The REG Procedure

MODEL Statement

< label: > MODEL dependents=<regressors> < / options > ;

After the keyword MODEL, the dependent (response) variables are specified, followed by an equal sign and the regressor variables. Variables specified in the MODEL statement must be numeric variables in the data set being analyzed. For example, if you want to specify a quadratic term for variable X1 in the model, you cannot use X1*X1 in the MODEL statement but must create a new variable (for example, X1SQUARE=X1*X1) in a DATA step and use this new variable in the MODEL statement. The label in the MODEL statement is optional.

Table 58.2 lists the options available in the MODEL statement. Equations for the statistics available are given in the " Model Fit and Diagnostic Statistics" section.

Table 58.2: MODEL Statement Options

Option	Description
Model Selection and Details of Selection
SELECTION=	specifies model selection method
BEST=	specifies maximum number of subset models displayed or output to the OUTEST= data set
DETAILS	produces summary statistics at each step
DETAILS=	specifies the display details for forward, backward, and stepwise methods
GROUPNAMES=	provides names for groups of variables
INCLUDE=	includes first n variables in the model
MAXSTEP=	specifies maximum number of steps that may be performed
NOINT	fits a model without the intercept term
PCOMIT=	performs incomplete principal component analysis and outputs estimates to the OUTEST= data set
SLE=	sets criterion for entry into model
RIDGE=	performs ridge regression analysis and outputs estimates to the OUTEST= data set
SLS=	sets criterion for staying in model
START=	specifies number of variables in model to begin the comparing and switching process
STOP=	stops selection criterion
Fit Statistics
ADJRSQ	computes adjusted R2
AIC	computes Akaike's information criterion
B	computes parameter estimates for each model
BIC	computes Sawa's Bayesian information criterion
CP	computes Mallows' Cp statistic
GMSEP	computes estimated MSE of prediction assuming multivariate normality
JP	computes Jp, the final prediction error
MSE	computes MSE for each model
PC	computes Amemiya's prediction criterion
RMSE	displays root MSE for each model
SBC	computes the SBC statistic
SP	computes Sp statistic for each model
SSE	computes error sum of squares for each model
Data Set Options
EDF	outputs the number of regressors, the error degrees of freedom, and the model R2 to the OUTEST= data set
OUTSEB	outputs standard errors of the parameter estimates to the OUTEST= data set
OUTSTB	outputs standardized parameter estimates to the OUTEST= data set. Use only with the RIDGE= or PCOMIT= option.
OUTVIF	outputs the variance inflation factors to the OUTEST= data set. Use only with the RIDGE= or PCOMIT= option.
PRESS	outputs the PRESS statistic to the OUTEST= data set
RSQUARE	has same effect as the EDF option
Regression Calculations
I	displays inverse of sums of squares and crossproducts
XPX	displays sums-of-squares and crossproducts matrix
Details on Estimates
ACOV	displays asymptotic covariance matrix of estimates assuming heteroscedasticity
COLLIN	produces collinearity analysis
COLLINOINT	produces collinearity analysis with intercept adjusted out
CORRB	displays correlation matrix of estimates
COVB	displays covariance matrix of estimates
PARTIALR2	displays squared semi-partial correlation coefficients using Type I sums of squares
PCORR1	displays squared partial correlation coefficients using Type I sums of squares
PCORR2	displays squared partial correlation coefficients using Type II sums of squares
SCORR1	displays squared semi-partial correlation coefficients using Type I sums of squares
SCORR2	displays squared semi-partial correlation coefficients using Type II sums of squares
SEQB	displays a sequence of parameter estimates during selection process
SPEC	tests that first and second moments of model are correctly specified
SS1	displays the sequential sums of squares
SS2	displays the partial sums of squares
STB	displays standardized parameter estimates
TOL	displays tolerance values for parameter estimates
VIF	computes variance-inflation factors
Predicted and Residual Values
CLB	computes % confidence limits for the parameter estimates
CLI	computes % confidence limits for an individual predicted value
CLM	computes % confidence limits for the expected value of the dependent variable
DW	computes a Durbin-Watson statistic
INFLUENCE	computes influence statistics
P	computes predicted values
PARTIAL	displays partial regression plots for each regressor
R	produces analysis of residuals
Display Options and Other Options
ALL	requests the following options: ACOV, CLB, CLI, CLM, CORRB, COVB, I, P, PCORR1, PCORR2, R, SCORR1, SCORR2, SEQB, SPEC, SS1, SS2, STB, TOL, VIF, XPX
ALPHA=	sets significance value for confidence and prediction intervals and tests
NOPRINT	suppresses display of results
SIGMA=	specifies the true standard deviation of error term for computing CP and BIC
SINGULAR=	sets criterion for checking for singularity

You can specify the following options in the MODEL statement after a slash (/).

