ࡱ>   wi bjbj$$ F|F|ل@@ EBj((()w00 0BBBBBBB$FgIB0G/0w000B()+D8880(p)B80B882j@TJZB )@ac4fA.BD0 E,B.1J81J\fB1JfB$00800000BB8000 E00001J000000000@ `:  EMBED Equation.COEE2 Chapter 1 Linear Regression with 1 Predictor Statistical Model  EMBED Equation.COEE2  where:  EMBED Equation.COEE2  is the (random) response for the ith case  EMBED Equation.COEE2  are parameters  EMBED Equation.COEE2  is a known constant, the value of the predictor variable for the ith case  EMBED Equation.COEE2  is a random error term, such that:  EMBED Equation.COEE2  The last point states that the random errors are independent (uncorrelated), with mean 0, and variance  EMBED Equation.COEE2 . This also implies that:  EMBED Equation.COEE2  Thus,  EMBED Equation.COEE2  represents the mean response when  EMBED Equation.COEE2  (assuming that is reasonable level of  EMBED Equation.COEE2 ), and is referred to as the Y-intercept. Also,  EMBED Equation.COEE2  represent the change in the mean response as  EMBED Equation.COEE2  increases by 1 unit, and is called the slope. Least Squares Estimation of Model Parameters In practice, the parameters  EMBED Equation.COEE2  and  EMBED Equation.COEE2  are unknown and must be estimated. One widely used criterion is to minimize the error sum of squares:  EMBED Equation.COEE2  This is done by calculus, by taking the partial derivatives of  EMBED Equation.COEE2  with respect to  EMBED Equation.COEE2  and  EMBED Equation.COEE2  and setting each equation to 0. The values of  EMBED Equation.COEE2  and  EMBED Equation.COEE2  that set these equations to 0 are the least squares estimates and are labelled  EMBED Equation.COEE2  and  EMBED Equation.COEE2 . First, take the partial derivates of  EMBED Equation.COEE2  with respect to  EMBED Equation.COEE2  and  EMBED Equation.COEE2 :  EMBED Equation.COEE2  Next, set these these 2 equations to 0, replacing  EMBED Equation.COEE2  and  EMBED Equation.COEE2  with  EMBED Equation.COEE2  and  EMBED Equation.COEE2  since these are the values that minimize the error sum of squares:  EMBED Equation.COEE2  These two equations are referred to as the normal equations (although, note that we have said nothing YET, about normally distributed data). Solving these two equations yields:  EMBED Equation.COEE2  where  EMBED Equation.COEE2  and  EMBED Equation.COEE2  are constants, and  EMBED Equation.COEE2  is a random variable with mean and variance given above:  EMBED Equation.COEE2  The fitted regression line, also known as the prediction equation is:  EMBED Equation.COEE2  The fitted values for the individual observations aye obtained by plugging in the corresponding level of the predictor variable ( EMBED Equation.COEE2 ) into the fitted equation. The residuals are the vertical distances between the observed values ( EMBED Equation.COEE2 ) and their fitted values ( EMBED Equation.COEE2 ), and are denoted as  EMBED Equation.COEE2 .  EMBED Equation.COEE2  Properties of the fitted regression line  EMBED Equation.COEE2  The residuals sum to 0  EMBED Equation.COEE2  The sum of the weighted (by  EMBED Equation.COEE2 ) residuals is 0  EMBED Equation.COEE2  The sum of the weighted (by  EMBED Equation.COEE2 ) residuals is 0 The regression line goes through the point ( EMBED Equation.COEE2 ) These can be derived via their definitions and the normal equations. Estimation of the Error Variance Note that for a random variable, its variance is the expected value of the squared deviation from the mean. That is, for a random variable  EMBED Equation.COEE2 , with mean  EMBED Equation.COEE2  its variance is:  EMBED Equation.COEE2  For the simple linear regression model, the errors have mean 0, and variance  EMBED Equation.COEE2 . This means that for the actual observed values  EMBED Equation.COEE2 , their mean and variance are as follows:  EMBED Equation.COEE2  First, we replace the unknown mean  EMBED Equation.COEE2  with its fitted value  EMBED Equation.COEE2 , then we take the average squared distance from the observed values to their fitted values. We divide the sum of squared errors by n-2 to obtain an unbiased estimate of  EMBED Equation.COEE2  (recall how you computed a sample variance when sampling from a single population).  EMBED Equation.COEE2  Common notation is to label the numerator as the error sum of squares (SSE).  EMBED Equation.COEE2  Also, the estimated variance is referred to as the error (or residual) mean square (MSE).  