Part II. Multiple Linear Regression

Part II Multiple Linear Regression 86

Chapter 7 Multiple Regression A multiple linear regression model is a linear model that describes how a y-variable relates to two or more xvariables (or transformations of x-variables). For example, suppose that a researcher is studying factors that might affect systolic blood pressures for women aged 45 to 65 years old. The response variable is systolic blood pressure (Y ). Suppose that two predictor variables of interest are age (X 1 ) and body mass index (X 2 ). The general structure of a multiple linear regression model for this situation would be Y = β 0 + β 1 X 1 + β 2 X 2 + ɛ. The equation β 0 + β 1 X 1 + β 2 X 2 describes the mean value of blood pressure for specific values of age and BMI. The error term (ɛ) describes the characteristics of the differences between individual values of blood pressure and their expected values of blood pressure. One note concerning terminology. A linear model is one that is linear in the beta coefficients, meaning that each beta coefficient simply multiplies an x-variable or a transformation of an x-variable. For instance y = β 0 + β 1 x + β 2 x 2 + ɛ is called a multiple linear regression model even though it describes a quadratic, curved, relationship between y and a single x-variable. 87

88 CHAPTER 7. MULTIPLE REGRESSION 7.1 About the Model Notation for the Population Model A population model for a multiple regression model that relates a y- variable to p 1 predictor variables is written as y i = β 0 + β 1 x i,1 + β 2 x i,2 +... + β p 1 x i,p 1 + ɛ i. (7.1) We assume that the ɛ i have a normal distribution with mean 0 and constant variance σ 2. These are the same assumptions that we used in simple regression with one x-variable. The subscript i refers to the i th individual or unit in the population. In the notation for the xvariables, the subscript following i simply denotes which x-variable it is. Estimates of the Model Parameters The estimates of the β coefficients are the values that minimize the sum of squared errors for the sample. The exact formula for this will be given in the next chapter when we introduce matrix notation. The letter b is used to represent a sample estimate of a β coefficient. Thus b 0 is the sample estimate of β 0, b 1 is the sample estimate of β 1, and so on. MSE = SSE n p estimates σ2, the variance of the errors. In the formula, n = sample size, p = number of β coefficients in the model and SSE = sum of squared errors. Notice that for simple linear regression p = 2. Thus, we get the formula for MSE that we introduced in that context of one predictor. In the case of two predictors, the estimated regression equation yields a plane (as opposed to a line in the simple linear regression setting). For more than two predictors, the estimated regression equation yields a hyperplane. STAT 501 D. S. Young

CHAPTER 7. MULTIPLE REGRESSION 89 Predicted Values and Residuals A predicted value is calculated as ŷ i = b 0 + b 1 x i,1 + b 2 x i,2 +... + b p 1 x i,p 1, where the b values come from statistical software and the x-values are specified by us. A residual (error) term is calculated as e i = y i ŷ i, the difference between an actual and a predicted value of y. A plot of residuals versus predicted values ideally should resemble a horizontal random band. Departures from this form indicates difficulties with the model and/or data. Other residual analyses can be done exactly as we did in simple regression. For instance, we might wish to examine a normal probability plot (NPP) of the residuals. Additional plots to consider are plots of residuals versus each x-variable separately. This might help us identify sources of curvature or nonconstant variance. Interaction Terms An interaction term is when there is a coupling or combined effect of 2 or more independent variables. Suppose we have a response variable (Y ) and two predictors (X 1 and X 2 ). Then, the regression model with an interaction term is written as Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 1 X 2 + ɛ. Suppose you also have a third predictor (X 3 ). model with all interaction terms is written as Then, the regression Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X 1 X 2 + β 5 X 1 X 3 + β 6 X 2 X 3 + β 7 X 1 X 2 X 3 + ɛ. In a model with more predictors, you can imagine how much the model grows by adding interactions. Just make sure that you have enough observations to cover the degrees of freedom used in estimating the corresponding regression coefficients! D. S. Young STAT 501

90 CHAPTER 7. MULTIPLE REGRESSION For each observation, their value of the interaction is found by multiplying the recorded values of the predictor variables in the interaction. In models with interaction terms, the significance of the interaction term should always be assessed first before proceeding with significance testing of the main variables. If one of the main variables is removed from the model, then the model should not include any interaction terms involving that variable. 7.2 Significance Testing of Each Variable Within a multiple regression model, we may want to know whether a particular x-variable is making a useful contribution to the model. That is, given the presence of the other x-variables in the model, does a particular x-variable help us predict or explain the y-variable? For instance, suppose that we have three x-variables in the model. The general structure of the model could be Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + ɛ. (7.2) As an example, to determine whether variable X 1 is a useful predictor variable in this model, we could test H 0 : β 1 = 0 H A : β 1 0. If the null hypothesis above were the case, then a change in the value of X 1 would not change Y, so Y and X 1 are not related. Also, we would still be left with variables X 2 and X 3 being present in the model. When we cannot reject the null hypothesis above, we should say that we do not need variable X 1 in the model given that variables X 2 and X 3 will remain in the model. In general, the interpretation of a slope in multiple regression can be tricky. Correlations among the predictors can change the slope values dramatically from what they would be in separate simple regressions. To carry out the test, statistical software will report p-values for all coefficients in the model. Each p-value will be based on a t-statistic calculated as t = (sample coefficient - hypothesized value) / standard error of coefficient. STAT 501 D. S. Young

CHAPTER 7. MULTIPLE REGRESSION 91 For our example above, the t-statistic is: t = b 1 0 s.e.(b 1 ) = b 1 s.e.(b 1 ). Note that the hypothesized value is usually just 0, so this portion of the formula is often omitted. 7.3 Examples Example 1: Heat Flux Data Set The data are from n = 29 homes used to test solar thermal energy. The variables of interest for our model are y = total heat flux, and x 1, x 2, and x 3, which are the focal points for the east, north, and south directions, respectively. There are two other measurements in this data set: another measurement of the focal points and the time of day. We will not utilize these predictors at this time. Table 7.1 gives the data used for this analysis. The regression model of interest is y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,3 + ɛ i. Figure 7.1(a) gives a histogram of the residuals. While the shape is not completely bell-shaped, it again is not suggestive of any severe departures from normality. Figure 7.1(b) gives a plot of the residuals versus the fitted values. Again, the values appear to be randomly scattered about 0, suggesting constant variance. The following provides the t-tests for the individual regression coefficients: ########## Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 389.1659 66.0937 5.888 3.83e-06 *** east 2.1247 1.2145 1.750 0.0925. north -24.1324 1.8685-12.915 1.46e-12 *** south 5.3185 0.9629 5.523 9.69e-06 *** --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 8.598 on 25 degrees of freedom D. S. Young STAT 501

92 CHAPTER 7. MULTIPLE REGRESSION Histogram of Residuals Residuals vs. Fitted Values Density 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Residuals 15 10 5 0 5 10 15 20 10 0 10 20 200 220 240 260 Residuals Fitted Values (a) (b) Figure 7.1: (a) Histogram of the residuals for the heat flux data set. (b) Plot of the residuals. Multiple R-Squared: 0.8741, Adjusted R-squared: 0.859 F-statistic: 57.87 on 3 and 25 DF, p-value: 2.167e-11 ########## At the α = 0.05 significance level, both north and south appear to be statistically significant predictors of heat flux. However, east is not (with a p-value of 0.0925). While we could claim this is a marginally significant predictor, we will rerun the analysis by dropping the east predictor. The following provides the t-tests for the individual regression coefficients for the newly suggested model: ########## Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 483.6703 39.5671 12.224 2.78e-12 *** north -24.2150 1.9405-12.479 1.75e-12 *** south 4.7963 0.9511 5.043 3.00e-05 *** --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 8.932 on 26 degrees of freedom STAT 501 D. S. Young

CHAPTER 7. MULTIPLE REGRESSION 93 Multiple R-Squared: 0.8587, Adjusted R-squared: 0.8478 F-statistic: 79.01 on 2 and 26 DF, p-value: 8.938e-12 ########## The residual plots still appear okay (they are not included here) and we obtain new estimates for our model (in the above). Some things to note from this final analysis are: The final sample multiple regression equation is ŷ i = 483.67 24.22x i,2 + 4.80x i,3. To use this equation for prediction, we substitute specified values for the two directions (i.e., north and south). We can interpret the slopes in the same way that we do for a straightline model, but we have to add the constraint that values of other variables remain constant. When the south position is held constant, the average flux temperature for a home decreases by 24.22 degrees for each 1 unit increase in the north position. When the north position is held constant, the average flux temperature for a home increases by 4.80 degrees for each 1 unit increase in the south position. The value of R 2 = 0.8587 means that the model (the two x-variables) explains 85.87% of the observed variation in a home s flux temperature. The value MSE = 8.9321 is the estimated standard deviation of the residuals. Roughly, it is the average absolute size of a residual. Example 2: Kola Project Data Set The Kola Project ran from 1993-1998 and involved extensive geological surveys of Finland, Norway, and Russia. The entire published data set consists of over 600 observations measured on 111 variables. Table 7.2 provides merely a subset of this data for three variables. The data is subsetted on the LITO variable for counts of 1. The sample size of this subset is n = 131. The investigators are interested in modeling the geological composition variable Cr INAA as a function of Cr and Co. A scatterplot of this data with D. S. Young STAT 501

94 CHAPTER 7. MULTIPLE REGRESSION the least squares plane is provided in Figure 7.2. In this 3D plot, observations above the plane (i.e., observations with positive residuals) are given by green points and observations below the plane (i.e., observations with negative residuals) are given by red points. The output for fitting a multiple linear regression model to this data is below: Residuals: Min 1Q Median 3Q Max -149.95-34.42-14.74 11.58 560.38 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 53.3483 11.6908 4.563 1.17e-05 *** Cr 1.8577 0.2324 7.994 6.66e-13 *** Co 2.1808 1.7530 1.244 0.216 --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 74.76 on 128 degrees of freedom Multiple R-squared: 0.544, Adjusted R-squared: 0.5369 F-statistic: 76.36 on 2 and 128 DF, p-value: < 2.2e-16 Note that Co is found to be not statistically significant. However, the scatterplot in Figure 7.2 clearly shows that the data is skewed to the right for each of the variables (i.e., the bulk of the data is clustered near the lower-end of values for each variable while there are fewer values as you increase along a given axis). In fact, a plot of the standardized residuals against the fitted values (Figure 7.3) indicates that a transformation is needed. Since the data appears skewed to the right for each of the variables, a log transformation on Cr INAA, Cr, and Co will be taken. The scatterplot in Figure 7.4 shows the results from this transformations along with the new least squares plane. Clearly, the transformation has done a better job linearizing the relationship. The output for fitting a multiple linear regression model to this transformed data is below: Residuals: Min 1Q Median 3Q Max -0.8181-0.2443-0.0667 0.1748 1.3401 STAT 501 D. S. Young

