Overdispersed Data in SAS for Regression Analysis

Analysis of Overdispersed Data in
SAS
Jessica Harwood, M.S.
Statistician, Center for Community Health
JHarwood@mednet.ucla.edu
Outline - Overdispersion
•
Definition, background, and causes of
overdispersion
•
Consequences of ignoring overdispersion
•
Accounting for overdispersion in regression
analysis in SAS
–
For count data
–
For binary data
•
Concluding remarks
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
2
Overdispersed data
•
Also known as “extra variation”
•
Arises when count or binary data exhibit
variances larger than those assumed
under the Poisson or binomial
distributions
3
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
Count data
•
Definition: non-negative integer values {0,
1, 2, 3, ...} arising from counting rather
than ranking
•
Example: the number of days a student is
absent in one school year
•
Commonly analyzed using Poisson
distribution, e.g., Poisson regression
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
4
Poisson Distribution
•
Poisson: number of occurrences of a
random event in an interval of time or
space.
•
Poisson regression 
 IRR (relative risk)
•
Natural model for count data
•
Disadvantage - strong assumption:
variance = mean
•
Overdispersion: variance > mean
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
5
•
Binary: 0 or 1
–
Example: ever tested for HIV (1) or not (0)
•
Grouped binary
–
Example: proportion tested for HIV
•
Commonly analyzed using binomial
distribution, e.g., logistic regression
          Binary: “tested_HIV”                            Grouped: “num_tested_HIV/num_subjects”
Binary Data
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
6
Binomial distribution
•
Binomial: the number of successes in a
sequence of random processes that results
in one of two mutually exclusive
outcomes
•
Overdispersion: variance larger than that
assumed under the binomial distribution
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
7
Causes of Overdispersion
•
Observed data rarely follow statistical
distributions exactly
•
The variance of count variables tends to
increase with the size of the counts
•
Correlated (ex: clustered) data
•
Heterogeneity among observations
•
Large number of 0’s
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
8
Consequences of Ignoring
Overdispersion
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
9
Checking for Overdispersion in
SAS – Count Data
•
PROC MEANS
–
variance > mean?
•
PROC GENMOD
–
“dist=negbin” 
 dispersion parameter
significant?
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
10
Example – Count Data
Differences in baseline depression between
intervention conditions in a RCT
•
Independent variable: INTV - 
intervention condition
•
1 = Randomized to intervention condition
•
0 = Randomized to control condition
•
Dependent variable: EPDS
 - Edinburgh Postnatal
Depression Scale; weighted count of depressive
symptoms felt in past week
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
11
HISTOGRAM OF EPDS
Example – Count Data
EPDS Score (range 0-30)
0
2
4
8
10
12
14
Percent
6
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
12
Example – Count Data
Check mean and variance for overdispersion
*Mean and variance;
proc
 
means
 
data
=base 
mean
 
var
;
 
var
 EPDS;
 
run
;
*Conditional mean and variance;
proc
 
means
 
data
=base 
mean
 
var
;
 
var
 EPDS;
 
class
 INTV;
 
run
;
_____SAS Code
                                 
   
            
                     SAS Output_____
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
13
*Poisson regression – ignore overdispersion;
proc
 
genmod
 
data 
= base;
 
model
 EPDS = INTV / 
dist
=poisson;
 
run
;
*Negative binomial regression – account for
overdispersion;
proc
 
genmod
 
data 
= base;
 
model
 EPDS = INTV / 
dist
=negbin;
 
run
;
Example – Count Data
SAS regression analysis
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
14
Dispersion parameter significantly different from
zero (see 95% CI):
–
Indicates significant over- (> 0) or under- (< 0)
dispersion
–
Use negative binomial rather than Poisson
Example – Count Data
Check for overdispersion: negative binomial
regression in PROC GENMOD
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
15
Example – Count Data
Results
•
P-values quite different
•
Different conclusions regarding similarity of intervention
conditions at baseline
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
16
Accounting for overdispersion in
SAS: count data
•
Negative binomial
•
Variance-adjustment models
–
Quasi-likelihood Estimation
–
Empirical (aka robust, sandwich) variance
estimation
–
Models for correlated data
•
Zero-inflated models
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
17
Negative binomial (NB)
•
Negative binomial distribution: variance
is larger than the mean 
 excellent model
for overdispersed count data
•
Negative binomial regression 
 relative
risk
•
Disadvantage: estimating extra parameter
(dispersion)
•
PROC GENMOD
•
PROC COUNTREG
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
18
SAS code: negative binomial
regression
proc
 
genmod
 
data 
= base;
 
model
 EPDS = INTV / 
dist
=negbin;
 
run
;
proc
 
countreg
 
data 
= base;
 
model
 EPDS = INTV / 
dist
=negbin;
 
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
19
SAS output: negative binomial
regression
Compare to Poisson regression:
INTV:  
Estimate=-0.0974;  
SE=0.0218;
  
P<.0001
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
20
SAS output: negative binomial regression
•
NB: Variance > mean
 