ACOV

displays the estimated asymptotic covariance matrix of the estimates under the hypothesis of heteroscedasticity. See the section " Testing for Heteroscedasticity" for more information.

ADJRSQ

computes R² adjusted for degrees of freedom for each model selected (Darlington 1968; Judge et al. 1980).

AIC

computes Akaike's information criterion for each model selected (Akaike 1969; Judge et al. 1980).

ALL

requests all these options: ACOV, CLB, CLI, CLM, CORRB, COVB, I, P, PCORR1, PCORR2, R, SCORR1, SCORR2, SEQB, SPEC, SS1, SS2, STB, TOL, VIF, and XPX.

ALPHA=number

sets the significance level used for the construction of confidence intervals for the current MODEL statement. The value must be between 0 and 1; the default value of 0.05 results in 95% intervals. This option affects the MODEL options CLB, CLI, and CLM; the OUTPUT statement keywords LCL, LCLM, UCL, and UCLM; the PLOT statement keywords LCL., LCLM., UCL., and UCLM.; and the PLOT statement options CONF and PRED. Specifying this option in the MODEL statement takes precedence over the ALPHA= option in the PROC REG statement.

B

is used with the RSQUARE, ADJRSQ, and CP model-selection methods to compute estimated regression coefficients for each model selected.

BEST=n

is used with the RSQUARE, ADJRSQ, and CP model-selection methods. If SELECTION=CP or SELECTION=ADJRSQ is specified, the BEST= option specifies the maximum number of subset models to be displayed or output to the OUTEST= data set. For SELECTION=RSQUARE, the BEST= option requests the maximum number of subset models for each size.

If the BEST= option is used without the B option (displaying estimated regression coefficients), the variables in each MODEL are listed in order of inclusion instead of the order in which they appear in the MODEL statement.

If the BEST= option is omitted and the number of regressors is less than 11, all possible subsets are evaluated. If the BEST= option is omitted and the number of regressors is greater than 10, the number of subsets selected is, at most, equal to the number of regressors. A small value of the BEST= option greatly reduces the CPU time required for large problems.

BIC

computes Sawa's Bayesian information criterion for each model selected (Sawa 1978; Judge et al. 1980).

CLB

requests the % upper- and lower-confidence limits for the parameter estimates. By default, the 95% limits are computed; the ALPHA= option in the PROC REG or MODEL statement can be used to change the -level.

CLI

requests the % upper- and lower-confidence limits for an individual predicted value. By default, the 95% limits are computed; the ALPHA= option in the PROC REG or MODEL statement can be used to change the -level. The confidence limits reflect variation in the error, as well as variation in the parameter estimates. See the " Predicted and Residual Values" section and Chapter 2, " Introduction to Regression Procedures," for more information.

CLM

displays the % upper- and lower-confidence limits for the expected value of the dependent variable (mean) for each observation. By default, the 95% limits are computed; the ALPHA= in the PROC REG or MODEL statement can be used to change the -level. This is not a prediction interval (see the CLI option) because it takes into account only the variation in the parameter estimates, not the variation in the error term. See the section " Predicted and Residual Values" and Chapter 2 for more information.

COLLIN

requests a detailed analysis of collinearity among the regressors. This includes eigenvalues, condition indices, and decomposition of the variances of the estimates with respect to each eigenvalue. See the " Collinearity Diagnostics" section.

COLLINOINT

requests the same analysis as the COLLIN option with the intercept variable adjusted out rather than included in the diagnostics. See the " Collinearity Diagnostics" section.

CORRB

displays the correlation matrix of the estimates. This is the (X'X)^-1 matrix scaled to unit diagonals.

