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Chapter 13: Multiple Regression Section 13.1: How Can We Use Several Variables to Predict a Response? 1 Learning Objectives 1. Regression Models 2. The Number of Explanatory Variables 3. Plotting Relationships 4. Interpretation of Multiple Regression Coefficients 5. Summarizing the Effect While Controlling for a Variable 6. Slopes in Multiple Regression and in Bivariate Regression 7. Importance of Multiple Regression 2 Learning Objective 1: Regression Models The model that contains only two variables, x and y, is called a bivariate model x y 3 Learning Objective 1: Regression Models Suppose there are two predictors, denoted by x1 and x2 This is called a multiple regression model x x y 1 1 2 2 4 Learning Objective 1: Multiple Regression Model The multiple regression model relates the mean µy of a quantitative response variable y to a set of explanatory variables x1, x2,… 5 Learning Objective 1: Multiple Regression Model Example: For three explanatory variables, the multiple regression equation is: x x x y 1 1 2 2 3 3 6 Learning Objective 1: Multiple Regression Model Example: The sample prediction equation with three explanatory variables is: yˆ a b x b x b x 1 1 2 2 3 3 7 Learning Objective 1: Example: Predicting Selling Price Using House and Lot Size The data set “house selling prices” contains observations on 100 home sales in Florida in November 2003 A multiple regression analysis was done with selling price as the response variable and with house size and lot size as the explanatory variables 8 Learning Objective 1: Example: Predicting Selling Price Using House and Lot Size Output from the analysis: 9 Learning Objective 1: Example: Predicting Selling Price Using House and Lot Size Prediction Equation: yˆ 10,536 53.8 x 2.84 x 1 2 where y = selling price, x1=house size and x2 = lot size 10 Learning Objective 1: Example: Predicting Selling Price Using House and Lot Size One house listed in the data set had house size = 1240 square feet, lot size = 18,000 square feet and selling price = $145,000 Find its predicted selling price: yˆ 10,536 53.8(1240) 2.84(18,000) 107,276 11 Learning Objective 1: Example: Predicting Selling Price Using House and Lot Size Find its residual: y yˆ 145,000 107,276 37,724 The residual tells us that the actual selling price was $37,724 higher than predicted 12 Learning Objective 2: The Number of Explanatory Variables You should not use many explanatory variables in a multiple regression model unless you have lots of data A rough guideline is that the sample size n should be at least 10 times the number of explanatory variables 13 Learning Objective 3: Plotting Relationships Always look at the data before doing a multiple regression Most software has the option of constructing scatterplots on a single graph for each pair of variables This is called a scatterplot matrix 14 Learning Objective 3: Plotting Relationships 15 Learning Objective 4: Interpretation of Multiple Regression Coefficients The simplest way to interpret a multiple regression equation looks at it in two dimensions as a function of a single explanatory variable We can look at it this way by fixing values for the other explanatory variable(s) 16 Learning Objective 4: Interpretation of Multiple Regression Coefficients Example using the housing data: Suppose we fix x1 = house size at 2000 square feet The prediction equation becomes: yˆ 10,536 53.8(2000) 2.84x 2 97,022 2.84x 2 17 Learning Objective 4: Interpretation of Multiple Regression Coefficients Since the slope coefficient of x2 is 2.84, the predicted selling price increases by $2.84 for every square foot increase in lot size when the house size is 2000 square feet For a 1000 square-foot increase in lot size, the predicted selling price increases by 1000(2.84) = $2840 when the house size is 2000 square feet 18 Learning Objective 4: Interpretation of Multiple Regression Coefficients Example using the housing data: Suppose we fix x2 = lot size at 30,000 square feet The prediction equation becomes: yˆ 10,536 53.8 x 2.84(30,000) 1 74,676 53.8x 1 19 Learning Objective 4: Interpretation of Multiple Regression Coefficients Since the slope coefficient of x1 is 53.8, for houses with a lot size of 30,000 square feet, the predicted selling price increases by $53.80 for every square foot increase in house size 20 Learning Objective 4: Interpretation of Multiple Regression Coefficients In summary, an increase of a square foot in house size has a larger impact on the selling price ($53.80) than an increase of a square foot in lot size ($2.84) We can compare slopes for these explanatory variables