Variance = mean + 
k 
*mean
2
•
SAS estimate of dispersion parameter 
k
: “Dispersion”,
“_Alpha”
•
If 
k 
significantly different from zero 
 use NB rather than
Poisson
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
21
Count data: Quasi-likelihood
Estimation (QLE)
•
QLE allows for adjusting variance without
specifying distribution exactly
•
Variances inflated by
–
Deviance/DOF (GENMOD: “dscale”)
–
Pearson’s Chi-Square/DOF
•
GENMOD: “pscale”
•
GLIMMIX: “random _residual_”
•
Poisson and negative binomial regression
(and logistic regression)
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
22
QLE – Example - SAS Code
Use “dscale” as the norm!
*Poisson regression- no adjustment for
overdispersion;
proc
 
genmod
 
data
=base;
model
 EPDS = INTV/ 
dist
=poisson;
run
;
*Poisson regression- adjust for overdispersion
using “DSCALE”;
proc
 
genmod
 
data
=base;
model
 EPDS = INTV/ 
dist
=poisson 
dscale
;
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
23
QLE- standard errors (SE) corrected
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
24
QLE- Poisson vs. NB
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
25
proc
 
genmod
 
data
=base;
model
 EPSD=INTV / 
dist
=poisson 
pscale
;
run
;
proc
 
glimmix
 
data
=base;
model
 EPSD=INTV / 
dist
=poisson 
s
;
random
 _residual_;
run
;
QLE – “PSCALE”- SAS Code
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
26
Count Data – QLE – In Sum
•
Use as the norm, in Poisson or NB
•
DSCALE better than PSCALE, especially
for low counts
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
27
Count Data:
Empirical Variance Estimation
•
Empirical (or 
robust
 or 
sandwich
) variance
estimation – account for extra variation by
using both empirical-based estimates and
model-based estimates in variance
estimation
•
Poisson and NB regression (and logistic
regression)
•
GENMOD: “REPEATED” statement
•
GLIMMIX: “EMPIRICAL” option
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
28
Empirical Variance Estimation –
GENMOD “REPEATED”
*PID = “Participant ID”, 1 observation per
PID;
proc
 
genmod
 
data
=base;
 
class
 PID;
 
model
 EPDS=INTV / 
dist
=poisson;
 
repeated subject 
= PID;
 
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
29
Compare to unadjusted Poisson regression:
INTV:  
Estimate=-0.0974;  
SE=0.0218;
  
P<.0001
Empirical Variance Estimation –
GLIMMIX “EMPIRICAL”
*PID = “Participant ID” 1 observation per
PID. “MBN” is small-sample bias correction;
proc
 
glimmix 
data
=base 
empirical
=mbn
;
 
class
 PID;
 
model
 EPDS=INTV / 
dist
=poisson 
s
;
 
random 
_residual_ 
/subject 
= PID;
 
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
30
Count Data: Correlated (ex:
clustered) data
•
Longitudinal data (clustering of repeated
measurements within subjects)
•
Nested data (clustering of multiple subjects
within groups)
•
Poisson, NB, or logistic regression
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
31
Count Data: Correlated (ex:
clustered) data
•
Generalized Linear Mixed Models (GLMM)
–
GLIMMIX “RANDOM INT” [Conditional model,
subject-specific inference]
–
GLIMMIX “RANDOM _RESIDUAL_” [Marginal
model, inference on population averages]
•
Generalized Estimating Equations (GEE)
[Marginal model]
–
GENMOD “REPEATED”
–
Small-sample bias correction in GLIMMIX with
“EMPIRICAL=mbn“ option
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
32
Clustered Data – GLMM
*Participants clustered by city. Marginal model;
proc
 
glimmix 
data
=base;
 
class
 city;
 
model
 EPDS=INTV / 
dist
=nb 
s
;
 
random 
_residual_ 
/ 
subject
=city 
type
=cs;
 
run
;
*Participants clustered by city. Conditional model;
proc
 
glimmix
 
data
=base;
 
class
 city;
 
model
 EPDS=INTV / 
dist
=nb 
s
;
 
random 
int /
subject
=city;
 
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
33
Clustered Data – GEE
*Participants clustered by city. “MBN” is
small-sample bias correction;
proc
 
glimmix 
data
=base 
empirical
=mbn
;
 
class
 city;
 
model
 EPDS=INTV / 
dist
=nb 
s
;
 
random 
_residual_ 
/subject 
=city;
 
run
;
proc
 
genmod
 
data
=base;
 
class
 city;
 
model
 EPDS=INTV / 
dist
=nb;
 
repeated subject 
= city / 
type
=cs;
 
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
34
Count Data - Zero-Inflated (ZI)
Models
•
ZI models appropriate when variable contains an
excess of zero values- sample heterogeneity
•
Assume sample contains two different populations:
“nonsusceptible” (always zero) subjects and
“susceptible” (not always zero) subjects
•
ZI regression - two regression models (each with
own explanatory variables):
–
Logit or probit regression - model the probability of
being “nonsusceptible”
–
Poisson/NB/logistic regression - model the mean for
the susceptible population
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
35
Count data - ZI in SAS
•
PROC GENMOD
•
PROC COUNTREG
•
Zero-inflated Poisson: “dist=ZIP”
•
Zero-inflated NB: “dist=ZINB”
–
Even after accounting for excess zeros, NB
may fit the remaining counts better than
Poisson
–
GENMOD ZINB: SAS version > 9.2
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
36
Example - ZI
•
Variable of interest - 
count
: number of fish
caught by groups of campers at a national
park
•
Explanatory variables:
–
Number of children in the group (
child
)
–
Whether or not the group brought a camper to
the park (
camper
)
–
Number of people in the group (
persons
)
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
37
 