EMBED Equation.COEE2  To obtain an estimate of the standard deviation (which is in the units of the data), we take the square root of the erro mean square.  EMBED Equation.COEE2 . A shortcut formula for the error sum of squares, which can cause problems due to round-off errors is:  EMBED Equation.COEE2  Some notation makes life easier when writing out elements of the regression model:  EMBED Equation.COEE2  Note that we will be able to obtain most all of the simple linear regression analysis from these quantities, the sample means, and the sample size.  EMBED Equation.COEE2  Normal Error Regression Model If we add further that the random errors follow a normal distribution, then the response variable also has a normal distribution, with mean and variance given above. The notation, we will use for the errors, and the data is:  EMBED Equation.COEE2  The density function for the ith observation is:  EMBED Equation.COEE2  The likelihood function, is the product of the individual density functions (due to the independence assumption on the random errors).  EMBED Equation.COEE2  The values of  EMBED Equation.COEE2  that maximize the likelihood function are referred to as maximum likelihood estimators. The MLEs are denoted as:  EMBED Equation.COEE2 . Note that the natural logarithm of the likelihood is maximized by the same values of  EMBED Equation.COEE2  that maximize the likelihood function, and its easier to work with the log likelihood function.  EMBED Equation.COEE2  Taking partial derivatives with respect to  EMBED Equation.COEE2  yields:  EMBED Equation.COEE2  Setting these three equations to 0, and placing hats on parameters denoting the maximum likelihood estimators, we get the following three equations:  EMBED Equation.COEE2  From equations 4a and 5a, we see that the maximum likelihood estimators are the same as the least squares estimators (these are the normal equations). However, from equation 6a, we obtain the maximum likelihood estimator for the error variance as:  EMBED Equation.COEE2  This estimator is biased downward. We will use the unbiased estimator  EMBED Equation.COEE2  throughout this course to estimate the error variance. Example LSD Concentration and Math Scores A pharmacodynamic study was conducted at Yale in the 1960s to determine the relationship between LSD concentration and math scores in a group of volunteers. The independent (predictor) variable was the mean tissue concentration of LSD in a group of 5 volunteers, and the dependent (response) variable was the mean math score among the volunteers. There were n=7 observations, collected at different time points throughout the experiment. Source: Wagner, J.G., Agahajanian, G.K., and Bing, O.H. (1968), Correlation of Performance Test Scores with Tissue Concentration of Lysergic Acid Diethylamide in Human Subjects, Clinical Pharmacology and Therapeutics, 9:635-638. The following EXCEL spreadsheet gives the data and pertinent calculations. Time (i)Score (Y)Conc (X)Y-YbarX-Xbar(Y-Ybar)**2(X-Xbar)**2(X-Xbar)(Y-Ybar)Yhatee**2178.931.1728.84286-3.162857831.910408210.0036653-91.2258367378.58280.34720.1205258.202.978.112857-1.36285765.818451021.85737959-11.0566653162.36576-4.165817.354367.473.2617.38286-1.072857302.16372241.15102245-18.6493224559.753017.71759.552437.474.69-12.617140.357143159.19229390.12755102-4.50612244946.86948-9.399588.35545.655.83-4.4371431.49714319.688236732.24143673-6.64303673536.598689.051381.926632.926.00-17.167141.667143294.71079392.77936531-28.6200795935.06708-2.14714.6099729.976.41-20.117142.077143404.69943674.31452245-41.7861795931.37319-1.40321.969Sum350.6130.33002078.18334322.4749429-202.4872429350.611E-14253.88Mean50.087142864.3328571b1-9.009466b089.123874MSE50.776266 The fitted equation is:  EMBED Equation.3  and the estimated error variance is  EMBED Equation.3 , with corresponding standard deviation  EMBED Equation.3 . A plot of the data and the fitted equation are given below, obtained from EXCEL.  