CHAPTER 7. MULTIPLE REGRESSION 95 Figure 7.2: 3D scatterplot of the Kola data set with the least squares plane. 100 200 300 400 500 600 2 0 2 4 6 Fitted Values Standardized Residuals Figure 7.3: The standardized residuals versus the fitted values for the raw Kola data set. D. S. Young STAT 501

96 CHAPTER 7. MULTIPLE REGRESSION Figure 7.4: 3D scatterplot of the Kola data set where the logarithm of each variable has been taken. 4.0 4.5 5.0 5.5 6.0 2 1 0 1 2 3 Fitted Values Standardized Residuals Figure 7.5: The standardized residuals versus the fitted values for the logtransformed Kola data set. STAT 501 D. S. Young

CHAPTER 7. MULTIPLE REGRESSION 97 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 2.65109 0.17630 15.037 < 2e-16 *** ln_cr 0.57873 0.08415 6.877 2.42e-10 *** ln_co 0.08587 0.09639 0.891 0.375 --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 0.3784 on 128 degrees of freedom Multiple R-squared: 0.5732, Adjusted R-squared: 0.5665 F-statistic: 85.94 on 2 and 128 DF, p-value: < 2.2e-16 There is also a noted improvement in the plot of the standardized residuals versus the fitted values (Figure 7.5). Notice that the log transformation of Co is not statistically significant as it has a high p-value (0.375). After omitting the log transformation of Co from our analysis, a simple linear regression model is fit to the data. Figure 7.6 provides a scatterplot of the data and a plot of the standardized residuals against the fitted values. These plots, combined with the following simple linear regression output, indicate a highly statistically significant relationship between the log transformation of Cr INAA and the log transformation of Cr. Residuals: Min 1Q Median 3Q Max -0.85999-0.24113-0.05484 0.17339 1.38702 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 2.60459 0.16826 15.48 <2e-16 *** ln_cr 0.63974 0.04887 13.09 <2e-16 *** --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 0.3781 on 129 degrees of freedom Multiple R-squared: 0.5705, Adjusted R-squared: 0.5672 F-statistic: 171.4 on 1 and 129 DF, p-value: < 2.2e-16 D. S. Young STAT 501

98 CHAPTER 7. MULTIPLE REGRESSION 2 3 4 5 3.5 4.0 4.5 5.0 5.5 6.0 6.5 ln(cr) ln(cr_inaa) (a) 4.0 4.5 5.0 5.5 6.0 2 1 0 1 2 3 Fitted Values Standardized Residuals (b) Figure 7.6: (a) Scatterplot of the Kola data set where the logarithm of Cr INAA has been regressed on the logarithm of Cr. (b) Plot of the standardized residuals for this simple linear regression fit. STAT 501 D. S. Young

CHAPTER 7. MULTIPLE REGRESSION 99 i Flux Insolation East South North Time 1 271.8 783.35 33.53 40.55 16.66 13.20 2 264.0 748.45 36.50 36.19 16.46 14.11 3 238.8 684.45 34.66 37.31 17.66 15.68 4 230.7 827.80 33.13 32.52 17.50 10.53 5 251.6 860.45 35.75 33.71 16.40 11.00 6 257.9 875.15 34.46 34.14 16.28 11.31 7 263.9 909.45 34.60 34.85 16.06 11.96 8 266.5 905.55 35.38 35.89 15.93 12.58 9 229.1 756.00 35.85 33.53 16.60 10.66 10 239.3 769.35 35.68 33.79 16.41 10.85 11 258.0 793.50 35.35 34.72 16.17 11.41 12 257.6 801.65 35.04 35.22 15.92 11.91 13 267.3 819.65 34.07 36.50 16.04 12.85 14 267.0 808.55 32.20 37.60 16.19 13.58 15 259.6 774.95 34.32 37.89 16.62 14.21 16 240.4 711.85 31.08 37.71 17.37 15.56 17 227.2 694.85 35.73 37.00 18.12 15.83 18 196.0 638.10 34.11 36.76 18.53 16.41 19 278.7 774.55 34.79 34.62 15.54 13.10 20 272.3 757.90 35.77 35.40 15.70 13.63 21 267.4 753.35 36.44 35.96 16.45 14.51 22 254.5 704.70 37.82 36.26 17.62 15.38 23 224.7 666.80 35.07 36.34 18.12 16.10 24 181.5 568.55 35.26 35.90 19.05 16.73 25 227.5 653.10 35.56 31.84 16.51 10.58 26 253.6 704.05 35.73 33.16 16.02 11.28 27 263.0 709.60 36.46 33.83 15.89 11.91 28 265.8 726.90 36.26 34.89 15.83 12.65 29 263.8 697.15 37.20 36.27 16.71 14.06 Table 7.1: The heat flux for homes data. D. S. Young STAT 501

100 CHAPTER 7. MULTIPLE REGRESSION X 1 X 2 Y X 1 X 2 Y X 1 X 2 Y X 1 X 2 Y 40.9 6.2 300 71.4 11.8 200 52.1 7.7 140 21.3 4.2 110 60.7 10.5 270 66.3 9 230 18 6.1 75 73.5 8.5 210 29.6 10.1 140 99.1 16.1 220 23.7 9.3 54 80.1 18.8 170 40.9 8.7 120 18.6 3.6 93 37.7 6.1 110 75.4 16.6 790 27.8 5.2 240 30.5 8.6 140 16.1 2.7 68 32.3 8.7 100 23.5 3.8 110 28.9 5.2 130 40 5.5 100 19.4 3.9 62 16.6 15.2 64 23 5.3 120 38.4 14.4 100 20 5.5 300 29.2 5.1 92 44.2 9.1 120 23.7 4.9 90 48.3 8.2 110 6.9 2 37 27.7 8 100 16.4 5.4 82 40 6.7 120 57.1 7.8 250 18.1 4.7 100 13.4 2.5 100 22.5 2.4 95 50 9.7 190 10 3.3 87 24 4.1 93 31.8 14.7 180 129 30.7 210 16.3 3.5 83 28.8 15.2 110 17.1 6.2 180 106 13.7 220 31.6 8.7 130 18.4 3.8 63 10.9 2 50 36.5 7.3 81 19.5 6.8 90 9.4 3.3 47 30.6 7.3 110 66.6 10.5 170 25.2 5.5 110 5.9 1.6 86 52.6 11.4 130 37.2 9.6 120 13.6 2.9 170 83.8 16.2 160 53.9 11.5 210 42 9.2 120 23.2 6.4 88 280 25.2 640 88.6 15.5 320 17.5 3.9 64 41.6 10.1 150 21.9 5 62 25.6 6.8 69 67.1 8.6 120 18 4 120 18.5 3.5 92 18.9 3.1 110 10.6 2.5 69 37.4 8.7 97 26.7 5.1 170 16.7 4.5 84 11.3 2 27 32.3 7.2 97 50.2 7 340 19.6 5.6 86 44 14.1 130 217 10 400 30.9 10 120 15.1 4.2 110 34.1 12.1 240 16.7 5 49 25.5 5.5 61 25.1 7 150 29.3 5.2 87 29.8 7.8 160 21.4 3.9 140 8.4 2 34 49.2 14 180 15.5 2.5 70 32.3 6.3 220 25.4 6.9 140 118 18.3 330 14 3 68 31.9 3.7 110 18.7 4.4 72 10.8 3.4 78 30 6.9 120 28.7 6.7 120 21.6 7.6 110 59.3 9.6 300 21.9 9 390 36.2 5.3 94 24 5.2 110 20 3.8 110 33 5.6 71 45.2 3.8 99 19.3 4.3 130 24.4 5.4 120 30.8 6.5 110 16.3 5.4 59 243 24.1 590 28.6 6.6 76 55.5 11.5 130 50 7.5 130 9.6 2.4 47 37.9 30.3 130 25.9 6.9 96 15.8 4.8 79 36.4 6 110 10.2 4.7 46 16.5 4.7 63 19.3 3.5 54 Table 7.2: The subset of the Kola data. Here X 1, X 2, and Y are the variables Cr, Co, and Cr INAA, respectively. STAT 501 D. S. Young

Chapter 8 Matrix Notation in Regression There are two main reasons for using matrices in regression. First, the notation simplifies the writing of the model. Secondly, and most importantly, matrix formulas provide the means by which statistical software calculates the estimated coefficients and their standard errors, as well as the set of predicted values for the observed sample. If necessary, a review of matrices and some of their basic properties can be found in Appendix B. 8.1 Matrices and Regression In matrix notation, the theoretical regression model for the population is written as Y = Xβ + ɛ. The four different items in the equation are: 1. Y is a n-dimensional column vector that vertically lists the y values: Y = Y 1 Y 2. Y n. 2. The X matrix is a matrix in which each row gives the x-variable data for a different observation. The first column equals 1 for all observations (unless doing a regression through the origin), and each column after 101

102 CHAPTER 8. MATRIX NOTATION IN REGRESSION the first gives the data for a different variable. There is a column for each variable, including any added interactions, transformations, indicators, and so on. The abstract formulation is: 1 X 1,1... X 1,p 1 1 X 2,1... X 2,p 1 X =....... 1 X n,1... X n,p 1 In the subscripting, the first value is the observation number and the second number is the variable number. The first column is always a column of 1 s. The X matrix has n rows and p columns. 3. β is a p-dimensional column vector listing the coefficients: β = β 0 β 1. β p 1. Notice the subscript for the numbering of the β s. As an example, for simple linear regression, β = (β 0 β 1 ) T. The β vector will contain symbols, not numbers, as it gives the population parameters. 4. ɛ is a n-dimensional column vector listing the errors: ɛ 1 ɛ 2 ɛ =. ɛ n. Again, we will not have numerical values for the ɛ vector. As an example, suppose that data for a y-variable and two x-variables is as given in Table 8.1. For the model y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,1 x i,2 + ɛ i, the matrices Y, X, β, and ɛ are as follows: STAT 501 D. S. Young

CHAPTER 8. MATRIX NOTATION IN REGRESSION 103 y i 6 5 10 12 14 18 x i,1 1 1 3 5 3 5 x i,2 1 2 1 1 2 2 Table 8.1: A sample data set. Y = 6 5 10 12 14 18, X = 1 1 1 1 1 1 2 2 1 3 1 3 1 5 1 5 1 3 2 6 1 5 2 10, β = β 0 β 1 β 2 β 3, ɛ = ɛ 1 ɛ 2 ɛ 3 ɛ 4 ɛ 5 ɛ 6. 1. Notice that the first column of the X matrix equals 1 for all rows (observations), the second column gives the values of x i,1, the third column lists the values of x i,2, and the fourth column gives the values of the interaction values x i,1 x i,2. 2. For the theoretical model, we do not know the values of the beta coefficients or the errors. In those two matrices (column vectors) we can only list the symbols for these items. 3. There is a slight abuse of notation that occurs here which often happens when writing regression models in matrix notation. I stated earlier how capital letters are reserved for random variables and lower case letters are reserved for realizations. In this example, capital letters have been used for the realizations. There should be no misunderstanding as it will usually be clear if we are in the context of discussing random variables or their realizations. Finally, using Calculus rules for matrices, it can be derived that the ordinary least squares estimates of the β coefficients are calculated using the matrix formula b = (X T X) 1 X T y, D. S. Young STAT 501

