1. Standard Statistical analysis Linear-, logistic- and Cox-regression
Hein Stigum
Presentation, data and programs at:
courses
2 h talk:
Part 1: start->robustness
Part 2: recap, robustness -> end
Jan-19
H.S.

2. Goal of analysis Adjusting removes confounding and some selection bias
Outcome
How much: Mean, risk, rate
Bi-variate analysis (crude or unadjusted)
Exposure
More among exposed: Diff. in mean, RR or OR
Multivariable analysis (adjusted, regression)
Adjusting removes confounding
and some selection bias
Jan-19
H.S.

3. Workflow DAG Scatter- and density plots Bivariate analysis Regression
Model estimation
Test of assumptions
Independent errors
Linear effects
No interactions
(Constant error variance)
Influence E
gest age D
birth weight C2
education C1
sex
Model fitting
Adjust if cofactor is confounder, problems cofacor has missing, cofactor has error, cofactor is in causal path
Test of assumptions
Stata: Estimation remains in memory. Post estimation commands
Normally do assumtions first, then influence.
Note leverage value of outlier
Jan-19
H.S.

4. Purpose of regression Estimation Prediction
Estimate association between exposure and outcome adjusted for other covariates
Prediction
Use an estimated model to predict the outcome given covariates in a new dataset
DAGs, bias, precision
Predictive power, model fit, R2
Counfounding matter in the first
Fit of the models matters in the last
Jan-19
H.S.

5. Outcome and regression-types
Numerical data
Continuous (weight) Linear regression
Count (partners) Poisson regression
Categorical data
Binary (0,1)
Risk Linear-risk-, log-risk model
Rate Cox, Poisson
Odds logistic
Ordinal (small,med,large) ologit
Multinomial (m. status) mlogit
1/1/2019
H.S.

6. ANALYSIS 1, CONTINUOUS OUTCOME
H.S.

7. Data: m:\pc\dokumenter\a_Courses\Master UiO\birth1.sav
SPSS from Kiosk
Data: m:\pc\dokumenter\a_Courses\Master UiO\birth1.sav
H.S.

8. Table 1 Outcome: Birth weight Exposure: Gestational age Covariates:
Jan-19
H.S.

9. Table 2 Outcome: Birth weight Exposure: Gestational age Adjusted for:
Education and sex
Change to crude and adjusted models
Jan-19
H.S.

10. DAG: Gestational age and BirthweightC2
education C1
sex E
gest age D
birth weight
Birth weight analysis
Continuous outcome
Plots by gestational age
Compare means
Linear regression
Jan-19
H.S.

11. Regression idea Jan-19 H.S. X=gest, Y=weigth
straigth line: y=constant +slope times x
beta0=constant or intercept, beta1=reg. coef. Or slope, y=dep. continous, x=..
Jan-19
H.S.

12. Model, measure and assumptions
Association measure
b1 = change in y for one unit increase in x1
Assumptions
Independent errors
Linear effects
Constant error variance
Influence
In GLM b-s and betas are coefficients
In some fields (and in SPSS) b-s are coefs and betas are standardizes coefs
Jan-19
H.S.

13. Outcome distributions by exposure
Linear regression
cutoff,
logistic regression
Linear regression or
Log-transform,
linear regression
Jan-19
H.S.

14. Scatter and density plots
Distribution of birth weight for low/high gestational age
Scatter of birth weight by gestational age
Look for deviations from linearity
and outliers
Look for shift in shape
Jan-19
H.S.

15. Bi-variate Weight by sex (continuous by binary)
ttest bw, by(sex) t-test
Weight by education (continuous by categorical-3)
anova bw, by(educ) one way anova
Weight by gest. age (continuous by continuous)
regress bw gest regression
ttest bw, by(gest2) cut in 2, t-test
Jan-19
H.S.

16. Bi-variate result
Jan-19
H.S.

17. Model 1: outcome+exposure
regress bw gest crude model
estimates store m1 store model results
Jan-19
H.S.

18. Model 2 and 3: Add covariates
regress bw gest i.educ sex add covariates
estimates table m1 m2 m3 compare coefs
Estimate association:
m1 is biased, m2=m3
m3 more precise?
m2: se(gest)=0.934
m3: se(gest)=0.926
m2: se(gest)=0.934
m3: se(gest)=0.926
estimates stats m1 m2 m3 compare fit
Prediction:
m3 is best
Jan-19
H.S.

19. Results so far
Westreich & Greenland, The table 2 fallacy, 2013
Jan-19
H.S.

20. Assumptions
Jan-19
H.S.

21. Test of assumptions Assumptions Plot (Test) Independent residuals:
Linear effects:
Constant variance:
No interactions
Plot
(Test)
discuss
plot residuals versus predicted y
predict res, residuals
predict pred, xb
scatter res pred
Dependent residuals if many children from same mother
estat hettest
p=0.2
no heteroskedasticity
Jan-19
H.S.

22. Violations of assumptions
Dependent residuals
robust(cluster) or mixed model
Non linear effects
Add square term or spline
Non-constant variance
Use robust variance estimation
regress y x, robust
Interactions
Add product (interaction) terms
Robust variance estimation with cluster variable: regress y x1 x2, vce(cluster village)
Linear mixed models: xtmixed
Robust variance: regress y x1 x2, robust
Jan-19
H.S.

23. Influence Measures of influence Jan-19 H.S.
Easier to understand influence when effects are linear and without interaction
Influence
Jan-19
H.S.

