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Lecture 21 – Thurs., Nov. 20. Review of Interpreting Coefficients and Prediction in Multiple Regression Strategy for Data Analysis and Graphics (Chapters 9.4 – 9.5) Specially Constructed Explanatory Variables (Chapter 9.3) Polynomial terms for curvature Interaction terms

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• Review of Interpreting Coefficients and Prediction in Multiple Regression

• Strategy for Data Analysis and Graphics (Chapters 9.4 – 9.5)

• Specially Constructed Explanatory Variables (Chapter 9.3)

• Polynomial terms for curvature

• Interaction terms

• Sets of indicator variables for nominal variables

• Multiple Linear Regression Model

• Interpretation of Coefficient : The change in the mean of Y that is associated with increasing Xj by one unit and not changing X1,…,Xj-1, Xj+1,…,Xp

• Interpretation holds even if X1,…,Xp are correlated.

• Same warning about extrapolation beyond the observed X1,…,Xp points as in simple linear regression.

• It is estimated that

• A 1 kg increase in body weight with gestation period and litter size held fixed is associated with a 0.90 g mean increase in brain weight [95% CI: (0.80,1.17)]

• A 1 day increase in gestation period with body weight and litter size held fixed is associated with a 1.81g mean increase in brain weight [95% CI : (1.10,2.51)]

• A 1 animal increase in litter size with body weight and gestation period held fixed is associated with a 27.65g mean increase in brain weight [95% CI: (-6.94, 62.23)]

• Estimated mean brain weight (=predicted brain weight) for a mammal which has a body weight of 3kg, a gestation period of 180 days and a litter size of 1

• Strategy for Data Analysis: Display 9.9 in Chapter 9.4

• Good graphical method for initial exploration of data is a matrix of pairwise scatterplots. To display this in JMP, click on Analyze, Multivariate and then put all the variables in Y, Columns.

• The scope of multiple linear regression can be dramatically expanded by using specially constructed explanatory variables:

• Powers of the explanatory variables Xjk can be used to model curvature in regression function.

• Indicator variables can be used to model the effect of nominal variables

• Products of explanatory variables can be used to model interactive effects of explanatory variables

• Linearity assumption in simple linear regression is violated. Transformations wouldn’t work because function isn’t monotonic.

• Multiple Linear Regression Model:

• Two ways to incorporate squared or higher polynomial terms for curvature in JMP

• Fit Model, create a variable rainfall2

• Fit Y by X, under red triangle next to Bivariate Fit of Yield by Rainfall, click Fit Polynomial then 2, Quadratic instead of Fit Line (a model with both a squared and cubed term can be fit by clicking 3, Cubic)

• Coefficients are not directly interpretable. Change in the mean of Y that is associated with a one unit increase in X depends on X

• Two variables are said to interact if the effect that one of them has on the mean response depends on the value of the other.

• An explanatory variable for interaction can be constructed as the product of the two explanatory variables that are thought to interact.

• Does the effect of light intesnity on mean number of flowers depend on the timing of light regime?

• Multiple linear regression model that has term for interaction:

• Model is equivalent to

• Change in mean of flowers for a one unit increase in light intensity depends on timing onset.

• Coefficients are not easily interpretable. Best method for communicating findings with interaction is table or graph of estimated means at various combinations of interacting variables.

• There is not much evidence of an interaction. The p-value for the test that the interaction coefficient is zero is 0.9096.

• A coded scatterplot is a scatterplot with different symbols to distinguish two or more groups

• Split the Y variable by the group identity variables (Click Tables, Split, then put Y variable in Split and Group Identity variable in Col ID).

• Graph, Overlay Plot, put the columns corresponding to the Y’s for the different group identity variables in Y and put the X variable (light intensity) in X.

• Model without interaction between time onset and light intensity is a “parallel regression lines” model

• Model with interaction is a “separate regression lines” model

• An analyst working for a fast food chain is asked to construct a multiple regression model to identify new locations that are likely to be profitable. The analyst has for a sample of 25 locations the annual gross revenue of the restaurant (y), the mean annual household income and the mean age of children in the area. Data in fastfoodchain.jmp

• Relationship between y and each explanatory variable might be quadratic because restaurants attract mostly middle-income households and children in the mid age ranges.

• Strong evidence of a quadratic relationship between revenue and age, revenue and income. Moderate evidence of an interaction between age and income.

Nominal Variables 9.5.2)

• To incorporate nominal variables in multiple regression analysis, we use indicator variables.

• Indicator variable to distinguish between two groups: The time onset (early vs. late is a nominal variable). To incorporate it into multiple regression analysis, we used indicator variable early which equals 1 if early, 0 if late.

• To incorporate nominal variables with more than two categories, we use multiple indicator variables. If there are k categories, we need k-1 indicator variables.

• A car dealer wants to predict the auction price of a car.

• The dealer believes that odometer reading and the car color are variables that affect a car’s price (data from sample of cars in auctionprice.JMP)

• Three color categories are considered:

• White

• Silver

• Other colors

• Note: Color is a nominal variable.

1 if the color is white

0 if the color is not white

I1 =

1 if the color is silver

0 if the color is not silver

I2 =

The category “Other colors” is defined by:

I1 = 0; I2 = 0

Auction Car Price Model 9.5.2)

• Solution

• the proposed model is

• The data

White car

Other color

Silver color

Price 9.5.2)

16996.48 - .0555(Odometer)

16791.48 - .0555(Odometer)

16701 - .0555(Odometer)

Odometer

Example: Auction Car Price The Regression Equation

From JMP we get the regression equation

PRICE = 16701-.0555(Odometer)+90.48(I-1)+295.48(I-2)

The equation for a

silver color car.

Price = 16701 - .0555(Odometer) + 90.48(0) + 295.48(1)

The equation for a

white color car.

Price = 16701 - .0555(Odometer) + 90.48(1) + 295.48(0)

Price = 6350 - .0278(Odometer) + 45.2(0) + 148(0)

The equation for an

“other color” car.

Example: Auction Car Price 9.5.2)The Regression Equation

From JMP we get the regression equation

PRICE = 16701-.0555(Odometer)+90.48(I-1)+295.48(I-2)

For one additional mile the auction price decreases by

5.55 cents.

A white car sells, on the average,

for \$90.48 more than a car of the “Other color” category

A silver color car sells, on the average,

for \$295.48 more than a car of the “Other color” category.

to infer that a white color car and

a car of “other color” sell for a

different auction price.

There is sufficient evidence

to infer that a silver color car

sells for a larger price than a

car of the “other color” category.

Example: Auction Car Price The Regression Equation

Xm18-02b

• Shorthand Notation for regression model with Nominal Variables. Use all capital letters for nominal variables

• Parallel Regression Lines model:

• Separate Regression Lines model:

Nominal Variables in JMP 9.5.2)

• It is not necessary to create indicator variables yourself to represent a nominal variable.

• Make sure that the nominal variable’s modeling type is in fact nominal.

• Include the nominal variable in the Construct Model Effects box in Fit Model

• JMP will create indicator variables. The brackets indicate the category of the nominal variable for which the indicator variable is 1.