 
57% of values are zero values
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
38
Example – ZI – SAS Code
proc
 
genmod
 
data
 = m.fish;
 
model
 count = child camper /
dist
=zip; 
 
 
zeromodel
 persons /
link
 = logit ;
 
run
;
proc
 
countreg
 
data
 = m.fish 
method
 = qn;
 
model
 count = child camper / 
dist
=zip;
 
zeromodel
 count ~ persons;
 
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
39
Example – ZI – SAS Output
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
40
Accounting for overdispersion in
SAS: binary data
•
Random-clumped binomial and beta-
binomial models
•
Zero-inflated binomial (ZIB)
•
Variance-adjustment models
–
Quasi-likelihood Estimation
–
Empirical (aka robust, sandwich) variance
estimation
–
Models for correlated data
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
41
Binary Data – BB and RCB
•
Beta-binomial (BB) and random-clumped
binomial (RCB)
•
Model physical mechanism behind
overdispersion
•
PROC NLMIXED
•
SAS 9.3 – PROC FMM (experimental)
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
42
 
 
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
43
Example – BB and RCB
•
n=337 nuclei
•
Each nucleus has 
m
=3 total number
of chromosome pairs
•
t
: number of chromosome pairs with
association at meiosis (t=0, 1, 2, 3)
•
If probability of association at
meiosis (
t/m
) is constant for all
nuclei and the same for all
chromosome pairs, then binomial
distribution appropriate
•
If not 
 RCB or BB
n=1
337
. . .
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
44
Example – BB and RCB Data
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
45
BB and RCB – PROC NLMIXED
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
46
Binary data: ZIB
•
ZI model – simultaneously model the probability of
being “always zero” and the probability of the event
of interest conditional on being in the “not always
zero” population
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
47
           Binary: “tox”                  Grouped: “num_tox/num_total”
Binary Data - QLE - Example
Cases of toxoplasmosis in 34 cities in El Salvador
tox: 
1=toxoplasmosis case, 0=no toxoplasmosis
rain
: annual rainfall (cm)
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
48
QLE – SAS Code
*Grouped binary (1 observation for each
city);
proc
 
genmod
 
data
=tox;
model
 num_tox/num_total = rain | rain |
 
rain/ 
dist
=bin 
dscale
;
run
;
*Binary–multiple observations per city;
proc
 
genmod
 
data
=test 
desc
;
model
 tox  = rain | rain | rain/
 
dist
=bin 
dscale
 
aggregate
=city;
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
49
QLE – SAS Output
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
50
Binary Data - Empirical Variance
*Grouped binary - one observation per city;
proc
 
genmod
 
data
=tox;
 
class
 city;
 
model
 num_tox/num_total = rain | rain |
    
rain/ 
dist
=bin ;
 
repeated
 
subject
=city;
 
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
51
Grouped: “num_tox/num_total”
           Binary: “tox”
Clustered Binary Data – multiple
observations per cluster
(cluster=city)
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
52
Clustered Binary Data – GEE
*Specify clustered by city and compound symmetry covariance
structure;
proc
 
genmod
 
data
=test 
desc
; 
 
 
class
 city;
 
model
 tox  = rain | rain | rain/ 
dist
=bin ;
 
repeated
 
subject
=city/
type
=cs;
 
run
;
*MBN small sample bias correction - specify clustered by city and
compound symmetry covariance structure;
proc
 
glimmix
 
data
=test 
empirical
=mbn; 
 
 
class
 city;
 
model
 tox = rain | rain | rain / 
dist
=bin  
s
;
 
random
 _residual_/
subject
=city 
type
=cs;
 
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
53
Clustered Binary Data – GLMM
*Random effects model – conditional model;
proc
 
glimmix
 
data
=test; 
  
 
class
 city;
 
model
 tox = rain | rain | rain / 
dist
=bin 
s
;
 
random
 int / 
subject
=city ;
 
run
;
*Marginal model with compound symmetry covariance structure;
proc
 
glimmix
 
data
=test; 
  
 
class
 city;
 
model
 tox = rain | rain | rain / 
dist
=bin 
s
;
 
random
 _residual_/
subject
=city 
type
=cs;
 