because their units of measurement are the same (square feet) Slopes cannot be compared when the units differ 21 Learning Objective 5: Summarizing the Effect While Controlling for a Variable The multiple regression model assumes that the slope for a particular explanatory variable is identical for all fixed values of the other explanatory variables 22 Learning Objective 5: Summarizing the Effect While Controlling for a Variable For example, the coefficient of x1 in the prediction equation: yˆ 10,536 53.8 x 2.84 x 1 2 is 53.8 regardless of whether we plug in x2 = 10,000 or x2 = 30,000 or x2 = 50,000 23 Learning Objective 5: Summarizing the Effect While Controlling for a Variable 24 Learning Objective 6: Slopes in Multiple Regression and in Bivariate Regression In multiple regression, a slope describes the effect of an explanatory variable while controlling effects of the other explanatory variables in the model 25 Learning Objective 6: Slopes in Multiple Regression and in Bivariate Regression Bivariate regression has only a single explanatory variable A slope in bivariate regression describes the effect of that variable while ignoring all other possible explanatory variables 26 Learning Objective 7: Importance of Multiple Regression One of the main uses of multiple regression is to identify potential lurking variables and control for them by including them as explanatory variables in the model 27 Chapter 13: Multiple Regression Section 13.2 Extending the Correlation and R-Squared for Multiple Regression 28 Learning Objectives 1. Multiple Correlation 2. R-squared 3. Properties of R2 29 Learning Objective 1: Multiple Correlation To summarize how well a multiple regression model predicts y, we analyze how well the observed y values correlate with the predicted yˆ values The multiple correlation is the correlation between the observed y values and the predicted yˆ values It is denoted by R 30 Learning Objective 1: Multiple Correlation For each subject, the regression equation provides a predicted value Each subject has an observed y-value and a predicted y-value 31 Learning Objective 1: Multiple Correlation The correlation computed between all pairs of observed y-values and predicted y-values is the multiple correlation, R The larger the multiple correlation, the better are the predictions of y by the set of explanatory variables 32 Learning Objective 1: Multiple Correlation The R-value always falls between 0 and 1 In this way, the multiple correlation ‘R’ differs from the bivariate correlation ‘r’ between y and a single variable x, which falls between -1 and +1 33 Learning Objective 2: R-squared For predicting y, the square of R describes the relative improvement from using the prediction equation instead of using the sample mean, y 34 Learning Objective 2: R-squared The error in using the prediction equation to predict y is summarized by the residual sum of squares: ( y yˆ ) 2 35 Learning Objective 2: R-squared The error in using y to predict y is summarized by the total sum of squares: ( y y) 2 36 Learning Objective 2: R-squared The proportional reduction in error is: ( y y ) ( y yˆ ) R ( y y) 2 2 2 2 37 Learning Objective 2: R-squared The better the predictions are using the regression equation, the larger R2 is For multiple regression, R2 is the square of the multiple correlation, R 38 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? For the 100 observations on y = selling price, x1 = house size, and x2 = lot size, a table, called the ANOVA (analysis of variance) table was created The table displays the sums of squares in the SS column 39 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? The R2 value can be created from the sums of squares in the table R 2 ( y y ) ( y yˆ ) 2 2 ( y y) 314,433- 90,756 0.711 90,756 2 40 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? Using house size and lot size together to predict selling price reduces the prediction error by 71%, relative to using y alone to predict selling price 41 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? Find and interpret the multiple correlation R R 0.711 0.84 2 There is a strong association between the observed and the predicted selling prices House size and lot size are very helpful in predicting selling prices 42 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? If we used a bivariate regression model to predict selling price with house size as the predictor, the r2 value would be 0.58 If we used a bivariate regression model to predict selling price with lot size as the predictor, the r2 value would be 0.51 43 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? The multiple regression model has R2 0.71, so it provides better predictions than either bivariate model 44 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? 