ࡱ> bd aZZ{` tIbjbjFF .,,tA""""d$$48d$"3333a4a4a4k8m8m8m8m8m8m8$9h <8a4a4a4a4a48338S7S7S7a433k8S7a4k8S7S7S73X$ Ј$"w4S7k8808S7<-7<S7<S7a4a4S7a4a4a4a4a488I7 a4a4a48a4a4a4a4< < Formulas and Relationships from Simple & Multiple Linear Regression I. Basics for Simple Linear Regression Let  EMBED Equation.DSMT4 be sample data from a bivariate normal population  EMBED Equation.DSMT4  (technically we have  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the sample size and will use the notation  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 ). Then we have the following sample statistics:  EMBED Equation.DSMT4  (sample mean for  EMBED Equation.DSMT4 )  EMBED Equation.DSMT4  (sample mean for  EMBED Equation.DSMT4 )  EMBED Equation.DSMT4 (sample variance for  EMBED Equation.DSMT4 )  EMBED Equation.DSMT4 (sample variance for  EMBED Equation.DSMT4 ) We will also use the following sums of squares:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Note: Sometimes we refer to  EMBED Equation.DSMT4 as  EMBED Equation.DSMT4 . We have the following relationships between the sample statistics and the sums of squares:  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  The sample covariance between  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 is defined by  EMBED Equation.DSMT4  The sample correlation coefficient is defined as  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . By plugging the formulas of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 into the formula for  EMBED Equation.DSMT4 we can easily derive that  EMBED Equation.DSMT4 . The least squares regression line for the data has the form  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Associated with the regression we have some additional sums of squares:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Where for each value of  EMBED Equation.DSMT4  in the sample data,  EMBED Equation.DSMT4 is the corresponding  EMBED Equation.DSMT4 coordinate and  EMBED Equation.DSMT4 is the predicted value from the regression line when  EMBED Equation.DSMT4 is used as the predictor (input) to the line. It is clear to see both symbolically and graphically that  EMBED Equation.DSMT4 . However, the more interesting result is that:  EMBED Equation.DSMT4  i.e.  EMBED Equation.DSMT4  This describes the total variation in  EMBED Equation.DSMT4 by the sum of the explained variation ( EMBED Equation.DSMT4 ) and the unexplained variation ( EMBED Equation.DSMT4 ). Remember, the least squares regression line is the line that fits the data in a way that minimizes the unexplained variation. Also note that if the data perfectly fits a line then  EMBED Equation.DSMT4 . One can derive that the square of the correlation coefficient can be written in terms of these sums of squares:  EMBED Equation.DSMT4  So here again we see that  EMBED Equation.DSMT4 measures how strongly the data fits a line since if the data is perfectly linear, then  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  is called the coefficient of determination. Another relationship that can be easily derived from the formula for  EMBED Equation.DSMT4  is that  EMBED Equation.DSMT4  We will define the standard error of the linear regression to be the following:  EMBED Equation.DSMT4  We will use this standard error to define the standard error for random variables associated with the linear regression and thus to test hypotheses about these random variables. We will also use the standard error of the linear regression to make inferences about the predicted values of  EMBED Equation.DSMT4 for a fixed value of  EMBED Equation.DSMT4 . II. Testing the correlation coefficient. The population correlation coefficient is represented by the symbol  EMBED Equation.DSMT4 . We will use the sample correlation coefficient,  EMBED Equation.DSMT4 , obtained from the sample  EMBED Equation.DSMT4  data, to test if there is a significant linear relationship between the population variables  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . If one assumes that  EMBED Equation.DSMT4 then it can be shown that the standard error for the random variable  EMBED Equation.DSMT4 is given by  EMBED Equation.DSMT4  and that the test statistic  EMBED Equation.DSMT4  has a student-t distribution with  EMBED Equation.DSMT4  degrees of freedom. Thus we can test the null hypotheses,  EMBED Equation.DSMT4  against appropriate alternative hypotheses using this test statistic. The standard error for EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , can also be written in terms of the standard error of linear regression and the sum of squares total:  EMBED Equation.DSMT4  III. Testing the slope of the regression line. The population regression line slope is represented by the symbol  EMBED Equation.DSMT4 . We will use the slope of the regression line that fits the sample EMBED Equation.DSMT4 data, EMBED Equation.DSMT4 , to make inferences about the slope of the regression line that would fit the population variables  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , i.e.  