Transcript Stata 3, Regression

Stata 3, Regression
Hein Stigum
Presentation, data and programs at:
http://folk.uio.no/heins/
Apr-15
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Agenda
• Linear regression
• GLM
• Logistic regression
• Binary regression
• (Conditional logistic)
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Linear regression
Birth weight
by
gestational age
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Regression idea
5000
model: y  b0  b1 x  e
4000
e  error,residual
2500
3000
b1  coefficient , effectof x
3500
4500
y = outcome
x = covariate
240
260
280
gestational age (days)
300
320
model with manycofactors: y  b0  b1 x1  b2 x2  e
x1 , x 2 = covariate
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Model and assumptions
• Model
y  0  1x1  2 x2   ,   N (0, 2 )
• Assumptions
– Independent errors
– Linear effects
– Constant error variance
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Association measure: RD
Model:
Start with:
y  β0  β1 x1  β2 x2
RD1  y x1  2  y x1 1
 β0  2 β1  β2 x2
 β0  1β1  β2 x2

Hence:
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β1
RD1  β1
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Purpose of regression
• Estimation
– Estimate association between outcome and
exposure adjusted for other covariates
• Prediction
– Use an estimated model to predict the
outcome given covariates in a new dataset
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Adjusting for confounders
True value
True value
Unadjusted estimate
Adjusted estimate
• Not adjust
– Cofactor is a collider
– Cofactor is in causal path
• May or may not adjust
– Cofactor has missing
– Cofactor has error
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Workflow
• Scatterplots
• Bivariate analysis
• Regression
– Model fitting
• Cofactors in/out
• Interactions
– Test of assumptions
• Independent errors
• Linear effects
• Constant error variance
– Influence (robustness)
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4000
3000
2000
weight
5000
6000
Scatterplot
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400
500
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700
gest
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Syntax
• Estimation
– regress y x1 x2
– xi: regress y x1 i.c1
linear regression
categorical c1
• Post estimation
– predict yf, xb
predict
• Manage models
– estimates store m1
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save model
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Model 1: outcome+exposure
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Model 2: Add counfounders
Estimate association:
m1=m2
Prediction: m2 is best
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”Dummies”
Assume educ is coded 1, 2, 3 for low, medium and high education
Choose low educ as reference
Make dummies for the two other categories:
generate medium=(educ==2) if educ<.
generate high =(educ==3) if educ<.
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Interaction
Model:
Start with:
y  β0  β1 x1  β2 x2  β3 x1sex
RD1  y x1 2  y x1 1
 β0  2 β1  β2 x2  β3 2 sex
 β0  1β1  β2 x2  β31sex

Hence:
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β1  β3sex
RD1  β1  β3sex
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Model 3: with interaction
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Test of assumptions
• Predict y and residuals
– predict y, xb
– predict res, resid
2000
0
-1000
– independent?
– linear?
– const. var?
1000
• Plot resid vs y
twoway (scatter res y )(qfitci res y)
3200
3400
3600
3800
Linear prediction
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Violations of assumptions
• Dependent residuals
.5
1
Mixed models: xtmixed
-1
200
220
240 260
gest
280
300
0
-1
-2
regress weigth gest sex, robust
res
• Non-constant variance
1
2
gen gest2=gest^2
regress weigth gest gest2 sex
-.5
0
• Non linear effects
3400
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3600
p
3700
18
3800
.2
Measures of influence
-.6
-.4
-.2
0
Remove obs 1, see change
remove obs 2, see change
1
2
10
Id
• Measure change in:
– Outcome (y)
– Deviance
– Coefficients (beta)
• Delta beta, Cook’s distance
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Points with high influence
.8
lvr2plot, mlabel(id)
.4
.2
Leverage
.6
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445
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.01
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Normalized residual squared
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Added variable plot: gestational age
2000
avplot gest, mlabel(id)
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293
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380
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143
440
404
256
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281
203
286
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136
468
172
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197
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542
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226
278
241
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457
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11
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410
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364
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65
375
434
42
175
91
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223
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69
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464
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497
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573
81
523
452
580
397
239
222
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505
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103
479
403
294
319
334
210
128
59
275
536
70
168
541
186
527
557
152
502
322
96
462
125
142
4
242
384
353
109
575
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389
515
76
181
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299
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312
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290
184
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385
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350
145
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360
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539
100
200
e( gest | X )
300
400
coef = 5.6762662, se = 1.1303444, t = 5.02
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Removing outlier
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6000
Influence
5000
Regression
without outlier
4000
Regression with outlier
2000
3000
Outlier
200
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300
400
500
Gestational age
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600
700
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Final model
y  0  1x1  2 x2   ,   N (0, )
2
Give meaning to constant term:
sum gest
generate gest2=gest-204
generate sex2=sex-1
regress weight gest2 sex2
estimates store m4
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/* find smallest value */
/* smallest gest=204 */
/* boys=0, girls=1 */
/* final model */
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Logistic regression
Being bullied
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Model and assumptions
• Model

ln(
)  0  1 x1   2 x2 ,   E ( y | x ), y | x ~ Binom ial
1 
• Assumptions
– Independent residuals
– Linear effects
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Association measure, Odds ratio
Model:
Start with:
  