COVB

displays the estimated covariance matrix of the estimates. This matrix is (X'X)^-1 s², where s² is the estimated mean squared error.

CP

computes Mallows' C_p statistic for each model selected (Mallows 1973; Hocking 1976). See the " Criteria Used in Model-Selection Methods" section for a discussion of the use of C_p.

DETAILS

DETAILS=name

specifies the level of detail produced when the BACKWARD, FORWARD or STEPWISE methods are used, where name can be ALL, STEPS or SUMMARY. The DETAILS or DETAILS=ALL option produces entry and removal statistics for each variable in the model building process, ANOVA and parameter estimates at each step, and a selection summary table. The option DETAILS=STEPS provides the step information and summary table. The option DETAILS=SUMMARY produces only the summary table. The default if the DETAILS option is omitted is DETAILS=STEPS.

DW

calculates a Durbin-Watson statistic to test whether or not the errors have first-order autocorrelation. (This test is appropriate only for time series data.) The sample autocorrelation of the residuals is also produced. See the section " Autocorrelation in Time Series Data."

EDF

outputs the number of regressors in the model excluding and including the intercept, the error degrees of freedom, and the model R² to the OUTEST= data set.

GMSEP

computes the estimated mean square error of prediction assuming that both independent and dependent variables are multivariate normal (Stein 1960; Darlington 1968). Note that Hocking's formula (1976, eq. 4.20) contains a misprint: "n-1" should read "n-2.")

GROUPNAMES='name1' 'name2' ...

provides names for variable groups. This option is available only in the BACKWARD, FORWARD, and STEPWISE methods. The group name can be up to 32 characters. Subsets of independent variables listed in the MODEL statement can be designated as variable groups. This is done by enclosing the appropriate variables in braces. Variables in the same group are entered into or removed from the regression model at the same time. However, if the tolerance of any variable (see the TOL option) in a group is less than the setting of the SINGULAR= option, then the variable is not entered into the model with the rest of its group. If the GROUPNAMES= option is not used, then the names GROUP1, GROUP2, ..., GROUPn are assigned to groups encountered in the MODEL statement. Variables not enclosed by braces are used as groups of a single variable.

For example,

 
   model y={x1 x2} x3 / selection=stepwise 
     groupnames='x1 x2' 'x3';

As another example,

 
   model y={ht wgt age} bodyfat / selection=forward 
     groupnames='htwgtage' 'bodyfat';

I

displays the (X'X)^-1 matrix. The inverse of the crossproducts matrix is bordered by the parameter estimates and SSE matrices.

INCLUDE=n

forces the first n independent variables listed in the MODEL statement to be included in all models. The selection methods are performed on the other variables in the MODEL statement. The INCLUDE= option is not available with SELECTION=NONE.

INFLUENCE

requests a detailed analysis of the influence of each observation on the estimates and the predicted values. See the " Influence Diagnostics" section for details.

JP

computes J_p, the estimated mean square error of prediction for each model selected assuming that the values of the regressors are fixed and that the model is correct. The J_p statistic is also called the final prediction error (FPE) by Akaike (Nicholson 1948; Lord 1950; Mallows 1967; Darlington 1968; Rothman 1968; Akaike 1969; Hocking 1976; Judge et al. 1980).

MSE

computes the mean square error for each model selected (Darlington 1968).

MAXSTEP=n

specifies the maximum number of steps that are done when SELECTION=FORWARD, SELECTION=BACKWARD or SELECTION=STEPWISE is used. The default value is the number of independent variables in the model for the forward and backward methods and three times this number for the stepwise method.

NOINT

suppresses the intercept term that is otherwise included in the model.

NOPRINT

suppresses the normal display of regression results. Note that this option temporarily disables the Output Delivery System (ODS); see Chapter 14, " Using the Output Delivery System," for more information.

OUTSEB

outputs the standard errors of the parameter estimates to the OUTEST= data set. The value SEB for the variable _TYPE_ identifies the standard errors. If the RIDGE= or PCOMIT= option is specified, additional observations are included and identified by the values RIDGESEB and IPCSEB, respectively, for the variable _TYPE_. The standard errors for ridge regression estimates and incomplete principal components (IPC) estimates are limited in their usefulness because these estimates are biased. This option is available for all model-selection methods except RSQUARE, ADJRSQ, and CP.

