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Mar-20 H.S. 1
Stata:Linear Regression
Stata 3, linear regression
Hein Stigum
Presentation, data and programs at:
http://folk.uio.no/heins/
courses
SYNTHETIC DATA EXAMPLEBirth weight by gestational age
Mar-20 H.S. 2
Mar-20 H.S. 3
Linear regression
Birth weight
by
gestational age
Mar-20 H.S. 4
Regression idea
residual error,e xofeffect ,tcoefficienb
covariate =xoutcome=y
:model
1
10
==
++= exbby
covariate = x,x :cofactorsmany with model
21
22110 exbxbby +++=
2500
3000
3500
4000
4500
5000
birth
wei
ght (
gram
)
250 260 270 280 290 300 310gestational age (days)
Mar-20 H.S. 5
Model, measure and assumptions
• Model
• Association measureb1 = change in y for one unit increase in x1
• Assumptions– Independent errors– Linear effects– Constant error variance
• Robustness– influence
),0(, 222110 seebbb Nxxy µ+++=
Mar-20 H.S. 6
Association measure
11
1
2210
2210
121
22110
12
11
βy
β xβββxβββ
yyy
xβxββy
xx
=D
=---++=
-=D
++=
==
Model:
Start with:
Hence:
Mar-20 H.S. 7
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
Outcome distributions by exposure
Exposed Unexposed
-3 0 1 4Outcome
Mar-20 H.S. 8
Exposed Unexposed
-3 -2 -1 0 1 2 3Outcome
Linear regression
Quantile regressionor
cutoff, logistic regression
0 2 4 6 8Outcome
Linear regressionor
transform,linear regression
Mar-20 H.S. 9
Workflow• DAG• Scatter- and densityplots• Bivariate analysis• Regression
– Model estimation– Test of assumptions
• Independent errors• Linear effects• Constant error variance
– Robustness • Influence
Egest age
Dbirth weight
C2education
C1sex
Scatter and density plotsScatter of birth weight by
gestational ageDistribution of birth weight for
low/high gestational age
Mar-20 H.S. 10
gest<280 days gest>=280 days
0 2000 4000 6000Birth weight (gr)
Look for deviations from linearityand outliers Look for shift in shape
3704962
020
0040
0060
00
Birth
wei
ght (
gr)
240 260 280 300 320 340Gestational age
Mar-20 H.S. 11
Syntax• Estimation
– regress y x1 x2 linear regression
– regress y c.age i.sex continuous age, categorical sex
– regress y c.age##i.sex main+interaction
• Compare models– estimates store m1 save model
– estimates table m1 m2 compare coefficients
– estimates stats m1 m2 compare model fit
• Post estimation– predict res, residuals predict residuals
Mar-20 H.S. 12
Model 1: outcome+exposure
regress bw gest crude model
estimates store m1 store model results
Mar-20 H.S. 13
Model 2 and 3: Add covariates
Estimate association:m1 is biased, m2=m3
Prediction: m3 is best
regress bw gest i.educ sex add covariatesestimates table m1 m2 m3 compare coefs
estimates stats m1 m2 m3 compare fit
m3 more precise?
Factor (categorical) variables
• Variable– educ = 1, 2, 3 for low, medium and high education
• Built in– i.educ use educ=1 as base (reference)
– ib3.educ use educ=3 as base (reference)
• Manual “dummies”– educ=1 as base, make dummies for 2 and 3– generate Medium =(educ==2) if educ<.– generate High =(educ==3) if educ<.
Mar-20 H.S. 14
Create meaningful constant
Expected birth weight at:
sexeduceducgestbwE ×+×+×+×+= 43210 32)( bbbbb
gr1572 0 -=b!
gest= 0, educ=1, sex=0, not meaningful
gest=280, educ=1, sex=0 gr342628010 =×+ bb!!
Expected birth weight:
Margins:margins, at(gest= 0 educ=1 sex=0) = -1572 not meaningful
margins, at(gest= 280 educ=1 sex=0) = 3426
Mar-20 H.S. 15
coeff 95% conf. Int.Birth weight at ref 3426 (3385 , 3467)Gestational age
per day 17.9 (16 , 20)Education
Low 0Medium 71.5 (25 , 118)High 99.1 (51 , 148)
SexBoy 0Girl -154.3 (-187 , -121)
Results so far
Mar-20 H.S. 16
Would normally check for interaction now!
