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Regressions offer a different approach to prediction compared to decision trees. Regressions, as parametric models, assume a specific association structure between inputs and target. By contrast, trees, as predictive algorithms, do not assume any association structure; they simply seek to isolate concentrations of cases with likevalued target measurements.
The regression approach to the model essentials in SAS Enterprise Miner is outlined over the following pages. Cases are scored using a simple mathematical prediction formula. One of several heuristic sequential selection techniques is used to pick from a collection of possible inputs and creates a series of models with increasing complexity. Fit statistics calculated from validation data select the optimal sequence model.
Regressions predict cases using a mathematical equation involving values of the input variables.
In standard linear regression, a prediction estimate for the target variable is formed from a simple linear combination of the inputs. The intercept centers the range of predictions, and the remaining parameter estimates determine the trend strength (or slope) between each input and the target. The simple structure of the model forces changes in predicted values to occur in only a single direction (a vector in the space of inputs with elements equal to the parameter estimates).
Intercept and parameter estimates are chosen to minimize the squared error between the predicted and observed target values (least squares estimation). The prediction estimates can be viewed as a linear approximation to the expected (average) value of a target conditioned on observed input values.
Linear regressions are usually deployed for targets with an interval measurement scale.
Logistic regressions are closely related to linear regressions. In logistic regression, the expected value of the target is transformed by a link function to restrict its value to the unit interval. In this way, model predictions can be viewed as primary outcome probabilities. A linear combination of the inputs generates a logit score, the log of the odds of primary outcome, in contrast to the linear regression’s direct prediction of the target.
The presence of the link function complicates parameter estimation. Least squares estimation is abandoned in favor of maximum likelihood estimation. The likelihood function is the joint probability density of the data treated as a function of the parameters. The maximum likelihood estimates the values of the parameters that maximize the probability of obtaining the training sample.
If your interest is ranking predictions, linear and logistic regressions will yield virtually identical results.
For binary prediction, any monotonic function that maps the unit interval to the real number line can be considered as a link. The logit link function is one of the most common. Its popularity is due, in part, to the interpretability of the model.
There are two equivalent ways to interpret a logistic regression model. Both relate changes in input measurements to changes in odds of primary outcome.
An odds ratio expresses the increase in primary outcome odds associated with a unit change in an input. It is obtained by exponentiating the parameter estimate of the input of interest.
A doubling amount gives the amount of change required for doubling the primary outcome odds. It is equal to log(2) ≈ 0.69 divided by the parameter estimate of the input of interest.
If the predicted logit scores remain in the range 2 to +2, linear and logistic regression models of binary targets are virtually indistinguishable. Balanced sampling (Chapter 7) often ensures this. Thus, the prevalence of balanced sampling in predictive modeling might, in fact, be a vestigial practice from a time when maximum likelihood estimation was computationally extravagant.
To demonstrate the properties of a logistic regression model, consider the twocolor prediction problem introduced in Chapter 3. As before, the goal is to predict the target color, based on location in the unit square.
The predictions can be decisions, rankings, or estimates. The logit equation produces a ranking or logit score. To get a decision, you need a threshold. The easiest way to get a meaningful threshold is to convert the prediction ranking to a prediction estimate. You can obtain a prediction estimate using a straightforward transformation of the logit score, the logistic function. The logistic function is simply the inverse of the logit function.
Parameter estimates are obtained by maximum likelihood estimation. These estimates can be used in the logit and logistic equations to obtain predictions. The plot on the right shows the prediction estimates from the logistic equation. One of the attractions of a standard logistic regression model is the simplicity of its predictions. The contours are simple straight lines. (In higher dimensions, they would be hyperplanes.) This enables a straightforward interpretation of the model using the odds ratios and doubling amounts shown at the bottom left. Unfortunately, simplicity can also lead to prediction bias (as scrutiny of the prediction contours suggests).
The last section of this chapter shows a way to extend the capabilities of logistic regression to address this possible bias.
To score a new case, the values of the inputs are plugged into the logit or logistic equation.
This action creates a logit score or prediction estimate. Typically, if the prediction estimate is greater than 0.5 (or equivalently, the logit score is positive), cases are usually classified to the primary outcome. (This assumes an equal misclassification cost. See Chapter 7.)
While the prediction formula would seem to be the final word in scoring a new case with a regression model, there are actually several additional issues that must be addressed.
What should be done when one of the input values used in the prediction formula is missing? You might be tempted to simply treat the missing value as zero and skip the term involving the missing value. While this approach can generate a prediction, this prediction is usually biased beyond reason.
How do you score cases with unusual values? Regression models make their best predictions for cases near the centers of the input distributions. If an input can have (on rare occasion) extreme or outlying values, the regression should respond appropriately.
What value should be used in the prediction formula when the input is not a number? Categorical inputs are common in predictive modeling. They did not present a problem for the rulebased predictions of decision trees, but regression predictions come from algebraic formulas that require numeric inputs. (You cannot multiply marital status by a number.) A method to include nonnumeric data in regression is needed.
What happens when the relationship between the inputs and the target (or rather logit of the target) is not a straight line? It is preferable to be able to build regression models in the presence of nonlinear (and even nonadditive) input target associations.
