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

Linear Regression

Hein Stigum

Presentation, data and programs at:

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

H.S.


Concepts

Linear regression

Concepts

H.S.


Outcome and regression types

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

Regression idea

H.S.


Measures and assumptions

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

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.


Analysis

Continuous outcome:Linear regression, Birth weight

Analysis

H.S.


Linear regression

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

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

Plot outcome by exposure, cont.

Linear effects?

Yes

H.S.


Bivariate analysis

Bivariate analysis

Outcome: birthweight

H.S.


Regression

Continuous outcome:Linear regression, Birth weight

Regression

H.S.


Categorical covariates

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

Model estimation

Syntax:

regress weight gest sex Parity1 Parity2_7

H.S.


Create meaningful constant

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

Model results

H.S.


Test of assumptions

Test of assumptions

  • Discuss

    • Independent residuals?

  • Plot residuals versus predicted y

    • Linear effects?

    • constantvariance?

  • H.S.


    Violations of assumptions

    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

    Influence

    H.S.


    Measures of influence

    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

    Delta beta for gestational age

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

    H.S.


    Removing outlier

    Removing outlier

    Full data

    Outlier removed

    One outlier affected two estimates

    Final model

    H.S.


    Summing up

    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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