104 CHAPTER 8. MATRIX NOTATION IN REGRESSION which minimizes the sum of squared errors e 2 = e T e = (Y Ŷ)T (Y Ŷ) = (Y Xb T (Y Xb), where b = (b 0 b 1... b p 1 ) T. As in the simple linear regression case, these regression coefficient estimators are unbiased (i.e., E(b) = β). The formula above is used by statistical software to calculate values of the sample coefficients. An important theorem in regression analysis (and Statistics in general) is the Gauss-Markov Theorem, which we alluded to earlier. Since we have the proper matrix notation in place, we will now formalize this very important result. Theorem 1 (Gauss-Markov Theorem) Suppose that we have the linear regression model Y = Xβ + ɛ, where E(ɛ i X) = 0 and E(ɛ i X) = σ 2 for all i = 1,..., n. Then ˆβ = b = (X T X) 1 X T Y is an unbiased estimator of β and has the smallest variance of all other unbiased estimates of β. Any estimator which is unbiased and has smaller variance than any other unbiased estimators is called a best linear unbiased estimator or BLUE. An important note regarding the matrix expressions introduced above is that and Ŷ = Xb = X(X T X) 1 X T Y = HY e = Y Ŷ = Y HY = (I n n H)Y, STAT 501 D. S. Young

CHAPTER 8. MATRIX NOTATION IN REGRESSION 105 where H = X(X T X) 1 X T is the n n hat matrix. H is important for several reasons as it appears often in regression formulas. One important implication of H is that it is a projection matrix, meaning that it projects the response vector, Y, as a linear combination of the columns of the X matrix in order to obtain the vector of fitted values, Ŷ. Also, the diagonal of this matrix contains the h j,j values we introduced earlier in the context of Studentized residuals, which is important when discussing leverage. 8.2 Variance-Covariance Matrix of b Two important characteristics of the sample multiple regression coefficients are their standard errors and their correlations with each other. The variancecovariance matrix of the sample coefficients b is a symmetric p p square matrix. Remember that p is the number of beta coefficients in the model (including the intercept). The rows and the columns of the variance-covariance matrix are in coefficient order (first row is information about b 0, second is about b 1, and so on). The diagonal values (from top left to bottom right) are the variances of the sample coefficients (written as Var(b i )). The standard error of a coefficient is the square root of its variance. An off-diagonal value is the covariance between two coefficient estimates (written as Cov(b i, b j )). The correlation between two coefficient estimates can be determined using the following relationship: correlation = covariance divided by product of standard deviations (written as Corr(b i, b j )). In regression, the theoretical variance-covariance matrix of the sample coefficients is V(b) = σ 2 (X T X) 1. Recall, the MSE estimates σ 2, so the estimated variance-covariance matrix of the sample beta coefficients is calculated as ˆV(b) = MSE(X T X) 1. D. S. Young STAT 501

106 CHAPTER 8. MATRIX NOTATION IN REGRESSION 100 (1 α)% confidence intervals are also readily available for β: b j ± t n p;1 α/2 ˆV(b) j,j, where ˆV(b) j,j is the j th diagonal element of the estimated variance-covariance matrix of the sample beta coefficients (i.e., the (estimated) standard error). Furthermore, the Bonferroni joint 100(1 α)% confidence intervals are: for j = 0, 1, 2,..., (p 1). b j ± t n p;1 α/(2p) ˆV(b) j,j, 8.3 Statistical Intervals The statistical intervals for estimating the mean or predicting new observations in the simple linear regression case can easily extend to the multiple regression case. Here, it is only necessary to present the formulas. First, let use define the vector of given predictors as X h = 1 X h,1 X h,2. X h,p 1. We are interested in either intervals for E(Y X = X h ) or intervals for the value of a new response y given that the observation has the particular value X h. First we define the standard error of the fit at X h given by: s.e.(ŷh) = MSE(X T h (X T X) 1 X h ). Now, we can give the formulas for the various intervals: 100 (1 α)% Confidence Interval: ŷ h ± t n p;1 α/2s.e.(ŷ h ). STAT 501 D. S. Young

CHAPTER 8. MATRIX NOTATION IN REGRESSION 107 Bonferroni Joint 100 (1 α)% Confidence Intervals: for i = 1, 2,..., q. ŷ hi ± t n p;1 α/(2q)s.e.(ŷ hi ), 100 (1 α)% Working-Hotelling Confidence Band: ŷ h ± pfp,n p;1 αs.e.(ŷ h ). 100 (1 α)% Prediction Interval: ŷ h ± t n p;1 α/2 MSE/m + [s.e.(ŷh )] 2, where m = 1 corresponds to a prediction interval for a new observation at a given X h and m > 1 corresponds to the mean of m new observations calculated at the same X h. Bonferroni Joint 100 (1 α)% Prediction Intervals: for i = 1, 2,..., q. ŷ hi ± t n p;1 α/(2q) MSE + [s.e.(ŷhi )] 2, Scheffé Joint 100 (1 α)% Prediction Intervals: ŷ hi ± qfq,n p;1 α(mse + [s.e.(ŷ h )] 2 ), for i = 1, 2,..., q. [100 (1 α)%]/[100 P %] Tolerance Intervals: One-Sided Intervals: (, ŷ h + K α,p MSE) and (ŷ h K α,p MSE, ) are the upper and lower one-sided tolerance intervals, respectively, where K α,p is found similarly as in the simple linear regression setting, but with n = (X T h (X T X) 1 X h ) 1. D. S. Young STAT 501

108 CHAPTER 8. MATRIX NOTATION IN REGRESSION Two-Sided Interval: ŷ h ± K α/2,p/2 MSE, where K α/2,p/2 is found similarly as in the simple linear regression setting, but with n as given above and f = n p, where p is the dimension of X h. 8.4 Example Example: Heat Flux Data Set (continued) Refer back to the heat flux data set where only north and south were used as predictors of insolation. The MSE for this model is equal to 79.7819. However, if we are interested in the full variance-covariance matrix and correlation matrix, then this must be calculated by hand by finding the (X T X) 1. Then, ˆV(b) = 79.7819 = 19.6229 0.5521 0.2918 0.5521 0.0472 0.0066 0.2918 0.0066 0.0113 1565.5532 44.0479 23.2797 44.0479 3.7657 0.5305 23.2797 0.5305 0.9046 Taking the square roots of the diagonal terms of this matrix gives you the values of s.e.(b 0 ), s.e.(b 1 ), and s.e.(b 2 ). We can also calculate the correlation matrix of b (denoted by r b ) for this data set: r b = = = Var(b 0 ) Var(b0 )Var(b 0 ) Cov(b 1,b 0 ) Var(b1 )Var(b 0 ) Cov(b 2,b 0 ) Var(b2 )Var(b 0 ) Cov(b 0,b 1 ) Var(b0 )Var(b 1 ) Var(b 1 ) Var(b1 )Var(b 1 ) Cov(b 2,b 1 ) Var(b2 )Var(b 1 ) Cov(b 0,b 2 ) Var(b0 )Var(b 2 ) Cov(b 1,b 2 ) Var(b1 )Var(b 2 ) Var(b 2 ) Var(b2 )Var(b 2 ). 1565.5532 44.0479 23.2797 (1565.5532)(1565.5532) (1565.5532)(3.7657) (1565.5532)(0.9046) 44.0479 3.7657 0.5305 (3.7657)(1565.5532) (3.7657)(3.7657) (3.7657)(0.9046) 23.2797 0.5305 0.9046 (0.9046)(1565.5532) (0.9046)(3.7657) 1 0.5737 0.6186 0.5737 1 0.2874 0.6186 0.2874 1. (0.9046)(0.9046) STAT 501 D. S. Young

CHAPTER 8. MATRIX NOTATION IN REGRESSION 109 r b is an estimate of the population correlation matrix ρ b. For example, Corr(b 1, b 2 ) = 0.2874, which implies there is a fairly low, negative correlation between the average change in flux for each unit increase in the south position and each unit increase in the north position. Therefore, the presence of the north position only slightly affects the estimate of the south s beta coefficient. The consequence is that it is fairly easy to separate the individual effects of these two variables. Note that we usually do not care about correlations concerning the intercept, b 0 since we usually wish to provide an interpretation concerning the x-variables. If all x-variables are uncorrelated with each other, then all covariances between pairs of sample coefficients that multiply x-variables will equal 0. This means that the estimate of one beta is not affected by the presence of the other x-variables. Many experiments are designed to achieve this property, but achieving it with real data is often a different story. The correlation matrix presented above should NOT be confused with the correlation matrix, r, constructed for each pairwise combination of the variables Y, X 1, X 2,..., X p 1 ; namely: 1 Corr(Y, X 1 )... Corr(Y, X p 1 ) Corr(X 1, Y ) 1... Corr(X 1, X p 1 ) r =....... Corr(X p 1, Y ) Corr(X p 1, X 1 )... 1 Note that all of the diagonal entries are 1 because the correlation between a variable and itself is a perfect (positive) association. This correlation matrix is what most statistical software reports and it does not always report r b. The interpretation of each entry in r is identical to the Pearson correlation coefficient interpretation presented earlier. Specifically, it provides the strength and direction of the association between the variables corresponding to the row and column of the respective entry. For this example, the correlation matrix is: r = 1 0.8488 0.1121 0.8488 1 0.2874 0.1121 0.2874 1 We can also calculate the 95% confidence intervals for the regression coefficients. First note that t 26,0.975 = 2.0555. The 95% confidence interval for β 1 is calculated using 24.2150 ± 2.0555 3.7657 and for β 2 it is calculated D. S. Young STAT 501.