run
;
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
54
Clustered Binary Data – SAS
Output
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
Least
conservative
p-values
from simple
logistic and
GEE without
small sample
bias
correction
In Sum
•
Overdispersion is a common issue when using
Poisson and binomial models
•
Overdispersion will increase false positive rates
•
For overdispersed data use:
–
Negative binomial rather than Poisson
–
RCB or BB rather than binomial
–
QLE: GENMOD “DSCALE”
–
Empirical variance: GENMOD “REPEATED”
–
Account for clustering - GLIMMIX or GENMOD
–
Zero-inflated models
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
56
Further Information
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
57
Plus:
•
Formal tests for
overdispersion and for
comparing models
•
GLOMM – Generalized
Linear Overdispersion
Mixed Models
Acknowledgements
•
CCH
•
CHIPTS Methods Core (sent me to JSM 2012!)
•
UCLA Biostatistics (I use my notes a lot!)
•
Morel JG, Neerchal NK. “Analysis of Overdispersed Data
using SAS.” Joint Statistical Meetings, San Diego, CA. July
31, 2012.
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
58
References and Resources
Horton NJ, Kim E, Saitz R. A cautionary note regarding count models of alcohol consumption
in randomized controlled trials. 
BMC Medical Research Methodology
 2007, 
7
:9.
Morel JG, Neerchal NK. “Analysis of Overdispersed Data using SAS.” Joint Statistical
Meetings, San Diego, CA. July 31, 2012.
Morel JG, Neerchal NK. 
Overdispersion Models in SAS
. Cary, NC: SAS Publishing; 2012.
Pedan, A. Analysis of count data using the SAS system. 26
th
 annual SAS Users Group
International conference, Paper 247-26. Long Beach, California 22-25 April 2001.
http://www2.sas.com/proceedings/sugi26/p247-26.pdf
Steventon JD, Bergerud WA, Ott PK. Analysis of Presence/Absence Data when
Absence is Uncertain (False Zeroes): An Example for the Northern Flying Squirrel
using SAS. Available at 
http://www.for.gov.bc.ca/hfd/pubs/docs/En/En74.pdf
. Published July
2005.
Wang W, Albert JM. Estimation of mediation effects for zero-inflated regression models.
Statistics in Medicine
 2012, 
31
(26): 3118–3132.
Zou G. A modified Poisson regression approach to prospective studies with binary data.
American Journal of Epidemiology
 2004, 
159
: 702-706.
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
59
References and Resources
UCLA: Statistical Consulting Group
•
Poisson regression
–
http://www.ats.ucla.edu/stat/sas/output/sas_poisson_output.htm
–
http://www.ats.ucla.edu/stat/sas/output/sas_poisson_output.htm
•
Negative binomial regression
–
http://www.ats.ucla.edu/stat/sas/dae/negbinreg.htm
–
http://www.ats.ucla.edu/stat/sas/output/sas_negbin_output.htm
•
ZIP regression
–
http://www.ats.ucla.edu/stat/sas/dae/zipreg.htm
–
http://www.ats.ucla.edu/stat/sas/output/sas_zip.htm
•
ZINB regression
–
http://www.ats.ucla.edu/stat/sas/dae/zinbreg.htm
–
http://www.ats.ucla.edu/stat/sas/output/sas_zinbreg.htm
•
Logistic regression: 
http://www.ats.ucla.edu/stat/sas/seminars/sas_logistic/logistic1.htm
PROC FMM:
•
SAS/STAT(R) 9.3 User's Guide:
http://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#stat
ug_fmm_a0000000313.htm
•
SAS code for fitting zero-inflated binomial / site occupancy models:
http://www.umesc.usgs.gov/staff/bios/bgray/code/zibsas.html
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
60
Thank you very much!
Questions?
Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
61
JHarwood@mednet.ucla.edu

								

								
								

										

												
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Explore the concept of overdispersion in count and binary data, its causes, consequences, and how to account for it in regression analysis using SAS. Learn about Poisson and binomial distributions, along with common techniques like Poisson regression and logistic regression. Gain insights into handling extra variation in your data for more accurate statistical modeling.


								
    
																		
  •   Overdispersed Data
    • 
																	
  •    Regression Analysis
    • 
																	
  •    SAS
    • 
																	
  •    Poisson Distribution
    • 
																	
  •    Binomial Distribution
    • 
																

								

								

									

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  1. Analysis of Overdispersed Data in  SAS Jessica Harwood, M.S. Statistician, Center for Community Health JHarwood@mednet.ucla.edu
    2. 
            
            
            
  2. Outline - Overdispersion     Definition, background, and causes of  overdispersion     Consequences of ignoring overdispersion     Accounting for overdispersion in regression  analysis in SAS     For count data     For binary data     Concluding remarks 2 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    3. 
            
            
            
  3. Overdispersed data     Also known as    extra variation        Arises when count or binary data exhibit  variances larger than those assumed  under the Poisson or binomial  distributions 3 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    4. 
            
            
            
  4. Count data     Definition: non-negative integer values {0,  1, 2, 3, ...} arising from counting rather  than ranking     Example: the number of days a student is  absent in one school year     Commonly analyzed using Poisson  distribution, e.g., Poisson regression 4 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    5. 
            
            
            
  5. Poisson Distribution     Poisson: number of occurrences of a  random event in an interval of time or  space.      Poisson regression     IRR (relative risk)     Natural model for count data     Disadvantage - strong assumption:  variance = mean     Overdispersion: variance > mean  5 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    6. 
            
            
            
  6. Binary Data     Binary: 0 or 1     Example: ever tested for HIV (1) or not (0)     Grouped binary     Example: proportion tested for HIV Binary:    tested_HIV                               Grouped:    num_tested_HIV/num_subjects    city tested_HIV Subject 1 1 1 1 1 2 1 0 3 1 0 4 2 1 1 2 0 2 2 0 3 city 1 2 num_tested_HIV num_subjects 2 1 4 3     Commonly analyzed using binomial  distribution, e.g., logistic regression 6 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    7. 
            