Output from various software packages is given below. Rules for standard errors and tests are given in the next chapter. We will mainly use SAS, EXCEL, and SPSS throughout the semester. EXCEL (Using Built-in Data Analysis Package) Data Cells Time (i)Score (Y)Conc (X)178.931.17258.22.97367.473.26437.474.69545.655.83632.926729.976.41 Regression Coefficients CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept89.123877.04754712.646085.49E-0571.00761107.2401Conc (X)-9.009471.503076-5.994020.001854-12.8732-5.14569 Fitted Values and Residuals ObservationPredicted Score (Y)Residuals178.58280.347202262.36576-4.16576359.753017.716987446.86948-9.39948536.598689.051315635.06708-2.14708731.37319-1.40319 SAS (Using PROC REG) Program (Bottom portion generates graphics quality plot for WORD) options nodate nonumber ps=55 ls=76; title Pharmacodynamic Study; title2 Y=Math Score X=Tissue LSD Concentration; data lsd; input score conc; cards; 78.93 1.17 58.20 2.97 67.47 3.26 37.47 4.69 45.65 5.83 32.92 6.00 29.97 6.41 ; run; proc reg; model score=conc / p r; run; symbol1 c=black i=rl v=dot; proc gplot; plot score*conc=1 / frame; run; quit; Program Output (Some output suppressed) Pharmacodynamic Study Y=Math Score X=Tissue LSD Concentration The REG Procedure Model: MODEL1 Dependent Variable: score Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 89.12387 7.04755 12.65 <.0001 conc 1 -9.00947 1.50308 -5.99 0.0019 Output Statistics Dep Var Predicted Std Error Std Error Student Obs score Value Mean Predict Residual Residual Residual 1 78.9300 78.5828 5.4639 0.3472 4.574 0.0759 2 58.2000 62.3658 3.3838 -4.1658 6.271 -0.664 3 67.4700 59.7530 3.1391 7.7170 6.397 1.206 4 37.4700 46.8695 2.7463 -9.3995 6.575 -1.430 5 45.6500 36.5987 3.5097 9.0513 6.201 1.460 6 32.9200 35.0671 3.6787 -2.1471 6.103 -0.352 7 29.9700 31.3732 4.1233 -1.4032 5.812 -0.241 Plot (Including Regression Line)  SPSS (Spreadsheet/Menu Driven Package) Output (Regression Coefficients Portion)  Plot of Data and Regression Line  STATVIEW (Spreadsheet/Menu Driven Package from SAS) Output (Regression Coefficients Portion) Graphic output  5) S-Plus (Also available in R) Program Commands x <- c(1.17, 2.97, 3.26, 4.69, 5.83, 6.00, 6.41) y <- c(78.93, 58.20, 67.47, 37.47, 45.65, 32.92, 29.97) plot (x,y) fit <- lm(y ~ x) abline (fit) summary (fit) Program Output Residuals: 1 2 3 4 5 6 7 0.3472 4.166 7.717 9.399 9.051 2.147 1.403 Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 89.1239 7.0475 12.6461 0.0001 x -9.0095 1.5031 -5.9940 0.0019 Residual standard error: 7.126 on 5 degrees of freedom Graphics Output  STATA Output (Regression Coefficients Portion) score Coef. Std. Err. t P>t [95% Conf. Interval] conc -9.009467 1.503077 -5.99 0.002 -12.87325 -5.145686 _cons 89.12388 7.047547 12.65 0.000 71.00758 107.2402 Graphics Output  Chapter 2 Inferences in Regression Analysis Rules Concerning Linear Functions of Random Variables (P. 1318) Let  EMBED Equation.3  be n random variables. Consider the function  EMBED Equation.3  where the coefficients  EMBED Equation.3  are constants. Then, we have:  EMBED Equation.3  EMBED Equation.3  When  EMBED Equation.3  are independent (as in the model in Chapter 1), the variance of the linear combination simplifies to:  EMBED Equation.3  When  EMBED Equation.3  are independent, the covariance of two linear functions  EMBED Equation.3  and  EMBED Equation.3  can be written as:  EMBED Equation.3  We will use these rules to obtain the distribution of the estimators  EMBED Equation.3  Inferences Concerning b1 Recall that the least squares estimate of the slope parameter,  EMBED Equation.3  , is a linear function of the observed responses  EMBED Equation.3 :  EMBED Equation.COEE2  Note that  EMBED Equation.COEE2 , so that the expected value of  EMBED Equation.3  is:  EMBED Equation.COEE2  Note that  EMBED Equation.COEE2  (why?), so that the first term in the brackets is 0, and that we can add  EMBED Equation.COEE2  to the last term to get:  EMBED Equation.COEE2  Thus,  EMBED Equation.COEE2  is an unbiased estimator of the parameter  EMBED Equation.COEE2 . To obtain the variance of  EMBED Equation.COEE2 , recall that  EMBED Equation.COEE2 . Thus:  EMBED Equation.COEE2  Note that the variance of  EMBED Equation.COEE2  decreases when we have larger sample sizes (as long as the added  EMBED Equation.COEE2  levels are not placed at the sample mean  EMBED Equation.COEE2 ). Since  EMBED Equation.COEE2  is unknown in practice, and must be estimated from the data, we obtain the estimated variance of the estimator  EMBED Equation.COEE2  by replacing the unknown  EMBED Equation.COEE2  with its unbiased estimate  EMBED Equation.COEE2 :  EMBED Equation.COEE2  with estimated standard error:  EMBED Equation.COEE2  Further, the sampling distribution of  EMBED