Lecture 20 Tues', Nov' 18th

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Lecture 20 Tues., Nov. 18th

• Multiple Regression
• Case Studies Chapter 9.1
• Regression Coefficients in the Multiple Linear
  Regression Model Chapter 9.2
• JMP Output Chapter 9.6.1
• Office Hours Today and Thursday after class,
  tomorrow (Wednesday) 11-12 instead of 130-230
  or by appointment.

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Midterm 2 Scores
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Multiple Regression

• Multiple Regression Seeks to estimate the mean
  of Y given multiple explanatory variables
  X1,,Xp, denoted by
• Examples
• Y1st year GPA, X1Math SAT, X2Verbal SAT,
  X3High School GPA
• YSales price of house, X1Square Footage,
  X2Number of Rooms
• Uses of Regression Analysis
• Describe association between mean of Y and
  X1,,Xp describe association between mean of Y
  and X1 after taking into account X2,,Xp.
• Passive Prediction Predict Y based on X1,,Xp.
• Control Predict what Y will be if you change
  X1,,Xp.

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Multiple Linear Regression Model

• There is a normally distributed subpopulation of
  responses for each combination of the
  explanatory variables with
• The observations are independent of one another.

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Case Study 9.1.1

• Meadowfoam is a small plant found growing in
  moist meadows in Northwest.
• Researchers conducted a randomized experiment to
  find out how to elevate meadowfoam production
• In a controlled growth chamber, they focused on
  the effects of two-light related factors light
  intensity and timing of onset of light treatment.
• Light intensity levels 150,300,450,600,750,900
• Timing of onset Early, Late

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Case Study 9.1.1. Cont.

• Variables
• Y average number of flowers per meadowfoam
  plant
• X1light intensity
• X21 if late timing, 0 if early timing
• Multiple Linear Regression Model

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Interpretation of Coefficients

• the change in the mean of y that is
  associated with a one unit increase in where
  is held fixed.
• the change in the mean of y that is
  associated with a one unit increase in where
  is held fixed.
• mean of y when

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Coefficients in Meadowfoam Study

• For meadowfoam study
• change in mean flowers per plant
  associated with 1 increase in
  light intensity for fixed time of onset
• change in mean flowers per plant associated
  with switching from late to early onset for fixed
  light intensity.

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Estimation of Multiple Linear Regression Model

• 
  
• The coefficients are estimated by choosing
  to make the sum of squared
  prediction errors as small as possible, i.e.,
  choose to minimize
• Predicted value of y given x1,,xp
• SD(YX1,,Xp), estimated by
  root mean square error

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Multiple Linear Regression in JMP

• Analyze, Fit Model
• Put response variable in Y
• Click on explanatory variables and then click Add
  under Construct Model Effects
• Click Run Model.

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JMP Output from Meadowfoam Study
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Reading JMP Output

• Estimated multiple linear regression model
• . Approximately 95 of flowers per
  plant will lie within 26.44 12.88 flowers per
  plant of
• p-values for coefficients indicate that there is
  strong evidence that higher light intensity is
  associated with less flowers per plant on average
  for fixed time onset and that early time onset is
  associated with more flowers per plant on average
  for fixed light intensity.

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Case Study 9.1.2

• What characteristics are associated with bigger
  brain size after accounting for body size, i.e.,
  what characteristics are associated with bigger
  brain size holding body size fixed?
• Ybrain weight, X1body weight, X2gestation
  period, X3litter size
• Multiple Linear Regression Model

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Interpretation in Randomized Experiments vs. Obs.
Studies

• Randomized Experiments Interpretation of an
  effect of an explanatory variable is
  straightforward and causation is implied.
  Example A 1-unit increase in light intensity
  causes the mean number of flowers to increase by
• Observational Studies Cannot make causal
  conclusions from statistical association. For
  any subpopulation of mammal species with the same
  body weight and litter size, a 1-day increase in
  the species gestation length is associated with
  a - gram increase in mean brain weight.
  Interpretation is only useful if subpopulation of
  mammals with fixed values of body weight and
  litter size, but varying gestation lengths, exist.

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Interpreting Coefficients

• Interpretation depends on what other Xs are
  included.
• measures rates of change in mean brain
  weight with changes in gestation length in
  population of all mammal species where body size
  is variable
• measures the rate of change in mean brain
  weight with changes in gestation length within
  subpopulations of fixed body size.