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Introduction to the multiple linear regression model

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Title: Introduction to the multiple linear regression model


1
Introduction to the multiple linear regression
model
  • Regression models with
  • more than one predictor (or term)

2
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.

3
Scatter plot matrix
4
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.

5
Marginal response plots
6
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.

7
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

8
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?

9
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.

10
Terms
Terms are functions of the predictor variables,
such as
Linear regression model as function of terms
11
Types of terms
  • The predictors themselves.
  • Powers of predictors.
  • Transformations of predictors.
  • Interactions.
  • Binary (or categorical) predictors.

12
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

13
Visualizing simple linear regression model with a
transformed predictor
14
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)

15
Visualizing a first order model with two
predictors
16
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

17
Visualizing a first order model with more than 2
predictors
18
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)

19
Visualizing a second order polynomial model with
one predictor
20
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

21
Visualizing second order polynomial model with 2
predictors
22
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

23
Visualizing a first order modelwith one binary
predictor
The regression equation is Weight - 2390 143
Gest - 245 Smoking
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