EMBED Equation.DSMT4 . If one assumes that  EMBED Equation.DSMT4 then it can be shown that the standard error for the random variable  EMBED Equation.DSMT4 is given by  EMBED Equation.DSMT4  and that the test statistic  EMBED Equation.DSMT4  has a student-t distribution with  EMBED Equation.DSMT4  degrees of freedom. Thus we can test the null hypotheses,  EMBED Equation.DSMT4  against appropriate alternative hypotheses using this test statistic. When conducting this test we are essentially determining if the independent effect of the variable EMBED Equation.DSMT4 is significant on the predicted (or criterion) variable  EMBED Equation.DSMT4 . The independent effect is measured by slope. That is a change in  EMBED Equation.DSMT4 produces a proportional change in  EMBED Equation.DSMT4 as measured by  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 . Now here is a very interesting result. You should be able to follow this derivation by seeing how we are using the formulas above:  EMBED Equation.DSMT4  Thus when testing  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 the value of the t-statistic will be the same! IV. Testing the coefficient of determination. Another test that we will use to test the significance of the linear regression is the  EMBED Equation.DSMT4 -test. The  EMBED Equation.DSMT4 -test is a test about the population coefficient of determination ( EMBED Equation.DSMT4 ). The null hypothesis for the  EMBED Equation.DSMT4 -test is  EMBED Equation.DSMT4  with alternative hypothesis  EMBED Equation.DSMT4 . The  EMBED Equation.DSMT4 distribution is used to test if two variances from independent populations are equal. This is done by testing the ratio of the two variances. The  EMBED Equation.DSMT4 distribution has two degrees of freedom parameters: the degrees of freedom of the numerator variance and the degrees of freedom of the denominator variance. In the case of linear regression we will use the following ratio for our  EMBED Equation.DSMT4 -test statistic:  EMBED Equation.DSMT4  Where  EMBED Equation.DSMT4 is the number of independent predictors (so far our  EMBED Equation.DSMT4 but this will change once we get to multiple regression) and  EMBED Equation.DSMT4 is the number of data points in the sample. The numerator of the  EMBED Equation.DSMT4 -test statistic is the variance in the regression ( EMBED Equation.DSMT4 ) and is also called the mean square regression. The denominator of the  EMBED Equation.DSMT4 -test statistic is the variance in the residual ( EMBED Equation.DSMT4 ) and is also called the mean square residual. We think of the  EMBED Equation.DSMT4 -test statistic as being the ratio of the explained variance to the unexplained variance. Since in regression our goal is to minimize the unexplained variance, then to have a significant result, we would expect the  EMBED Equation.DSMT4 -test statistic to be greater than 1. Consider the case when  EMBED Equation.DSMT4 (i.e. simple (or single variable) linear regression). Then  EMBED Equation.DSMT4 . Another way to think of what the  EMBED Equation.DSMT4 -test statistic measures is to consider a comparison of how well the regression line ( EMBED Equation.DSMT4 ) fits the data versus how well the line  EMBED Equation.DSMT4  fits the data. The more the regression line explains the variance in  EMBED Equation.DSMT4 (i.e. the higher the ratio of the variance in the regression to the variance in the residual), the more significant the result. Note that the line  EMBED Equation.DSMT4  has slope equal to zero, so the  EMBED Equation.DSMT4 -test is essentially a test about the slope of the regression line (i.e. if the slope of the regression line is zero then the  EMBED Equation.DSMT4 -test statistic will be zero since  EMBED Equation.DSMT4 ). Recall the  EMBED Equation.DSMT4 -test statistic for testing  EMBED Equation.DSMT4  against appropriate alternative hypotheses:  EMBED Equation.DSMT4  Thus we have  EMBED Equation.DSMT4  So the  EMBED Equation.DSMT4 -test statistic is the square of the  EMBED Equation.DSMT4 -test statistic for the slope of the regression line. Thus it turns out that in single variable regression, using the  EMBED Equation.DSMT4 -test statistic is equivalent to testing if the slope of the regression line is zero! When doing simple linear regression, if you check your computer output for the  EMBED Equation.DSMT4 -value for the  EMBED Equation.DSMT4 -test statistic and for the  EMBED Equation.DSMT4 -test statistic for the slope, they will be the same value! There will be an analogy to this test in multivariable regression! V. Basics for Multiple Regression Let  EMBED Equation.DSMT4 be sample data from a multivariate normal population  EMBED Equation.DSMT4  (technically we have  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the sample size and will use the notation  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 ). We will again perform linear regression on the data: i.e. we will use the data to find  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 such that  EMBED Equation.DSMT4  As before we will consider the sum of the squares of the residuals:  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4 is from the data (i.e.,  EMBED Equation.DSMT4 ) and  EMBED Equation.DSMT4  is the predicted value of  EMBED Equation.DSMT4  using the inputs  EMBED Equation.DSMT4 . The values that we get