            
            
  7. Binomial distribution     Binomial: the number of successes in a  sequence of random processes that results  in one of two mutually exclusive  outcomes     Overdispersion: variance larger than that  assumed under the binomial distribution 7 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    8. 
            
            
            
  8. Causes of Overdispersion     Observed data rarely follow statistical  distributions exactly     The variance of count variables tends to  increase with the size of the counts     Correlated (ex: clustered) data     Heterogeneity among observations      Large number of 0   s 8 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    9. 
            
            
            
  9. Consequences of Ignoring  Overdispersion Overdispersion (observed variance larger than  that assumed by model) Standard Errors  Underestimated P-Values Underestimated  (insignificant associations appear  significant) Type I Error Inflated  (higher false positive rates) Erroneous Inference 9 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    10. 
            
            
            
  10. Checking for Overdispersion in  SAS     Count Data     PROC MEANS     variance > mean?       PROC GENMOD        dist=negbin        dispersion parameter  significant? 10 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    11. 
            
            
            
  11. Example  Count Data Differences in baseline depression between  intervention conditions in a RCT     Independent variable: INTV - intervention condition     1 = Randomized to intervention condition     0 = Randomized to control condition     Dependent variable: EPDS - Edinburgh Postnatal  Depression Scale; weighted count of depressive  symptoms felt in past week 11 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    12. 
            
            
            
  12. Example  Count Data HISTOGRAM OF EPDS 14 12 10 8 Percent 6 4 2 0 EPDS Score (range 0-30) 12 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    13. 
            
            
            
  13. Example  Count Data Check mean and variance for overdispersion _____SAS Code   SAS Output_____ *Mean and variance; procmeans data=base mean var;  var EPDS;  run; Analysis Variable : EPDS Mean Variance 11.17  47.34  *Conditional mean and variance; procmeans data=base mean var;  var EPDS;  class INTV;  run; Analysis Variable : EPDS INTV N Obs Mean Variance   11.35        48.31  0 533   11.01        46.52  1 611 13 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    14. 
            
            
            
  14. Example  Count Data SAS regression analysis *Poisson regression     ignore overdispersion; procgenmod data = base; model EPDS = INTV / dist=poisson; run; *Negative binomial regression     account for  overdispersion; procgenmod data = base; model EPDS = INTV / dist=negbin; run; 14 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    15. 
            
            
            
  15. Example  Count Data Check for overdispersion: negative binomial  regression in PROC GENMOD Dispersion parameter significantly different from  zero (see 95% CI):      Indicates significant over- (> 0) or under- (< 0)  dispersion     Use negative binomial rather than Poisson Negative binomial regression- account for overdispersion Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard  Wald 95%  Confidence  Limits 2.0306 -0.227 0.9185 Wald  Chi- Square 1973.7 <.0001 2.18 0.1394 Pr >  ChiSq Error 1 1 1 2.1244 -0.0974 1.0192 0.0478 0.0659 0.0514 2.2181 0.0318 1.1198 Intercept INTV Dispersion 15 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    16. 
            
            
            
  16. Example  Count Data Results     P-values quite different     Different conclusions regarding similarity of intervention  conditions at baseline EPDS: Poisson regression- ignore overdispersion Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard  Wald 95%  Confidence  Limits 2.094 -0.14 1 Wald  Chi- Square 18805 <.0001 19.95 <.0001 Pr >  ChiSq Error 1 1 0 2.1244 -0.0974 0.0155 0.0218 2.1547 -0.055 Intercept INTV Scale EPDS: Negative binomial regression- account for overdispersion Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard  1 0 1 Wald 95%  Confidence  Limits 2.0306 -0.227 0.9185 Wald  Chi- Square 1973.7 <.0001 2.18 0.1394 Pr >  ChiSq Error 1 1 1 2.1244 -0.0974 1.0192 0.0478 0.0659 0.0514 2.2181 0.0318 1.1198 Intercept INTV Dispersion 16 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    17. 
            
            
            
  17. Accounting for overdispersion in  SAS: count data     Negative binomial      Variance-adjustment models     Quasi-likelihood Estimation     Empirical (aka robust, sandwich) variance  estimation      Models for correlated data     Zero-inflated models 17 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    18. 
            
            
            
  18. Negative binomial (NB)      Negative binomial distribution: variance  is larger than the mean     excellent model  for overdispersed count data     Negative binomial regression     relative  risk     Disadvantage: estimating extra parameter  (dispersion)     PROC GENMOD      PROC COUNTREG 18 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    19. 
            
            
            
  19. SAS code: negative binomial  regression procgenmod data = base; model EPDS = INTV / dist=negbin; run; proccountreg data = base; model EPDS = INTV / dist=negbin; run; 19 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    20. 
            
