Coefficient of DeterminationPartial As the R provides information about the SSR(X ,X ,X ), there are also2"#$ Coefficients of PARTIAL Determination : this measures “how muchvariation a variable accounts for out of the variation available to thatvariable when it enters". This gives a proportional measure of thecontribution of each variable after all other variables are in the model. eg. Y = b bX bX bX + e! " "3 # #3 $ $3 3 Take X#What is the Coefficient of Partial Determination? 1) How much did the variable account for? (after other variables partial)p SSR(X |X ,X ) = 1.502621#"$ 2) What SS was available to it when it entered the model. SSE(X ,X ) = SSE(X ,X ,X ) + SSR(X |X ,X )"$ "#$ #"$ = 61.443 +230.62548 = 292.06848 Partial R = r = = 0.7896281 = 78.96281%22X|X X.230.62548292.06848#" $ß#"$ These calculations are available from SAS PROC REG with the / PCORR2option on the MODEL statement.SAS will also produce a Partial Correlation of the TYPE I SS.Output from PROC REGParameter Estimates Parameter StandardStandardizedVariable DF Estimate Error t Value Pr > |t|Intercept 1 17.84693 2.00188 8.92 <.0001X1 1 1.10313 0.32957 3.35 0.0032X2 1 0.32152 0.03711 8.66 <.0001X3 1 1.28894 0.29848 4.32 0.0003 Squared Standardized Semi-partialVariable DF Type I SS Type II SS Estimate Corr Type IINTERCEP 1 37446 244.171679 0.00000000 .X1 1 306.732328 34.418508 0.26023468 0.44501687X2 1 263.794445 230.625476 0.65915439 0.38272125X3 1 57.290222 57.290222 0.30693999 0.08311845 Squared Squared Squared Partial Semi-partial PartialVariable DF Corr Type I Corr Type II Corr Type II ToleranceINTERCEP 1 . . . .X1 1 0.44501687 0.04993545 0.35904408 0.73735836X2 1 0.68960879 0.33459866 0.78962809 0.77010493X3 1 0.48251214 0.08311845 0.48251214 0.88224762Standardized Regression CoefficientsThis technique addresses two aspects of estimating values"51) There is some potential difficulty with rounding errors in the calculations,particularly for the (X X) matrix calculations.w"- These roundoff errors are aggravated by (1) more variables in the model, (2)multicolinearity and (3) b values of very different magnitudes. Standardized regression coefficients can help with the last problem.2) The magnitude of the regression coefficients cannot be compared. Since the regression coefficients have units which are , they will varyY unitsX unitwith the units of X and Y. eg. If different people do the same study and various investigators takemeasurements on X in (1) inches, (2) feet, (3) meters and (4) mm, then"the same study will very different values for b ." The same is true of if a dependent variable (Y) is measured in (1) dollars,(2) thousands of dollars, or (3) median family income units (multiples ofabout 18 thousand). As a result of these scaling factors, the regression coefficients have aninterpretation in terms of the regression coefficients, but the regressioncoefficients will differ for different units, and must be examined withinthe context of those units. Standardized Regression Coefficients, however, have no units, but their sizecan be interpreted as a measure of impact or importance of each variableon the calculation of the predicted value.There are several ways to calculated Standardized Regression Coefficients 1) The variables can be “standardized" prior to doing the regression Y = w1n1YY–sÈ3Y X = w351n1XX–sÈ35 55x where s and s are ordinary standard deviationsYx5 regression on these variables gives the standardized regression model Y = X + X + X + 33ww ww ww"#$"#$"""%iii where = 0"!2) If the matrix calculations are done with the standardized values of X and Y,then the X X and and X Y matrices areww X X = X Y = 1r rr1 r rrr 1rrww"# " :"#" :" ] #:"ß" :"ß#]"]:"Ô×Ô×ÖÙÖÙÖÙÖÙÕØÕØááãáã,2,,Note that there is no row for the interceptso, another way to get the standardized regression coefficients is to calculated thematrix formula for B = (X X) X Y using the correlation matrices, orw"w-B=(R ) Rw\\ \]-13) There is also a relationship between the standardized regression coefficient andthe ordinary least squares regression coefficient. The relationship is = ""5w5ss]\The interpretation of the standardized regression coefficient is as a measureof relative impact on the calculations or as relative importance of thevariable to the model. The size of the variable is not longer influenced by units, and standardizedregression coefficients are unitless. The SIGN of the regression coefficient is retained, so negative and positiveeffects can still be interpreted.Example : The standard deviations are given by (for the mathematician example) s = = = 5.47429]ÊÊDY 38135.26n-1 232( Y) (948)22n243D s = \ÊDXn-12(X)2n355D for X$ s = = = 1.303$ÊÊDX 899.49n-1 232(X)2n24(128.6)23$$D = so = = 1.2889 = 0.30694"" ""$$ww$$   sss s 5.4721.303]\\] all values are available from the X X, X Y and Y Y matricesww wInterpretation 1) Size of value (magnitude, regardless of sign) is important. This is anindicator of "importance", or impact in the calculation of the predictedvalue. This would generally agree with observations and evaluationsmade by P>|t| and SSII and Partial R , but not always.2 2) The SIGN is important, and will match the sign on the regressioncoefficient.Effect of Correlation among the X variables51) If the X variables are uncorrelated, then they will describe a certain variation5whether alone or in concert with other variables, and the variables describe the same variation no matter which othervariables they are adjusted for. Also the regression coefficients will stay the same, will be stable There are several ways of creating this type of design. a) Orthogonal variables : result from transformation which extract theattributes of a variable while retaining a 0 correlation with