45 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? The single predictor in the data set that is most strongly associated with y is the house’s real estate tax assessment (r2 = 0.679) When we add house size as a second predictor, R2 goes up from 0.679 to 0.730 As other predictors are added, R2 continues to go up, but not by much 46 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? R2 does not increase much after a few predictors are in the model When there are many explanatory variables but the correlations among them are strong, once you have included a few of them in the model, R2 usually doesn’t increase much more when you add additional ones 47 Learning Objective 2: Example: How Well Can We Predict House Selling Prices? This does not mean that the additional variables are uncorrelated with the response variable It merely means that they don’t add much new power for predicting y, given the values of the predictors already in the model 48 Learning Objective 3: Properties of R2 The previous example showed that R2 for the multiple regression model was larger than r2 for a bivariate model using only one of the explanatory variables A key factor of R2 is that it cannot decrease when predictors are added to a model 49 Learning Objective 3: Properties of R2 R2 falls between 0 and 1 The larger the value, the better the explanatory variables collectively predict y R2 =1 only when all residuals are 0, that is, when all regression predictions are prefect R2 = 0 when the correlation between y and each explanatory variable equals 0 50 Learning Objective 3: Properties of R2 R2 gets larger, or at worst stays the same, whenever an explanatory variable is added to the multiple regression model The value of R2 does not depend on the units of measurement 51 Chapter 13: Multiple Regression Section 13.3: How Can We Use Multiple Regression to Make Inferences? 52 Learning Objectives 1. 2. 3. 4. 5. 6. 7. Inferences about the Population Inferences about Individual Regression Parameters Significance Test about a Multiple Regression Parameter Confidence Interval for a Multiple Regression Parameter Estimating Variability Around the Regression Equation Do the Explanatory Variables Collectively Have an Effect? Summary of F Test That All Beta Parameters = 0 53 Learning Objective 1: Inferences about the Population Assumptions required when using a multiple regression model to make inferences about the population: The regression equation truly holds for the population means This implies that there is a straight-line relationship between the mean of y and each explanatory variable, with the same slope at each value of the other predictors 54 Learning Objective 1: Inferences about the Population Assumptions required when using a multiple regression model to make inferences about the population: The data were gathered using randomization The response variable y has a normal distribution at each combination of values of the explanatory variables, with the same standard deviation 55 Learning Objective 2: Inferences about Individual Regression Parameters Consider a particular parameter, β1 If β1= 0, the mean of y is identical for all values of x1, at fixed values of the other explanatory variables So, H0: β1= 0 states that y and x1 are statistically independent, controlling for the other variables This means that once the other explanatory variables are in the model, it doesn’t help to have x1 in the model 56 Learning Objective 3: Significance Test about a Multiple Regression Parameter 1. Assumptions: Each explanatory variable has a straightline relation with µy with the same slope for all combinations of values of other predictors in the model Data gathered with randomization Normal distribution for y with same standard deviation at each combination of values of other predictors in model 57 Learning Objective 3: Significance Test about a Multiple Regression Parameter 2. Hypotheses: H0: β1= 0 Ha: β1≠ 0 When H0 is true, y is independent of x1, controlling for the other predictors 58 Learning Objective 3: Significance Test about a Multiple Regression Parameter 3. Test Statistic: b 0 t se 1 59 Learning Objective 3: Significance Test about a Multiple Regression Parameter 4. P-value: Two-tail probability from tdistribution of values larger than observed t test statistic (in absolute value) The t-distribution has: df = n – number of parameters in the regression equation 60 Learning Objective 3: Significance Test about a Multiple Regression Parameter 