            
            
  20. SAS output: negative binomial  regression PROC GENMOD Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard  Error Wald 95%  Confidence  Limits Wald  Chi- Square 1973.73 <.0001 2.18 0.1394 Pr >  ChiSq 1 1 1 2.1244 -0.0974 1.0192 0.0478 2.0306 2.2181 0.0659-0.2265 0.0318 0.0514 0.9185 1.1198 Intercept INTV Dispersion PROC  COUNTREG Parameter Estimates Parameter DF Estimate Standard  t Value Approx Error Pr > |t| 1 1 1 2.1244 -0.0974 1.0192 0.0478 0.0659 0.0514 44.43 <.0001 -1.48 0.1394 19.84 <.0001 Intercept INTV _Alpha Compare to Poisson regression:  INTV:  Estimate=-0.0974;  SE=0.0218;P<.0001 20 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    21. 
            
            
            
  21. SAS output: negative binomial regression     NB: Variance > mean Variance = mean + k *mean     SAS estimate of dispersion parameter k:    Dispersion   ,     _Alpha        If k significantly different from zero     use NB rather than  Poisson 2 PROC GENMOD Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard  Error Wald 95%  Confidence  Limits Wald Chi- Square Pr >  ChiSq 1 1 1 2.1244 -0.0974 1.0192 0.0478 2.0306 2.2181 0.0659 -0.2265 0.0318 0.0514 0.9185 1.1198 1973.73 <.0001 2.18 0.1394 Intercept INTV Dispersion PROC COUNTREG Parameter Estimates Estimate Standard  Parameter DF t Value Approx Error Pr > |t| <.0001 0.1394 <.0001 1 1 1 2.1244 -0.0974 1.0192 0.0478 0.0659 0.0514 44.43 -1.48 19.84 Intercept INTV _Alpha 21 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    22. 
            
            
            
  22. Count data: Quasi-likelihood  Estimation (QLE)     QLE allows for adjusting variance without  specifying distribution exactly     Variances inflated by      Deviance/DOF (GENMOD:    dscale   )     Pearson   s Chi-Square/DOF     GENMOD:    pscale       GLIMMIX:    random _residual_        Poisson and negative binomial regression  (and logistic regression) 22 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    23. 
            
            
            
  23. QLE  Example - SAS Code Use    dscale    as the norm! *Poisson regression- no adjustment for  overdispersion; procgenmod data=base; model EPDS = INTV/ dist=poisson; run; *Poisson regression- adjust for overdispersion  using    DSCALE   ; procgenmod data=base; model EPDS = INTV/ dist=poisson dscale; run; 23 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    24. 
            
            
            
  24. QLE- standard errors (SE) corrected Poisson - unadjusted variances Criteria For Assessing Goodness Of Fit Criterion DF Value Deviance 1056 7783.18 Analysis Of Maximum Likelihood Parameter Estimates Value/DF 7.3704 Parameter DF Estimate Standard  Wald 95%  Confidence  Limits Wald  Chi- Square 18805 <.0001 19.95 <.0001 Pr >  ChiSq Error 1 1 0 2.1244 -0.0974 0.0155 0.0218 -0.1 -0.055 0 2.1 2.155 Intercept INTV Scale Note: The scale parameter was held fixed. 1 1 1 Poisson-QLE using "DSCALE" - SE inflated by the square root of Deviance/DOF Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 1056 7783.18 7.3704 Analysis Of Maximum Likelihood Parameter Estimates  =    7.3704 = 2.7148  Parameter DF Estimate Standard  Wald 95%  Confidence  Limits Wald  Chi- Square 2551.4 <.0001 2.71 0.0999 Pr >  ChiSq Error 1 1 0 2.1244 -0.0974 2.7149 0.0421 0.0592 -0.2 0 2 2.207 0.019 2.715 Intercept INTV Scale Note: The scale parameter was estimated by the square root of DEVIANCE/DOF. 2.7 24 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    25. 
            
            
            
  25. QLE- Poisson vs. NB Poisson - unadjusted variances Criteria For Assessing Goodness Of Fit Criterion DF Deviance 1056 7783.177 Analysis Of Maximum Likelihood Parameter  Negative binomial - unadjusted variances Criteria For Assessing Goodness Of Fit Criterion DF Value Deviance 1056 1220.558 Analysis Of Maximum Likelihood Parameter  Value Value/DF 7.3704 Value/DF 1.1558 Parameter INTV DF Estimate -0.0974 SE 0.0218 <.0001 P-Val Parameter INTV DF Estimate -0.0974 SE 0.0659 P-Val 0.1394 1 1 Poisson-QLE using "DSCALE" -  SE inflated by the square root of   Deviance/DOF =    7.3704 = 2.7148  Criteria For Assessing Goodness Of Fit Criterion DF Value Deviance 1056 7783.177 Analysis Of Maximum Likelihood Parameter  NB -QLE using "DSCALE" -  SE inflated by the square root of  Deviance/DOF =    1.1558 = 1.0751  Criteria For Assessing Goodness Of Fit Criterion DF Value Deviance 1056 1220.558 Analysis Of Maximum Likelihood Parameter  Value/DF 7.3704 Value/DF 1.1558 Parameter INTV Note: The scale parameter was estimated by the  square root of DEVIANCE/DOF. DF Estimate -0.0974 SE 0.0592 P-Val 0.0999 Parameter INTV Note: The covariance matrix was multiplied by a  factor of DEVIANCE/DOF. DF Estimate -0.0974 SE 0.0708 P-Val 0.1692 1 1 25 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    26. 
            
