Introduction to the multiple linear regression model

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Introduction to the multiple linear regression
model

• Regression models with
• more than one predictor (or term)

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Is brain and body size predictive of
intelligence?

• Sample of n 38 college students
• Response (Y) intelligence based on PIQ
  (performance) scores from the (revised) Wechsler
  Adult Intelligence Scale.
• Potential predictor (x1) Brain size based on MRI
  scans (given as count/10,000).
• Potential predictor (x2) Height in inches.
• Potential predictor (x3) Weight in pounds.

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Scatter plot matrix
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Scatter plot matrix

• Tells us about 2D marginal relationships between
  each pair of variables without regard to other
  variables.
• The challenge is how the 2D relationships relate
  to how the response y depends on all 3 predictors
  simultaneously.

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Marginal response plots
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Marginal response plots

• Scatter plot of response y vs. each predictor.
• Suggest how response y depends on each predictor
  without regard to other predictors.
• Provide a visual lower bound for the
  goodness-of-fit that can be achieved by the full
  regression model.

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A potential multiple linear regression model

• where
• Yi is intelligence (PIQ) of student i
• xi1 is brain size (MRI) of student i
• xi2 is height (Height) of student i
• xi3 is weight (Weight) of student i

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Potential research questions

• Which predictors explain some of the variation in
  PIQ?
• What is the effect of brain size on PIQ?
• What is the PIQ of an individual with a given
  brain size, height, and weight?

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Predictors

• As before, the x variable. Also, called
  explanatory variables or independent variables.
• Most often numerical measurements, such as age,
  weight, length, and temperature.
• But, can be categorical, such as gender, race,
  and species.

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Terms
Terms are functions of the predictor variables,
such as
Linear regression model as function of terms
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Types of terms

• The predictors themselves.
• Powers of predictors.
• Transformations of predictors.
• Interactions.
• Binary (or categorical) predictors.

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Simple linear regression model with a
transformed predictor

• where
• Yi is proportion of items correctly recalled for
  person i
• xi is time since person i initially memorized
  the list

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Visualizing simple linear regression model with a
transformed predictor
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A first order model with two predictors

• where
• Yi is life of power cell i (number of cycles)
• xi1 is charge rate of power cell i (amperes)
• xi2 is ambient temperature of power cell i
  (celsius)

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Visualizing a first order model with two
predictors
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A first order model with more than 2 predictors

• where
• Yi is intelligence (PIQ) of student i
• xi1 is brain size (MRI) of student i
• xi2 is height (Height) of student i
• xi3 is weight (Weight) of student i

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Visualizing a first order model with more than 2
predictors
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A second order polynomial model with one
predictor

• where
• Yi is length of bluegill (fish) i (in mm)
• xi is age of bluegill (fish) i (in years)

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Visualizing a second order polynomial model with
one predictor
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A second order polynomial model with 2 predictors

• where
• Yi is grade point average of student i
• xi1 is verbal test score of student i
• xi2 is math test score of student i

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Visualizing second order polynomial model with 2
predictors
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A first order modelwith one binary predictor

• where
• Yi is birth weight of baby i
• xi1 is length of gestation of baby i
• xi2 1, if mother smokes and xi2 0, if not

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Visualizing a first order modelwith one binary
predictor
The regression equation is Weight - 2390 143
Gest - 245 Smoking