5. Conclusion: Interpret P-value in context; compare to significance level if decision needed 61 Learning Objective 3: Example: What Helps Predict a Female Athlete’s Weight? The “College Athletes” data set comes from a study of 64 University of Georgia female athletes The study measured several physical characteristics, including total body weight in pounds (TBW), height in inches (HGT), the percent of body fat (%BF) and age 62 Learning Objective 3: Example: What Helps Predict a Female Athlete’s Weight? The results of fitting a multiple regression model for predicting weight using the other variables: 63 Learning Objective 3: Example: What Helps Predict a Female Athlete’s Weight? Interpret the effect of age on weight in the multiple regression equation: Let yˆ predicted weight, x1 height, x2 % body fat, and x3 age Then yˆ 97.7 3.43x1 1.36 x2 0.96 x3 64 Learning Objective 3: Example: What Helps Predict a Female Athlete’s Weight? The slope coefficient of age is -0.96 For athletes having fixed values for x1 and x2, the predicted weight decreases by 0.96 pounds for a 1-year increase in age, and the ages vary only between 17 and 23 65 Learning Objective 3: Example: What Helps Predict a Female Athlete’s Weight? Run a hypothesis test to determine whether age helps to predict weight, if you already know height and percent body fat 66 Learning Objective 3: Example: What Helps Predict a Female Athlete’s Weight? 1. Assumptions Met?: The 64 female athletes were a convenience sample, not a random sample Caution should be taken when making inferences about all female college athletes 67 Learning Objective 3: Example: What Helps Predict a Female Athlete’s Weight? 2. Hypotheses: 3. H0: β3= 0 Ha: β3≠ 0 Test statistic: b 0 0.960 t 1.48 se 0.648 3 68 Learning Objective 3: Example: What Helps Predict a Female Athlete’s Weight? 4. P-value: This value is reported in the output as 0.14 5. Conclusion: • The P-value of 0.14 does not give much evidence against the null hypothesis that β3 = 0 Age does not significantly predict weight if we already know height and % body fat 69 Learning Objective 4: Confidence Interval for a Multiple Regression Parameter A 95% confidence interval for a β slope parameter in multiple regression equals: Estimated slope t (se) .025 The t-score has: df = (n - # of parameters in the model) Assumptions are the same as for the t test 70 Learning Objective 4: Confidence Interval for a Multiple Regression Parameter Construct and interpret a 95% CI for β3, the effect of age while controlling for height and % body fat b t ( se) 0.96 2.00(0.648) 3 .025 0.96 1.30 (2.3,0.3) 71 Learning Objective 4: Confidence Interval for a Multiple Regression Parameter At fixed values of x1 and x2, we infer that the population mean of weight changes very little (and maybe not at all) for a 1 year increase in age The confidence interval contains 0 Age may have no effect on weight, once we control for height and % body fat 72 Learning Objective 5: Estimating Variability Around the Regression Equation A standard deviation parameter, σ, describes variability of the observations around the regression equation Its sample estimate is: s Residual SS df ( y yˆ ) 2 n (# of parameters in reg. eq.) 73 Learning Objective 5: Example: Estimating Variability of Female Athletes’ Weight Anova Table for the “college athletes” data set: 74 Learning Objective 5: Example: Estimating Variability of Female Athletes’ Weight For female athletes at particular values of height, % of body fat, and age, estimate the standard deviation of their weights Begin by finding the Mean Square Error: residual SS 6131.0 s 102.2 df 60 2 Notice that this value (102.2) appears in the MS column in the ANOVA table 75 Learning Objective 5: Example: Estimating Variability of Female Athletes’ Weight The standard deviation is: s 102.2 10.1 This value is also displayed in the ANOVA table For athletes with certain fixed values of height, % body fat, and age, the weights vary with a standard deviation of about 10 pounds 76 Learning Objective 5: Example: Estimating Variability of Female Athletes’ Weight If the conditional distributions of weight are approximately bell-shaped, about 95% of the weight values fall within about 2s = 20 pounds of the true regression line 77 Learning Objective 5: Do the Explanatory Variables Collectively Have an Effect? Example: With 3 predictors in a model, we can check this by testing: H : 0 0 1 2 3 H : At least one parameter 0 a 78 Learning Objective 5: Do the Explanatory Variables Collectively Have an Effect? The test statistic for H0 is denoted by F Mean square for regression F Mean square error 79 Learning Objective 