Publication date: 11/29/2021

Use the Advanced Controls options to adjust various aspects of the model fitting process. A number of controls relate to the grid for the tuning parameter.

#### Tuning Parameter

The solution paths for the Lasso and Ridge Estimation Methods depend on a single tuning parameter. The solution path for the Elastic Net depends on a tuning parameter for the penalty on the likelihood as well as the Elastic Net Alpha. The penalty on the likelihood for the Elastic Net is a weighted sum of the penalties associated with the Lasso and Ridge Estimation Methods. The Elastic Net Alpha determines the weights of these two penalties. See Statistical Details for Estimation Methods and Statistical Details for Advanced Controls.

When the tuning parameter is zero, the solution is unpenalized and maximum likelihood estimates are obtained. As the tuning parameter increases, the penalty increases.

The solution is the set of parameter estimates that minimizes the penalized negative log-likelihood function relative to the selected validation method. The current solution is designated by the solid red vertical line in the Solution Path Plots.

Note: The value of the tuning parameter increases as the Magnitude of Scaled Parameter Estimates in the Solution Path Plot decreases. Estimates close to the MLE are associated with large magnitudes and estimates that are heavily penalized are associated with small magnitudes.

It is important to be mindful of the following:

When the tuning parameter is too small, the data are typically overfit and result in models with high variance.

When the tuning parameter is too large, the data are typically underfit.

#### The Tuning Parameter Grid

To obtain a solution, the tuning parameter is increased over a fine grid.

For the Lasso, Elastic Net with Elastic Net Alpha specified, and Ridge, the value of the tuning parameter that gives the solution is the one that provides the best fit over the grid of tuning parameters.

Note: Elastic Net Alpha is set to 0.99 by default.

If you do not set a value for the Elastic Net Alpha, the value of alpha is also increased over a fine grid. For a fixed value of the tuning parameter, alpha is varied until ten consecutive values of alpha fail to improve upon the best fit as determined by the validation method. This process is repeated for the entire grid of tuning parameter values. The final values of the tuning parameter and alpha are the values that provide the best fit over the grid of tuning parameters.

The grid of tuning parameter values ranges from zero, in most cases, to the smallest value for which all of the non-intercept terms are zero. Define the smallest value of the tuning parameter for which all non-intercept terms are zero to be its upper bound. The lower bound for the tuning parameter is zero except in the following two cases where it is set to 0.0001:

If the design matrix is singular, the maximum likelihood estimates cannot be computed. The lower bound of 0.0001 allows estimates close to the MLEs to be computed.

If the selected distribution is binomial or multinomial, the lower bound of 0.0001 helps prevent separation.

Enforce effect heredity

Requires lower-order effects to enter the model before their related higher order effects. In most cases, this means that X2 is not in the model unless X is in the model. For estimation methods other than Forward Selection, however, it is possible for X2 to enter the model and X to leave the model in the same step. If the data table contains a DOE script, this option is enabled, but it is off by default.

Elastic Net Alpha

Sets the α parameter for the Elastic Net. This α parameter determines the mix of the l1 and l2 penalty tuning parameters in estimating the Elastic Net coefficients. The default value is α = 0.99, which sets the coefficient on the l1 penalty to 0.99 and the coefficient on the l2 penalty to 0.01. This option is available only when Elastic Net is selected as the Estimation Method. See Statistical Details for Estimation Methods.

Number of Grid Points

Specifies the number of grid points between the lower and upper bounds for the tuning parameter. At each grid point value, parameter estimates for that value of the tuning parameter are obtained. The default value is 150 grid points.

Minimum Penalty Fraction

Indicates the minimum value for the ratio of the lower bound of the tuning parameter to its upper bound. When the lower bound for the tuning parameter is 0, the solution provides the MLE. In cases where you do not want to include the MLE or solutions very close to it, you can set the Minimum Penalty Fraction to a nonzero value. For the Double Lasso estimation method, the specified value of this option is used only in the first stage of the fit. When there is a singularity in the design matrix, the default value is 0.0001. Otherwise, the default value is 0.

Grid Scale

Provides options for choosing the distribution of the grid scale. You can choose between a linear, square root, or log scale. Grid points equal in number to the specified Number of Grid Points are distributed according to the selected scale between the lower and upper bounds of the tuning parameter. The default grid scale is square root. See Statistical Details for Advanced Controls.

First Stage Solution

Provides options for choosing the solution in the first stage of the Double Lasso and Two Stage Forward Selection. By default, the solution that is the best fit according to the specified Validation Method is selected and is the solution initially shown (Best Fit). You can choose to initially display models with larger or smaller l1 norm values that lie in the green or yellow zones. For example, if you choose Smallest in Yellow Zone, the initially displayed solution is the model in the yellow zone that has the smallest l1 norm. See Comparable Model Zones.

Max Number of Effects

Specifies the maximum number of effects to consider in models for the Best Subset estimation method. You can use this to limit the number of computations needed to fit the model. The default value is 10.

Initial Displayed Solution

Provides options for choosing the solution that is initially displayed as the current model in the Solution Path report. The current model is identified by a solid vertical line. See Current Model Indicator. The best fit solution is identified by a dotted vertical line. By default, the displayed solution is the one that is considered the best fit according to the specified Validation Method.

You can choose to initially display models with larger or smaller l1 norm values that still lie in the green or yellow zones. For example, if you choose Smallest in Yellow Zone, the initially displayed solution is the model in the yellow zone that has the smallest l1 norm. See Comparable Model Zones.