110 CHAPTER 8. MATRIX NOTATION IN REGRESSION using 4.7963 ± 2.0555 0.9046. Thus, we are 95% confident that the true population regression coefficients for the north and south focal points are between (-28.2039, -20.2262) and (2.8413, 6.7513), respectively. STAT 501 D. S. Young

Chapter 9 Indicator Variables We next discuss how to include categorical predictor variables in a regression model. A categorical variable is a variable for which the possible outcomes are nameable characteristics, groups or treatments. Some examples are gender (male or female), highest educational degree attained (secondary school, college undergraduate, college graduate), blood pressure medication used (drug 1, drug 2, drug 3), etc. We use indicator variables to incorporate a categorical x-variable into a regression model. An indicator variable equals 1 when an observation is in a particular group and equals 0 when an observation is not in that group. An interaction between an indicator variable and a quantitative variable exists if the slope between the response and the quantitative variable depends upon the specific value present for the indicator variable. 9.1 The Leave One Out Method When a categorical predictor variable has k categories, it is possible to define k indicator variables. However, as explained later, we should only use k 1 of them as predictor variables in the regression model. Let us consider an example where we are analyzing data for a clinical trial done to compare the effectiveness of three different medications used to treat high blood pressure. n = 90 participants are randomly divided into three groups of 30 patients and each group is assigned a different medication. The response variable is the reduction in diastolic blood pressure in a 3 month period. In addition to the treatment variables, two other predictor variables 111

112 CHAPTER 9. INDICATOR VARIABLES will be X 1 =age and X 2 =body mass index. We are examining three different treatments so we can define the following three indicator variables for the treatment: X 3 = 1 if patient used treatment 1, 0 otherwise X 4 = 1 if patient used treatment 2, 0 otherwise X 5 = 1 if patient used treatment 3, 0 otherwise. On the surface, it seems that our model should be the following overparameterized model, a model that requires us to make a modification in order to estimate coefficients: y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,3 + β 4 x i,4 + β 5 x i,5 + ɛ i. (9.1) The difficulty with this model is that the X matrix has a linear dependency, so we can t estimate the individual coefficients (technically, this is because there will be an infinite number of solutions for the betas). The dependency stems from the fact that X i,3 + X i,4 + X i,5 = 1 for all observations because each patient uses one (and only one) of the treatments. In the X matrix, the linear dependency is that the sum of the last three columns will equal the first column (all 1 s). This scenario leads to what is called collinearity and we investigate this in the next chapter. One solution (there are others) for avoiding this difficulty is the leave one out method. The leave one out method has the general rule that whenever a categorical predictor variable has k categories, it is possible to define k indicator variables, but we should only use k 1 of them to describe the differences among the k categories. For the overall fit of the model, it does not matter which set of k 1 indicators we use. The choice of which k 1 indicator variables we use, however, does affect the interpretation of the coefficients that multiply the specific indicators in the model. In our example with three treatments (and three possible indicator variables), we might leave out the third indicator giving us this model: y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,3 + β 4 x i,4 + ɛ i. (9.2) For the overall fit of the model, it would work equally well to leave out the first indicator and include the other two or to leave out the second and include the first and third. STAT 501 D. S. Young

CHAPTER 9. INDICATOR VARIABLES 113 9.2 Coefficient Interpretations The interpretation of the coefficients that multiply indicator variables is tricky. The interpretation for the individual betas with the leave one out method is that a coefficient multiplying an indicator in the model measures the difference between the group defined by the indicator in the model and the group defined by the indicator that was left. Usually, a control or placebo group is the one that is left out. Let us consider our example again. We are predicting decreases in blood pressure in response to X 1 =age, X 2 =body mass, and which of three different treatments a person used. The variables X 3 and X 4 are indicators of the treatment, as defined above. The model we will examine is y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,3 + β 4 x i,4 + ɛ i. To see what is going on, look at each treatment separately by substituting the appropriately defined values of the two indicators into the equation. For treatment 1, by definition X 3 = 1 and X 4 = 0 leading to y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 (1) + β 4 (0) + ɛ i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 + ɛ i. For treatment 2, by definition X 3 = 0 and X 4 = 1 leading to y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 (0) + β 4 (1) + ɛ i = β 0 + β 1 x i,1 + β 2 x i,2 + β 4 + ɛ i. For treatment 3, by definition X 3 = 0 and X 4 = 0 leading to y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 (0) + β 4 (0) + ɛ i = β 0 + β 1 x i,1 + β 2 x i,2 + ɛ i. Now compare the three equations to each other. The only difference between the equations for treatments 1 and 3 is the coefficient β 3. The only difference between the equations for treatments 2 and 3 is the coefficient β 4. This leads to the following meanings for the coefficients: β 3 = difference in mean response for treatment 1 versus treatment 3, assuming the same age and body mass. D. S. Young STAT 501

114 CHAPTER 9. INDICATOR VARIABLES β 4 = difference in mean response for treatment 2 versus treatment 3, assuming the same age and body mass. Here the coefficients are measuring differences from the third treatment. With the leave one out method, a coefficient multiplying an indicator in the model measures the difference between the group defined by the indicator in the model and the group defined by the indicator that was left. IMPORTANT CAUTIONS: Notice that the coefficient that multiplies an indicator variable in the model does not retain the meaning implied by the definition of the indicator. It is common for students to wrongly state that a coefficient measures the difference between that group and the other groups. That is WRONG! It is also incorrect to say only that a coefficient multiplying an indicator measures the effect of being in that group. An effect has to involve a comparison - with the leave one out method it is a comparison to the group associated with the indicator left out. One application where many indicator variables (or binary predictors) are used is in conjoint analysis, which is a marketing tool that attempts to capture a respondent s preference given the presence or absence of various attribute levels. The X matrix is called a dummy matrix as it consists of only 1 s and 0 s. The response is then regressed on the indicators using ordinary least squares and researchers attempt to quantify items like identification of different market segments, predict profitability, or predict the impact of a new competitor. One additional note is that, in theory, with a linear dependence there are an infinite number of suitable solutions for the betas (as will be seen with multicollinearity). With the leave one out method, we are picking one with a particular meaning and then the resulting coefficients measure differences from the specified group. A method, often used in courses focused strictly on ANOVA or Design of Experiments, offers a different meaning for what we estimate. There it will be more common to parameterize in a way so that a coefficient measures how a group differs from an overall average. 9.3 Testing Overall Group Differences To test the overall significance of a categorical predictor variable, we use a general linear F -test procedure (which is developed in detail later). We form the reduced model by dropping the indicator variables from the model. STAT 501 D. S. Young

CHAPTER 9. INDICATOR VARIABLES 115 More technically, the null hypothesis is that the coefficients multiplying the indicator all equal 0. For our example with three treatments of high blood pressure and additional x-variables age and body mass, the details for doing an overall test of treatment differences are: Full model is: y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,3 + β 4 x i,4 + ɛ i. Null hypothesis is: H 0 : β 3 = β 4 = 0. Reduced model is: y i = β 0 + β 1 x i,1 + β 2 x i,2 + ɛ i. 9.4 Interactions To examine a possible interaction between a categorical predictor and a quantitative predictor, include product variables between each indicator and the quantitative variable. As an example, suppose we thought there could be an interaction between the body mass variable (X 2 ) and the treatment variable. This would mean that we thought that treatment differences in blood pressure reduction depend on the specific value of body mass. The model we would use is: y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,3 + β 4 x i,4 + β 5 x i,2 x i,3 + β 6 x i,2 x i,4 + ɛ i. To test whether there is an interaction, the null hypothesis is H 0 : β 5 = β 6 = 0. We would use the general linear F test procedure to carry out the test. The full model is the interaction model given three lines above. The reduced model is now: y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,3 + β 4 x i,4 + ɛ i. A visual way to assess if there is an interaction is by using an interaction plot. An interaction plot is created by plotting the response versus the quantitative predictor and connecting the successive values according to the grouping of the observations. Recall that an interaction between factors occurs when the change in response from lower levels to higher levels of one factor is not quite the same as going from lower levels to higher levels of another factor. Interaction plots allow us to compare the relative strength of the effects across factors. What results is one of three possible trends: D. S. Young STAT 501

116 CHAPTER 9. INDICATOR VARIABLES The lines could be (nearly) parallel, which indicates no interaction. This means that the change in the response from lower levels to higher levels for each factor is roughly the same. The lines intersect within the scope of the study, which indicates an interaction. This means that the change in the response from lower levels to higher levels of one factor is noticeably different than the change in another factor. This type of interaction is called a disordinal interaction. The lines do not intersect within the scope of the study, but the trends indicate that if we were to extend the levels of our factors, then we may see an interaction. This type of interaction is called an ordinal interaction. Figure 9.1 illustrates each type of interaction plot using a mock data set pertaining to the mean tensile strength measured at three different speeds of 3 different processes. The upper left plot illustrates the case where no interaction is present because the change in mean tensile strength is similar for each process as you increase the speed (i.e., the lines are parallel). The upper right plot illustrates an interaction because as the speeds increase, the change in mean tensile strength is noticeably different depending on which process is being used (i.e., the lines cross). The bottom right plot illustrates an ordinal interaction where no interaction is present within the scope of the range of speeds studied, but if these trends continued for higher speeds, then we may see an interaction (i.e., the lines may cross). It should also be noted that just because lines cross, it does not necessarily imply the interaction is statistically significant. Lines which appear nearly parallel, yet cross at some point, may not yield a statistically significant interaction term. If two lines cross, the more different the slopes appear and the more data that is available, then the more likely the interaction term will be significant. 9.5 Relationship to ANCOVA When dealing with categorical predictors in regression analysis, we often say that we are performing a regression with indicator variables or a regression STAT 501 D. S. Young

CHAPTER 9. INDICATOR VARIABLES 117 No Interaction Disordinal Interaction Response 20 40 60 80 100 Treatment 1 Treatment 2 Treatment 3 Response 10 20 30 40 50 60 Treatment 1 Treatment 2 Treatment 3 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Predictor 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Predictor (a) (b) Ordinal Interaction Response 10 20 30 40 50 60 Treatment 1 Treatment 2 Treatment 3 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Predictor (c) Figure 9.1: (a) A plot of no interactions amongst the groups (notice how the lines are nearly parallel). (b) A plot of a disordinal interaction amongst the groups (notice how the lines intersect). (c) A plot of an ordinal interaction amongst the groups (notice how the lines don t intersect, but if we were to extrapolate beyond the predictor limits, then the lines would likely cross). D. S. Young STAT 501