Equation.COEE2  is normal, that is:  EMBED Equation.COEE2  since under the current model,  EMBED Equation.COEE2  is a linear function of independent, normal random variables  EMBED Equation.3 . Making use of theory from mathematical statistics, we obtain the following result that allows us to make inferences concerning  EMBED Equation.COEE2 :  EMBED Equation.3  where t(n-2) represents Student s t-distribution with n-2 degrees of freedom. Confidence Interval for b1 As a result of the fact that  EMBED Equation.3 , we obtain the following probability statement:  EMBED Equation.3  where  EMBED Equation.3  is the (a/2)100th percentile of the t-distribution with n-2 degrees of freedom. Note that since the t-distribution is symmetric around 0, we have that  EMBED Equation.3 . Traditionally, we obtain the table values corresponding to  EMBED Equation.3 , which is the value of that leaves an upper tail area of a/2. The following algebra results in obtaining a (1-a)100% confidence interval for b1:  EMBED Equation.3  This leads to the following rule for a (1-a)100% confidence interval for b1:  EMBED Equation.3  Some statistical software packages print this out automatically (e.g. EXCEL and SPSS). Other packages simply print out estimates and standard errors only (e.g. SAS). Tests Concerning b1 We can also make use of the of the fact that  EMBED Equation.COEE2  to test hypotheses concerning the slope parameter. As with means and proportions (and differences of means and proportions), we can conduct one-sided and two-sided tests, depending on whether a priori a specific directional belief is held regarding the slope. More often than not (but not necessarily), the null value for b1 is 0 (the mean of Y is independent of X) and the alternative is that b1 is positive (1-sided), negative (1-sided), or different from 0 (2-sided). The alternative hypothesis must be selected before observing the data. 2-sided tests Null Hypothesis:  EMBED Equation.3  EMBED Equation.3  Alternative (Research Hypothesis):  EMBED Equation.3  Test Statistic:  EMBED Equation.3  Decision Rule: Conclude HA if  EMBED Equation.3 , otherwise conclude H0 P-value:  EMBED Equation.3  All statistical software packages (to my knowledge) will print out the test statistic and P-value corresponding to a 2-sided test with b10=0. 1-sided tests (Upper Tail) Null Hypothesis:  EMBED Equation.3  EMBED Equation.3  Alternative (Research Hypothesis):  EMBED Equation.3  Test Statistic:  EMBED Equation.3  Decision Rule: Conclude HA if  EMBED Equation.3 , otherwise conclude H0 P-value:  EMBED Equation.3  A test for positive association between Y and X (HA:b1>0) can be obtained from standard statisical software by first checking that b1 (and thus t*) is positive, and cutting the printed P-value in half. 1-sided tests (Lower Tail) Null Hypothesis:  EMBED Equation.3  EMBED Equation.3  Alternative (Research Hypothesis):  EMBED Equation.3  Test Statistic:  EMBED Equation.3  Decision Rule: Conclude HA if  EMBED Equation.3 , otherwise conclude H0 P-value:  EMBED Equation.3  A test for negative association between Y and X (HA:b1<0) can be obtained from standard statisical software by first checking that b1 (and thus t*) is negative, and cutting the printed P-value in half. Inferences Concerning b0 Recall that the least squares estimate of the intercept parameter,  EMBED Equation.3  , is a linear function of the observed responses  EMBED Equation.3 :  EMBED Equation.3  Recalling that  EMBED Equation.3 :  EMBED Equation.3  Thus, b0 is an unbiased estimator or the parameter b0. Below, we obtain the variance of the estimator of b0.  EMBED Equation.3  Note that the variance will decrease as the sample size increases, as long as X values are not all placed at the mean. Further, the sampling distribution is normal under the assumptions of the model. The estimated standard error of b0 replaces s2 with its unbiased estimate s2=MSE and taking the square root of the variance.  