Title: UCSF - Simple Linear Regression

Slide 1: Simple Linear Regression

• HERS: Randomized clinical trial n=2763
• Post menopausal women with hx of MI
• Randomized to placebo v. hormone therapy
• Outcome: Second MI
• Wealth of baseline data
• HDL associated w/ waist circumference?
• Sample of 221 subjects - baseline data

Slide 2: Why regression?

• Take as given that mean is a good summary
• How does mean HDL depend on waist circumference?
• Other methods are not ideal
• Try forming groups based on value of waist circumference...

Slide 3: Grouping Approach

• Advantages:
  • Simplicity
  • Interpretable
• Disadvantages:
  • Choice of groups
  • More groups: resolution but variability
  • Maybe important differences w/in groups
  • Hard to describe

Slide 4: Linear Regression

• No groups
• Mean changes continuously with predictor
• Uses all the data to predict mean at any point (borrows strength)
• Can work around linear assumption
• Has a straightforward interpretation!

Slide 5: Linear Regression Mean

Slide 6: Example

• Study of 52 HIV+ individuals
• Recruited from the SFGH Neurology clinic
• Studied cross-sectionally
• Not on anti retroviral therapy
• HIV-RNA determined in plasma and CSF
• How are the two related?

Slide 7: Scatterplot Smoother

• Nonparametric method
• Draws and connects a series of local lines
• Result: flexible smooth curve
• Mean of y as a function of x non-linear regression line
• Useful tool for exploring association

Slide 8: Suppose - Let's consider 4 equal-sized groups

• Break the data into a series of groups Data
• Fit a line to each of those groups
• "Connect" them Four local lines
• What would that look like?
• It would be kind a like a smooth Four local lines Lowess Smooth

Slide 9: Scatterplot Smoother

• Many methods for smoothing.
• LOWESS (locally weighted scatterplot smoothing) is the most popular
• Depends on inputs how smooth to make the line
• Oversmooth: linear fit
• Undersmooth: connects the dots
• Programs have defaults (80% smoothing)
• Methods work by "local" regression

Slide 10: STATA

• twoway scatter csfrna hivrna scatterplot
• lowess csfrna hivrna lowess curve for data
• Menu: Graphics, Twoway Graphs

Slide 11: Next Example

• Based on a 19 subject subsample of the HERS data
• Makes it easier to visualize data
• Effects of the outliers is more vivid

Slide 12: Scatterplot + Fitted

Slide 13: Least Squares

Slide 14: Effect of Outlier

• Large if....
• Predictor value is far from mean
• Large residual in regression
• Relatively few values
• Same outlier has little effect in big dataset

Slide 15: Interpreting Regression

Slide 16: Intercept/Slope

Slide 17: Questions a Regression Can Answer

Slide 18: Answers - Question 1

Slide 19: Answers - Question 2

Slide 20: Answers - Question 3

Slide 21: Answers - Question 3

Slide 22: Sample Paragraph

Slide 23: Good Paragraph

• Doesn't focus solely on significance
• Quotes slope and 95% CI
• Perhaps R-squared
• Don't bother with sum of squares

Slide 24: Confounding

• Is the effect of waist on HDL causal? -1cm change in person translate to +.45?
• Maybe waist circ. reflecting different populations? with different diet and exercise patterns?
• How can we isolate the effect of waist? adjusting for diet and exercise! *Solution... Multiple Linear Regression

Slide 25: Summary

• Linear regression is a powerful tool for interpreting associations
• Extends familiar methods (t-test, ANOVA)
• Series of assumptions' (Lines, normality) all of which can be relaxed