Linear regression
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Linear Regression. Hein Stigum Presentation, data and programs at: http://folk.uio.no/heins/ courses. Linear regression. Concepts. Outcome and regression types. Numerical data Discrete number of partners Continuous Weight Categorical data Nominal disease/ no disease Ordinal

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Linear Regression

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Linear Regression

Hein Stigum

Presentation, data and programs at:

http://folk.uio.no/heins/courses

H.S.


Linear regression

Concepts

H.S.


Outcome and regression types

  • Numerical data

    • Discrete

      • number of partners

    • Continuous

      • Weight

  • Categorical data

    • Nominal

      • disease/ no disease

    • Ordinal

      • small/ medium/ large

  • Poisson regression

  • Linear regression

  • Logistic regression

  • Ordinal regression

H.S.


Regression idea

H.S.


Measures and Assumptions

  • Adjusted effects

    • b1 is the increase in weight per day of gestational age

    • b1 is adjusted for b2

  • Assumptions

    • Independent errors

    • Linear effects

    • Constant error variance

  • Robustness

    • influence

H.S.


Workflow

  • DAG

  • Plots: distribution and scatter

  • Bivariate analysis

  • Regression

    • Model estimation

    • Test of assumptions

      • Independent errors

      • Linear effects

      • Constant error variance

    • Robustness

      • Influence

Discuss

Plot

Plot

H.S.


Continuous outcome:Linear regression, Birth weight

Analysis

H.S.


C2

parity

C1

sex

E

gest age

D

birth weight

DAGs

AssociationsBivariate (unadjusted)

Causal effectsMultivariable (adjusted)

Draw your assumptions before your conclusions

H.S.


Plot outcome by exposure

Effects on linear regression:

OK

Be clear on the research question:

overall birth weight: linear regression

low birth weight:logistic regression

linear and logistic can give opposite results

May lead to non-constant error variance

May have high influential outliers

H.S.


Plot outcome by exposure, cont.

Linear effects?

Yes

H.S.


Bivariate analysis

Outcome: birthweight

H.S.


Continuous outcome:Linear regression, Birth weight

Regression

H.S.


2 categories

OK, but know the coding

3+ categories

Use “dummies”

“Dummies” are 0/1 variables used to create contrasts

Want 3 categories for parity: 0, 1 and 2-7 children

Choose 0 as reference

Make dummies for the two other categories

Categorical covariates

generate Parity1 =(parity==1) if parity<.

generate Parity2_7 =(parity>=2) if parity<.

H.S.


Model estimation

Syntax:

regress weight gest sex Parity1 Parity2_7

H.S.


Create meaningful constant

Expected birth weight at:

gest= 0, sex=0, parity=0

gest=280, sex=1, parity=0

Alternative: center variables

gen gest280=gest-280gest280 has a meaningful zero at 280 days

gen sex0=sex-1 sex0 has a meaningful zero at boys


Model results

H.S.


Test of assumptions

  • Discuss

    • Independent residuals?

  • Plot residuals versus predicted y

    • Linear effects?

    • constantvariance?

  • H.S.


    Violations of assumptions

    • Dependent residuals

      Use linear mixed models

    • Non linear effects

      Add square term

      Or use piecewise linear

    • Non-constant variance

      Use robust variance estimation

    H.S.


    Influence

    H.S.


    Measures of influence

    • Measure change in:

      • Predicted outcome

      • Deviance

      • Coefficients (beta)

        • Delta beta

    Remove obs 1, see change

    remove obs 2, see change

    H.S.


    Delta beta for gestational age

    If obs nr 539 is removed, beta will change from 6 to 16

    H.S.


    Removing outlier

    Full data

    Outlier removed

    One outlier affected two estimates

    Final model

    H.S.


    Summing up

    • DAGs

      • Guide analysis

    • Plots

      • Unequal variance, non-linearity, outliers

    • Bivariate analysis

    • Linear regression

      • Fit model

      • Check assumptions

      • Check robustness

      • Make meaningful constant

    H.S.


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