EMBED Equation.3  Note that  EMBED Equation.3 , allowing for inferences concerning the intercept parameter b0 when it is meaningful, namely when X=0 is within the range of observed data. Confidence Interval for b0  It is also useful to obtain the covariance of b0 and b1, as they are only independent under very rare circumstances:  EMBED Equation.3  In practice,  EMBED Equation.3  is usually positive, so that the intercept and slope estimators are usually negatively correlated. We will use the result shortly. Considerations on Making Inferences Concerning b0 and b1 Normality of Error Terms If the data are approximately normal, simulation results have shown that using the t-distribution will provide approximately correct significance levels and confidence coefficients for tests and confidence intervals, respectively. Even if the distribution of the errors (and thus Y) is far from normal, in large samples the sampling distributions of b0 and b1 have sampling distributions that are approximately normal as results of central limit theorems. This is sometimes referred to as asymptotic normality. Interpretations of Confidence Coefficients and Error Probabilities Since X levels are treated as fixed constants, these refer to the case where we repeated the experiment many times at the current set of X levels in this data set. In this sense, its easier to interpret these terms in controlled experiments where the experimenter has set the levels of X (such as time and temperature in a laboratory type setting) as opposed to observational studies, where nature determines the X levels, and we may not be able to reproduce the same conditions repeatedly. This will be covered later. Spacing of X Levels The variances of b0 and b1 (for given n and s2) decrease as the X levels are more spread out, since their variances are inversely related to  EMBED Equation.3 . However, there are reasons to choose a diverse range of X levels for assessing model fit. This is covered in Chapter 4. Power of Tests The power of a statistical test refers to the probability that we reject the null hypothesis. Note that when the null hypothesis is true, the power is simply the probability of a Type I error (a). When the null hypothesis is false, the power is the probability that we correctly reject the null hypothesis, which is 1 minus the probability of a Type II error (p=1-b), where p denotes the power of the test and b is the probability of a Type II error (failing to reject the null hypothesis when the alternative hypothesis is true). The following procedure can be used to obtain the power of the test concerning the slope parameter with a 2-sided alternative. Write out null and alternative hypotheses:  EMBED Equation.3  Obtain the noncentrality measure, the standardized distance between the true value of b1 and the value under the null hypothesis (b10):  EMBED Equation.3  Choose the probability of a Type I error (a=0.05 or a=0.01) Determine the degrees of freedom for error: df = n-2 Refer to Table B.5 (pages 1346-7), identifying a (page), d (row) and error degrees of freedom (column). The table provides the power of the test under these parameter values. Note that the power increases within each tables as the noncentrality measure increases for a given degrees of freedom, and as the degrees of freedom increases for a given noncentrality measure. Confidence Interval for E{Yh}=b0+b1Xh When we wish to estimate the mean at a hypothetical X value (within the range of observed X values), we can use the fitted equation at that value of X=Xh as a point estimate, but we haveto include the uncertainty in the regression estimators to construct a confidence interval for the mean. Parameter:  EMBED Equation.3  Estimator:  EMBED Equation.3  We can obtain the variance of the estimator (as a function of X=Xh) as follows:  EMBED Equation.3  Estimated standard error of estimator:  EMBED Equation.3   EMBED Equation.3  which can be used to construct confidence intervals for the mean response at specific X levels, and tests concerning the mean (tests are rarely conducted). (1-a)100% Confidence Interval for E{Yh}:  EMBED Equation.3  Predicting a Future Observation When X is Known If  EMBED Equation.3  were known, we d know that the distribution of responses when X=Xh is normal with mean  EMBED Equation.3  and standard deviation  EMBED Equation.3 . Thus, making use of the normal distribution (and equivalently, the empirical rule) we know that if we took a sample item from this distribution, it is very likely that the value fall within 2 standard deviations of the mean. That is, we would know that the probability that the sampled item lies within the range  EMBED Equation.3  is approximately 0.95. In practice, we dont know the mean  EMBED Equation.3  or the standard deviation  EMBED Equation.3 . However, we just constructed