movies Name:

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1 movies Name: Contents movies USRevenue ~ Budget + Opening + Theaters + Opinion USRevenue ~ Opening + Opinion comparison of the two models comparison of nested linear models another linear model movies People in the movie industry would like very much to be able to quickly predict the total US revenue of a recently released movie. Construct a linear model for this purpose based on the following information: a. budget the movie s budget b. opening total revenue from the opening weekend c. theaters the number of theaters in which the movie was shown during the opening weekend d. opinion the movie s rating on the Internet Movie Database at the end of its first week The data to be used comes from a random sample of 35 movies which premiered in 28. [EPS, ex , pp ] references: - Moore, et al., EPS, ex , pp IMDb, Wikipedia Load library library(ggplot2) Import the data. data <- read.csv("movies.csv", header=true) head(data) Title Rating Genre Budget USRevenue 1 Madagascar: Escape 2 Africa PG Animation Sex and the City R Comedy The Ruins R Horror Stop-Loss R Drama The Curious Case of Benjamin Button PG-13 Drama Redbelt R Action Opening LOpening Theaters Opinion Profit

2 Distribution of U SRevenue ggplot(data, aes(usrevenue)) + geom_histogram(fill = "wheat", color = "saddlebrown") 8 6 count 4 2 Scatterplots. 2 4 USRevenue Plot U SRevenue against each of the four predictors. These curves are called loess curves (= local regression) because they are built up from short segments of local regression lines. ggplot(data, aes(budget, USRevenue)) + geom_point(pch = 2, color = "darkred") + geom_smooth(method = 'loess') 2

3 6 4 USRevenue Budget ggplot(data, aes(opening, USRevenue)) + geom_point(pch = 2, color = "darkred") + geom_smooth(method = 'loess') 3

4 4 USRevenue Opening ggplot(data, aes(theaters, USRevenue)) + geom_point(pch = 2, color = "darkred") + geom_smooth(method = 'loess') 4

5 4 USRevenue Theaters ggplot(data, aes(opinion, USRevenue)) + geom_point(pch = 2, color = "darkred") + geom_smooth(method = 'loess') 5

6 6 4 USRevenue Opinion USRevenue ~ Budget + Opening + Theaters + Opinion movies.lm <- lm(usrevenue ~ Budget + Opening + Theaters + Opinion, data=data) options(show.signif.stars = FALSE) summary(movies.lm) Call: lm(formula = USRevenue ~ Budget + Opening + Theaters + Opinion, data = data) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) Budget Opening < 2e-16 Theaters Opinion Residual standard error: on 3 degrees of freedom Multiple R-squared:.987, Adjusted R-squared:.9782 F-statistic: on 4 and 3 DF, p-value: < 2.2e-16 6

7 Plot y vs. ŷ. y <- data$usrevenue y.hat <- predict(movies.lm) lm.data<- data.frame(y.hat, y) ggplot(lm.data, aes(y.hat, y)) + geom_point(pch = 2, color = "darkred") + geom_smooth(method = "lm") 4 y y.hat Measure the strength of this relationship. r1 <- cor(y.hat, y) r1 [1] USRevenue ~ Opening + Opinion Are all of the variables equally useful in this full linear model? Construct a new linear model with fewer predictors. How does this new model compare with the original model? movies.lm2 <- lm(usrevenue ~ Opening + Opinion, data=data) summary(movies.lm2) Call: lm(formula = USRevenue ~ Opening + Opinion, data = data) 7

8 Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-5 Opening < 2e-16 Opinion Residual standard error: on 32 degrees of freedom Multiple R-squared:.9794, Adjusted R-squared:.9781 F-statistic: on 2 and 32 DF, p-value: < 2.2e-16 Plot y vs. ŷ. y <- data$usrevenue y.hat <- predict(movies.lm2) lm.data<- data.frame(y.hat, y) ggplot(lm.data, aes(y.hat, y)) + geom_point(pch = 2, color = "darkred") + geom_smooth(method = "lm") 4 y y.hat Measure the strength of this relationship. r2 <- cor(y.hat, y) r2 8

9 [1] comparison of the two models Compare these two linear models on the basis of their R 2, s e, and r statistics. Here r is the correlation between y and ŷ. Is one model better than the other or are they fairly similar? R.sq s.e r movies.lm movies.lm Check the matrix of correlation coefficients to see to what extent the variables that were removed from the original model are correlated with the variables that remain in the new model. What do you find? cor(data[, c(5, 4, 6, 8:9)]) USRevenue Budget Opening Theaters Opinion USRevenue Budget Opening Theaters Opinion comparison of nested linear models. Assume a linear model of the form ŷ = β + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 We would like to test the hypothesis that two of these coefficients are. For this case, that means testing H : USRevenue Opening + Opinion H a : USRevenue Budget + Opening + T heaters + Opinion The anova function performs this test. The test statistic has an F distribution. In the following case, we fail to reject the null hypothesis. The conclusion, in the context of multiple regression, is that the two variables x 1 and x 3 do not contribute significantly to the determination of ŷ when x 2 and x 4 are in the model. anova(movies.lm2, movies.lm) Analysis of Variance Table Model 1: USRevenue ~ Opening + Opinion Model 2: USRevenue ~ Budget + Opening + Theaters + Opinion Res.Df RSS Df Sum of Sq F Pr(>F)

10 x.max <- 5 y.max <-.75 f.val < f.df1 <- 2 f.df2 <- 3 f.p.value <- round(1 - pf(f.val, df1=f.df1, df2=f.df2), 3) title <- "F Test" gg.draw.f(x.max, y.max, f.val, f.df1, f.df2, f.p.value, title) F Test.6 F(df 1 = 2, df 2 = 3).4 y.2 p.value =.367. F = x alpha <-.5 p.value <- f.p.value reject.h <- p.value <= alpha reject.h [1] FALSE another linear model Create another linear model from the original by removing just one of the two variables which seem to be contributing little additional usable information given the presence of the other variables in the model. Use anova and the two linear models to test the hypothesis that the corresponding β is. Interpret your findings. Linear model. Remove one of the predictor variables. 1

11 # movies.lm3 <- lm(usrevenue ~ Budget + Opening + Theaters + Opinion, data=data) # summary(movies.lm3) HT Change the letter i to the index of the variable you removed. Call ANOVA on the nested models. # anova(movies.lm3, movies.lm) H : β i = H : β i Change the values in this image to reflect your findings. x.max <- 5 y.max <-.75 f.val < f.df1 <- 2 f.df2 <- 3 f.p.value <- round(1 - pf(f.val, df1=f.df1, df2=f.df2), 3) title <- "F Test" gg.draw.f(x.max, y.max, f.val, f.df1, f.df2, f.p.value, title) F Test.6 F(df 1 = 2, df 2 = 3).4 y.2 p.value =.367. F = x alpha <-.5 p.value <- f.p.value 11

12 reject.h <- p.value <= alpha reject.h [1] FALSE Interpret your results. 12