118 CHAPTER 9. INDICATOR VARIABLES with interactions (if we are interested in testing for interactions with indicator variables and other variables). However, in the design and analysis of experiments literature, this model is also used, but with a slightly different motivation. Various experimental layouts using ANOVA tables are commonly used in the design and analysis of experiments. These ANOVA tables are constructed to compare the means of several levels of one or more treatments. For example, a one-way ANOVA can be used to compare six different dosages of blood pressure pills and the mean blood pressure of individuals who are taking one of those six dosages. In this case, there is one factor with six different levels. Suppose further that there are four different races represented in this study. Then a two-way ANOVA can be used since we have two factors - the dosage of the pill and the race of the individual taking the pill. Furthermore, an interaction term can be included if we suspect that the dosage a person is taking and the race of the individual have a combined effect on the response. As you can see, you can extend to the more general n-way ANOVA (with or without interactions) for the setting with n treatments. However, dealing with n > 2 can often lead to difficulty in interpreting the results. One other important thing to point out with ANOVA models is that, while they use least squares for estimation, they differ from how categorical variables are handled in a regression model. In an ANOVA model, there is a parameter estimated for the factor level means and these are used for the linear model of the ANOVA. This differs slightly from a regression model which estimates a regression coefficient for, say, n 1 indicator variables (assuming there are n levels of the categorical variable and we are using the leave-one-out method). Also, ANOVA models utilize ANOVA tables, which are broken down by each factor (i.e., you would look at the sums of squares for each factor present). ANOVA tables for regression models simply test if the regression model has at least one variable which is a significant predictor of the response. More details on these differences are better left to a course on design of experiments. When there is also a continuous variable measured with each response, then the n-way ANOVA model needs to reflect the continuous variable. This model is then referred to as an Analysis of Covariance (or ANCOVA) model. The continuous variable in an ANCOVA model is usually called the covariate or sometimes the concomitant variable. One difference in how an ANCOVA model is approached is that an interaction between the covariate and each factor is always tested first. The reason why is because an STAT 501 D. S. Young

CHAPTER 9. INDICATOR VARIABLES 119 ANCOVA is conducted to investigate the overall relationship between the response and the covariate while assuming this relationship is true for all groups (i.e., for all treatment levels). If, however, this relationship does differ across the groups, then the overall regression model is inaccurate. This assumption is called the assumption of homogeneity of slopes. This is assessed by testing for parallel slopes, which involves testing the interaction term between the covariate and each factor in the ANCOVA table. If the interaction is not statistically significant, then you can claim parallel slopes and proceed to build the ANCOVA model. If the interaction is statistically significant, then the regression model used is not appropriate and an AN- COVA model should not be used. As an example of how to write ANCOVA models, first consider the oneway ANCOVA setting. Suppose we have i = 1,..., I treatments and each treatment has j = 1,..., J i pairs of continuous variables measured (i.e., (x i,1, y i,1 ),..., (x i,ji, y i,ji )). Then the one-way ANCOVA model is written as y i,j = α i + βx i,j + ɛ i,j, where α i is the mean of the i th treatment level, β is the common regression slope, and the ɛ i,j are iid normal with mean 0 and variance σ 2. So note that the test of parallel slopes concerns testing if β is the same for all slopes versus if it is not the same for all slopes. A high p-value indicates that we have parallel slopes (or homogeneity of slopes) and can therefore use an ANCOVA model. 9.6 Coded Variables In the early days when computing power was limited, coding of the variables accomplished simplifying the linear algebra and thus allowing least squares solutions to be solved manually. Many methods exist for coding data, such as: Converting variables to two values (e.g., {-1, 1} or{0, 1}). Converting variables to three values (e.g., {-1, 0, 1}). Coding continuous variables to reflect only important digits (e.g., if the costs of various nuclear programs range from $100,000 to $150,000, D. S. Young STAT 501

120 CHAPTER 9. INDICATOR VARIABLES coding can be done by dividing through by $100,000, resulting in the range being from 1 to 1.5). The purpose of coding is to simplify the calculation of (X T X) 1 in the various regression equations, which was especially important when this had to be done by hand. It is important to note that the above methods are just a few possibilities and that there are no specific guidelines or rules of thumb for when to code data. Today when (X T X) 1 is calculated with computers, there may be a significant rounding error in the linear algebra manipulations if the difference in the magnitude of the predictors is large. Good statistical programs assess the probability of such errors, which would warrant using coded variables. When coding variables, one should be aware of different magnitudes of the parameter estimates compared to those for the original data. The intercept term can change dramatically, but we are concerned with any drastic changes in the slope estimates. In order to protect against additional errors due to the varying magnitudes of the regression parameters, you can compare plots of the actual data and the coded data and see if they appear similar. 9.7 Examples Example 1: Software Development Data Set Suppose that data from n = 20 institutions is collected on similar software development projects. The data set includes Y = number of man-years required for each project, X 1 = number of application subprograms developed for the project, and X 2 = 1 if an academic institution developed the program or 0 if a private firm developed the program. The data is given in Table 9.1. Suppose we wish to estimate the number of man-years necessary for developing this type of software for the purpose of contract bidding. We also suspect a possible interaction between the number of application subprograms developed and the type of institution. Thus, we consider the multiple regression model y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,1 x i,2 + ɛ i. So first, we fit the above model and assess the significance of the interaction term. STAT 501 D. S. Young

CHAPTER 9. INDICATOR VARIABLES 121 ########## Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) -23.8112 20.4315-1.165 0.261 subprograms 0.8541 0.1066 8.012 5.44e-07 *** institution 35.3686 26.7086 1.324 0.204 sub.inst -0.2019 0.1556-1.297 0.213 --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 38.42 on 16 degrees of freedom Multiple R-Squared: 0.8616, Adjusted R-squared: 0.8356 F-statistic: 33.2 on 3 and 16 DF, p-value: 4.210e-07 ########## The above gives the t-tests for these predictors. Notice that only the predictor of application subprograms (i.e., X 1 ) is statistically significant, so we should consider dropping the interaction term for starters. Number of Man Years 0 50 100 150 200 250 300 350 Academic Institution Private Firm 0 100 200 300 400 Number of Subprograms Figure 9.2: An interaction plot where the grouping is by institution. An interaction plot can also be used to justify use of an interaction term. Figure 9.2 provides the interaction plot for this data set. This plot seems to D. S. Young STAT 501

122 CHAPTER 9. INDICATOR VARIABLES indicate a possible (disordinal) interaction. A test of this interaction term yields p = 0.213 (see the earlier output). Even though the interaction plot indicates a possible interaction, the actual interaction term is deemed not statistically significant and thus we can drop it from the model. i Subprograms Institution Man-Years 1 135 0 52 2 128 1 58 3 221 0 207 4 82 1 95 5 401 0 346 6 360 1 244 7 241 0 215 8 130 0 112 9 252 1 195 10 220 0 54 11 112 0 48 12 29 1 39 13 57 0 31 14 28 1 57 15 41 1 20 16 27 1 33 17 33 1 19 18 7 0 6 19 17 0 7 20 94 1 56 Table 9.1: The software development data set. We next provide the analysis without the interaction term. Though the results are not shown here, a test of each predictor shows that the subprograms predictor is statistically significant (p = 0.000), while the institution predictor is not statistically significant (p = 0.612). This then tells us that there is no statistically significant difference in man-years for this type of software development between academic institutions and private firms. However, the number of subprograms is still a statistically significant predictor. So the final model should be a simple linear regression model with subprograms as the predictor and man-years as the response. The final estimated regression STAT 501 D. S. Young

CHAPTER 9. INDICATOR VARIABLES 123 equation is ŷ i = 3.47742 + 0.75088x i,1, which can be found from the following output: ########## Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) -3.47742 13.12068-0.265 0.794 subprograms 0.75088 0.07591 9.892 1.06e-08 *** --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 38.38 on 18 degrees of freedom Multiple R-Squared: 0.8446, Adjusted R-squared: 0.836 F-statistic: 97.85 on 1 and 18 DF, p-value: 1.055e-08 ########## Example 2: Steam Output Data (continued) Consider coding the steam output data by rounding the temperature to the nearest integer value ending in either 0 or 5. For example, a temperature of 57.5 degrees would be rounded up to 60 degrees while a temperature of 76.8 degrees would be rounded down to 75 degrees. While you would probably not utilize coding on such an easy data set where magnitude is not an issue, it is utilized here just for illustrative purposes. Figure 9.3 compares the scatterplots of this data set with the original temperature value and the coded temperature value. The plots look comparable, suggesting that coding could be used here. Recall that the estimated regression equation for the original data was ŷ i = 13.6230 0.0798x i. The estimated regression equation for the coded data is ŷ i = 13.7765 0.0824x i, which is also comparable. D. S. Young STAT 501

124 CHAPTER 9. INDICATOR VARIABLES Steam Data Steam Data Steam Usage (Monthly) 7 8 9 10 11 12 Steam Usage (Monthly) 7 8 9 10 11 12 30 40 50 60 70 Uncoded Temperature (Fahrenheit) (a) 30 40 50 60 70 Coded Temperature (Fahrenheit) (b) Figure 9.3: Comparing scatterplots of the atmospheric pressure data with the original temperature (a) and with the temperature coded (b). A line of best fit for each is also shown. STAT 501 D. S. Young

Chapter 10 Multicollinearity Recall that the columns of a matrix are linearly dependent if one column can be expressed as a linear combination of the other columns. A matrix theorem is that if there is a linear dependence among the columns of X, then (X T X) 1 does not exist. This means that we can t determine estimates of the beta coefficients since the formula for determining the estimates involves (X T X) 1. In multiple regression, the term multicollinearity refers to the linear relationships among the x-variables. Often, the use of this term implies that the x-variables are correlated with each other, so when the x-variables are not correlated with each other, we might say that there is no multicollinearity. 10.1 Sources and Effects of Multicollinearity There are various sources for multicollinearity. For example, in the data collection phase an investigator may have drawn the data from such a narrow subspace of the independent variables that collinearity appears. Physical constraints, such as design limits, may also impact the range of some of these independent variables. Model specification (such as defining more variables than observations or specifying too many higher-ordered terms/interactions) and outliers can both lead to collinearity. When there is no multicollinearity among x-variables, the effects of the individual x-variables can be estimated independently of each other (although we will still want to do a multiple regression). When multicollinearity is present, the estimated coefficients are correlated (confounding) with each 125

126 CHAPTER 10. MULTICOLLINEARITY other. This creates difficulty when we attempt to interpret how individual x-variables affect y. Along with this correlation, multicollinearity has a multitude of other ramifications on our analysis, including: inaccurate regression coefficient estimates, inflated standard errors of the regression coefficient estimates, deflated t-tests for significance testing of the regression coefficients, false nonsignificance determined by the p-values, and degradation of model predictability. In designed experiments with multiple x-variables, researchers usually choose the value of the x-variables so that there is no multicollinearity. In observational studies (sample surveys), it is nearly always the case that the x-variables will be correlated. 10.2 Detecting and Correcting Multicollinearity We introduce three primary ways for detecting multicollinearity - two of which are fairly straight-forward to implement, while the third method is actually a variety of measures based on the eigenvalues and eigenvectors of the standardized design matrix. Method 1: Pairwise Scatterplots For the first method, we can visually inspect the data by doing pairwise scatterplots of the independent variables. So if you have p 1 independent variables, then you should inspect all ( ) p 1 2 pairwise scatterplots. You will be looking for any plots that seem to indicate a linear relationship between pairs of independent variables. Method 2: VIF Second, we can use a measure of multicollinearity called the variance inflation factor (VIF ). This is defined as 1 V IF j =, 1 Rj 2 STAT 501 D. S. Young

