Simple Linear Regression

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Simple Linear Regression

• Statistics 700
• Week of November 27

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Example for Illustration

• The human body takes in more oxygen when
  exercising than when it is at rest. To deliver
  oxygen to the muscles, the heart must beat
  faster. Heart rate is easy to measure, but
  measuring oxygen uptake requires elaborate
  equipment. If oxygen uptake (VO2) can be
  accurately predicted from heart rate (HR), the
  predicted values may replace actually measured
  values for various research purposes.
  Unfortunately, not all human bodies are the same,
  so no single prediction equation works for all
  people. Researchers can, however, measure both HR
  and VO2 for one person under varying sets of
  exercise conditions and calculate a regression
  equation for predicting that persons oxygen
  uptake from heart rate.

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Data From An Individual

• Goals in this illustration
• Scatterplot linear relationship or not?
• Obtain the best-fitting line using least-squares.
• To test whether the model is significant or not.
• To obtain a confidence interval for the
  regression coefficient.
• To obtain predictions.

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The Scatterplot
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Simple Linear Regression Model

• 1. Conditional on Xx, the response variable Y
  has mean equal to m(x) a bx.
• 2. a is the y-intercept while b is the slope of
  the regression line, which could be interpreted
  as the change in the mean value per unit change
  in the independent variable.
• 3. For each X x, the conditional distribution
  of Y is normal with mean m(x) and variance s2.
• 4. Y1, Y2, , Yn are independent of each other.
• Shorthand
• Yi a bxi ei with ei IID N(0,s2)

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Results of Regression Analysis (using Minitab)
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Fitted Line on the Scatterplot
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