Part II. Multiple Linear Regression

Transcription

1 Part II Multiple Linear Regression 86

2 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

3 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, 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

4 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, 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

5 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

6 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) e-06 *** east north e-12 *** south e-06 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 25 degrees of freedom D. S. Young STAT 501

7 92 CHAPTER 7. MULTIPLE REGRESSION Histogram of Residuals Residuals vs. Fitted Values Density Residuals 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: , Adjusted R-squared: F-statistic: 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 ). 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) e-12 *** north e-12 *** south e-05 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 26 degrees of freedom STAT 501 D. S. Young

8 CHAPTER 7. MULTIPLE REGRESSION 93 Multiple R-Squared: , Adjusted R-squared: F-statistic: 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 = x i, x 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 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 = means that the model (the two x-variables) explains 85.87% of the observed variation in a home s flux temperature. The value MSE = 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 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

9 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 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-05 *** Cr e-13 *** Co Signif. codes: 0 *** ** 0.01 * Residual standard error: on 128 degrees of freedom Multiple R-squared: 0.544, Adjusted R-squared: F-statistic: 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 STAT 501 D. S. Young

10 CHAPTER 7. MULTIPLE REGRESSION 95 Figure 7.2: 3D scatterplot of the Kola data set with the least squares plane 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

11 96 CHAPTER 7. MULTIPLE REGRESSION Figure 7.4: 3D scatterplot of the Kola data set where the logarithm of each variable has been taken 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

12 CHAPTER 7. MULTIPLE REGRESSION 97 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** ln_cr e-10 *** ln_co Signif. codes: 0 *** ** 0.01 * Residual standard error: on 128 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 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 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** ln_cr <2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 129 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 129 DF, p-value: < 2.2e-16 D. S. Young STAT 501

13 98 CHAPTER 7. MULTIPLE REGRESSION ln(cr) ln(cr_inaa) (a) 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

14 CHAPTER 7. MULTIPLE REGRESSION 99 i Flux Insolation East South North Time Table 7.1: The heat flux for homes data. D. S. Young STAT 501

15 100 CHAPTER 7. MULTIPLE REGRESSION X 1 X 2 Y X 1 X 2 Y X 1 X 2 Y X 1 X 2 Y 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

16 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

17 102 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 = 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

18 CHAPTER 8. MATRIX NOTATION IN REGRESSION 103 y i x i, x i, Table 8.1: A sample data set. Y = , X = , β = β 0 β 1 β 2 β 3, ɛ = ɛ 1 ɛ 2 ɛ 3 ɛ 4 ɛ 5 ɛ 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

19 104 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

20 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

21 106 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

22 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

23 108 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 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) = = 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 ) ( )( ) ( )(3.7657) ( )(0.9046) (3.7657)( ) (3.7657)(3.7657) (3.7657)(0.9046) (0.9046)( ) (0.9046)(3.7657) (0.9046)(0.9046) STAT 501 D. S. Young

24 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 ) = , 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 = We can also calculate the 95% confidence intervals for the regression coefficients. First note that t 26,0.975 = The 95% confidence interval for β 1 is calculated using ± and for β 2 it is calculated D. S. Young STAT 501.

25 110 CHAPTER 8. MATRIX NOTATION IN REGRESSION using ± Thus, we are 95% confident that the true population regression coefficients for the north and south focal points are between ( , ) and (2.8413, ), respectively. STAT 501 D. S. Young

26 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

27 112 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

28 CHAPTER 9. INDICATOR VARIABLES 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

29 114 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

30 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

31 116 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

32 CHAPTER 9. INDICATOR VARIABLES 117 No Interaction Disordinal Interaction Response Treatment 1 Treatment 2 Treatment 3 Response Treatment 1 Treatment 2 Treatment Predictor Predictor (a) (b) Ordinal Interaction Response Treatment 1 Treatment 2 Treatment 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

33 118 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

34 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

35 120 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