            
            
  26. QLE PSCALE- SAS Code procgenmod data=base; model EPSD=INTV / dist=poisson pscale; run; Criterion Pearson Chi-Square Analysis Of Maximum Likelihood Parameter Estimates Criteria For Assessing Goodness Of Fit DF Value 7752.096 Value/DF 1056 7.341 Standard  Error 0.0420 0.0591 0 Wald 95% Confidence  Limits 2.0421 -0.2131 2.7094 Wald Chi- Square 2561.66 2.72 Parameter Intercept INTV Scale Note: The scale parameter was estimated by the square root of Pearson's Chi-Square/DOF. DF 1 1 0 Estimate 2.1244 -0.0974 2.7094 Pr > ChiSq <.0001 0.0992 2.2066 0.0184 2.7094 procglimmix data=base;  model EPSD=INTV / dist=poisson s; random _residual_; run; Fit Statistics Pearson Chi-Square Pearson Chi-Square / DF 7752 7.34 Parameter Estimates Standard  Error 2.1244 0.0420 -0.0974 0.0591 7.341 Effect Intercept INTV Residual Estimate DF 1056 1056 . t Value Pr > |t| 50.61 -1.65 . <.0001 0.0995 . . 26 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    27. 
            
            
            
  27. Count Data  QLE  In Sum     Use as the norm, in Poisson or NB     DSCALE better than PSCALE, especially  for low counts 27 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    28. 
            
            
            
  28. Count Data: Empirical Variance Estimation     Empirical (or robust or sandwich) variance  estimation     account for extra variation by  using both empirical-based estimates and  model-based estimates in variance  estimation      Poisson and NB regression (and logistic  regression)      GENMOD:    REPEATED    statement     GLIMMIX:    EMPIRICAL    option   28 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    29. 
            
            
            
  29. Empirical Variance Estimation  GENMOD    REPEATED    *PID =    Participant ID   , 1 observation per  PID; procgenmod data=base; class PID; model EPDS=INTV / dist=poisson; repeated subject = PID; run; Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Parameter Estimate Standard  Error Intercept 2.1244 0.0415 2.0431 INTV -0.0974 95% Confidence  Limits Z Pr > |Z| 2.2056 0.0182 51.25<.0001 -1.65 0.0986 0.059 -0.2129 Compare to unadjusted Poisson regression:  INTV:  Estimate=-0.0974;  SE=0.0218;P<.0001 29 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    30. 
            
            
            
  30. Empirical Variance Estimation  GLIMMIX    EMPIRICAL    *PID =    Participant ID    1 observation per  PID.    MBN    is small-sample bias correction; procglimmix data=base empirical=mbn; class PID; model EPDS=INTV / dist=poisson s; random _residual_ /subject = PID; run; Effect Estimate Standard  Solutions for Fixed Effects DF t Value Pr > |t| Error 0.04153 0.05907 2.1244 -0.09738 1056 1056 51.16<.0001 -1.65 0.0995 Intercept INTV 30 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    31. 
            
            
            
  31. Count Data: Correlated (ex:  clustered) data     Longitudinal data (clustering of repeated  measurements within subjects)     Nested data (clustering of multiple subjects  within groups)     Poisson, NB, or logistic regression 31 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    32. 
            
            
            
  32. Count Data: Correlated (ex:  clustered) data     Generalized Linear Mixed Models (GLMM)     GLIMMIX    RANDOM INT    [Conditional model,  subject-specific inference]     GLIMMIX    RANDOM _RESIDUAL_    [Marginal  model, inference on population averages]     Generalized Estimating Equations (GEE)  [Marginal model]     GENMOD    REPEATED        Small-sample bias correction in GLIMMIX with     EMPIRICAL=mbn    option  32 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    33. 
            
            
            
  33. Clustered Data  GLMM *Participants clustered by city. Marginal model; procglimmix data=base; class city; model EPDS=INTV / dist=nb s; random _residual_ / subject=city type=cs; run; *Participants clustered by city. Conditional model; procglimmix data=base; class city; model EPDS=INTV / dist=nb s; random int /subject=city; run; 33 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    34. 
            
            
            
  34. Clustered Data  GEE *Participants clustered by city.    MBN    is  small-sample bias correction; procglimmix data=base empirical=mbn; class city; model EPDS=INTV / dist=nb s; random _residual_ /subject =city; run; procgenmod data=base; class city; model EPDS=INTV / dist=nb; repeated subject = city / type=cs; run; 34 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    35. 
            
            
            
  35. Count Data - Zero-Inflated (ZI)  Models     ZI models appropriate when variable contains an  excess of zero values- sample heterogeneity      Assume sample contains two different populations:     nonsusceptible    (always zero) subjects and     susceptible    (not always zero) subjects     ZI regression - two regression models (each with  own explanatory variables):      Logit or probit regression - model the probability of  being    nonsusceptible        Poisson/NB/logistic regression - model the mean for  the susceptible population 35 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    36. 
            
