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 X j by one unit and not changing X 1,…,X j-1, X j+1,…,X p Interpretation holds even if X 1,…,X p are correlated. Same warning about extrapolation beyond the observed X 1,…,X p 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 X j k 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 rainfall 2 –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.

Interaction in Meadowfoam 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.

Interaction in Meadowfoam There is not much evidence of an interaction. The p-value for the test that the interaction coefficient is zero is

Displaying Interaction – Coded Scatterplots (Section 9.5.2) 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.

I 1 = 1 if the color is white 0 if the color is not white I 2 = 1 if the color is silver 0 if the color is not silver The category “Other colors” is defined by: I 1 = 0; I 2 = 0 Indicator Variables in Auction Car Prices

Solution –the proposed model is –The data White car Other color Silver color Auction Car Price Model

Odometer Price Price = (Odometer) (0) (1) Price = (Odometer) (1) (0) Price = (Odometer) (0) + 148(0) (Odometer) (Odometer) (Odometer) The equation for an “other color” car. The equation for a white color car. The equation for a silver color car. From JMP we get the regression equation PRICE = (Odometer)+90.48(I-1) (I-2) Example: Auction Car Price The Regression Equation

From JMP we get the regression equation PRICE = (Odometer)+90.48(I-1) (I-2) 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 $ more than a car of the “Other color” category. For one additional mile the auction price decreases by 5.55 cents. Example: Auction Car Price The Regression Equation

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. Xm18-02b Example: Auction Car Price The Regression Equation

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.