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Polynomial Effects

You use a polynomial parameter to define effects that are constructed as multiples of powers of numeric variables. The variables specified in the vars subparameter must be numeric. A design matrix column is generated for each term of the specified polynomial. By default, each of these terms is treated as a separate effect for the purpose of model building. For example, the following two action parameters (displayed in the CASL language) yield the identical analysis: 

   polynomial={name='MyPoly', vars={'x1','x2','x3'}, degree='2'},

   effects={ {vars={'x1','x2','x3'}, interaction='BAR', maxinteract='2'},
             {vars={'x1','x1'}, interaction='CROSS'},
             {vars={'x2','x2'}, interaction='CROSS'},
             {vars={'x3','x3'}, interaction='CROSS'}}}

The subparameters that you can specify are described in the Syntax section of the specific action chapters. The following list provides more detail about these subparameters using the CASL-language.

degree=n

specifies the degree of the polynomial, where n must be a positive integer and is typically a small integer, such as 1, 2, or 3. By default, degree is 1.

details

when set to 'True,' displays a table that shows the details of the specified polynomial, including the number of terms generated. If you also specify the standardize subparameter, then a table that shows the standardization details is also produced.

mDegree=n

specifies the maximum degree of any variable in a term of the polynomial, where n must be a positive integer. The default is the degree of the specified polynomial. For example, the following CASL-language statement generates the terms x 1, x 2, x 1 squared, x 1 x 2, x 2 squared, x 1 squared x 2, x 1 x 2 squared and x 1 squared x 2 squared: 

polynomial={name='MyPoly', vars={x1 x2}, degree=4, mdegree=2};
noseparate

when set to 'True,' treats the polynomial as a single effect that has multiple degrees of freedom. The value of the name subparameter that you specify is used as the constructed effect name, and the labels of the terms are used as labels of the corresponding parameters.

standardize={stdize-opts}

standardizes the variables that define the polynomial. The stdize-opts include the prefix, method, and options subparameters.

By default, the standardized variables receive prefix "s_" in the variable names. You can use the prefix subparameter to control this prefix as follows:

prefix='none' | quoted-string

specifies the prefix that is appended to standardized variables when forming the term labels. You can specify the following values:

quoted-string

specifies the prefix

NONE

does not apply a prefix

You can use the method subparameter to control how the center and scale are estimated. You can specify the following methods:

moments

estimates the center by the variable mean and the scale by the standard deviation. If a weight variable is specified using a weight parameter, the weights are not used in the computation of the mean and standard deviation, and observations that have invalid weights are ignored. Only observations that are used in performing the analysis are used for the standardization.

mrange

estimates the center by the midpoint of the variable range and the scale as half the variable range. Any observation that has a missing value for any regressor used in the model is ignored when the range of variables in a polynomial effect is computed. Observations that have valid regressor values but missing or invalid values of frequency variables, weight variables, or dependent variables are used in computing variable ranges. By default, method is RANGE.

wmoments

is the same as MOMENTS except that weighted means and weighted standard deviations are used.

Let

StartLayout 1st Row 1st Column n 2nd Column equals 3rd Column normal n normal u normal m normal b normal e normal r normal o normal f normal o normal b normal s normal e normal r normal v normal a normal t normal i normal o normal n normal s normal u normal s normal e normal d normal i normal n normal t normal h normal e normal a normal n normal a normal l normal y normal s normal i normal s 2nd Row 1st Column w 2nd Column equals 3rd Column normal w normal e normal i normal g normal h normal t normal v normal a normal r normal i normal a normal b normal l normal e 3rd Row 1st Column f 2nd Column equals 3rd Column normal f normal r normal e normal q normal u normal e normal n normal c normal y normal v normal a normal r normal i normal a normal b normal l normal e 4th Row 1st Column x 2nd Column equals 3rd Column normal v normal a normal r normal i normal a normal b normal l normal e normal t normal o normal b normal e normal s normal t normal a normal n normal d normal a normal r normal d normal i normal z normal e normal d 5th Row 1st Column x Subscript left-parenthesis n right-parenthesis 2nd Column equals 3rd Column Max Subscript i equals 1 Superscript n Baseline left-parenthesis x Subscript i Baseline right-parenthesis 6th Row 1st Column x Subscript left-parenthesis 1 right-parenthesis 2nd Column equals 3rd Column Min Subscript i equals 1 Superscript n Baseline left-parenthesis x Subscript i Baseline right-parenthesis 7th Row 1st Column upper F 2nd Column equals 3rd Column normal s normal u normal m normal o normal f normal f normal r normal e normal q normal u normal e normal n normal c normal i normal e normal s 8th Row 1st Column Blank 2nd Column equals 3rd Column normal upper Sigma Subscript i equals 1 Superscript n Baseline f Subscript i 9th Row 1st Column normal upper W normal upper F 2nd Column equals 3rd Column normal s normal u normal m normal o normal f normal w normal e normal i normal g normal h normal t normal e normal d normal f normal r normal e normal q normal u normal e normal n normal c normal i normal e normal s 10th Row 1st Column Blank 2nd Column equals 3rd Column normal upper Sigma Subscript i equals 1 Superscript n Baseline w Subscript i Baseline f Subscript i EndLayout

Table 4 shows how the center and scale are computed for each of the supported methods.


You can control whether the standardization is to center, scale, or both center and scale by specifying the options subparameter as follows:

center

centers but does not scale the variables. For a variable x,

centerscale

centers and scales the variables. This is the default. For a variable x,

none

performs no standardization.

scale

scales but does not center the variables. For a variable x,

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Last updated: October 29, 2020