  β0  β1 x1  β2 x2
ln
 1- 
 Oddsx1 2 

ln (OR1 )  ln
 Oddsx 1 
1


ln (Oddsx1 2 )  ln (Oddsx1 1 )
 β0  2 β1  β2 x2
 β0  1β1  β2 x2

Hence:
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β1
OR1  exp(β1 )
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Syntax
• Estimation
– logistic y x1 x2
– xi: logistic y x1 i.c1
logistic regression
categorical c1
• Post estimation
– predict yf, pr
predict probability
• Manage models
– estimates store m1
– est table m1, eform
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save model
show OR
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Workflow
• Bivariate analysis
• Regression
– Model fitting
• Cofactors in/out
• Interactions
– Test of assumptions
• Independent errors
• Linear effects
– Influence (robustness)
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Bivariate
Generate dummies
gen Island=
gen Norway=
gen Finland=
gen Denmark=
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(country==2) if country<.
(country==3)
(country==4)
(country==5)
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Model 1: outcome and exposure
Alternative commands:
xi:logistic bullied i.country
use xi: i.var for categorical variables
xi:logistic bullied i.country , coef
coefs instead of OR's
xi:logistic bullied i.country if sex!=. & age!=. do if sex and age not missing
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Model 2: Add confounders
Estimate associations: m1=m2
Predict:
m2 best
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Interaction
Model:
Start with:
  
  β0  β1 x1  β2 sex  β3 x1sex
ln
 1- 
 Oddsx1 2 

ln (OR1 )  ln
 Oddsx 1 
1


ln (Oddsx1  2 )  ln (Oddsx1 1 )
 β0  2 β1  β2 sex  β3 2sex
 β0  1β1  β2 sex  β31sex

Hence:
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β1  β3 sex
OR1  exp(β1  β3 sex)
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Model 3: interaction
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Test of assumptions
• Linear effects (of age)
.4
0
.2
coefficient
.6
.8
– findit lincheck
search and install
– lincheck xi:logistic bullied age I.country sex
5
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age
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Points with high influence
restore best model
probability (mu in our notation)
delta-beta (one value, not one per estimate)
delta-beta plot
0
.2
.4
Pregibon's dbeta
.6
.8
estimates restore m2
predict p, p
predict db, db
scatter db p
.05
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.1
.15
.2
Pr(bullied)
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.25
.3
36
Removing 2 observations
Conclusion:
Robust results
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Generalized Linear Models
Being bullied
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Designs and measures
Natural association
measures
Design
Frequency measures
Cross-sectional
Prevalence
Cohort
Incidence risk
Incidence rate
Case Control
Traditional CC
Case Cohort
Nested Case Control
Risk Ratio, Risk Difference
Risk Ratio, Risk Difference
Rate Ratio
Odds Ratio
Rate Ratio
Rate Ratio
Models
GLM
Survival
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Measures
RR, RD, OR
Rate Ratio
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2500 3000 3500 4000 4500 5000
Generalized Linear Models, GLM
Linear regression
250
270
290
310
.6
.8
1
gestational age (days)
0
.2
.4
risk
Logistic regression
20
40
age
60
80
10
15
0
0
5
Poisson regression
0
20
Apr-15
40
60
80
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GLM: Distribution and link
• Distribution family
– Given by data
– Influence p-value, CI
Normal
Binomial
Poisson
– May chose
– Shape (=link-1)
– Scale
Identity
Logit
Log
Additive
Multi.
Multi.
– Association measure
RD
OR
RR
• Link function
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Distribution and link examples
Data
Distribution
Standard models
Continuous Normal
Count
Poisson
0/1
Binomial
Other models
0/1
Binomial
0/1
Binomial
Link
Measure
Name
Identity
Log
Logit
RR
OR
Linear
Poisson
Logistic
Log
Identity
RR
RD
Log binomial
Linear binomial
OBS: not for traditional case control data
Link: Identity  linear model  additive scale
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Being bullied, 3 models
glm bullied Island Norway Finland Denmark sex age, family(binomial) link(logit)
glm bullied Island Norway Finland Denmark sex age, family(binomial) link(log)
glm bullied Island Norway Finland Denmark sex age, family(binomial) link(identity)
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Convergence problems
Stop
• If glm does not converge, use:
– poisson y x1 x2, irr robust
– regress y x1 x2,
robust
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RR
RD
44
Association measure, RR
Model:
Start with:
ln    β0  β1 x1  β2 x2  ...
  x1 2 

ln (RR1 )  ln
  x 1 
 1 
ln ( x1 2 )  ln ( x1 1 )
 β0  2 β1  β2 x2  ...
 β0  1 β1  β2 x2  ...

Hence:
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β1
RR1  exp(β1 )
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Association measure: RD
Model:
Start with:
  β0  β1 x1  β2 x2  ...
RD1   x1 2   x1 1
 β0  2 β1  β2 x2  ...
 β0  1β1  β2 x2  ...

Hence:
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β1
RD1  β1
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30
The importance of scale
20
Females
Multiplicative scale
Absolute increase
Relative increase
Females: 30-20=10
Males: 20-10=10
Females: 30/20=1.5
Males: 20/10=2.0
Conclusion:
Same increase for
males and females
Conclusion:
More increase for
males
RD
RR
0
10
Males
Additive scale
T1
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T2
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Stata regression commands
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• Regression with simple error structure
– regress linear regression (also heteroschedastic errors)
– nl
non linear least squares
• GLM
– logistic logistic regression
– poisson Poisson regression
– binreg binary outcome, OR, RR, or RD effect measures
• Conditional logistc
– clogit
for matched case-control data
• Multiple outcome
– mlogit
– ologit
multinomial logit (not ordered)
ordered logit
• Regression with complex error structure
– xtmixed linear mixed models
– xtlogit random effect logistic
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• Estimation
Syntax
– regress y x1 x2
– logistic y x1 x2
– xi:regress y x1 i.x2
linear regression
logistic regression
categorical x2
• Manage results
– estimates store m1
– estimates table m1 m2
– estimates stats m1 m2
store results
table of results
statistics of results
• Post estimation
– predict y, xb
– predict res, resid
– lincom b0+2*b3
linear prediction
residuals
linear combination
• Help
– help logistic postestimation
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