ASSUMPTIONS
Mar-20 H.S. 17
Mar-20 H.S. 18
Test of assumptions• Assumptions
– Independent residuals:– Linear effects:– Constant variance:
-300
0-2
000
-100
00
1000
2000
2500 3000 3500 4000 4500Linear prediction
estat hettestp=0.9 no heteroskedasticity
discuss
plot residuals versus predicted y
predict res, residualspredict pred, xbscatter res pred
Mar-20 H.S. 19
Violations of assumptions• Dependent residuals
Use mixed models or GEE
• Non linear effectsAdd square term or spline
• Non-constant varianceUse robust variance estimation
-1-.5
0.5
1
200 220 240 260 280 300gest
-2-1
01
2res
3400 3500 3600 3700 3800p
ROBUSTNESSMeasures of influence
Mar-20 H.S. 20
Mar-20 H.S. 21
Influence idea
outlier
regression without outlier
regression with outlier
020
0040
0060
00
Birth
wei
ght (
gr)
250 300 350 400Gestational age (days)
Mar-20 H.S. 22
Measures of influence
• Measure change in:– Predicted outcome – Deviance– Coefficients (beta)
• Delta beta
Remove obs 1, see changeremove obs 2, see change
-.6-.4
-.20
.2
Influence
1 2 10Id
1 2 34 56
78
9101112131415 16171819 2021
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34 353637383940
41424344
454647484950
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68697071727374 7576
77787980 818283 84
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49995000
0.0
05.0
1.0
15Le
vera
ge
0 .002 .004 .006 .008Normalized residual squared
Mar-20 H.S. 23
Leverage versus residuals2
lvr2plot, mlabel(id)
high influence
370
4962
-.8-.6
-.4-.2
0.2
Dfb
eta
gest
0 1000 2000 3000 4000 5000id
beta(gest)= 17.9
Delta-beta for gestational age
Mar-20 H.S. 24
dfbeta(gest)scatter _dfbeta_1 id
OBS, variable specific
If obs nr 370 is removed, beta will change from 17.9 to 18.6
Mar-20 H.S. 25
Removing outlier
regress bw gest i.educ sex if id!=370est store m4est table m3 m4, b(%8.1f)
Removing outlier
Mar-20 H.S. 26
Full model N=5000 Outlier removed N=4999
One outlier affected several estimates
Final model
coeff 95% conf. Int.Birth weight at ref 3426 (3385 , 3467)Gestational age
per day 17.9 (16 , 20)Education
Low 0Medium 71.5 (25 , 118)High 99.1 (51 , 148)
SexBoy 0Girl -154.3 (-187 , -121)
coeff 95% conf. Int.Birth weight at ref 3433 (3391 , 3474)Gestational age
per day 18.5 (17 , 20)Education
Low 0Medium 64.2 (18 , 110)High 88.6 (40 , 137)
SexBoy 0Girl -152.7 (-185 , -120)
Help
• Linear regression– help regress
• syntax and options– help regress postestimation
• dfbeta• estat hettest• lvr2plot• predict• margins
Mar-20 H.S. 27
NON-LINEAR EFFECTSbw2
Mar-20 H.S. 28
bw2: Non-linear effects
Mar-20 H.S. 29
1000
2000
3000
4000
5000
6000
Birth
wei
ght (
gr)
240 260 280 300 320Gestational age
Handle:add
polynomialor
spline
Non-linear effects: polynomial
Mar-20 H.S. 30
regress bw2 c.gest##c.gest i.educ sex 2. order polynomial in gest
margins, at(gest=(250(10)310)) predicted bw2 by gestmarginsplot plot
2500
3000
3500
4000
Line
ar P
redi
ctio
n
250 260 270 280 290 300 310Gestational age
Predictive Margins with 95% CIs
Non-linear effects: spline• Qubic spline
• Plot
• Linear spline
Mar-20 H.S. 31
mkspline g=gest, cubic nknots(4) make spline with 4 knotsregress bw2 g1 g2 g3 i.educ sex regression with spline
gen igest=5*round(gest/5) 5-year integer values of gest margins, over(igest) predicted bw by gestmarginsplot
mkspline g1 280 g2=gest make linear spline with knot at 280regress bw2 g1 g2 i.educ sex regression with spline
2500
3000
3500
4000
Line
ar P
redi
ctio
n
250 260 270 280 290 300 310igest
Predictive Margins with 95% CIs
INTERACTIONbw3
Mar-20 H.S. 32
Interaction definitions
• Interaction: combined effect of two variables
• Scale– Linear models additive
• y=b0+b1x1+b2x2 both x1 and x2 = b1+b2
– Logistic, Poisson, Cox multiplicative
• Interaction– deviation from additivity (multiplicativity)
Û– effect of x1 depends on x2
Mar-20 H.S. 33
bw3: Interaction (only linear effects)
• Add interaction terms
• Show results
Mar-20 H.S. 34
regress bw3 c.gest##i.sex i.educ gest-sex interaction
margins, dydx(gest) at(sex=0) effect of gest for boysmargins, dydx(gest) at(sex=1) effect of gest for girls
Summing up 1• Build model
– regress bw gest crude model– est store m1 store– regress bw gest i.educ sex full model– est store m2– est table m1 m2 compare coefficients
• Interaction– regress bw3 c.gest##i.sex i.educ test interaction– margins, dydx(gest) at(sex=0) gest for boys
• Assumptions– predict res, residuals residuals– predict pred, xb predicted– scatter res pred plot
Mar-20 H.S. 35
Summing up 2
• Non-linearity (linear spline)– mkspline g1 280 g2=gest spline with knot at 280– regress bw2 g1 g2 i.educ sex regression with spline
• Robustness– dfbeta(gest) delta-beta– scatter _dfbeta_1 id plot versus id
Mar-20 H.S. 36