24. Influence idea (different data)
delta beta*se=-6.8
Have taken N down to 1000 and moved outlier out to gest=400
Beta changes from 10 to 17 when removing influential outlier
Est change=Delta beta*se(gest)=-3.5*2.1=-6.8
Influence=“leverage*residual”
Jan-19
H.S.

25. Delta-beta for gestational age
dfbeta(gest) create delta-beta
scatter _dfbeta_1 id plot vs id-variable
OBS, variable specific
If obs nr 370 is removed, beta will change se’s
-0.7*0.9 = -0.6 gr
17.9-(-0.6)=18.5 gr
Jan-19
H.S.

26. Removing outlier regress bw gest i.educ sex if id!=370 est store m4
est table m3 m4, b(%8.1f)
Est change=delta-beta*se=-0.7*0.9=-0.6
Jan-19
H.S.

27. Removing outlier cont. Full model N=5000 Outlier removed N=4999
One outlier affected several estimates
Final model
Report only the effect of exposure
Jan-19
H.S.

28. bw2
Non-linear effects
Jan-19
H.S.

29. bw2: Non-linear effects
scatter bw2 gest or
scatter res pred
Handle:
add
polynomial or
spline
Weak non-linearity in exposure: would care
Weak non-linearity in confounder: may not care
Jan-19
H.S.

30. Non-linear effects: polynomial
regress bw2 c.gest##c.gest i.educ sex 2. order polynomial in gest
margins, at(gest=(250(10)310)) predicted bw2 by gest
marginsplot plot
Could also ask for the effect of gest on bw2:
Margins, dydx(gest) at(gest=(250(10)310))
Jan-19
H.S.

31. Non-linear effects: cubic spline
Plot
List of knots display
mkspline g=gest, cubic nknots(4) make spline with 4 knots (g1,g2,g3)
regress 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 gest *
marginsplot plot
Package:
findit postrcspline
ssc hot author(Buis) Statistical Software Components
* findit postrcspline
Jan-19
H.S.

32. Non-linear effects: linear spline
Plot (as previous)
mkspline g1 280 g2=gest make linear spline with knot at 280
regress bw2 g1 g2 i.educ sex regression with spline
Jan-19
H.S.

33. Interaction only linear effects
bw3
Interaction only linear effects
Jan-19
H.S.

34. 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
both x1 and x2 = OR1*OR2
Interaction
deviation from additivity (multiplicativity) 
effect of x1 depends on x2
Jan-19
H.S.

35. bw3: Interaction (only linear effects)
Add interaction terms
Show results
regress bw3 c.gest##i.sex i.educ main + gest-sex interaction
margins, dydx(gest) at(sex=0) effect of gest for boys
margins, dydx(gest) at(sex=1) effect of gest for girls
Jan-19
H.S.

36. Summing up DAG Scatter- and density plots Bivariate analysis
Regression
Model estimation
Test of assumptions
Independent errors
Linear effects
No interactions
Constant error variance
Influence E
gest age D
birth weight C2
education C1
sex
Model fitting
Adjust if cofactor is confounder, problems cofacor has missing, cofactor has error, cofactor is in causal path
Test of assumptions
Stata: Estimation remains in memory. Post estimation commands
Normally do assumtions first, then influence.
Note leverage value of outlier
Jan-19
H.S.

37. Summing up 1 Build model Interaction Assumptions
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
Jan-19
H.S.

38. Summing up 2 Non-linearity (linear spline) Robustness
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
Jan-19
H.S.

39. ANALYSIS 2, BINARY OUTCOME
SPSS from Kiosk
Data: m:\pc\dokumenter\a_Courses\Master UiO\Smoking and Alzheimer.sav
ANALYSIS 2, BINARY OUTCOME
H.S.

40. Table 1
Outcome: Alzheimer
Exposure: Smoking
Covariates:
Jan-19
H.S.

41. Table 2 Outcome: Alzheimer Exposure: Smoking Adjusted for:
Age, Physical activity and Education
Change to crude and adjusted models
Jan-19
H.S.

42. Logistic model and assumptions
Independent residuals
Linear effects (on the log-odds scale)
No interactions (on the multiplicative scale)
Inverse link
in the plots
Mu=proportion with disease in each group
Influence: use db
Logistic.pdf: Natalia Sarkisian
1 January 2019
H.S.

43. Association measure, Odds ratio
Model:
Start with:
OR: variable and two values
OR for x1 (2 vs 1)
Linear effect on log odds level
Hence:
1 January 2019
H.S.

44. Short: need to know Assume Association measure Scale
Linear effects on the log-odds scale
Association measure
OR=eb
Scale
Multiplicative exposed to both x1 and x2 : OR1*OR2
1 January 2019
H.S.

45. Purpose of regression Estimation Prediction
Estimate association between exposure and outcome adjusted for other covariates
Prediction
Use an estimated model to predict the outcome given covariates in a new dataset
DAGs, bias, precision
Predictive power, model fit, R2
Counfounding matter in the first
Fit of the models matters in the last
Jan-19
H.S.

46. Workflow DAG Bivariate analysis Regression Model fitting
Exposure
+ Confounders
Test of assumptions
Independent errors
Linear effects (on the log odds scale)
No interactions (on the multiplicative scale)
Influence
Model fitting
Adjust if cofactor is confounder, problems cofacor has missing, cofactor has error, cofactor is in causal path
Test of assumptions
Stata: Estimation remains in memory. Post estimation commands
1 January 2019
H.S.

47. Go to syntax
1 January 2019
H.S.

48. The end
1 January 2019
H.S.