CHAPTER 10. MULTICOLLINEARITY 127 where Rj 2 is the coefficient of determination obtained by regressing X j on the remaining independent variables. A common rule of thumb is that if V IF j = 1, then there is no collinearity, if 1 < V IF j < 5, then there is possibly some moderate collinearity, and if V IF j 5, then there is a strong indication of a collinearity problem. Most of the time, we will shoot for values as close to 1 as possible and that usually will be sufficient. The bottom line is that the higher the V IF, the more likely multicollinearity is an issue. Sometimes, the tolerance is also reported. The tolerance is simply the inverse of the V IF (i.e., T ol j = V IF 1 j ). In this case, the lower the T ol, the more likely multicollinearity is an issue. If multicollinearity is suspected after doing the above, then a couple of things can be done. First, reassess the choice of model and determine if there are any unnecessary terms and remove them. You may wish to start by removing the one you most suspect first, because this will then drive down the V IF s of the remaining variables. Next, check for outliers and see what effects some of the observations with higher residuals have on the analysis. Remove some (or all) of the suspected outliers and see how that effects the pairwise scatterplots and V IF values. You can also standardize the variables which involves simply subtracting each variable by it s mean and dividing by it s standard deviation. Thus, the standardized X matrix is given as: X 1,1 X 1 X 1,2 X 2 X s X1 s X2... 1,p 1 X p 1 s Xp 1 X 2,1 X 1 X 2,2 X 2 X X = 1 s n 1 X1 s X2... 2,p 1 X p 1 s Xp 1.,..... X n,1 X 1 X n,2 X 2 s X1 s X2... X n,p 1 X p 1 s Xp 1 which is a n (p 1) matrix, and the standardized Y vector is given as: Y = 1 n 1 which is still a n-dimensional vector. Here, s Xj = n Y 1 Ȳ s Y Y 2 Ȳ s Ỵ. Y n Ȳ s Y, i=1 (X i,j X j ) 2 n 1 D. S. Young STAT 501

128 CHAPTER 10. MULTICOLLINEARITY for j = 1, 2,..., (p 1) and s Y = n i=1 (Y i Ȳ )2. n 1 Notice that we have removed the column of 1 s in forming X, effectively reducing the column dimension of the original X matrix by 1. Because of this, we no longer can estimate an intercept term (b 0 ), which may be an important part of the analysis. Thus, proceed with this method only if you believe the intercept term adds little value to explaining the science behind your regression model! When using the standardized variables, the regression model of interest becomes: Y = X β + ɛ, where β is now a (p 1)-dimensional vector of standardized regression coefficients and ɛ is an n-dimensional vector of errors pertaining to this standardized model. Thus, the ordinary least squares estimates are b = (X T X ) 1 X T Y = r 1 XX r XY, where r XX is the (p 1) (p 1) correlation matrix of the predictors and r XY is the (p 1)-dimensional vector of correlation coefficients between the predictors and the response. Because b is a function of correlations, this method is called a correlation transformation. Sometimes, it may be enough to just simply center the variables by their respective means in order to decrease the V IF s. Note the relationship between the quantities introduced above and the correlation matrix r from earlier: ( 1 r T r = XY r XY r XX Method 3: Eigenvalue Methods Finally, the third method for identifying potential multicollinearity concerns a variety of measures utilizing eigenvalues and eigenvectors. First, note that the eigenvalue λ j and the corresponding (p 1)-dimensional orthonormal eigenvectors ξ j are solutions to the system of equations: X T X ξ j = λ j ξ j, ). STAT 501 D. S. Young

CHAPTER 10. MULTICOLLINEARITY 129 for j = 1,..., (p 1). Since the λ j s are normalized, it follows that ξ T j X T X ξ j = λ j. Therefore, if λ j 0, then X ξ j 0; i.e., the columns of X are approximately linearly dependent. Thus, since the sum of the eigenvalues must equal the number of predictors (i.e., (p 1)), then very small λ j s (say, near 0.05) are indicative of collinearity. Another criterion commonly used is to declare multicollinearity is present when p 1 j=1 λ 1 j > 5(p 1). Moreover, the entries of the corresponding ξ j s indicate the nature of the linear dependencies; i.e., large elements of the eigenvectors identify the predictor variables that comprise the collinearity. A measure of the overall multicollinearity of the variables can be obtained by computing what is called the condition number of the correlation matrix (i.e., r) which is defined as λ (p 1) /λ (1), such that λ (1) and λ (p 1) are the minimum and maximum eigenvalues, respectively. Obviously this quantity is always greater than 1, so a large number is indicative of collinearity. Empirical evidence suggests that a value less than 15 typically means weak collinearity, values between 15 and 30 is evidence of moderate collinearity, while anything over 30 is evidence of strong collinearity. Condition numbers for the individual predictors can also be calculated. This is accomplished by taking c j = λ (p 1) /λ j for each j = 1,..., (p 1). When data is centered and scaled, then c j 100 indicates no collinearity, 100 < c j < 1000 indicates moderate collinearity, while c j 1000 indicates strong collinearity for predictor X j. When the data is only scaled (i.e., for regression through the origin models), then collinearity will always be worse. Thus, more relaxed limits are usually used. For example, a common rule of thumb is to use 5 times the limits mentioned above; namely, c j 500 indicates no collinearity, 500 < c j < 5000 indicates moderate collinearity, while c j 5000 indicates strong collinearity for predictor X j. It should be noted that there are many heuristic ways other than those described above to assess multicollinearity with eigenvalues and eigenvectors. 1 Moreover, it should be noted that some observations can have an undue influence on these various measures of collinearity. These observations are called collinearity-influential observations and care should be taken with how 1 For example, one such technique involves taking the square eigenvector relative to the square eigenvalue and then seeing what percentage each quantity in this (p 1)-dimensional vector explains of the total variation for the corresponding regression coefficient. D. S. Young STAT 501

130 CHAPTER 10. MULTICOLLINEARITY these observations are handled. You can typically use some of the residual diagnostic measures (e.g., DFITs, Cook s D i, DFBETAS, etc.) for identifying potential collinearity-influential observations since there is no established or agreed-upon method for classifying such observations. Finally, there are also some more advanced regression procedures that can be performed in the presence of multicollinearity. Such methods include principal components regression and ridge regression. These methods are discussed later. 10.3 Examples Example 1: Muscle Mass Data Set Suppose that data from n = 6 individuals is collected on their muscle mass. The data set includes Y = muscle mass, X 1 = age, and two possible (yet redundant) indicator variables for gender: 1. X 2 = 1 if the person is male (M), and 0 if female (F). 2. X 3 = 1 if the person is female (F), and 0 if male (M). The data are given in Table 10.1. mass 60 50 70 42 50 45 age 40 45 43 60 60 65 sex M F M F M F Table 10.1: The muscle mass data set. Suppose that we (mistakenly) attempt to use the model y i = β 0 + β 1 x i,1 + β 2 x i,2 + β 3 x i,3 + ɛ i. Notice that we used both indicator variables in the model. For these data and this model, 1 40 1 0 1 45 0 1 X = 1 43 1 0 1 60 0 1. 1 60 1 0 1 65 0 1 STAT 501 D. S. Young

CHAPTER 10. MULTICOLLINEARITY 131 The sum of the last two columns equals the first column for every row in the X matrix. This is a linear dependence, so parameter estimates cannot be calculated because (X T X) 1 does not exist. In practice, the usual solution is to drop one of the indicator variables from the model. Another solution is to drop the intercept (thus dropping the first column of X above), but that is not usually done. For this example, we can t proceed with a multiple regression analysis because there is perfect collinearity with X 2 and X 3. Sometimes, a generalized inverse can be used (which requires more of a discussion beyond the scope of this course) or if you attempt to do an analysis on such a data set, the software you are using may zero out one of the variables that is contributing to the collinearity and then proceed to do an analysis. However, this can lead to errors in the final analysis. Example 2: Heat Flux Data Set (continued) Let us return to the heat flux data set. Let our model include the east, south, and north focal points, but also incorporate time and insolation as predictors. First, let us run a multiple regression analysis which includes these predictors. ########## Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 325.43612 96.12721 3.385 0.00255 ** east 2.55198 1.24824 2.044 0.05252. north -22.94947 2.70360-8.488 1.53e-08 *** south 3.80019 1.46114 2.601 0.01598 * time 2.41748 1.80829 1.337 0.19433 insolation 0.06753 0.02899 2.329 0.02900 * --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 8.039 on 23 degrees of freedom Multiple R-Squared: 0.8988, Adjusted R-squared: 0.8768 F-statistic: 40.84 on 5 and 23 DF, p-value: 1.077e-10 ########## We see that time is not a statistically significant predictor of heat flux and, in fact, east has become marginally significant. D. S. Young STAT 501

132 CHAPTER 10. MULTICOLLINEARITY However, let us now look at the V IF values: ########## east north south time insolation 1.355448 2.612066 3.175970 5.370059 2.319035 ########## Notice that the V IF for time is fairly high (about 5.37). This is somewhat a high value and should be investigated further. So next, the pairwise scatterplots are given in Figure 10.1 (we will only look at the plots involving the time variable since that is the variable we are investigating). Notice how there appears to be a noticeable linear trend between time and the south focal point. There also appears to be some sort of curvilinear trend between time and the north focal point as well as between time and insolation. These plots, combined with the V IF for time, suggests to look at a model without the time variable. After removing the time variable, we obtain the new V IF values: ########## east north south insolation 1.277792 1.942421 1.206057 1.925791 ########## Notice how removal of the time variable has sharply decreased the V IF values for the other variables. The regression coefficient estimates are: ########## Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 270.21013 88.21060 3.063 0.00534 ** east 2.95141 1.23167 2.396 0.02471 * north -21.11940 2.36936-8.914 4.42e-09 *** south 5.33861 0.91506 5.834 5.13e-06 *** insolation 0.05156 0.02685 1.920 0.06676. --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 8.17 on 24 degrees of freedom Multiple R-Squared: 0.8909, Adjusted R-squared: 0.8727 F-statistic: 48.99 on 4 and 24 DF, p-value: 3.327e-11 ########## STAT 501 D. S. Young

CHAPTER 10. MULTICOLLINEARITY 133 11 12 13 14 15 16 31 32 33 34 35 36 37 38 Time East (a) 11 12 13 14 15 16 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 Time North (b) 11 12 13 14 15 16 32 34 36 38 40 Time South (c) 11 12 13 14 15 16 600 650 700 750 800 850 900 Time Insolation (d) Figure 10.1: Pairwise scatterplots of the variable time versus (a) east, (b) north, (c) south, and (d) insolation. Notice how there appears to be some sort of relationship between time and the predictors north, south, and insolation. D. S. Young STAT 501