OUTSTB

outputs the standardized parameter estimates as well as the usual estimates to the OUTEST= data set when the RIDGE= or PCOMIT= option is specified. The values RIDGESTB and IPCSTB for the variable _TYPE_ identify ridge regression estimates and IPC estimates, respectively.

OUTVIF

outputs the variance inflation factors (VIF) to the OUTEST= data set when the RIDGE= or PCOMIT= option is specified. The factors are the diagonal elements of the inverse of the correlation matrix of regressors as adjusted by ridge regression or IPC analysis. These observations are identified in the output data set by the values RIDGEVIF and IPCVIF for the variable _TYPE_.

P

calculates predicted values from the input data and the estimated model. The display includes the observation number, the ID variable (if one is specified), the actual and predicted values, and the residual. If the CLI, CLM, or R option is specified, the P option is unnecessary. See the section " Predicted and Residual Values" for more information.

PARTIAL

requests partial regression leverage plots for each regressor. See the " Influence Diagnostics" section for more information.

PARTIALR2 < ( < TESTS > < SEQTESTS > ) >

See the SCORR1 option.

PC

computes Amemiya's prediction criterion for each model selected (Amemiya 1976; Judge et al. 1980).

PCOMIT=list

requests an IPC analysis for each value m in the list. The procedure computes parameter estimates using all but the last m principal components. Each value of m produces a set of IPC estimates, which is output to the OUTEST= data set. The values of m are saved by the variable _PCOMIT_, and the value of the variable _TYPE_ is set to IPC to identify the estimates. Only nonnegative integers can be specified with the PCOMIT= option.

If you specify the PCOMIT= option, RESTRICT statements are ignored. The PCOMIT= option is ignored if you use the SELECTION= option in the MODEL statement.

PCORR1

displays the squared partial correlation coefficients using Type I Sum of Squares (SS). This is calculated as SS/(SS+SSE), where SSE is the error Sum of Squares.

PCORR2

displays the squared partial correlation coefficients using Type II sums of squares. These are calculated the same way as with the PCORR1 option, except that Type II SS are used instead of Type I SS.

PRESS

outputs the PRESS statistic to the OUTEST= data set. The values of this statistic are saved in the variable _PRESS_. This option is available for all model-selection methods except RSQUARE, ADJRSQ, and CP.

R

requests an analysis of the residuals. The results include everything requested by the P option plus the standard errors of the mean predicted and residual values, the studentized residual, and Cook's D statistic to measure the influence of each observation on the parameter estimates. See the section " Predicted and Residual Values" for more information.

RIDGE=list

requests a ridge regression analysis and specifies the values of the ridge constant k (see the " Computations for Ridge Regression and IPC Analysis" section). Each value of k produces a set of ridge regression estimates that are placed in the OUTEST= data set. The values of k are saved by the variable _RIDGE_, and the value of the variable _TYPE_ is set to RIDGE to identify the estimates.

Only nonnegative numbers can be specified with the RIDGE= option. Example 58.10 illustrates this option.

If you specify the RIDGE= option, RESTRICT statements are ignored. The RIDGE= option is ignored if you use the SELECTION= option in the MODEL statement.

RMSE

displays the root mean square error for each model selected.

RSQUARE

has the same effect as the EDF option.

SBC

computes the SBC statistic for each model selected (Schwarz 1978; Judge et al. 1980).

SCORR1 < ( < TESTS > < SEQTESTS > ) >

displays the squared semi-partial correlation coefficients using Type I sums of squares. This is calculated as SS/SST, where SST is the corrected total SS. If the NOINT option is used, the uncorrected total SS is used in the denominator. The optional arguments TESTS and SEQTESTS request F-tests, p-values, and cumulative R-Square values as variables are sequentially added to a model. The F-test values are computed as the Type I sum of squares for the variable in question divided by a mean square error. If you specify the TESTS option, the denominator MSE is the residual mean square for the full model specified in the MODEL statement. If you specify the SEQTESTS option, the denominator MSE is the residual mean square for the model containing all the independent variables that have been added to the model up to and including the variable in question. The TESTS and SEQTESTS options are not supported if you specify model selection methods, or the RIDGE or PCOMIT options. Note that the PARTIALR2 option is a synonym for the SCORR1 option.

