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Lecture 6 Notes. Note: I will e-mail homework 2 tonight. It will be due next Thursday. The Multiple Linear Regression model (Chapter 4.1) Inferences from multiple regression analysis (Chapter 4.2)

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Lecture 6 notes
Lecture 6 Notes

  • Note: I will e-mail homework 2 tonight. It will be due next Thursday.

  • The Multiple Linear Regression model (Chapter 4.1)

  • Inferences from multiple regression analysis (Chapter 4.2)

  • In multiple regression analysis, we consider more than one independent variable x1,…,xK . We are interested in the conditional mean of y given x1,…,xK .

Automobile example
Automobile Example

  • A team charged with designing a new automobile is concerned about the gas mileage that can be achieved. The design team is interested in two things:

    (1) Which characteristics of the design are likely to affect mileage?

    (2) A new car is planned to have the following characteristics: weight – 4000 lbs, horsepower – 200, cargo – 18 cubic feet, seating – 5 adults. Predict the new car’s gas mileage.

  • The team has available information about gallons per 1000 miles and four design characteristics (weight, horsepower, cargo, seating) for a sample of cars made in 1989. Data is in car89.JMP.

Best single predictor
Best Single Predictor

  • To obtain the correlation matrix and pairwise scatterplots, click Analyze, Multivariate Methods, Multivariate.

  • If we use simple linear regression with each of the four independent variables, which provides the best predictions?

Best single predictor1
Best Single Predictor

  • Answer: The simple linear regression that has the highest R2 gives the best predictions because recall that

  • Weight gives the best predictions of GPM1000Hwy based on simple linear regression.

  • But we can obtain better predictions by using more than one of the independent variables.

Multiple linear regression model
Multiple Linear Regression Model

  • Assumptions about :

    • The expected value of the disturbances is zero for each ,

    • The variance of each is equal to ,i.e.,

    • The are normally distributed.

    • The are independent.

Point estimates for multiple linear regression model
Point Estimates for Multiple Linear Regression Model

  • We use the same least squares procedure as for simple linear regression.

  • Our estimates of are the coefficients that minimize the sum of squared prediction errors:

  • Least Squares in JMP: Click Analyze, Fit Model, put dependent variable into Y and add independent variables to the construct model effects box.

Root mean square error
Root Mean Square Error

  • Estimate of :

  • = Root Mean Square Error in JMP

  • For simple linear regression of GP1000MHWY on Weight, . For multiple linear regression of GP1000MHWY on weight, horsepower, cargo, seating,

Residuals and root mean square errors
Residuals and Root Mean Square Errors

  • Residual for observation i = prediction error for observation i =

  • Root mean square error = Typical size of absolute value of prediction error

  • As with simple linear regression model, if multiple linear regression model holds

    • About 95% of the observations will be within two RMSEs of their predicted value

  • For car data, about 95% of the time, the actual GP1000M will be within 2*3.54=7.08 GP1000M of the predicted GP1000M of the car based on the car’s weight, horsepower, cargo and seating.

Inferences about regression coefficients
Inferences about Regression Coefficients

  • Confidence intervals: confidence interval for :

    Degrees of freedom for t equals n-(K+1). Standard error of , , found on JMP output.

  • Hypothesis Test:

    Decision rule for test: Reject H0 if or


    p-value for testing is printed in JMP output under Prob>|t|.

Inference examples
Inference Examples

  • Find a 95% confidence interval for ?

  • Is seating of any help in predicting gas mileage once horsepower, weight and cargo have been taken into account? Carry out a test at the 0.05 significance level.

Partial slopes vs marginal slopes
Partial Slopes vs. Marginal Slopes

  • Multiple Linear Regression Model:

  • The coefficient is a partial slope. It indicates the change in the mean of y that is associated with a one unit increase in while holding all other variables fixed.

  • A marginal slope is obtained when we perform a simple regression with only one X, ignoring all other variables. Consequently the other variables are not held fixed.

Partial slopes vs marginal slopes example
Partial Slopes vs. Marginal Slopes Example

  • In order to evaluate the benefits of a proposed irrigation scheme in a certain region, suppose that the relation of yield Y to rainfall R is investigated over several years.

  • Data is in rainfall.JMP.

Rainfall is estimated to be beneficial once temperature is held fixed.

Multiple regression provides a better picture of the benefits of

an irrigation scheme because temperature would be held fixed in

an irrigation scheme.