The available estimation methods can be grouped into techniques with no selection and no penalty, step-based model selection techniques, and penalized regression techniques.

The Maximum Likelihood, Standard Least Squares, and Logistic Regression methods fit the entire model that is specified in the Fit Model launch window. No variable selection is performed. These models can serve as baselines for comparison to other methods.

Note: Only one of Maximum Likelihood, Standard Least Squares, and Logistic Regression is available for a given report. The name of this estimation method depends on the Distribution specified in the Fit Model launch window.

The Backward Elimination, Forward Selection, Pruned Forward Selection, Best Subset, and Two Stage Forward Selection methods are based on variables entering or leaving the model at each step. However, they do not impose a penalty on the regression coefficients.

The Dantzig Selector, Lasso, Elastic Net, Ridge, and Double Lasso methods are penalized regression techniques. They shrink the size of regression coefficients and reduce the variance of the estimates, in order to improve predictive ability of the model.

Note: When your data are highly collinear, the adaptive versions of Lasso and Elastic Net might not provide good solutions. This is because the adaptive versions presume that the MLE provides a good estimate. The Adaptive option is not recommended in such cases.

Two types of penalties are used in these techniques:

• the l1 penalty, which penalizes the sum of the absolute values of the regression coefficients

• the l2 penalty, which penalizes the sum of the squares of the regression coefficients

The default Estimation Method for observational data is the Lasso. If the data table contains a DOE script and no singularities, the default Estimation Method is Forward Selection with the Effect Heredity option enabled. If the data table contains a DOE script and a singularity in the design matrix, the default Estimation Method is Two-Stage Forward Selection with the Effect Heredity option enabled.

The following methods are available for model fitting:

Estimation Methods with No Selection and No Penalty

Maximum Likelihood

Computes maximum likelihood estimates (MLEs) for model parameters. No penalty is imposed. Maximum Likelihood is the only estimation method available for Quantile Regression. If you specified a Validation column in the Fit Model launch window, the maximum likelihood model is fit to the Training set. A maximum likelihood model report appears by default, as long as the following conditions are met:

– There are no linear dependencies among the predictors.

– There are more observations than predictors.

– There are no more than 250 predictors.

The Maximum Likelihood option gives you a way to construct classical models for the response distributions supported by the Generalized Regression personality. In addition, a model based on maximum likelihood can serve as a baseline for model comparison.

When the specified Distribution is Normal or Binomial, the Maximum Likelihood method is called Standard Least Squares or Logistic Regression, respectively.

Standard Least Squares

When the Normal distribution is specified, the Maximum Likelihood estimation method is replaced with the Standard Least Squares estimation method. The default report is a Standard Least Squares report that gives the usual standard least squares results.

Logistic Regression

When the Binomial distribution is specified, the Maximum Likelihood estimation method is replaced with the Logistic Regression estimation method. The default report is a Logistic Regression report. The logistic results are identical to maximum likelihood results.

Step-Based Estimation Methods

Note: Step-based estimation methods are not available when the specified Distribution is Multinomial.

Backward Elimination

Computes parameter estimates using backward elimination regression. The model chosen provides the best solution relative to the selected Validation Method. Backward elimination starts by including all parameters in the model and removing one effect at each step until reaching the intercept-only model. At each step, the Wald tests for each parameter is used to determine which parameter is removed.

Caution: The horizontal axis of the Solution Path for Backward Elimination is the reverse of the same axis in other estimation methods. Therefore, as you move left to right in the Solution Path for the Backward Elimination estimation method, terms are being removed from the model, rather than added.

Forward Selection

Computes parameter estimates using forward stepwise regression. At each step, the effect with the most significant score test is added to the model. The model chosen is the one that provides the best solution relative to the selected Validation Method.

When there are interactions and the Effect Heredity option is enabled, compound effects are handled in the following manner. If the effect with the most significant score test at a given step is one that would violate effect heredity, then a compound effect is created. The compound effect contains the effect with the most significant score test as well as any other inactive effects that are needed to satisfy effect heredity. If the compound effect has the most significant score test, then all of the effects in the compound effect are added to the model.

Pruned Forward Selection

Computes parameter estimates using a mixture of forward and backward steps. The algorithm starts with an intercept-only model. At the first step, the effect with the most significant score test is added to the model. After the first step, the algorithm considers the following three possibilities at each step:

1. From the effects not in the model, add the effect that has the most significant score test.

2. From the effects in the model, remove the effect that has the least significant Wald test.

3. Do both of the above actions in a single step.

To choose the action taken at each step, the algorithm uses the specified Validation Method. For example, if the Validation Method is BIC, the algorithm chooses the action that results in the smallest BIC value. When there are interactions and the Effect Heredity option is enabled, compound effects are considered for adding effects, but they are not considered for removing effects.

When the model becomes saturated, the algorithm attempts a backward step to check if that improves the model. The maximum number of steps in the algorithm is 5 times the number of parameters. The model chosen is the one that provides the best solution relative to the selected Validation Method.

Pruned Forward Selection is an alternative to the Mixed Step option in the Stepwise Regression personality. However, it does not use the p-value to determine which variables enter or leave the model.

Tip: The Early Stopping option is not recommended for the Pruned Forward Selection Estimation Method.