a (1-a)100% Confidence Interval for E{Yh}, and we have an estimate of  EMBED Equation.3  (s). Intuitively, we can approximately use the logic of the previous paragraph (with the estimate of  EMBED Equation.3 ) across the range of believable values for the mean. Then our prediction interval spans the lower tail of the normal curve centered at the lower bound for the mean to the upper tail of the normal curve centered at the upper bound for the mean. See Figure 2.5 on page 64 of the text book. The prediction error is for the new observation is the difference between the observed value and its predicted value:  EMBED Equation.3 . Since the data are assumed to be independent, the new (future) value is independent of its predicted value, since it wasn t used in the regression analysis. The variance of the prediction error can be obtained as follows:  EMBED Equation.3  and an unbiased estimator is:  EMBED Equation.3  (1-a)100% Prediction Interval for New Observation When X=Xh  EMBED Equation.3  It is a simple extension to obtain a prediction for the mean of m new observations when X=Xh. The sample mean of m observations is  EMBED Equation.3  and we get the following variance for for the error in the prediction mean:  EMBED Equation.3  and the obvious adjustment to the prediction interval for a single observation. (1-a)100% Prediction Interval for the Mean of m New Observations When X=Xh  EMBED Equation.3  Confidence Band for the Entire Regression Line (Working-Hotelling Method)  EMBED Equation.3  Analysis of Variance Approach to Regression Consider the total deviations of the observed responses from the mean:  EMBED Equation.3 . When these terms are all squared and summed up, this is referred to as the total sum of squares (SSTO).  EMBED Equation.3  The more spread out the observed data are, the larger SSTO will be. Now consider the deviation of the observed responses from their fitted values based on the regression model:  EMBED Equation.3 . When these terms are squared and summed up, this is referred to as the error sum of squares (SSE). Weve already encounterd this quantity and used it to estimate the error variance.  EMBED Equation.3  When the observed responses fall close to the regression line, SSE will be small. When the data are not near the line, SSE will be large. Finally, there is a third quantity, representing the deviations of the predicted values from the mean. Then these deviations are squared and summed up, this is referred to as the regression sum of squares (SSR).  EMBED Equation.3  The error and regression sums of squares sum to the total sum of squares:  EMBED Equation.3  which can be seen as follows:  EMBED Equation.3  The last term was 0 since  EMBED Equation.3 , Each sum of squares has associated with degrees of freedom. The total degrees of freedom is dfT = n-1. The error degrees of freedom is dfE = n-2. The regression degrees of freedom is dfR = 1. Note that the error and regression degrees of freedom sum to the total degrees of freedom:  EMBED Equation.3 . 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It can be shown that the expected values of the mean squares are:  EMBED Equation.3  Note that these expected mean squares are the same if and only if b1=0. The Analysis of Variance is reported in tabular form: SourcedfSSMSFRegression1SSRMSR=SSR/1F=MSR/MSEErrorn-2SSEMSE=SSE/(n-2)C Totaln-1SSTO F Test of b1 = 0 versus b1 ( 0 As a result of Cochran s Theorem (stated on page 76 of text book), we have a test of whether the dependent variable Y is linearly related to the predictor variable X. This is a very specific case of the t-test described previously. Its full utility will be seen when we consider multiple predictors. The test proceeds as follows: Null hypothesis:  EMBED Equation.3  Alternative (Research) Hypothesis:  EMBED Equation.3  Test Statistic:  EMBED Equation.3  Rejection Region:  EMBED Equation.3  P-value:  EMBED Equation.3  Critical values of the F-distribution (indexed by numerator and denominator degrees of freedom) are given in Table B.4, pages 1340-1345. Note that this is a very specific version of the t-test regarding the slope parameter, specifically a 2-sided test of whether the slope is 0. Mathematically, the tests are identical:  EMBED Equation.3  Note that:  EMBED Equation.3  Thus: Further, the critical values are equivalent:  EMBED Equation.3 , check this from the two