Lecture 21 – Thurs., Nov. 20

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# Lecture 21 Thurs., Nov. 20 - PowerPoint PPT Presentation

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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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
• Sets of indicator variables for nominal variables
Interpreting Coefficients
• 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.
Coefficients in Mammal Study
• 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)]
Prediction from Multiple Regression
• 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 and Graphics
• 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.
Specially Constructed Explanatory Variables
• 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
Curved Regression Functions
• Linearity assumption in simple linear regression is violated. Transformations wouldn’t work because function isn’t monotonic.
Squared Term for Curvature
• Multiple Linear Regression Model:
Terms for Curvature
• 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
Interaction Terms
• 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
Coded Scatterplots in JMP
• 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.
Parallel vs. Separate Regression Lines
• Model without interaction between time onset and light intensity is a “parallel regression lines” model
• Model with interaction is a “separate regression lines” model
Polynomials and Interactions Example
• 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.
fastfoodchain.jmp results
• Strong evidence of a quadratic relationship between revenue and age, revenue and income. Moderate evidence of an interaction between age and income.
Nominal Variables
• 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.
Nominal Variables with More than Two Categories
• 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.
Nominal Explanatory Variables Example: Auction Car Prices
• 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.
Indicator Variables in Auction Car Prices

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
• Solution
• the proposed model is
• The data

White car

Other color

Silver color

Price

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 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.

There is insufficient evidence

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 Nominal Variables
• 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
• 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.