134 CHAPTER 10. MULTICOLLINEARITY So notice how now east is statistically significant while insolation is marginally significant. If we proceeded to drop insolation from the model, then we would be back to the analysis we did earlier in the chapter. This illustrates how dropping or adding a predictor to a model can change the significance of other predictors. We will return to this when we discuss stepwise regression. STAT 501 D. S. Young

Chapter 11 ANOVA II As in simple regression, the ANOVA table for a multiple regression model displays quantities that measure how much of the variability in the y-variable is explained and how much is not explained by the x-variables. The calculation of the quantities involved is nearly identical to what we did in simple regression. The main difference has to do with the degrees of freedom quantities. The basic structure is given in Table 11.1 and the explanations follow. Source df SS MS F Regression p 1 n i=1 (ŷ i ȳ) 2 MSR MSR/MSE Error n p n i=1 (y i ŷ i ) 2 MSE Total n 1 n i=1 (y i ȳ) 2 Table 11.1: ANOVA table for any regression. The sum of squares for total is SSTO = n i=1 (y i ȳ) 2, which is the sum of squared deviations from the overall mean of y and df T = n 1. SSTO is a measure of the overall variation in the y-values. In matrix notation, SSTO = Y Ȳ 1 2. The sum of squared errors is SSE = n i=1 (y i ŷ i ) 2, which is the sum of squared observed errors (residuals) for the observed data. SSE is a measure of the variation in y that is not explained by the regression. For multiple linear regression, df E = n p, where p = number of beta coefficients in the model (including the intercept β 0 ). As an example, 135

136 CHAPTER 11. ANOVA II if there are four x-variables then p = 5 (the betas multiplying the x-variables plus the intercept). In matrix notation, SSE = Y Ŷ 2. The mean squared error is MSE = SSE df E the variance of the errors. = SSE n p, which estimates σ2, The sum of squares due to the regression is SSR = SSTO SSE, and it is a measure of the total variation in y that can be explained by the regression with the x-variable. Also, df R = df T df E. For multiple regression, df R = (n 1) (n p) = p 1. In matrix notation, SSR = Ŷ Ȳ 1 2. The mean square for the regression is MSR = SSR df R = SSR p 1. 11.1 Uses of the ANOVA Table 1. The F -statistic in the ANOVA given in Table 11.1 can be used to test whether the y-variable is related to one or more of the x-variables in the model. Specifically, F = MSR/MSE is a test statistic for H 0 : β 1 = β 2 =... = β p 1 = 0 H A : at least one β i 0 for i = 1,..., p 1. The null hypothesis means that the y-variable is not related to any of the x-variables in the model. The alternative hypothesis means that the y-variable is related to one or more of the x-variables in the model. Statistical software will report a p-value for this test statistic. The p- value is calculated as the probability to the right of the calculated value of F in an F distribution with p 1 and n p degrees of freedom (often written as F p 1,n p;1 α ). The usual decision rule also applies here in that if p < 0.05, reject the null hypothesis. If that is our decision, we conclude that y is related to at least one of the x-variables in the model. 2. MSE is the estimate of the error variance σ 2. Thus s = MSE estimates the standard deviation of the errors. 3. As in simple regression, R 2 = SSTO-SSE, but here it is called the coefficient of multiple determination. R 2 is interpreted as the SSTO pro- STAT 501 D. S. Young

CHAPTER 11. ANOVA II 137 portion of variation in the observed y values that is explained by the model (i.e., by the entire set of x-variables in the model). 11.2 The General Linear F -Test The general linear F -test procedure is used to test any null hypothesis that, if true, still leaves us with a linear model (linear in the β s). The most common application is to test whether a particular set of β coefficients are all equal 0. As an example, suppose we have a response variable (Y ) and 5 predictor variables (X 1, X 2,..., X 5 ). Then, we might wish to test H 0 : β 1 = β 3 = β 4 = 0 H A : at least one of {β 1, β 3, β 4 } 0. The purpose for testing a hypothesis like this is to determine if we could eliminate variables X 1, X 3, and X 4 from a multiple regression model (an action implied by the statistical truth of the null hypothesis). The full model is the multiple regression model that includes all variables under consideration. The reduced model is the regression model that would result if the null hypothesis is true. The general linear F -statistic is F = SSE(reduced) SSE(full) df E (reduced) df E (full). MSE(full) Here, this F -statistic has degrees of freedom df 1 = df E (reduced) df E (full) and df 2 = df E (full). With the rejection region approach and a 0.05 significance level, we reject H 0 if the calculated F is greater than the tabled value F df1,df 2 ;1 α, which is the 95 th percentile of the appropriate F df1,df 2 -distribution. A p-value is found as the probability that the F -statistic would be as large or larger than the calculated F. To summarize, the general linear F -test is used in settings where there are many predictors and it is desirable to see if only one or a few of the predictors can adequately perform the task of estimating the mean response and prediction of new observations. Sometimes the full and reduced sums of squares (that we introduced above) are referred to as extra sum of squares. D. S. Young STAT 501

138 CHAPTER 11. ANOVA II 11.3 Extra Sums of Squares The extra sums of squares measure the marginal reduction in the SSE when one or more predictor variables are added to the regression model given that other predictors are already in the model. In probability theory, we write A B which means that event A happens GIVEN that event B happens (the vertical bar means given ). We also utilize this notation when writing extra sums of squares. For example, suppose we are considering two predictors, X 1 and X 2. The SSE when both variables are in the model is smaller than when only one of the predictors is in the model. This is because when both variables are in the model, they both explain additional variability in Y which drives down the SSE compared to when only one of the variables is in the model. This difference is what we call the extra sums of squares. For example, SSR(X 1 X 2 ) = SSE(X 2 ) SSE(X 1, X 2 ), which measures the marginal effect of adding X 1 to the model, given that X 2 is already in the model. An equivalent expression is to write SSR(X 1 X 2 ) = SSR(X 1, X 2 ) SSR(X 2 ), which can be viewed as the marginal increase in the regression sum of squares. Notice (for the second formulation) that the corresponding degrees of freedom is (3-1)-(2-1)=1 (because the df for SSR(X 1, X 2 ) is (3-1)=2 and the df for SSR(X 2 ) is (2-1)=1). Thus, MSR(X 1 X 2 ) = SSR(X 1 X 2 ). 1 When more predictors are available, then there are a vast array of possible decompositions of the SSR into extra sums of squares. One generic formulation is if you have p predictors, then SSR(X 1,..., X j,..., X p ) = SSR(X j ) + SSR(X 1 X j ) +... + SSR(X j 1 X 1,..., X j 2, X j ) + SSR(X j+1 X 1,..., X j ) +... + SSR(X p X 1,..., X p 1 ). In the above, j is just being used to indicate any one of the p predictors. You can also calculate the marginal increase in the regression sum of squares STAT 501 D. S. Young

CHAPTER 11. ANOVA II 139 when adding more than one predictor. One generic formulation is SSR(X 1,..., X j X j+1,..., X p ) = SSR(X 1 X j+1,..., X p ) + SSR(X 2 X 1, X j+1,..., X p ) +... + SSR(X j 1 X 1,..., X j 2, X j+1,..., X p ) + SSR(X j X 1,..., X j 1, X j+1,..., X p ). Again, you could imagine many such possibilities. The primary use of extra sums of squares is in the testing of whether or not certain predictors can be dropped from the model (i.e., the general linear F -test). Furthermore, they can also be used to calculate a version of R 2 for such models, called the partial R 2. 11.4 Lack of Fit Testing in the Multiple Regression Setting Formal lack of fit testing can also be performed in the multiple regression setting; however, the ability to achieve replicates can be more difficult as more predictors are added to the model. Note that the corresponding ANOVA table (Table 11.2) is similar to that introduced for the simple linear regression setting. However, now we have the notion of p regression parameters and the number of replicates (m) refers to the number of unique X vectors. In other words, each predictor must have the same value for two observations for it to be considered a replicate. For example, suppose we have 3 predictors for our model. The observations (40, 10, 12) and (40, 10, 7) are unique levels for our X vectors, whereas the observations (10, 5, 13) and (10, 5, 13) would constitute a replicate. Formal lack of fit testing in multiple regression can be difficult due to sparse data, unless the experiment was designed properly to achieve replicates. However, other methods can be employed for lack of fit testing when you do not have replicates. Such methods involve data subsetting. The basic approach is to establish criteria by introducing indicator variables, which in turn creates coded variables (as discussed earlier). By coding the variables, you can artificially create replicates and then you can proceed with lack of fit testing. Another approach with data subsetting is to look at central regions of the data (i.e., observations where the leverage is less than (1.1) p/n) and treat this as a reduced data set. Then compare this reduced fit to the full fit D. S. Young STAT 501

140 CHAPTER 11. ANOVA II Source df SS MS F Regression p 1 SSR MSR MSR/MSE Error n p SSE MSE Lack of Fit m p SSLOF MSLOF MSLOF/MSPE Pure Error n m SSPE MSPE Total n 1 SSTO Table 11.2: ANOVA table for multiple linear regression which includes a lack of fit test. (i.e., the fit with all of the data), for which the formulas for a lack of fit test can be employed. Be forewarned that these methods should only be used as exploratory methods and they are heavily dependent on what sort of data subsetting method is used. 11.5 Partial R 2 Suppose we have set up a general linear F -test. Then, we may be interested in seeing what percent of the variation in the response cannot be explained by the predictors in the reduced model (i.e., the model specified by H 0 ), but can be explained by the rest of the predictors in the full model. If we obtain a large percentage, then it is likely we would want to specify some or all of the remaining predictors to be in the final model since they explain so much variation. The way we formally define this percentage is by what is called the partial R 2 (or it is also called the coefficient of partial determination). Specifically, suppose we have three predictors: X 1, X 2, and X 3. For the corresponding multiple regression model (with response Y ), we wish to know what percent of the variation is explained by X 2 and X 3 which is not explained by X 1. In other words, given X 1, what additional percent of the variation can be explained by X 2 and X 3? Note that here the full model will include all three predictors, while the reduced model will only include X 1. After obtaining the relevant ANOVA tables for these two models, the STAT 501 D. S. Young

CHAPTER 11. ANOVA II 141 partial R 2 is as follows: R 2 Y,2,3 1 = SSR(X 2, X 3 X 1 ) SSE(X 1 ) = SSE(X 1) SSE(X 1, X 2, X 3 ) SSE(X 1 ) SSE(reduced) SSE(full) =. SSE(reduced) Then, this gives us the proportion of variation explained by X 2 and X 3 that cannot be explained by X 1. Note that the last line of the above equation is just demonstrating that the partial R 2 has a similar form to the R 2 that we calculated in the simple linear regression case. More generally, consider partitioning the predictors X 1, X 2,..., X k into two groups, A and B, containing u and k u predictors, respectively. The proportion of variation explained by the predictors in group B that cannot be explained by the predictors in group A is given by RY,B A 2 = SSR(B A) SSE(A) SSE(A) SSE(A, B) =. SSE(A) These partial R 2 values can also be used to calculate the power for the corresponding general linear F -test. The power of this test is calculated by first finding the tabled 100 (1 α) th percentile of the F u,n k 1 -distribution. Next we calculate F u,n k 1;1 α (δ), which is the 100 (1 α) th percentile of a non-central F u,n k 1 -distribution with non-centrality parameter δ. The non-centrality parameter is calculated as: ( R 2 Y,A,B RY,B 2 δ = n 1 RY,A,B 2 Finally, the power is simply the probability that the calculated F u,n k 1;1 α (δ) value is greater than the calculated F u,n k 1;1 α value under the F u,n k 1 (δ)- distribution. D. S. Young STAT 501 ).