5: Do the Explanatory Variables Collectively Have an Effect? When H0 is true, the expected value of the F test statistic is approximately 1 When H0 is false, F tends to be larger than 1 The larger the F test statistic, the stronger the evidence against H0 80 Learning Objective 6: Summary of F Test That All Beta Parameters = 0 1. Assumptions: Multiple regression equation holds, data gathered randomly, normal distribution for y with same standard deviation at each combination of predictors 81 Learning Objective 6: Summary of F Test That All Beta Parameters = 0 2. H0 : 1 2 0 Ha : At least one parameter 0 3. Test statistic: Mean square for regression F Mean square error 82 Learning Objective 6: Summary of F-Test That All Beta Parameters = 0 4. P-value: Right-tail probability above observed F-test statistic value from F distribution with: df1 = number of explanatory variables df2 = n – (number of parameters in regression equation) 83 Learning Objective 6: Summary of F-Test That All Beta Parameters = 0 5. Conclusion: The smaller the P-value, the stronger the evidence that at least one explanatory variable has an effect on y If a decision is needed, reject H0 if Pvalue ≤ significance level, such as 0.05 84 Learning Objective 6: Example: The F-Test for Predictors of Athletes’ Weight For the 64 female college athletes, the regression model for predicting y = weight using x1 = height, x2 = % body fat and x3 = age is summarized in the ANOVA table on the next slide 85 Learning Objective 6: Example: The F-Test for Predictors of Athletes’ Weight 86 Learning Objective 6: Example: The F-Test for Predictors of Athletes’ Weight Use the output in the ANOVA table to test the hypothesis: H : 0 0 1 2 3 H : At least one parameter 0 a 87 Learning Objective 6: Example: The F-Test for Predictors of Athletes’ Weight The observed F statistic is 40.48 The corresponding P-value is 0.000 We can reject H0 at the 0.05 significance level We conclude that at least one predictor has an effect on weight 88 Learning Objective 6: Example: The F-Test for Predictors of Athletes’ Weight The F-test tells us that at least one explanatory variable has an effect If the explanatory variables are chosen sensibly, at least one should have some predictive power The F-test result tells us whether there is sufficient evidence to make it worthwhile to consider the individual effects, using t-tests 89 Learning Objective 6: Example: The F-Test for Predictors of Athletes’ Weight The individual t-tests identify which of the variables are significant (controlling for the other variables) 90 Learning Objective 6: Example: The F-Test for Predictors of Athletes’ Weight If a variable turns out not to be significant, it can be removed from the model In this example, ‘age’ can be removed from the model 91 Chapter 13: Multiple Regression Section 13.4: Checking a Regression Model Using Residual Plots 92 Learning Objectives 1. Assumptions for Inference with a Multiple Regression Model 2. Checking Shape and Detecting Unusual Observations 3. Plotting Residuals against Each Explanatory Variable 93 Learning Objective 1: Assumptions for Inference with a Multiple Regression Model • • • The regression equation approximates well the true relationship between the predictors and the mean of y The data were gathered randomly y has a normal distribution with the same standard deviation at each combination of predictors 94 Learning Objective 2: Checking Shape and Detecting Unusual Observations To test Assumption 3 (the conditional distribution of y is normal at any fixed values of the explanatory variables): Construction a histogram of the standardized residuals The histogram should be approximately bellshaped Nearly all the standardized residuals should fall between -3 and +3. Any residual outside these limits is a potential outlier 95 Learning Objective 2: Example: Residuals for House Selling Price For the house selling price data, a MINITAB histogram of the standardized residuals for the multiple regression model predicting selling price by the house size and the lot size was created and is displayed on the following slide 96 Learning Objective 2: Example: Residuals for House Selling Price 97 Learning Objective 2: Example: Residuals for House Selling Price The residuals are roughly bell shaped about 0 They fall between about -3 and +3 No severe nonnormality is indicated 98 Learning Objective 3: Plotting Residuals against Each Explanatory Variable Plots of residuals against each explanatory variable help us check for potential problems with the regression model Ideally, the residuals should fluctuate randomly about 0 There should be no obvious change in trend or change in variation as the values of the explanatory variable increases 99 Learning Objective 3: Plotting Residuals against Each Explanatory Variable 100 Chapter 13: Multiple Regression Section 13.5: How Can Regression Include Categorical Predictors? 