MODEL Statement

The TPSPLINE Procedure

MODEL Statement

MODEL dependents = < regression variables > (smoothing variables) < /options > ;

The MODEL statement specifies the dependent variables, the independent regression variables, which are listed with no parentheses, and the independent smoothing variables, which are listed inside parentheses.

The regression variables are optional. At least one smoothing variable is required, and it must be listed after the regression variables. No variables can be listed in both the regression variable list and the smoothing variable list.

If you specify more than one dependent variable, PROC TPSPLINE calculates a thin-plate smoothing spline estimate for each dependent variable, using the regression variables and smoothing variables specified on the right-hand side.

If you specify regression variables, PROC TPSPLINE fits a semiparametric model using the regression variables as the linear part of the model.

You can specify the following options in the MODEL statement.

ALPHA=number

specifies the significance level $\alpha$ of the confidence limits on the final thin-plate smoothing spline estimate when you request confidence limits to be included in the output data set. Specify number as a value between 0 and 1. The default value is 0.05. See the "OUTPUT Statement" section for more information on the OUTPUT statement.

DF=number

specifies the degrees of freedom of the thin-plate smoothing spline estimate, defined as

$df=trace(A(\lambda))$

where $A(\lambda)$ is the hat matrix. Specify number as a value between zero and the number of unique design points.

DISTANCE=number

D=number

defines a range such that if two data points (x_i, z_i) and (x_j, z_j) satisfy

$max_k | x_{ik}-x_{jk}| \leq D/2$

then these data points are treated as replicates, where x_i are the smoothing variables and z_i are the regression variables.

You can use the DISTANCE= option to reduce the number of unique design points by treating nearby data as replicates. This can be useful when you have a large data set. The default value is 0.

LAMBDA0=number

specifies the smoothing parameter, $\lambda_0$ , to be used in the thin-plate smoothing spline estimate. By default, PROC TPSPLINE uses the $\lambda$ parameter that minimizes the GCV function for the final fit. The LAMBDA0= value must be positive.

LAMBDA=list-of-values

specifies a set of values for the $\lambda$ parameter. PROC TPSPLINE returns a GCV value for each $\lambda$ point that you specify. You can use the LAMBDA= option to study the GCV function curve for a set of values for $\lambda$ . All values listed in the LAMBDA= option must be positive.

LOGNLAMBDA0=number

LOGNL0=number

specifies the smoothing parameter $\lambda_0$ on the $log10(n\lambda)$ scale. If you specify both the LOGNL0= and LAMBDA0= options, only the value provided by the LOGNL0= option is used. By default, PROC TPSPLINE uses the $\lambda$ parameter that minimizes the GCV function for the estimate.

LOGNLAMBDA=list-of-values

LOGNL=list-of-values

specifies a set of values for the $\lambda$ parameter on the $log10(n\lambda)$ scale. PROC TPSPLINE returns a GCV value for each $\lambda$ point that you specify. You can use the LOGNLAMBDA= option to study the GCV function curve for a set of $\lambda$ values. If you specify both the LOGNL= and LAMBDA= options, only the list of values provided by LOGNL= option is used.

In some cases, the LOGNL= option may be prefered over the LAMBDA= option. Because the LAMBDA= value must be positive, a small change in that value can result in a major change in the GCV value. If you instead specify $\lambda$ on the log₁₀ scale, the allowable range is enlarged to include negative values. Thus, the GCV function is less sensitive to changes in LOGNLAMBDA.

M=number

specifies the order of the derivative in the penalty term. The M= value must be a positive integer. The default value is the max(2, INT(d/2)+1), where d is the number of smoothing variables.

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