142 CHAPTER 11. ANOVA II 11.6 Partial Leverage and Partial Residual Plots Next we establish a way to visually assess the relationship of a given predictor to the response when accounting for all of the other predictors in the multiple linear regression model. Suppose we have p 1 predictors X 1,..., X p 1 and that we are trying to assess each predictor s relationship with a response Y given the other predictors are already in the model. Let r Y[j] denote the residuals that result from regressing Y on all of the predictors except X j. Moreover, let r X[j] denote the residuals that result from regressing X j on all of the remaining p 2 predictors. A partial leverage regression plot (also referred to as an added variable plot, adjusted variable plot, or individual coefficients plot) is constructed by plotting r Y[j] on the y-axis and r X[j] on the x-axis. 1 Then, the relationship between these two sets of residuals is examined to provide insight into X j s contribution to the response given the other p 2 predictors are in the model. This is a helpful exploratory measure if you are not quite sure about what type of relationship (e.g., linear, quadratic, logarithmic, etc.) that the response may have with a particular predictor. More formally, partial leverage is used to measure the contribution of the individual predictor variables to the leverage of each observation. That is, if h i,i is the i th diagonal entry of the hat matrix, the partial leverage is a measure of how h i,i changes as a variable is added to the regression model. The partial leverage is computed as: (h j) i = rx 2 [j],i n. k=1 r2 x [j],k If you do a simple linear regression of r Y[j] on r X[j], you will see that the OLS line goes through the origin. This is because the OLS line goes through the point ( r X[j], r Y[j] ) and each of these means is 0. Thus, a regression through the origin can be fit to this data. Moreover, the slope from this regression through the origin fit is equal to the slope for X j if it were included in the full model where Y is regressed on all p 1 predictors. We have given a great deal of attention to various residual measures as well as plots to assess the adequacy of our fitted model. We can also 1 Note that you can produce p 1 partial leverage regression plots (i.e., one for each predictor). STAT 501 D. S. Young

CHAPTER 11. ANOVA II 143 use another type of residual to check the assumption of linearity for each predictor. Partial residuals are residuals that have not been adjusted for a particular predictor variable (say, X j ). Suppose, we partition the X matrix such that X = (X j, X j ). For this formulation, X j is the same as the X matrix, but with the vector of observations for the predictor X j omitted (i.e., this vector of values is X j ). Similarly, let us partition the vector of estimated regression coefficients as b = (b T j, b j ) T. Then, the set of partial residuals for the predictor X j would be e j = Y X j b j = Y Ŷ + Ŷ X j b j = e (X j b j Xb) = e + b j X j. Note in the above that b j is just a univariate quantity, so b j X j is still an n-dimensional vector. Finally, a plot of e j versus X j has slope b j. The more the data deviates from a straight-line fit for this plot (which is sometimes called a component plus residual plot), the greater the evidence that a higher-ordered term or transformation on this predictor variable is necessary. Note also that the vector e would provide the residuals if a straight-line fit were made to these data. 11.7 Examples Example 1: Heat Flux Data Set (continued) Refer back to the raw data given in Table 7.1. We will again look at the full model which includes all of the predictor variables (even though we have shown which predictors should likely be removed from the model). The response is still y = total heat flux, while the predictors are x 1 = insolation recording, x 2 = east focal point, x 3 = south focal point, x 4 = north focal point, and x 5 = time of the recordings. So the full model will be Y = Xβ + ɛ, where β = (β 0 β 1 β 2 β 3 β 4 β 5 ) T, Y is a 29-dimensional response vector, X is a 29 5-dimensional design matrix, and ɛ is a 29-dimensional error vector. First, here is the ANOVA for the model with all of the predictors: D. S. Young STAT 501

144 CHAPTER 11. ANOVA II ########## Analysis of Variance Table Response: flux Df Sum Sq Mean Sq F value Pr(>F) Regression 5 13195.5 2639.1 40.837 1.077e-10 *** Residuals 23 1486.4 64.6 --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 ########## Now, using the results from earlier (which seem to indicate a model including only the north and south focal points as predictors) let us test the following hypothesis: H 0 : β 1 = β 2 = β 5 = 0 H A : at least one of {β 1, β 2, β 5 } 0. In other words, we only want our model to include the south (x 3 ) and north (x 4 ) focal points. We see that MSE(full) = 64.63, SSE(full) = 1486.40, and df E (full) = 23. Next we calculate the ANOVA table for the above null hypothesis: ########## Analysis of Variance Table Response: flux Df Sum Sq Mean Sq F value Pr(>F) Regression 2 12607.6 6303.8 79.013 8.938e-12 *** Residuals 26 2074.3 79.8 --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 ########## The ANOVA for this analysis shows that SSE(reduced) = 2074.33 and df E (reduced) = 26. Thus, the F -statistic is: F = 2074.33 1486.40 26 23 64.63 = 3.027998, STAT 501 D. S. Young

CHAPTER 11. ANOVA II 145 which follows a F 3,23 distribution. The p-value (i.e., the probability of getting an F -statistic as extreme or more extreme than 3.03 under a F 3,23 distribution) is 0.0499. Thus we just barely claim statistical significance and conclude that at least one of the other predictors (insolation, east focal point, and time) is a statistically significant predictor of heat flux. We can also calculate the power of this F -test by using the partial R 2 values. Specifically, R 2 Y,1,2,3,4,5 = 0.8987602 RY,3,4 2 = 0.8587154 0.8987602 0.8587154 δ = (29) 1 0.8987602 = 11.47078. Therefore, P (F > 3.027998) under an F 3,23 (11.47078)-distribution gives the power, which is as follows: ########## Power F-Test 0.7455648 ########## Notice that this is not very powerful (i.e., we usually hope to attain a power of at least 0.80). Thus the probability of committing a Type II error is somewhat high. We can also calculate the partial R 2 for this testing situation: RY,1,2,5 3,4 2 = SSE(X 3, X 4 ) SSE(X 1, X 2, X 3, X 4, X 5 ) SSE(X 3, X 4 ) 2074.33 1486.4 = 2074.33 = 0.2834. This means that insolation, the east focal point, and time explain about 28.34% of the variation in heat flux that could not be explained by the north and south focal points. Example 2: Simulated Data for Partial Leverage Plots Suppose we have a response variable, Y, and two predictors, X 1 and X 2. Let us consider three settings: D. S. Young STAT 501

146 CHAPTER 11. ANOVA II 1. Y is a function of X 1 ; 2. Y is a function of X 1 and X 2 ; and 3. Y is a function of X 1, X 2, and X 2 2. Setting (3) is a quadratic regression model which falls under the polynomial regression framework, which we discuss in greater detail later. Figure 11.1 shows the partial leverage regression plots for X 1 and X 2 for each of these three settings. In Figure 11.1(a), we see how the plot indicates that there is a strong linear relationship between Y and X 1 when X 2 is in the model, but this is not the case between Y and X 2 when X 1 is in the model (Figure 11.1(b)). Figures 11.1(c) and 11.1(d) shows that there is a strong linear relationship between Y and X 1 when X 2 is in the model as well as between Y and X 2 when X 1 is in the model. Finally, Figure 11.1(e) shows that there is a linear relationship between Y and X 1 when X 2 is in the model, but there is an indication of a quadratic (i.e., curvilinear) relationship between Y and X 2 when X 1 is in the model. For setting (2), the fitted model when Y is regressed on X 1 and X 2 is Ŷ = 9.92 + 6.95X 1 4.20X 2 as shows in the following output: Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 9.9174 2.4202 4.098 0.000164 *** X1 6.9483 0.3427 20.274 < 2e-16 *** X2-4.2044 0.3265-12.879 < 2e-16 *** --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 6.058 on 47 degrees of freedom Multiple R-squared: 0.915, Adjusted R-squared: 0.9114 F-statistic: 253 on 2 and 47 DF, p-value: < 2.2e-16 The slope from the regression through the origin fit of r Y[1] on r X[1] is 6.95 while the slope from the regression through the origin fit of r Y[2] on r X[2] is -4.20. The output from both of these fits is given below: Coefficients: Estimate Std. Error t value Pr(> t ) r_x.1 6.9483 0.3357 20.7 <2e-16 *** STAT 501 D. S. Young

CHAPTER 11. ANOVA II 147 4 2 0 2 4 40 20 0 20 40 r X[1] r Y[1] (a) 4 2 0 2 4 10 5 0 5 10 r X[2] r Y[2] (b) 4 2 0 2 4 40 20 0 20 40 r X[1] r Y[1] (c) 4 2 0 2 4 20 10 0 10 20 r X[2] r Y[2] (d) 4 2 0 2 4 50 0 50 100 150 r X[1] r Y[1] (e) 4 2 0 2 4 200 100 0 100 200 300 400 r X[2] r Y[2] (f) Figure 11.1: Scatterplots of (a) r Y[1] versus r X[1] and (b) r Y[2] versus r X[2] for setting (1), (c) r Y[1] versus r X[1] and (d) r Y[2] versus r X[2] for setting (2), and (e) r Y[1] versus r X[1] and (f) r Y[2] versus r X[2] for setting (3). D. S. Young STAT 501

148 CHAPTER 11. ANOVA II --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 5.933 on 49 degrees of freedom Multiple R-squared: 0.8974, Adjusted R-squared: 0.8953 F-statistic: 428.5 on 1 and 49 DF, p-value: < 2.2e-16 -------------------------------------------------------------- Coefficients: Estimate Std. Error t value Pr(> t ) r_x.2-4.2044 0.3197-13.15 <2e-16 *** --- Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1 Residual standard error: 5.933 on 49 degrees of freedom Multiple R-squared: 0.7792, Adjusted R-squared: 0.7747 F-statistic: 172.9 on 1 and 49 DF, p-value: < 2.2e-16 Note that while the slopes from these partial leverage regression routines are the same as their respective slopes in the full multiple linear regression routine, the other statistics regarding hypothesis testing are not the same since, fundamentally, different assumptions are made for the respective tests. STAT 501 D. S. Young