for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 are such that  EMBED Equation.DSMT4 is minimized (note this is the same criteria we had for single variable (i.e. simple) regression). The formulas for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 are somewhat complicated and for now we will compute these via computer technology. Now, just as in simple linear regression, we can determine how well the regression equation explains the variance in  EMBED Equation.DSMT4 by considering the coefficient of determination:  EMBED Equation.DSMT4  Where  EMBED Equation.DSMT4 is defined as in simple linear regression and we have the relationship of  EMBED Equation.DSMT4 still true! Testing the coefficient of determination in multiple regression. Again we will use the EMBED Equation.DSMT4 -test. The null hypothesis for the  EMBED Equation.DSMT4 -test is  EMBED Equation.DSMT4  with alternative hypothesis EMBED Equation.DSMT4 . Our EMBED Equation.DSMT4 -test statistic will again be:  EMBED Equation.DSMT4  Where  EMBED Equation.DSMT4 is the number of independent predictors and  EMBED Equation.DSMT4 is the number of data points in the sample. The numerator of the  EMBED Equation.DSMT4 -test statistic is the variance in the regression ( EMBED Equation.DSMT4 ) and is also called the mean square regression. The denominator of the  EMBED Equation.DSMT4 -test statistic is the variance in the residual ( EMBED Equation.DSMT4 ) and is also called the mean square residual. We think of the  EMBED Equation.DSMT4 -test statistic as being the ratio of the explained variance to the unexplained variance. Since in regression our goal is to minimize the unexplained variance and have most of the variance in  EMBED Equation.DSMT4 explained by the regression equation, then to have a significant result, we would expect the  EMBED Equation.DSMT4 -test statistic to be greater than 1. Note that an alternative formula for the  EMBED Equation.DSMT4 -test statistic is  EMBED Equation.DSMT4  Can you show this is true?? In the case of simple linear regression we saw that the above hypothesis test was equivalent to the hypothesis test:  EMBED Equation.DSMT4  with  EMBED Equation.DSMT4 . In the case of multiple regression, the above hypothesis test is equivalent to the following hypothesis test:  EMBED Equation.DSMT4  with alternative hypothesis  EMBED Equation.DSMT4  At least one coefficient is not equal to zero. Where  EMBED Equation.DSMT4  are the slopes of the regression equation that fits the population. So acceptance of the null hypothesis implies that the explanatory (i.e. predictor, i.e. independent) variables do not have any significant impact on explaining the variance in the dependent variable,  EMBED Equation.DSMT4 . So to estimate the average value of  EMBED Equation.DSMT4 one would not take into account the values of the independent variables and thus use the sample mean  EMBED Equation.DSMT4  as a point estimate. Rejection of the null hypothesis does not necessarily mean that all of the coefficients are significantly different from zero but that at least one of the coefficients is significantly different than zero. Rejection does imply that the regression equation is useful for predicting values of  EMBED Equation.DSMT4 given values of the independent variables. In particular, given values of  EMBED Equation.DSMT4 , a point estimate for the average value of  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4  (i.e. we would take into account the variances of the  EMBED Equation.DSMT4  from their averages to compute the average value of  EMBED Equation.DSMT4 . Once we have determined that the regression equation significantly explains the variance in  EMBED Equation.DSMT4 then we can run a  EMBED Equation.DSMT4 -test on each of the independent variable coefficients to test if they are significantly different from zero. We will then be interested in seeing if we can get a better model by reducing the full model by dropping independent variables whose coefficients are not significantly different than zero. More statistics associated with multiple regression: Then we have the following sample statistics:  EMBED Equation.DSMT4  (sample mean for each predictor variable EMBED Equation.DSMT4 )  EMBED Equation.DSMT4  (sample mean for  EMBED Equation.DSMT4 )  EMBED Equation.DSMT4 (sample variance for each predictor variable EMBED Equation.DSMT4 )  EMBED Equation.DSMT4 (sample variance for  EMBED Equation.DSMT4 ) We will also use the following sums of squares:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Note: Sometimes we refer to  EMBED Equation.DSMT4 as  EMBED Equation.DSMT4 . We have the following relationships between the sample statistics and the sums of squares:  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  for each predictor variable EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  The sample covariance between  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 is defined by  EMBED Equation.DSMT4  We will also define the covariance between  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  as follows:  EMBED Equation.DSMT4  Note that when  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  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