Stata 3, Linear Regression v3

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

SYNTHETIC DATA EXAMPLEBirth weight by gestational age

04/18/23 H.S. 2

04/18/23 H.S. 3

Linear regression

Birth weight

by

gestational age

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!

ASSUMPTIONS

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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

ROBUSTNESSMeasures of influence

04/18/23 H.S. 20

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

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0.2

Influ

ence

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49995000

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



Low 0Medium 71.5 (25 , 118)High 99.1 (51 , 148)

SexBoy 0Girl -154.3 (-187 , -121)



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

NON-LINEAR EFFECTSbw2

04/18/23 H.S. 28

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

INTERACTIONbw3

04/18/23 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)

– 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

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

04/18/23 H.S. 36

Stata 3, Linear Regression v3

Documents

educuse educ

linear regressionstata

model fitpost

estimated model

outcome distributions

lowhigh gestational

sexcontinuous age

categorical sexregress