            
            
  36. Count data - ZI in SAS     PROC GENMOD     PROC COUNTREG     Zero-inflated Poisson:    dist=ZIP        Zero-inflated NB:    dist=ZINB        Even after accounting for excess zeros, NB  may fit the remaining counts better than  Poisson     GENMOD ZINB: SAS version > 9.2 36 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    37. 
            
            
            
  37. Example - ZI     Variable of interest - count: number of fish  caught by groups of campers at a national  park     Explanatory variables:      Number of children in the group (child)     Whether or not the group brought a camper to  the park (camper)     Number of people in the group (persons) 37 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    38. 
            
            
            
  38. 57% of values are zero values 38 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    39. 
            
            
            
  39. Example  ZI  SAS Code procgenmod data = m.fish;  model count = child camper /dist=zip;  zeromodel persons /link = logit ;  run;  proccountreg data = m.fish method = qn;  model count = child camper / dist=zip;  zeromodel count ~ persons;  run; 39 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    40. 
            
            
            
  40. Example  ZI  SAS Output GENMOD Analysis Of Maximum Likelihood Parameter Estimates DF Estimate Standard  Error Parameter Wald 95%  Confidence  Limits 1.4302 -1.239 0.6505 1 Wald  Chi- Square 348.96<.0001 108.78<.0001 79.35<.0001 Pr > Chi Sq 1 1 1 0 1.5979 -1.0428 0.834 0.0855 1.7655 -0.847 1.0175 Intercept child camper Scale Analysis Of Maximum Likelihood Zero Inflation Parameter Estimates Parameter DF Estimate Standard  0.1 0.0936 1 0 1 Wald 95%  Confidence  Limits 0.5647 -0.884 Wald  Chi- Square 12.04 11.99 Pr > Chi Sq Error 1 1 1.2974 -0.5643 0.3739 0.163 2.0302 -0.245 0.0005 0.0005 Intercept persons COUNTREG Parameter Estimates DF Estimate Standard  Parameter t Value Approx Error Pr > |t| 1 1 1 1 1 1.59789 -1.04284 0.83402 1.29744 -0.56435 0.08554 0.09999 0.09363 0.37385 0.16296 18.68<.0001 -10.43<.0001 8.91<.0001 3.47 -3.46 Intercept child camper Inf_Intercept Inf_persons 0.0005 0.0005 40 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    41. 
            
            
            
  41. Accounting for overdispersion in  SAS: binary data     Random-clumped binomial and beta- binomial models     Zero-inflated binomial (ZIB)     Variance-adjustment models     Quasi-likelihood Estimation     Empirical (aka robust, sandwich) variance  estimation      Models for correlated data 41 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    42. 
            
            
            
  42. Binary Data  BB and RCB     Beta-binomial (BB) and random-clumped  binomial (RCB)      Model physical mechanism behind  overdispersion     PROC NLMIXED     SAS 9.3     PROC FMM (experimental) 42 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    43. 
            
            
            
  43. 43 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    44. 
            
            
            
  44. Example  BB and RCB . . . n=1 337     n=337 nuclei     Each nucleus has m=3 total number  of chromosome pairs     t: number of chromosome pairs with  association at meiosis (t=0, 1, 2, 3)     If probability of association at  meiosis (t/m) is constant for all  nuclei and the same for all  chromosome pairs, then binomial  distribution appropriate     If not     RCB or BB  44 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    45. 
            
            
            
  45. Example  BB and RCB Data 45 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    46. 
            
            
            
  46. BB and RCB  PROC NLMIXED 46 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    47. 
            
            
            
  47. Binary data: ZIB     ZI model     simultaneously model the probability of  being    always zero    and the probability of the event  of interest conditional on being in the    not always  zero    population 47 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    48. 
            
            
            
  48. Binary Data - QLE - Example Cases of toxoplasmosis in 34 cities in El Salvador tox: 1=toxoplasmosis case, 0=no toxoplasmosis rain: annual rainfall (cm) Binary:    tox                     Grouped:    num_tox/num_total    city tox Subject 1 1 1 1 1 2 1 0 3 1 0 4 2 1 1 2 0 2 2 0 3 city 1 2 num_tox 2 1 num_total 4 3 48 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    49. 
            
            
            
  49. QLE  SAS Code *Grouped binary (1 observation for each  city); procgenmod data=tox; model num_tox/num_total = rain | rain |  rain/ dist=bin dscale; run; *Binary   multiple observations per city; procgenmod data=test desc; model tox  = rain | rain | rain/  dist=bin dscale aggregate=city; run; 49 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    50. 
            
            
            
  50. QLE  SAS Output Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard  Error Wald 95%  Confidence  Limits Wald  Chi- Square Pr > Ch iSq 1 1 1 1 0 0.0994 -0.449 -0.187 0.2134 1.4449 0.1473 -0.189 0.3882 0.2242 -0.888 -0.009 0.1322 -0.447 0.0719 0.092 0.033 0.3938 01.4449 1.4449 0.46 0.4999 4 0.0454 2.01 0.1568 5.38 0.0204 Intercept rain rain*rain rain*rain*rain Scale Note: The scale parameter was estimated by the square root of  DEVIANCE/DOF. 50 Jessica Harwood                                   CHIPTS Methods Seminar 1/8/2013
    51. 
            
            
            
							

							

					

					
					
					

						
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