1. Regression diagnostics
Hein Stigum
Presentation, data and programs at:
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2. Agenda Linear regression diagnostics (Logistic regression)
Assumtions
Robust results
(Logistic regression)
(Poisson regression)
Time:
Linear: min
Logistic, binary, conditional: 60 min
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3. Birth weight by gestational age
Linear regression
Birth weight by
gestational age
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4. Workflow Scatterplots Bivariate analysis Regression Model fitting
Cofactors in/out
Interactions
Test of assumptions
Independent errors
Linear effects
Constant error variance
Influence (robustness)
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
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5. Scatterplot Jan-19 2 January 2019 H.S. H.S. 5
pregnancy=280 days is normal
N=518
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6. Results Outcome: birthweight Covariates: gestational age, sex, parity
Model: linear regression
OBS: synthetic data
Study birthweigth
Vary with sex, mothers age and espesially gest age (length of pregnancy in days)
Descriptive (bivar analysis)
Expected =3531 for boys, parity=0 gest=280
To get average must take /2+2/5*230+2*17=3580
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7. Model diagnostics Model Assumptions Robustness
Independent errors (residuals)
Linear effects
Constant error variance
Robustness
Y must be normal?
Normal Y-skewed X
Skewed Y-Normal X
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8. Checking assumptions
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9. 1. Independent residuals
No diagnostic tool
Possible violations
Pupils nested in schools: weak correlations
Repeated measurement: strong correlations
Models
Adjust for clustering
Linear mixed models
GEE
Birth weight example: possible violations: repeated births by the same mother
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10. 2. Linear effects Save residuals and predicted values
Plot resid vs pred
If non-linear:
Plot resid vs cont. vars
Add square term
or cut in categories
Add gest^2, linearity now OK
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11. Linear effect test Model 1: only linear terms Significant means
non-linearity
Model 2: linear terms+square term
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12. 3. Constant residual variance
Plot resid vs pred
If non-constant variance:
Robust regression
Weighted regression
SPSS:
log transform,
use poisson regression
look for missing cofactor
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13. Constant variance test
Significant means
non-const. var.
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14. Weighted regression Estimate residual variance Weights=1/variance
Effects
Takes care of heteroskedasticity
“robustification”
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15. Summary of assumptions
Dependent residuals
Mixed models: xtmixed
Non linear effects
gen gest2=gest^2
regress weigth gest gest2 sex
Non-constant variance
regress weigth gest sex, robust
Linear mixed models: xtmixed
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16. Checking robustness , Measures of influence
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17. Measures of influence Measure change in: Predicted (y) Deviance
Remove obs 1, see change
remove obs 2, see change
Measure change in:
Predicted (y)
Deviance
Coefficients (beta)
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18. Influence idea Outlierness Leverage Influence Residuals
Distance from x-mean
Influence
Combination
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19. Leverage versus residuals2
“Adjusted” scatterplot
Added variable plot
(partial regression leverage)
Look at:
321: high lev, med resid
111: low lev, high resid
Lack the ability to see 2 points with opposite large effects, as in delta beta
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20. Delta beta (for gestational age)
Pro: Advantage: direct measure of the coef we are interested in, both pos and neg directions
Con: Disadvatage: one measure for each covariate
Scaled?
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21. Delta fitted value, Dfits
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22. Summary: Robustness, influence
Linear regression sensitive!
Look for influential points
Leverage versus residual plots
Added variable plots
Delta-beta
Rerun regression without influential points and look for change in:
coefficients
constant term
p-values
Found influential point in a dataset with N=30 000! (MoBa, exercise)
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23. Logistic, Poisson regression
Assumptions
Independent errors as before
Linear effects as before
Constant error variance no!
Robustness
Linear not robust!
Poisson medium robust
Logistic fairly robust
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