Statistical Fridays

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Statistical Fridays

• J C Horrow, MD, MSSTAT
• Clinical Professor, Anesthesiology
• Drexel University College of Medicine

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Previous Session Review

• Students t test.
• Frequency Data.
• Chi-square contingency tables.

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Session Outline Regression

• Regression v. Correlation
• The regression model
• Types of regression
• How to do linear regression
• Features of well-performed regression
• How to examine regression data

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Regression v. Correlation

• Correlation observational data
• Regression cause-effect (experimental)
• Do not imply a cause-effect relationship with
  observational data

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The regression model

• SAMPLE (xi,yi).
• Dependent variable x. May have gt1
• Response variable y. May have gt1
• MODEL y x ?
• where ? describes the error ? N(0,?2).
• Models can be very complicated

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Types of Regression

• SIMPLE one dependent variable
• MULTIPLE several dependent variables
• LINEAR x variables appear to 1st power
• y ?0 ?1x
• QUADRATIC y ?0 ?1x ?2x2
• LOGISTIC outcome is dichotamous (0,1)

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Simple Linear Regression

• Obtain data pairs (xi,yi).
• Plot your data should look linear.
• Xi measured without error
• Yi measured with common error
• ? N(0,?2).
• Minimize S(?0,?1) ? ?2 w.r.t. ?0,?1

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Simple Linear Regression
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Simple Linear Regression
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Simple Linear Regression

• Obtain data pairs (xi,yi).
• Plot your data should look linear.
• Xi measured without error
• Yi measured with common error
• ? N(0,?2).
• Minimize S(?0,?1) ? ?2 w.r.t. ?0,?1

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Simple Linear Regression

• Find minimum by taking derivitives
• ?S/??00 and ?S/??10.
• Get 2 equations, 2 unknowns. Solve
• ?1 Sxy/Sxx whereSxy ? (xiyi) (?xi)(?yi)/n
  andSxx ?(xi2) (?xi)2/n
• Then ?0 ybar - ?1(xbar)

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No relationship !!
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Features of well-performed regression

• Test ?1 against 0 (no relationship)
• Can do this because we know its variance
• Test assumptions of
• Linearity
• Homoschedasticity Var(?i)?2 for all i
• ?i N(0,?2)
• Plot residuals (yi yhati)

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How to Examine Regression Data

• Check r2 value
• if gt0.70, then fit is good
• if lt0.60, very suspicious
• Look for influential points
• Usually at extremes of dependent range

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Example of an influential point
Slope from 9.9 to 9.0
Omit Point
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Example of an influential point
Slope from 9.9 to 5.2
Move Point
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Multiple Regression

• Lots of explanatory variables
• Y X1 X2 X3 Xk ?
• Art as well as science
• All possible regressions (2k possibilities)
• Forward selection
• Backward elimination
• Stepwise

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Multiple Regression

• Fewer explanatory variables are better
• Stepwise gt Backward gt Forward
• Check final model for common error ?2
• Best model has smallest error ?2
• Beware multi-collinearity
• Age as surrogate for decr renal function
• Weight as surrogate for diabetes mellitus

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Logistic Regression Results

• Outcome variable is an event (yes/no)
• Measured as incidence
• Can be simple or multiple
• Results as p-value and as odds-ratio
• O.R. point estimate and confidence interval
• C.I. Includes 1.0 ? not significant (pNS)

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Odds Ratios v. Hazard Ratios

• Odds Ratios
• Relate to event incidences ()
• Measured variable is occurrence of event (y/n)
• Hazard Ratios
• Relate to event rates ( per time)
• Measured variable is time to event survival
  analysis

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Session Review Regression

• Regression v. Correlation
• The regression model
• Types of regression
• How to do linear regression
• Features of well-performed regression
• How to examine regression data