101 Learning Objectives 1. Indicator Variables 2. Is There Interaction? 102 Learning Objective 1: Indicator Variables Regression models can specify categories of a categorical explanatory variable using artificial variables, called indicator variables The indicator variable for a particular category is binary It equals 1 if the observation falls into that category and it equals 0 otherwise 103 Learning Objective 1: Indicator Variables In the house selling prices data set, the city region in which a house is located is a categorical variable The indicator variable x for region is x = 1 if house is in NW (northwest region) x = 0 if house is not in NW 104 Learning Objective 1: Indicator Variables The coefficient β of the indicator variable x is the difference between the mean selling prices for homes in the NW and for homes not in the NW: y (1) , if house is in NW (so x 1) y (0) , if house is not in NW (so x 0) y for NW y for other regions 105 Learning Objective 1: Example: Including Region in Regression for House Selling Price Output from the regression model for selling price of home using house size and region 106 Learning Objective 1: Example: Including Region in Regression for House Selling Price Find and plot the lines showing how predicted selling price varies as a function of house size, for homes in the NW and for homes not in the NW 107 Learning Objective 1: Example: Including Region in Regression for House Selling Price The regression equation from the MINITAB output is: yˆ 15,258 78.0 x 30,569 x 1 2 108 Learning Objective 1: Example: Including Region in Regression for House Selling Price For homes not in the NW, x2 = 0 The prediction equation then simplifies to: yˆ 15,258 78.0 x 30,569(0) 1 - 15,258 78.0x 1 109 Learning Objective 1: Example: Including Region in Regression for House Selling Price For homes in the NW, x2 = 1 The prediction equation then simplifies to: yˆ 15,258 78.0 x 30,569(1) 1 15,311 78.0x 1 110 Learning Objective 1: Example: Including Region in Regression for House Selling Price 111 Learning Objective 1: Example: Including Region in Regression for House Selling Price Both lines have the same slope, 78 For homes in the NW and for homes not in the NW, the predicted selling price increases by $78 for each square-foot increase in house size The figure portrays a separate line for each category of region (NW, not NW) 112 Learning Objective 1: Example: Including Region in Regression for House Selling Price The coefficient of the indicator variable is 30569 For any fixed value of house size, we predict that the selling price is $30,569 higher for homes in the NW 113 Learning Objective 1: Example: Including Region in Regression for House Selling Price The line for homes in the NW is above the line for homes not in the NW The predicted selling price is higher for homes in the NW The P-value of 0.000 for the test for the coefficient of the indicator variable suggests that this difference is statistically significant 114 Learning Objective 2: Is there Interaction? For two explanatory variables, interaction exists between them in their effects on the response variable when the slope of the relationship between µy and one of them changes as the value of the other changes 115 Learning Objective 2: Example: Interaction in effects on House Selling Price Suppose the actual population relationship between house size and mean selling price is: y 15,000 100x1 for homes in the NW y 12,000 25x1 for homes in other regions Then the slope for the effect of x1 differs for the two regions - there is interaction between house size and region in their effects on selling price 116 Learning Objective 2: Example: Interaction in effects on House Selling Price 117 Learning Objective 2: Example: Interaction in effects on House Selling Price To allow for interaction with two explanatory variables, one quantitative and one categorical, you can fit a separate regression line with a different slope between the two quantitative variables for each category of the categorical variable. 118 Chapter 13: Multiple Regression Section 13.6: How Can We Model A Categorical Response? 