ࡱ> >@=S @+bjbj )hxxl    8LLD 52t( . . . ./<.,14$96 Y8@4 @4`4```  .` .``T!{, - $ @ ?--405O-8`8-`  Linear Least Squares Regression I. Why linear regression? A. Its simple. B. It fits many functions pretty well. C. Many nonlinear functions can be transformed to linear. II. Why least squares regression? A. Because it works better than the alternatives in many cases. B. Because it is easy to work with mathematically. III. Derivation of the least-squares parameters of a line A. Equation of a line:  EMBED Equation.3  (a is the y intercept, b is the slope B. We wish to minimize the sum of squared deviations of estimated y values ( EMBED Equation.3 ) from the actual y values (y):  EMBED Equation.3  C. To simplify the mathematics, we can transform the xi's to be difference scores by subtracting their mean: xi=xi- EMBED Equation.2  Hence, the (xi= 0. D. Substitute equation for line into summation above.  EMBED Equation.3 = EMBED Equation.3  1. To find the a (intercept) and b (slope) that will make this expression the smallest, take the partial derivatives of the expression, set them equal to zero, and solve the equations.  EMBED Equation.3  = ( 2(-1)(yi-a-bxi)= 0 -2 ((yi-a-bxi) = 0 distributive rule: -2(w+u)=-2w+(-2u) ( (yi-a-bxi) = 0 divide both sides by -2 (yi - Na - b(xi = 0 carry out the summation (yi - Na = 0 recall that Sxi= 0 ( yi = Na add Na to both sides ( yi/N = a divide by N ( yi/N =  EMBED Equation.3   EMBED Equation.3  = ( 2(-xi)(yi-a-bxi) = 0 -2 (( xiyi -axi -bxi2)= 0 distributive rule: -2(w+u)=-2w+(-2u) ( (xiyi -axi -bxi2) = 0 divide both sides by -2 (xiyi -(axi -(bxi2 = 0 carry out the summation (xiyi -a(xi -b(xi2 = 0 distributive rule again ( xiyi - b( xi2 = 0 recall that (xi= 0 ( xiyi = b( xi2 add b(xi2 to both sides ( xiyi/( xi2 = b divide by (xi2 IV. Goodness of fit of regression model A. Decomposition of variation Total SS = SS regression + SS residual Total variation around mean of Y= variation explained by line + unexplained variation around line.  EMBED Equation.3 =  EMBED Equation.3  +  EMBED Equation.3  B. Proportion of variance explained by line 1.  EMBED Equation.2  =  EMBED Equation.2  =  EMBED Equation.3  C. Estimate of variance s2 s2=  EMBED Equation.2  where k=number of predictors (in linear regression=1). Note: 2 df are lost, 1 for each parameter estimated (no variance with only 2 points) D. F-ratio testing variance explained by line  EMBED Equation.2  =  EMBED Equation.2  dfreg = k = # of predictors; dfres = N-k-1 V. Testing parameters of regression model A. Purpose The estimated a and b are sample estimates of the true population parameters. Want to determine how close the sample estimates are to the true population parameters. Confidence intervals give a range of values within which the population parameters will lie with a stated degree of certainty ((). Tests of significance ask whether for some given degree of certainty, the population value of a parameter may be different from some given value (usually 0). B. Standard Assumptions 1. Given a regression model: yi=a + bxi + ei 2. yi are independently and identically distributed with variance s2. 3. ei are independently and identically distributed with mean 0 and variance s2. 4. xi are fixed (not random variables). C. Statistics for b 1. Derivation of variance of b b = Sxiyi/Sxi2 see above = S(xi/Sxi2)yi distributive rule = Swiyi rewrite equation; let wi=xi/Sxi2 so, b is a linear combination of random variables var(b) = Swi2var(yi) variance of linear combination of random variables = Swi2s2 by assumption 2 above = Sxi2 s2 replacing wi by its equivalent (Sxi2)2 = s2 simplifying by canceling one Sxi2 Sxi2 2. Standard error of b sb =  EMBED Equation.2  =  EMBED Equation.2  3. T-test for b t = EMBED Equation.2  df=N-k-1 4. Confidence interval for b = b t(/2  EMBED Equation.2  5. Note: large variation in x will yield smaller sb and larger t. With small variation in x, estimates are unstable. D. The constant is rarely of interest. When it is, similar tests can be performed. Note that the constant is simply the regression coefficient for a predictor that does not vary in the data. The variance of a = s2/n. VI. Problems A. Outliers 1. There may be observations for which the relation(s) between the criterion and the predictor(s) are not summarized well by the regression equation. B. Heteroscedasticity 1. The variance around the regression line may not be constant. Hence, the equation predicts better in some ranges than in others. C. Curvilinearity 1. The regression line may systematically underestimate in some ranges and overestimate in others because the relation between the criterion and the predictor(s) is not linear. D. Autocollinearity 1. The observations (and the residuals) may be correlated (frequently a problem with time series data) yielding inaccurate parameter estimates that may appear to be more precise than they really are. E. Nonlinearity 1. The relation(s) between the predictor(s) and the criterion may be nonlinear. For example: y= a + bx +bx2 + e y= a + bln(x) + e 2. Note that in some cases, relations that are theoretically nonlinear may be transformed into linear relations. a. Example - learning theory transformed to linear ti = abxi a > 0 [positive] 0 2 requires inspection. b. Some authors suggest examining studentized residuals because the residuals may not be homoscedastic. The studentized residual is calculated by dividing each residual by its estimated standard deviation. The estimated standard deviation of a residual is defined as:  EMBED Equation.2  EMBED Equation.2  where s=standard error of the estimate (see above). The studentized residuals can then be tested against a t distribution with N-k-1 degrees of freedom. Note that in practice this is not an appropriate test because the t tests are not independent. However, it is a way of locating unusually large residuals. 2. Leverage a. Some authors suggest examining the leverage of an observation, defined as the quantity in brackets above: hi =  EMBED Equation.2  A large leverage is an indication that the value of the predictor is far from the mean of that predictor. 3. Cooks D (Cook, 1977) a. Leverage does not measure the influence of an observation on the criterion. Cooks D does:  EMBED Equation.2  where esi2 = studentized residual for observation i; hi=leverage for observation i. Approximate tests of the significance of D are available, but usually one simply looks for Ds that are large relative to the others in the dataset. 4. Delete suspect observations a. One should be able to delete points without substantial effect on parameters. If one cannot, then either the number of observations is too small or the observation in question is an outlier. b. Some computer programs routinely report DFBETAi which is the regression coefficient that would result were observation i to be deleted. PAGE 4 PAGE 5 (.)*=>?@`atuvwTUhijklmú}j4A UVjCJEHUj#A UV jSCJjD6 CJEHUjD6 UVmHnHuCJEHjCJEHUjA UVj$CJEHUjA UVjCJEHUjA UV jCJU>*CJCJ5CJ1 !<LtGH`ySTABqd^^^$a$#+>+?+BCVWXY\]efkluvxy~ ! "          : < > @ D F l n p r 찧j6CJEHUj_A UVjT CJEHUjA UV CJOJQJCJEH jSCJjR CJEHUjA UVCJ jCJUjCJEHUBqr    v B D   I J | d^ pd^pd^pd "pd^pd^r x z       ! % & - J K L M N O Q R T U W X Z [ \ } ~      CJEHCJEH jSCJCJ^| }   ? @ i &  & F^ & F^^^pd^p         : ; < = ? 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