SCORR2 < ( TESTS ) >

displays the squared semi-partial correlation coefficients using Type II sums of squares. These are calculated the same way as with the SCORR1 option, except that Type II SS are used instead of Type I SS. The optional TEST argument requests F-tests, p-values, and cumulative R-Square values as variables are sequentially added to a model. The F-test values are computed as the Type II sum of squares for the variable in question divided by the residual mean square for the full model specified in the MODEL statement. The TESTS option is not supported if you specify model selection methods, or the RIDGE or PCOMIT options.

SELECTION=name

specifies the method used to select the model, where name can be FORWARD (or F), BACKWARD (or B), STEPWISE, MAXR, MINR, RSQUARE, ADJRSQ, CP, or NONE (use the full model). The default method is NONE. See the " Model-Selection Methods" section for a description of each method.

SEQB

produces a sequence of parameter estimates as each variable is entered into the model. This is displayed as a matrix where each row is a set of parameter estimates.

SIGMA=n

specifies the true standard deviation of the error term to be used in computing the CP and BIC statistics. If the SIGMA= option is not specified, an estimate from the full model is used. This option is available in the RSQUARE, ADJRSQ, and CP model-selection methods only.

SINGULAR=n

tunes the mechanism used to check for singularities. Specifying this option in the MODEL statement takes precedence over the SINGULAR= option in the PROC REG statement. The default value is machine dependent but is approximately 1E-7 on most machines. This option is rarely needed. Singularity checking is described in the " Computational Methods" section.

SLENTRY=value

SLE=value

specifies the significance level for entry into the model used in the FORWARD and STEPWISE methods. The defaults are 0.50 for FORWARD and 0.15 for STEPWISE.

SLSTAY=value

SLS=value

specifies the significance level for staying in the model for the BACKWARD and STEPWISE methods. The defaults are 0.10 for BACKWARD and 0.15 for STEPWISE.

SP

computes the S_p statistic for each model selected (Hocking 1976).

SPEC

performs a test that the first and second moments of the model are correctly specified. See the section " Testing for Heteroscedasticity" for more information.

SS1

displays the sequential sums of squares (Type I SS) along with the parameter estimates for each term in the model. See Chapter 11, " The Four Types of Estimable Functions," for more information on the different types of sums of squares.

SS2

displays the partial sums of squares (Type II SS) along with the parameter estimates for each term in the model. See the SS1 option also.

SSE

computes the error sum of squares for each model selected.

START=s

is used to begin the comparing-and-switching process in the MAXR, MINR, and STEPWISE methods for a model containing the first s independent variables in the MODEL statement, where s is the START value. For these methods, the default is START=0.

For the RSQUARE, ADJRSQ, and CP methods, START=s specifies the smallest number of regressors to be reported in a subset model. For these methods, the default is START=1.

The START= option cannot be used with model-selection methods other than the six described here.

STB

produces standardized regression coefficients. A standardized regression coefficient is computed by dividing a parameter estimate by the ratio of the sample standard deviation of the dependent variable to the sample standard deviation of the regressor.

STOP=s

causes PROC REG to stop when it has found the "best" s-variable model, where s is the STOP value. For the RSQUARE, ADJRSQ, and CP methods, STOP=s specifies the largest number of regressors to be reported in a subset model. For the MAXR and MINR methods, STOP=s specifies the largest number of regressors to be included in the model.

The default setting for the STOP= option is the number of variables in the MODEL statement. This option can be used only with the MAXR, MINR, RSQUARE, ADJRSQ and CP methods.

TOL

produces tolerance values for the estimates. Tolerance for a variable is defined as 1-R², where R² is obtained from the regression of the variable on all other regressors in the model. See the section " Collinearity Diagnostics" for more detail.

VIF

produces variance inflation factors with the parameter estimates. Variance inflation is the reciprocal of tolerance. See the section " Collinearity Diagnostics" for more detail.

XPX

displays the X'X crossproducts matrix for the model. The crossproducts matrix is bordered by the X'Y and Y'Y matrices.

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