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04/18/23 H.S. 1
Stata: Linear Regression
Stata 3, linear regression
Hein Stigum
Presentation, data and programs at:
http://folk.uio.no/heins/
courses
04/18/23 H.S. 4
Regression idea
residual error,e
xofeffect ,tcoefficienb
covariate =x
outcome=y
:model
1
10
exbby
covariate = x,x
:cofactorsmany with model
21
22110 exbxbby
2500
3000
3500
4000
4500
5000
birt
h w
eigh
t (gr
am
)
250 260 270 280 290 300 310gestational age (days)
04/18/23 H.S. 5
Model, measure and assumptions
• Model
• Association measure1 = change in y for one unit increase in x1
• Assumptions– Independent errors
– Linear effects
– Constant error variance
• Robustness– influence
),0(, 222110 Nxxy
04/18/23 H.S. 6
Association measure
11
1
2210
2210
121
22110
1
211
βy
β
xβββ
xβββ
yyy
xβxββy
xx
Model:
Start with:
Hence:
04/18/23 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
04/18/23 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
04/18/23 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 plots
Scatter of birth weight by gestational age
Distribution of birth weight for low/high gestational age
04/18/23 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
00B
irth
wei
ght (
gr)
240 260 280 300 320 340Gestational age
04/18/23 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
04/18/23 H.S. 12
Model 1: outcome+exposure
regress bw gest crude model
estimates store m1 store model results
04/18/23 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<.
04/18/23 H.S. 14
Create meaningful constant
Expected birth weight at:
sexeduceducgestbwE 43210 32)(
gr1572 0
gest= 0, educ=1, sex=0, not meaningful
gest=280, educ=1, sex=0 gr342628010
Expected birth weight:
Margins:margins, at(gest= 0 educ=1 sex=0) = -1572 not meaningful
margins, at(gest= 280 educ=1 sex=0) = 3426
04/18/23 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
04/18/23 H.S. 16
Would normally check for interaction now!
04/18/23 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
04/18/23 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-.
50
.51
200 220 240 260 280 300gest
-2-1
01
2re
s
3400 3500 3600 3700 3800p
04/18/23 H.S. 21
Influence idea
outlier
regression without outlier
regression with outlier
020
0040
0060
00B
irth
wei
ght (
gr)
250 300 350 400Gestational age (days)
04/18/23 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
-.2
0.2
Influ
ence
1 2 10Id
1 2 34 56
7
8910111213
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22
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68697071727374 7576
77787980 818283 84
85
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9596
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0.0
05
.01
.01
5L
eve
rag
e
0 .002 .004 .006 .008Normalized residual squared
04/18/23 H.S. 23
Leverage versus residuals2
lvr2plot, mlabel(id)
high influ
ence
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
04/18/23 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
04/18/23 H.S. 25
Removing outlier
regress bw gest i.educ sex if id!=370est store m4est table m3 m4, b(%8.1f)
Removing outlier
04/18/23 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
04/18/23 H.S. 27
bw2: Non-linear effects
04/18/23 H.S. 29
1000
2000
3000
4000
5000
6000
Birt
h w
eigh
t (gr
)
240 260 280 300 320Gestational age
Handle:add
polynomialor
spline
Non-linear effects: polynomial
04/18/23 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
25
00
30
00
35
00
40
00
Lin
ea
r P
red
ictio
n
250 260 270 280 290 300 310Gestational age
Predictive Margins with 95% CIs
Non-linear effects: spline• Qubic spline
• Plot
• Linear spline
04/18/23 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
250
03
000
350
04
000
Lin
ear
Pre
dict
ion
250 260 270 280 290 300 310igest
Predictive Margins with 95% CIs
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)
– effect of x1 depends on x2
04/18/23 H.S. 33
bw3: Interaction (only linear effects)
• Add interaction terms
• Show results
04/18/23 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
04/18/23 H.S. 35