if you have multiple predictors and interactions in your
model, SAS can systematically select significant predictors
using forward selection, backwards selection, or stepwise
selection.
In forward selection, SAS starts with no predictors in the
model. It then selects the predictor with the smallest pvalue
and adds it to the model. It then selects another predictor
from the remaining variables with the smallest pvalue and adds
it to the model. It continues doing this until no more
predictors have pvalues less than 0.05.
In backwards selection, SAS starts with all of the predictors in
the model and eliminates the non-significant predictors one at
a time, refitting the model between each elimination. It stops
once all the predictors remaining in the model are statistically
significant.
Raul Cruz-Cano, HLTH653
Spring 2013
Forward Selection in SAS
We will let SAS select a model for us out of the
three predictors: white, male, white*male.
Type the following code into SAS:
PROC LOGISTIC DATA = intercourse descending;
weight count;
MODEL intercourse = white male white*male/selection = forward lackfit;
RUN;
Raul Cruz-Cano, HLTH653
Spring 2013
Output from Forward Selection: “white”
is added to the model
“male” is added to the model
No more predictors are found to be statistically significant
The Final Model:
Hosmer and Lemeshow GOF Test: The model is a good fit
Multilevel Modeling (refresher)
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Multi-level modeling takes into account the hierarchical structure
of the data (e.g. decedents clustered within occupations as in our
data).
Such data structure is subject to intra-class correlation, whereby
individuals within the same group are more alike than individuals
across groups.
Analysis that ignores this intra-class correlation may
underestimate the standard error of the regression coefficient of
the aggregate risk factor, leading to overestimation of the
significance of the risk factor.
To illustrate the above point, we conducted our analysis using two
approaches
Raul Cruz-Cano, HLTH653
Spring 2013
1st Approach
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Fit a multiple logistic regression model on the combined
data with PROC LOGISTIC.
The dependent variable is death from injury (yes/no);
the risk factor of interest is exposure to hazardous
equipment at work (high/low);
confounders included are gender, race (white/black/other),
age (continuous, centered) and a quadratic term for age.
This model ignores the hierarchical structure of the data,
and treats aggregate exposure as if it was measured at
individual level. The model is expressed by the following
equation
 p
log it ( pij )  log  ij
 1 p
ij


2
    Exposurei  Genderij  Raceij  1 Ageij   2 Ageij

Raul Cruz-Cano, HLTH653
Spring 2013
1st Approach
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pij is the expected probability of death from injury for the
jth individual of the ith occupation conditional on the
predictor variables
 pij
log it ( pij )  log 
 1 p
ij


