ESTIMATING THE REGRESSION LINE in Java

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ESTIMATING THE REGRESSION LINE
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tion, it turns out that if we let the conditional distribution of (a, {3) for each given value of a be a bivariate normal distribution and let the distribution of 1/a 2 be a gamma distribution, then this constitutes a conjugate prior distribution for the parameter vector (a, {3, a). By using a conjugate prior distribution we will be able to obtain explicit formulas for the expectations of a and {3 under the posterior distribution.
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Quantifying Prior Information. The decision to use a prior distribution from the family of conjugate distributions is not based on prior information but simply on convenience. Our prior information about the parameters enters the procedure when we select a particular conjugate distribution. For instance, we might quantify our prior information by specifying the expectations, standard deviations and correlation of a and {3 given a and the expectation and standard deviation of 1/a 2 If we are only interested in calculating estimates of a and {3, as in this section, it is not necessary to specify the expectation and standard deviation of 1/a 2 Rather than translate our prior information into a distribution for a and {3, it may be easier, and is equivalent, to translate it into a prior distribution for JL and {3, where JL = a + {3xm and xm is a "middle" x-value. Note that JL is the height of the regression line near the middle of the data. It would often be reasonable to regard our prior information about JL to be independent of our prior information about {3, the slope of the regression line. One case in which this assumption of independence is especially suitable is when our prior information comes from a previous least-squares analysis of similar data. In a normal simple linear regression model, if x m is the average x-value, then ilLS and ffiLS are independent.
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Bayes Estimates. Let JL = a + {3x; that is, take xm to be the average x-value in the current data set. Suppose that, based on previous information, we specify a conjugate prior distribution in which, conditional on a given value of a, the expectations of JL and {3 are elL and ef3 and their standard deviations are clLa and cf3a. Let the prior distributions of JL and {3 be independent. The Bayes estimates of a and {3 are given by the expectations of a and {3 under the posterior distribution:
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is the least-squares estimate calculated from the current data.
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Reasonableness of Formulas (7.3). These formulas show how past knowledge and current data are combined. The Bayes estimate of {3 is a weighted average of e(3' the prior expected value of {3 based on past knowledge, and ~LS' the least-squares estimate of {3 based on the current data. The weights are sensible in so far as more weight is put on the current data when there are more data, so that L:(x; - i)2 is larger, or when the prior knowledge is more imprecise, that is, c(3 is larger. The Bayes estimate of J.L is a similar weighted average of the prior expected value based on past knowledge and the least-squares estimate based on the current data. (Note that ji = ~ LSJ The Rainfall Data. Results are available from a previous regression analysis of annual rainfall in Seattle and Portland for the years 1950 through 1979. Let the subscript "0" identify these previous results. The numbers we need for our prior distribution are a o = 5.513, ~o = 0.8961, SD(a o) = 1.140u, SD(~o) = 0.02986u, and the correlation Corr(ao, ~o) = - 0.9871. (These numbers do not completely specify the prior distribution, but they are all we need for the purpose of estimating a and {3.) In order to use formulas (7.3), we let J.L = a + 35.14{3, where 35.14 is the average rainfall in Portland for the current data. Then ~o = 5.513 + (35.14)(0.8961) = 37.00 and SD(~o) = 0.1977u. The number 0.1977 is the square root of (1.140)2 + (35.14)2(0.02986)2 + 2(35.14X1.140XO.02986X -0.9871). To specify a prior distribution for the parameters it makes sense to let J.L have expectation 37.00 and standard deviation 0.1977u (conditional on u) and let {3 have expectation 0.8961 and standard deviation 0.02986u. In order to illustrate the use of (7.3) we suppose that the prior distributions of J.L and {3 are independent. Note, however, that the estimates ~o and ~o are not exactly independent because the average x-value in the previous data is not exactly 35.14. Next, from the current data, we calculate ji = 35.82, ~LS = 0.5063, and L(x; - i)2 = 497.2. Now we can use formulas (7.3) to obtain the Bayes estimates of a and {3. We calculate
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+ 497.2] = 0.6929
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~ = (0.6929)(0.8961) + (1 - 0.6929)(0.5063) = 0.7764
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+ 11] = 0.6993
= (0.6993)(37.00) + (1 - 0.6993)(35.82) = 36.65
a = 36.65 -
(0.7764)(35.14) = 9.363