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All Possible Regressions and Statistics for Comparing Models Animal Science 500 Lecture No. 12 October 12, 2010 IOWA STATE UNIVERSITY Department of Animal Science Example analysis The RSQUARE Procedure RECALL The RSQUARE procedure selects optimal subsets of independent variables in a multiple regression analysis IOWA STATE UNIVERSITY Department of Animal Science Example analysis PROC RSQUARE options; MODEL dependents = independents / options; (options can appear in either PROC RSQUARE or any MODEL statement). SELECT = n specific maximum number of subset models INCLUDE = I requests that the first I variables after the equal sign be included in every regression SIGMA = n specifies the true standard deviation of the error term ADJRSQ computes R2 adjusted for degrees of freedom CP computes MALLOWS’ Cp statistic IOWA STATE UNIVERSITY Department of Animal Science Example analysis PROC RSQUARE options; MODEL dependents = independents / options; (options can appear in either PROC RSQUARE or any MODEL statement). PROC RSQUARE DATA=name OUTEST=EST ADJRSQ MSE CP; SELECT=n; MODEL = variable list; IOWA STATE UNIVERSITY Department of Animal Science Example analysis PROC PRINT DATA=EST; PROC PLOT; PLOT _CP_*_P_ = ‘C’ _P_*_P_ = ‘P’ / OVERLAY; PLOT _MSE_*_P_ = ‘M’; Run; Quit IOWA STATE UNIVERSITY Department of Animal Science PROC STEPWISE The STEPWISE procedure provides five methods for stepwise regression. General form: PROC STEPWISE; MODEL dependents = independents / options; Run; Quit; ** Assumes that you have at least one dependent variable and 2 or more independent variables. If only one independent variable exists then you are just doing a simple regression of x on y or y on x. IOWA STATE UNIVERSITY Department of Animal Science Types of Regression Uses of PROC REG for standard problems: 1. PROC REG; model y = x; /* simple linear regression */ 2. PROC REG; model y = x; weight w; /* weighted linear regression */ 3. PROC REG; /* multiple regression */ model y = x1 x2 x3; IOWA STATE UNIVERSITY Department of Animal Science PROC REG General form: PROC REG; MODEL dependents = independents / options; Options available include: NOINT – regression with no intercept FORWARD A forward selection analysis starts out with no predictors in the model. Each predictor that that was chosen by the user is evaluated with respect to see how much the R 2 is increased by adding it to the model. The predictor that increases the R2 will be added if it meets the statistical conditions for entry With SAS the statistical conditions is the significance level for the increase in the R2 produced by addition of the predictor. If no predictor meets the condition, the analysis stops. If a predictor is added, then the second step involves re-evaluating all of the available predictors which have not yet been entered into the model. If any satisfy the statistical condition for entry, the predictor increasing the R2 the greatest is added. This process is continued until no predictors remain that could enter. IOWA STATE UNIVERSITY Department of Animal Science PROC REG General form: PROC REG; MODEL dependents = independents / options; Options available include: BACKWARD In a backwards elimination analysis we start out with all of the predictors in the model. At each step we evaluate the predictors which are in the model and eliminate any that meet the criterion for removal. STEPWISE Stepwise selection begins similar to forwards selection. However at each “step” variables that are in the model are first evaluated for removal. Those meeting removal criteria are evaluated to see which would lower the R 2, the least. How does this work where a variable enters and then might leave later? If two predictors ultimately enter the model, one may be removed because they are well correlated and removing one impacts the R 2 very little if at all. IOWA STATE UNIVERSITY Department of Animal Science PROC REG General form: PROC REG; MODEL dependents = independents / options; Options available include: MAXR The maximum R2 option does not settle on a single model. Instead, it tries to find the "best" one-variable model, the "best" two-variable model, and so forth. , MAXR starts out by finding the single variable model producing the greatest R2 After finding the one variable MAXR then another variable is added until it finds the variable that increases the R2 the most. It continues this process until it stops where the addition of another variable is no better than the previous (i.e. adding the 4th variable did not significantly improve the R2 compared to the 3 variable model for example. The difference between the STEPWISE and MAXR options is that all switches are evaluated before any switch is made in the MAXR method . Using the STEPWISE option, the "worst" variable may be removed without considering what adding the "best" remaining variable might accomplish. IOWA STATE UNIVERSITY Department of Animal Science PROC REG General form: PROC REG; MODEL dependents = independents / options; Options available include: MINR The MINR option closely resembles the MAXR method. However, the switch chosen with the MINR option is switch that produces the smallest increase in R2. In a way approaching the “best” model in reverse compared to MAXR. IOWA STATE UNIVERSITY Department of Animal Science PROC REG General form: PROC REG; MODEL dependents = independents / options; Options available include: SLE=value This option sets some criterion for entry into the model. This can be defined by the user by meeting some level of change or Δ to the R2 SLS=value This option sets some criterion for staying or remaining in the model. This can be defined by the user by meeting some level of change or Δ to the R2 to stay in the model. IOWA STATE UNIVERSITY Department of Animal Science PROC REG The default statistical levels for each type of regression analysis is different unless it is changed by the user: The defaults are: BACKWARD = 0.10 FORWARD = 0.10 STEPWISE = 0.15 User can set it by using the SLSTAY option for example / SLSTAY=.05. IOWA STATE UNIVERSITY Department of Animal Science Significance Tests for the Regression Coefficients 1. Finding the significance of the parameter estimates by using the F or t test (will see in a couple of slides) 2. R2 = R-Square is the proportion of variation in the dependent variable (Y) that can be explained by the predictors (X variables) in the regression model. 3. Adjusted R2 Predictors could be added to the model which would continue to improve the ability of the predictors to explain the dependent variable. Some of the improvement in the R-Square would be simply due to chance variation. The adjusted R-Square attempts to yield a more honest value to estimate R-Square. = 1-(1-R2) (n-1)/(n-p-1) where R2 = the unadjusted R2 n = the number of number of observations, and p = the number of predictors IOWA STATE UNIVERSITY Department of Animal Science Significance Tests for the Regression Coefficients The Mallows’ Cp statistic CP (Cp) = SSE / σ2 + 2p – n where SSE = error sums of squares σ2 = the estimate of pure error variance from the SIGMA = option for from fitting the full model p = the number of parameters including the intercept, and n = the number of observations IOWA STATE UNIVERSITY Department of Animal Science F and T tests for significance for overall model F Model variance error variance R2 / p (1 R 2 ) /( n p 1) where p number of parameters n sample size t F (n 2) * r 2 1 r2 IOWA STATE UNIVERSITY Department of Animal Science