We extended the additive model in two variables to the interaction model by adding a third term to the equation.


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1 Quadratic Models We extended the additive model in two variables to the interaction model by adding a third term to the equation. Similarly, we can extend the linear model in one variable to the quadratic model by adding a second term to the equation: E(Y ) = β 0 + β 1 x + β 2 x 2. This a special case of the twovariable model with x 1 = x and x 2 = x 2. E(Y ) = β 0 + β 1 x 1 + β 2 x 2 1 / 16 Multiple Linear Regression Quadratic Models
2 Example: immune system and exercise x = maximal oxygen uptake (VO 2 max, ml/(kg min)); y = immunoglobulin level (IgG, mg/dl); data for 30 subjects (AEROBIC.txt). Get the data and plot them: aerobic < read.table("text/exercises&examples/aerobic.txt", header = TRUE) plot(aerobic[, c("maxoxy", "IGG")]) Slight curvature suggests a linear model may not fit. 2 / 16 Multiple Linear Regression Quadratic Models
3 Check the linear model: plot(lm(igg ~ MAXOXY, aerobic)) Graph of residuals against fitted values shows definite curvature. Fit and summarize the quadratic model: aerobiclm < lm(igg ~ MAXOXY + I(MAXOXY^2), aerobic) summary(aerobiclm) 3 / 16 Multiple Linear Regression Quadratic Models
4 Output Call: lm(formula = IGG ~ MAXOXY + I(MAXOXY^2), data = aerobic) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) ** MAXOXY e05 *** I(MAXOXY^2) **  Signif. codes: 0 *** ** 0.01 * Residual standard error: on 27 degrees of freedom Multiple Rsquared: , Adjusted Rsquared: Fstatistic: on 2 and 27 DF, pvalue: < 2.2e16 4 / 16 Multiple Linear Regression Quadratic Models
5 The quadratic term I(MAXOXY^2) is significant, so we reject the null hypothesis that the linear model is acceptable. The quadratic term is negative, which is consistent with the concavity of the curve. The other two tratios test irrelevant hypotheses, because the quadratic term is important. Extrapolation: the fitted curve has a maximum at MAXOXY = and declines for higher MAXOXY, which seems unlikely to represent the real relationship. 5 / 16 Multiple Linear Regression Quadratic Models
6 An alternative analysis The graph of IGG against log(maxoxy) is more linear: with(aerobic, plot(log(maxoxy), IGG)) aerobiclm2 < lm(igg ~ log(maxoxy), aerobic) summary(aerobiclm2) with(aerobic, plot(maxoxy, IGG)) with(aerobic, lines(sort(maxoxy), fitted(aerobiclm)[order(maxoxy)], col = "blue")) with(aerobic, lines(sort(maxoxy), fitted(aerobiclm2)[order(maxoxy)], col = "red")) The fitted curve continues to increase indefinitely, but with diminishing slope. 6 / 16 Multiple Linear Regression Quadratic Models
7 Output Call: lm(formula = IGG ~ log(maxoxy), data = aerobic) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e15 *** log(maxoxy) < 2e16 ***  Signif. codes: 0 *** ** 0.01 * Residual standard error: on 28 degrees of freedom Multiple Rsquared: 0.934, Adjusted Rsquared: Fstatistic: on 1 and 28 DF, pvalue: < 2.2e16 7 / 16 Multiple Linear Regression Quadratic Models
8 More Complex Models ST 430/514 Complete secondorder model When the firstorder model E(Y ) = β 0 + β 1 x 1 + β 2 x 2 is inadequate, the interaction model E(Y ) = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 may be better, but sometimes a complete secondorder model is needed: E(Y ) = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + β 4 x β 5 x / 16 Multiple Linear Regression More Complex Models
9 Example: cost of shipping packages Get the data and plot them: express < read.table("text/exercises&examples/express.txt", header = TRUE) pairs(express) Fit the complete secondorder model and summarize it: expresslm < lm(cost ~ Weight * Distance + I(Weight^2) + I(Distance^2), express) summary(expresslm) plot(expresslm) 9 / 16 Multiple Linear Regression More Complex Models
10 Output ST 430/514 Call: lm(formula = Cost ~ Weight * Distance + I(Weight^2) + I(Distance^2), data = express) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 8.270e e Weight e e ** Distance 4.021e e I(Weight^2) 8.975e e *** I(Distance^2) 1.507e e Weight:Distance 7.327e e e08 ***  Signif. codes: 0 *** ** 0.01 * Residual standard error: on 14 degrees of freedom Multiple Rsquared: , Adjusted Rsquared: Fstatistic: on 5 and 14 DF, pvalue: 5.371e / 16 Multiple Linear Regression More Complex Models
11 Qualitative Variables A qualitative variable (or factor) is one that indicates membership of different categories. E.g., a person s gender = male or female: a qualitative variable with two levels, indicating membership of one of two categories. E.g., package type = Fragile, Semifragile, or Durable: three levels, corresponding to three categories. 11 / 16 Multiple Linear Regression More Complex Models
12 We code a qualitative variable using indicator (dummy) variables: Choose one level to use as a base or reference level, say male or Durable. For each other level, create a variable { 1 if this item is in this category x j = 0 otherwise. For gender, there is only one other category, so the only indicator variable is { 1 for a female x = 0 for a male. 12 / 16 Multiple Linear Regression More Complex Models
13 For packages, there are two other categories, so the indicator variables are { 1 for a Fragile package x Fragile = 0 otherwise, { 1 for a Semifragile package x Semifragile = 0 otherwise, For any item, at most one of the indicator variables is nonzero, indicating a nonbase category; if they are all zero, the item belongs to the base category. 13 / 16 Multiple Linear Regression More Complex Models
14 Example: shipment cost of packages, by type. Get the data and plot them: cargo < read.table("text/exercises&examples/cargo.txt", header = TRUE) plot(cost ~ CARGO, cargo) Fit and summarize the model: cargolm < lm(cost ~ CARGO, cargo) summary(cargolm) 14 / 16 Multiple Linear Regression More Complex Models
15 Output Call: lm(formula = COST ~ CARGO, data = cargo) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) * CARGOFragile e05 *** CARGOSemiFrag **  Signif. codes: 0 *** ** 0.01 * Residual standard error: on 12 degrees of freedom Multiple Rsquared: , Adjusted Rsquared: Fstatistic: on 2 and 12 DF, pvalue: / 16 Multiple Linear Regression More Complex Models
16 Note that the intercept is the fitted value for CARGOFragile = 0 and CARGOSemiFrag = 0; that is, for Durable packages. The coefficients of CARGOFragile and CARGOSemiFrag measure the differences between those categories and Durable. The overall model F test is the same as the analysis of variance test: cargoaov < aov(cost ~ CARGO, cargo) summary(cargoaov) Output Df Sum Sq Mean Sq F value Pr(>F) CARGO *** Residuals Signif. codes: 0 *** ** 0.01 * / 16 Multiple Linear Regression More Complex Models
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