tables. Thus, the tests are equivalent. General Linear Test Approach This is a very general method of testing hypotheses concerning regression models. We first consider the the simple linear regression model, and testing whether Y is linearly associated with X. We wish to test  EMBED Equation.3  vs  EMBED Equation.3 . Full Model This is the model specified under the alternative hypothesis, also referrred to as the unrestricted model. Under simple linear regression with normal errors, we have:  EMBED Equation.3  Using least squares (and maximum likelihood) to estimate the model parameters ( EMBED Equation.3 ), we obtain the error sum of squares for the full model:  EMBED Equation.3  Reduced Model This the model specified by the null hypothesis, also referred to as the restricted model. Under simple linear regression with normal errors, we have:  EMBED Equation.3  Under least squares (and maximum likelihood) to estimate the model parameter, we obtain  EMBED Equation.3 as the estimate of b0, and have  EMBED Equation.3 as the fitted value for each observation. We when get the following error sum of squares under the reduced model:  EMBED Equation.3  Test Statistic The error sum of squares for the full model will always be less that or equal to the error sum of squares for reduced model, by definition of least squares. The test statistic will be:  EMBED Equation.3  where  EMBED Equation.3  are the error degrees of freedom for the full and reduced models. We will use this method throughout course. For the simple linear regression model, we obtain the following quantities:  EMBED Equation.3  thus the F-Statistic for the General Linear Test can be written:  EMBED Equation.3  Thus, for this particular null hypothesis, the general linear test generalizes to the F-test. Descriptive Measures of Association Along with the slope, Y-intercept, and error variance; several other measures are often reported. Coefficient of Determination (r2) The coefficient of determination measures the proportion of the variation in Y that is explained by the regression on X. It is computed as the regression sum of squares divided by the total (corrected) sum of squares. Values near 0 imply that the regression model has done little to explain variation in Y, while values near 1 imply that the model has explained a large portion of the variation in Y. If all the data fall exactly on the fitted line, r2=1. The coefficient of determination will lie beween 0 and 1.  EMBED Equation.3  Coefficient of Correlation (r) The coefficient of correlation is a measure of the strength of the linear association between Y and X. It will always be the same sign as the slope estimate (b1), but it has several advantages: In some applications, we cannot identify a clear dependent and independent variable, we just wish to determine how two variables vary together in a population (peoples heights and weights, closing stock prices of two firms, etc). Unlike the slope estimate, the coefficient of correlation does not depend on which variable is labeled as Y, and which is labeled as X. The slope estimate depends on the units of X and Y, while the correlation coefficient does not. The slope estimate has no bound on its range of potential values. The correlation coefficient is bounded by 1 and +1, with higher values (in absolute value) implying stronger linear association (it is not useful in measuring nonlinear association which may exist, however).  EMBED Equation.3  where sgn(b1) is the sign (positive or negative) of b1, and  EMBED Equation.3  are the sample standard deviations of X and Y, respectively. Issues in Applying Regression Analysis When using regression to predict the future, the assumption is that the conditions are the same in future as they are now. Clearly any future predictions of economic variables such as tourism made prior to September 11, 2001 would not be valid. Often when we predict in the future, we must also predict X, as well as Y, especially when we arent controlling the levels of X. Prediction intervals using methods described previously will be too narrow (that is, they will overstate confidence levels). Inferences should be made only within the range of X values used in the regression analysis. We have no means of knowing whether a linear association continues outside the range observed. That is, we should not extrapolate outside the range of X levels observed in experiment. Even if we determine that X and Y are associated based on the t-test and/or F-test, we cannot conclude that changes in X cause changes in Y. Finding an association is only one step in demonstrating a causal relationship. When multiple tests and/or confidence intervals are being made, we must adjust our confidence levels. 