119 Learning Objectives 1. Modeling a Categorical Response Variable 2. Examples of Logistic Regression 3. The Logistic Regression Model 120 Learning Objective 1: Modeling a Categorical Response Variable When y is categorical, a different regression model applies, called logistic regression 121 Learning Objective 2: Examples of Logistic Regression A voter’s choice in an election (Democrat or Republican), with explanatory variables: annual income, political ideology, religious affiliation, and race Whether a credit card holder pays their bill on time (yes or no), with explanatory variables: family income and the number of months in the past year that the customer paid the bill on time 122 Learning Objective 3: The Logistic Regression Model Denote the possible outcomes for y as 0 and 1 Use the generic terms failure (for outcome = 0) and success (for outcome =1) The population mean of the scores equals the population proportion of ‘1’ outcomes (successes) That is, µy = p The proportion, p, also represents the probability that a randomly selected subject has a success outcome 123 Learning Objective 3: The Logistic Regression Model The straight-line model is usually inadequate A more realistic model has a curved S- shape instead of a straight-line trend 124 Learning Objective 3: The Logistic Regression Model A regression equation for an S-shaped curve for the probability of success p is: ( x ) e p 1 e ( x ) 125 Learning Objective 3: Example: Annual Income and Having a Travel Credit Card An Italian study with 100 randomly selected Italian adults considered factors that are associated with whether a person possesses at least one travel credit card The table on the next page shows results for the first 15 people on this response variable and on the person’s annual income (in thousands of euros) 126 Learning Objective 3: Example: Annual Income and Having a Travel Credit Card 127 Learning Objective 3: Example: Annual Income and Having a Travel Credit Card Let x = annual income and let y = whether the person possesses a travel credit card (1 = yes, 0 = no) 128 Learning Objective 3: Example: Annual Income and Having a Travel Credit Card Substituting the α and β estimates into the logistic regression model formula yields: ( 3.52 0.105 x ) e pˆ 1 e ( 3.52 0.105 x ) 129 Learning Objective 3: Example: Annual Income and Having a Travel Credit Card Find the estimated probability of possessing a travel credit card at the lowest and highest annual income levels in the sample, which were x = 12 and x = 65 130 Learning Objective 3: Example: Annual Income and Having a Travel Credit Card For x = 12 thousand euros, the estimated probability of possessing a travel credit card is: 3.52 0.105 ( 12 ) 2.26 e e pˆ 1 e 1 e 0.104 0.09 1.104 3.521.05 ( 12 ) 2.26 131 Learning Objective 3: Example: Annual Income and Having a Travel Credit Card For x = 65 thousand euros, the estimated probability of possessing a travel credit card is: 3.52 0.105 ( 65 ) 3.305 e e pˆ 1 e 1 e 27.2485 0.97 28.2485 3.521.05 ( 65 ) 3.305 132 Learning Objective 3: Example: Annual Income and Having a Travel Credit Card Annual income has a strong positive effect on having a credit card The estimated probability of having a travel credit card changes from 0.09 to 0.97 as annual income changes over its range 133 Learning Objective 3: Example: Estimating Proportion of Students Who’ve Used Marijuana A three-variable contingency table from a survey of senior high-school students is shown on the next slide The students were asked whether they had ever used: alcohol, cigarettes or marijuana 134 Learning Objective 3: Example: Estimating Proportion of Students Who’ve Used Marijuana 135 Learning Objective 3: Example: Estimating Proportion of Students Who’ve Used Marijuana Let y indicate marijuana use, coded: (1 = yes, 0 = no) Let x1 be an indicator variable for alcohol use (1 = yes, 0 = no) Let x2 be an indicator variable for cigarette use (1 = yes, 0 = no) 136 Learning Objective 3: Example: Estimating Proportion of Students Who’ve Used Marijuana 137 Learning Objective 3: Example: Estimating Proportion of Students Who’ve Used Marijuana The logistic regression prediction equation is: 5.31 2.99 x1 2.85 x2 e pˆ 1 e 5.31 2.99 x1 2.85 x2 138 Learning Objective 3: Example: Estimating Proportion of Students Who’ve Used Marijuana For those who have not used alcohol or cigarettes, x1= x2 = 0 and: 5.31 2.99 ( 0 ) 2.85 ( 0 ) e pˆ 1 e 5.31 2.99 ( 0 ) 2.85 ( 0 ) 0.005 139 Learning Objective 3: Example: Estimating Proportion of Students Who’ve Used Marijuana For those who have used alcohol and cigarettes, x1= x2 = 1 and: 5.31 2.99 ( 1 ) 2.85 ( 1 ) e pˆ 1 e 5.31 2.99 ( 1 ) 2.85 ( 1 ) 0.628 140 Learning Objective 3: Example: Estimating Proportion of Students Who’ve Used Marijuana The probability that students have tried marijuana seems to depend greatly on whether they’ve used alcohol and cigarettes 141