2
    Exposurei  Genderij  Raceij  1 Ageij   2 Ageij

proc logistic data=noms.combined descending;
class exposure gender race;
model injury = exposure gender race age age*age;
run;
Raul Cruz-Cano, HLTH653
Spring 2013
Multilevel Example
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Allison, 2006
The sample consists of 1151 girls from the National Longitudinal Survey of
Youth who were interviewed annually for nine years, beginning in 1979.
For this initial example, we’ll only use data from year 1 and year 5.
The response variable POV has a value of 1 if the girl’s household was in
poverty (as defined by U.S. federal standards) in each of the years,
otherwise 0.
The predictor variables are:
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AGE: Age in years at the first interview
BLACK: 1 if respondent is black, otherwise 0
MOTHER: 1 if respondent currently had a least one child, otherwise 0
SPOUSE: 1 if respondent is currently living with a spouse, otherwise 0
INSCHOOL: 1 if respondent is currently enrolled in school, otherwise 0
HOURS: Hours worked during the week of the survey
Raul Cruz-Cano, HLTH653
Spring 2013
Multilevel Example
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5755 observations, five for each of the 1151 girls
The CLASS statement declares YEAR to be a categorical
variable, with the highest year (year 5) being the reference
category.
The STRATA statement says that each girl is a separate
stratum, which has the consequence of grouping together
the five observations for each girl in the process of
constructing the likelihood function.
PROC LOGISTIC DATA=teenyrs5 DESC;
CLASS year;
MODEL pov = year mother spouse inschool hours;
STRATA id;
RUN;
In PROC LOGISTIC there is no CLUSTER, just CLASS and STRATA
Multilevel Example
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In the “Analysis of Maximum of Likelihood Estimates” panel, we see that motherhood
and school enrollment increase the risk of poverty while living with a husband and
working more hours reduce the risk.
The last panel gives the odds ratios.
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We see that motherhood increases the odds of poverty by an estimated 79 percent.
Living with a husband cuts the odds approximately in half.
Each additional hour of employment per week reduces the odds by about 2 percent.
Keep in mind that these estimates control for all stable characteristics of the girls, including such things as race,
intelligence, place of birth and parent’s education
Raul Cruz-Cano, HLTH653
Spring 2013
Multilevel Example
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The next model, for example, includes the interaction between MOTHER
and BLACK.
PROC LOGISTIC DATA=teenyrs5 DESC;
CLASS year;
MODEL pov = year mother spouse inschool hours mother*black;
STRATA id;
RUN;
Raul Cruz-Cano, HLTH653
Spring 2013
Multilevel Example
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The interaction is statistically significant at the .05 level.
For nonblack girls, the effect of motherhood is to increase the odds of
poverty by a factor of exp(.9821)=2.67.
For black girls, on the other hand, the effect of motherhood is to increase
the odds of poverty by a factor of exp(.9821-.5989)= 1.47.
Thus, motherhood has a larger effect on poverty status among nonblack
girls than among black girls.
Raul Cruz-Cano, HLTH653
Spring 2013
SAS Weigted Example
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A random sample
300 students from each of the classes: freshman, sophomore,
junior, and senior classes. data WebSurvey;
proc format;
value Design 1='A' 2='B' 3='C';
value Rating 1='dislike very much'
2='dislike'
3='neutral'
4='like'
5='like very much';
value Class 1='Freshman' 2='Sophomore'
3='Junior' 4='Senior';
run;
data Enrollment;
format Class Class.;
input Class _TOTAL_;
datalines;
1 3734
2 3565
3 3903
4 4196
;
Raul
run;
format Class Class. Design Design. Rating Rating. ;
do Class=1 to 4;
do Design=1 to 3;
do Rating=1 to 5;
input Count @@;
output;
end;
end;
end;
datalines;
10 34 35 16 15 8 21 23 26 22 5 10 24 30 21
1 14 25 23 37 11 14 20 34 21 16 19 30 23 12
19 12 26 18 25 11 14 24 33 18 10 18 32 23 17
8 15 35 30 12 15 22 34 9 20 2 34 30 18 16
;
run;
data WebSurvey;
set WebSurvey;
if Class=1 then Weight=3734/300;
if Class=2 then Weight=3565/300;
if Class=3 then Weight=3903/300;
if Class=4 then Weight=4196/300;
run;
Cruz-Cano, HLTH653
Spring 2013
PROC Logistic
proc logistic data=WebSurvey;
freq Count;
class Design;
model Rating (ref='neutral') = Design ;
weight Weight;
run;
Raul Cruz-Cano, HLTH653
Spring 2013
PROC surveylogistic
If you want “better” results..
proc surveylogistic data=WebSurvey total=Enrollment;
freq Count;
class Design;
model Rating (ref='neutral') = Design;
stratum Class;
weight Weight;
run;
For the Ratings for Design B vs. Design C compare
1. The point estimete
2. 95% Confidence Interval
Raul Cruz-Cano, HLTH653
Spring 2013
More to come…
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There are also mixed effects logistic
models…which will be studied later
Raul Cruz-Cano, HLTH653
Spring 2013
References
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Paul D. Allison, Fixed Effects Regression Methods In SAS, SUGI 31
Proceedings (2006), paper 184-31
Jia Li, Toni Alterman, James A. Deddens, Analysis of Large
Hierarchical Data with Multilevel Logistic Modeling Using PROC
GLIMMIX In SAS, SUGI 31 Proceedings (2006), paper 151-31
David L. Cassell, (2006) “Wait Wait, Don't Tell Me… You're Using
the Wrong Proc! SUGI31. Paper 193-31.
Raul Cruz-Cano, HLTH653
Spring 2013