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Equation Equation.39ql]  2 {pred}= 2 {Y h "'Y ^  h }= 2 {Y h }+ 2 {'Y ^CompObj)fObjInfo+Equation Native ,_1123145899FPYaPYa  h }= 2 + 2 1n+(X h "X) 2 (X i "X) 2i=1n " []= 2 1+1n+(X h "X) 2 (X i "X) 2i=1n " []Ole 8CompObj9fObjInfo;Equation Native <" FMicrosoft Equation 3.0 DS Equation Equation.39q`T s 2 {pred}=MSE1+1n+(X h "X) 2 (X i "X) 2i=1n " [] FMicrosoft Equation 3.0 DS Equation Equation.39q9T ''Y ^  h t(/2;n"2) MSE1+1n+(X h_1123155916FPYaPYaOle ACompObjBfObjInfoDEquation Native EU_1123155816FPYaPYaOle KCompObjLf "X) 2 (X i "X) 2i=1n " []  FMicrosoft Equation 3.0 DS Equation Equation.39q(l]  2 mObjInfoNEquation Native OD_1123156030FPYaPYaOle Q FMicrosoft Equation 3.0 DS Equation Equation.39q#l] s 2 {predmean}=MSE1m+1n+(X h "CompObjRfObjInfoTEquation Native U?_1123158876FPYaPYaX) 2 (X i "X) 2i=1n " [] FMicrosoft Equation 3.0 DS Equation Equation.39q Fl] ''Y ^  h t(/2;n"2) MOle ZCompObj[fObjInfo]Equation Native ^bSE1m+1n+(X h "X) 2 (X i "X) 2i=1n " []  FMicrosoft Equation 3.0 DS Equation Equation.39q_1123247286FPYaPYaOle dCompObjefObjInfogEquation Native h_1123159028FPYaPYaOle lCompObjmfl] 'Y ^  h Ws{'Y ^  h }W= 2F(1";2,n"2)  FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfooEquation Native pF_1123159243FPYaPYaOle r *l] Y i "Y FMicrosoft Equation 3.0 DS Equation Equation.39q {l] SSTO=(Y i "Y) 2iCompObjsfObjInfouEquation Native v_1123159419FPYaPYa=1n " FMicrosoft Equation 3.0 DS Equation Equation.39q l] Y i "'Y ^  i =Y i "(b 0 +b 1 X i )=eOle yCompObjzfObjInfo|Equation Native } i FMicrosoft Equation 3.0 DS Equation Equation.39q l] SSE=(Y i "'Y ^  i ) 2i=1n "_1123159764FPYaPYaOle CompObjfObjInfoEquation Native _1123160105FPYaPYaOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q l] SSR=('Y ^  i "Y) 2i=1n "ObjInfoEquation Native _1123160293FPYa@aOle CompObjfObjInfoEquation Native U_1123161222F@a@a FMicrosoft Equation 3.0 DS Equation Equation.39q 9l] SSTO=SSR+SSE FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native   0T Y i "Y=Y i "Y+'Y ^  i "'Y ^  i =(Y i "'Y ^  i )+('Y ^  i "Y)!(Y i "Y) 2 =[(Y i "'Y ^  i )+('Y ^  i "Y)] 2 =(Y i "'Y ^  i ) 2 +('Y ^  i "Y) 2 +2(Y i "'Y ^  i )('Y ^  i "Y)!SSTO=(Y i "Y) 2i=1n " =(Y i "'Y ^  i ) 2 +('Y ^  i "Y) 2 +2(Y i "'Y ^  i )('Y ^  i "Y)[] i=1n " =(Y i "'Y ^  i ) 2i=1n " +('Y ^  i "Y) 2i=1n " +2(Y i "'Y ^  i )('Y ^  i "Y) i=1n " =(Y i "'Y ^  i ) 2i=1n " +('Y ^  i "Y) 2i=1n " +2e i (b 0 +b 1 X i "Y) i=1n " =(Y i "'Y ^  i ) 2i=1n " +('Y ^  i "Y) 2i=1n " +2b 0 e i +b 1i=1n " e i X i "Ye ii=1n " i=1n " []=(Y i "'Y ^  i ) 2i=1n " +('Y ^  i "Y) 2i=1n " +2(0)=(Y i "'Y ^  i ) 2i=1n " +('Y ^  i "Y) 2i=1n " =SSE+SSR FMicrosoft Equation 3.0 DS Equation Equation.39q i  e i =e i X i =0  "  "_1123161854F@a@aOle CompObjfObjInfoEquation Native _1123162350F@a@aOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q 5l] n"1=1+(n"2) FMicrosoft Equation 3.0 DS EqObjInfoEquation Native Q_1123162669 F@a@aOle CompObj fObjInfo Equation Native _1123162826 F@a@auation Equation.39q kl] MSR=SSR1MSE=SSEn"2 FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj fObjInfoEquation Native  l] E{MSE}= 2 E{MSR}= 2 + 12 (X i "X) 2i=1n " FMicrosoft Equation 3.0 DS Equation Equation.39q_11231644568F@a@aOle CompObjfObjInfo =l] H 0 : 1 =0 FMicrosoft Equation 3.0 DS Equation Equation.39q 9 H A :Equation Native Y_1123164524F@a@aOle CompObjfI2ObjInfoEquation Native U_1123164589F@a@aOle  1 `"0 FMicrosoft Equation 3.0 DS Equation Equation.39q B TS:F*=MSRMSECompObjfObjInfoEquation Native ^_1123164669!F@a@aOle CompObj "fObjInfo#Equation Native m FMicrosoft Equation 3.0 DS Equation Equation.39q Q@ RR:F*e"F(1";1,n"2) FMicrosoft Equation 3.0 DS Equation Equation.39q_1123164740)&F@a@aOle CompObj%'fObjInfo(      !"#$%&'()*+,-./0123456789:;<=>?BEFILORUXY\_`cfghilopsx{~ A < P{F(1,n"2)e"F*} FMicrosoft Equation 3.0 DS Equation Equation.39q Ql] t*=b Equation Native ]_1123168042.3+F@a@aOle CompObj*,fObjInfo-Equation Native m_11231673680F@a@aOle 1 "0s{b 1 }=(X i "X)(Y i "Y)  " (X i "X) 2 "  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