Best Subset

Computes parameter estimates by increasing the number of active effects in the model at each step. In each step, the model is chosen among all possible models with a number of effects given by the step number. The values on the horizontal axes of the Solution Path plots represent the number of active effects in the model. Step 0 corresponds to the intercept-only model. Step 1 corresponds to the best model of the ones that contain only one active effect. The steps continue up to the value of Max Number of Effects specified in the Advanced Controls in the Model Launch report. See Advanced Controls.

Tip: The Best Subset Estimation Method is computationally intensive. It is not recommended for large problems.

Two Stage Forward Selection

(Available only when there are second- or higher-order effects in the model.) Computes parameter estimates in two stages. In the first stage, a forward stepwise regression model is run on the main effects to determine which to retain in the model. In the second stage, a forward stepwise regression model is run on all of the higher-order effects that are composed entirely of the main effects chosen in the first stage. This method assumes strong effect heredity.

Terms that are not retained from the first stage still appear in the Parameter Estimates reports as zeroed terms. However, they are ignored in the fitting of the second stage model. Terms that are selected in the first stage are not forced into the second stage; they are available for selection in the second stage.

Penalized Estimation Methods

Dantzig Selector

(Available only when the specified Distribution is Normal and the No Intercept option is not selected.) Computes parameter estimates by applying an l1 penalty using a linear programming approach. See Candes and Tao (2007). The Dantzig Selector is useful for analyzing the results of designed experiments. For orthogonal problems, the Dantzig Selector and Lasso give identical results. See Dantzig Selector.

Lasso

Computes parameter estimates by applying an l1 penalty. Due to the l1 penalty, some coefficients can be estimated as zero. Thus, variable selection is performed as part of the fitting procedure. In the ordinary Lasso, all coefficients are equally penalized.

Adaptive Lasso

Computes parameter estimates by penalizing a weighted sum of the absolute values of the regression coefficients. The weights in the l1 penalty are determined by the data in such as way as to guarantee the oracle property (Zou 2006). This option uses the MLEs to weight the l1 penalty. MLEs cannot be computed when the number of predictors exceeds the number of observations or when there are strict linear dependencies among the predictors. If MLEs for the regression parameters cannot be computed, a generalized inverse solution or a ridge solution is used for the l1 penalty weights. See Adaptive Methods.

The Lasso and the adaptive Lasso options generally choose parsimonious models when predictors are highly correlated. These techniques tend to select only one of a group of correlated predictors. High-dimensional data tend to have highly correlated predictors. For this type of data, the Elastic Net might be a better choice than the Lasso. See Lasso Regression.

Elastic Net

Computes parameter estimates by applying both an l1 penalty and an l2 penalty. The l1 penalty ensures that variable selection is performed. The l2 penalty improves predictive ability by shrinking the coefficients as ridge does.

Adaptive Elastic Net

Computes parameter estimates using an adaptive l1 penalty as well as an l2 penalty. This option uses the MLEs to weight the l1 penalty. MLEs cannot be computed when the number of predictors exceeds the number of observations or when there are strict linear dependencies among the predictors. If MLEs for the regression parameters cannot be computed, a generalized inverse solution or a ridge solution is used for the l1 penalty weights. You can set a value for the Elastic Net Alpha in the Advanced Controls panel. See Adaptive Methods.

The Elastic Net tends to provide better prediction accuracy than the Lasso when predictors are highly correlated. (In fact, both Ridge and the Lasso are special cases of the Elastic Net.) In terms of predictive ability, the adaptive Elastic Net often outperforms both the Elastic Net and the adaptive Lasso. The Elastic Net has the ability to select groups of correlated predictors and to assign appropriate parameter estimates to the predictors involved. See Elastic Net.

Note: If you select an Elastic Net fit and set the Elastic Net Alpha to missing, the algorithm computes the Lasso, Elastic Net, and Ridge fits, in that order. If a fit is time intensive, a progress bar appears. When you click Accept Current Estimates, the calculation stops and the reported parameter estimates correspond to the best model fit at that point. The progress bar indicates when the algorithm is fitting Lasso, Elastic Net, and Ridge. You can use this information to decide when to click Accept Current Estimates.

Ridge

Computes parameter estimates using ridge regression. Ridge regression is a biased regression technique that applies an l2 penalty and does not result in zero parameter estimates. It is useful when you want to retain all predictors in your model. See Ridge Regression.

Double Lasso

Computes parameter estimates in two stages. In the first stage, a Lasso model is fit to determine the terms to be used in the second stage. In the second stage, a Lasso model is fit using the terms from the first stage. The Solution Path results and the parameter estimate reports that appear are for the second-stage fit. If none of the variables enters the model in the first stage, there is no second stage, and the results of the first stage appear in the report.

The Double Lasso is especially useful when the number of observations is less than the number of predictors. By breaking the variable selection and shrinkage operations into two stages, the Lasso in the second stage is less likely to overly penalize the terms that should be included in the model. The double lasso is similar to the relaxed lasso. The relaxed lasso is described in Hastie et al. (2009, p. 91).

Adaptive Double Lasso

Computes parameter estimates in two stages. In the first stage, an adaptive Lasso model is fit to determine the terms to be used in the second stage. In the second stage, an adaptive Lasso model is fit using the terms from the first stage. The second stage considers only the terms that are included in the first stage model and uses weights based on the parameter estimates in the first stage. You can choose the method of calculating the weights using the Adaptive Penalty Weights option in the Advanced Controls. See Advanced Control Options. The results that are shown are for the second-stage fit. If none of the variables enters the model in the first stage, there is no second stage, and the results of the first stage appear in the report. See Adaptive Methods.