The above questions are presented as scoringrelated issues. They are also problems for model construction. (Maximum likelihood estimation requires iteratively scoring the training data.) The first of these, handling missing values, is dealt with immediately. The remaining issues are addressed, in turn, at the end of this chapter.
The default method for treating missing values in most regression tools in SAS Enterprise Miner is completecase analysis. In completecase analysis, only those cases without any missing values are used in the analysis.
Even a smattering of missing values can cause an enormous loss of data in high dimensions. For instance, suppose that each of the k input variables is missing at random with probability a. In this situation, the expected proportion of complete cases is as follows:
_{}
Therefore, a 1% probability of missing (a=.01) for 100 inputs leaves only 37% of the data for analysis, 200 leaves 13%, and 400 leaves 2%. If the missingness were increased to 5% (a=.05), then <1% of the data would be available with 100 inputs.
The purpose of predictive modeling is scoring new cases. How would a model built on the complete cases score a new case if it had a missing value? To decline to score new incomplete cases would be practical only if there were a very small number of missing values.
Missing values arise for a variety of
reasons. For example, the time since last donation to a card campaign is
meaningless if someone did not donate to a card campaign. In the
Missing value replacement strategies fall into one of two categories.
Synthetic distribution methods use a onesizefitsall approach to handle missing values. Any case with a missing input measurement has the missing value replaced with a fixed number. The net effect is to modify an input’s distribution to include a point mass at the selected fixed number. The location of the point mass in synthetic distribution methods is not arbitrary. Ideally, it should be chosen to have minimal impact on the magnitude of an input’s association with the target. With many modeling methods, this can be achieved by locating the point mass at the input’s mean value.
Estimation methods eschew the onesizefitsall approach and provide tailored imputations for each case with missing values. This is done by viewing the missing value problem as a prediction problem. That is, you can train a model to predict an input’s value from other inputs. Then, when an input’s value is unknown, you can use this model to predict or estimate the unknown missing value. This approach is best suited for missing values that result from a lack of knowledge, that is, nomatch or nondisclosure, but it is not appropriate for notapplicable missing values. (An exercise at the end of this chapter demonstrates using decision trees to estimate missing values.)
Because predicted response might be different for cases with a missing input value, a binary imputation indicator variable is often added to the training data. Adding this variable enables a model to adjust its predictions in the situation where missingness itself is correlated with the target.
The demonstrations in this chapter assume that you completed the demonstrations in Section 4.1.
As discussed above, regression requires that a case have a complete set of input values for both training and scoring.
Rightclick the
Explore  AAEM.
There are several inputs with a noticeable frequency of missing values, for example, Age and Income Group
There are several ways to proceed:
Do nothing. If there are very few cases with missing values, this is a viable option. The difficulty with this approach comes when the model must predict a new case that contains a missing value. Omitting the missing term from the parametric equation usually produces an extremely biased prediction.
Impute a synthetic value for the missing value. For example, if an interval input contains a missing value, replace the missing value with the mean of the nonmissing values for the input. This eliminates the incomplete case problem but modifies the input’s distribution. This can bias the model predictions.
Making the missing value imputation process part of the modeling process allays the modified distribution concern. Any modifications made to the training data are also made to the validation data and the remainder of the modeling population. A model trained with the modified training data will not be biased if the same modifications are made to any other data set that the model might encounter (and the data has a similar pattern of missing values).
Create a missing indicator for each input in the data set. Cases often contain missing values for a reason. If the reason for the missing value is in some way related to the target variable, useful predictive information is lost.
The missing indicator is 1 when the corresponding input is missing and 0 otherwise. Each missing indicator becomes an input to the model. This enables modeling of the association between the target and a missing value on an input.
To address
missing values in the
Select the Modify tab.
Drag an Impute tool into the diagram workspace.
Connect the Data Partition node to the Impute node.
Select the Impute node and examine the Properties panel.
The defaults of the Impute node are as follows:
For interval inputs, replace any missing values with the mean of the nonmissing value
For categorical inputs, replace any missing values with the most frequent category.
Select Indicator Variable Unique.
Select Indicator Variable Role Input.
With these settings, each input with missing values generates a new input. The new input named IMP_original_input_name will have missing values replaced by a synthetic value and nonmissing values copied from the original input. In addition, new inputs named M_original_input_name will be added to the training data to indicate the synthetic data values.
Run the Impute node and review the Results window. Three inputs had missing values.
With all missing values imputed, the entire training data set is available for building the logistic regression model. In addition, a method is in place for scoring new cases with missing values. (See Chapter 8.)
There are several tools in SAS Enterprise Miner to fit regression or regressionlike models. By far, the most commonly used (and, arguably, the most useful) is the simply named Regression tool.
Select the Model tab.
Drag a Regression tool into the diagram workspace.
Connect the Impute node to the Regression node.
The Regression node can create several types of regression models, including linear and logistic. The type of default regression type is determined by the target’s measurement level.
Run the Regression node and view the results. The Results  Regression window opens.
Maximize the Output window.
Lines 513 of the Output window summarize the roles of variables used (or not) by the Regression node.
Variable Summary
ROLE LEVEL COUNT
INPUT BINARY 5
INPUT INTERVAL 20
INPUT NOMINAL 3
REJECTED INTERVAL 1
TARGET BINARY 1
The fit model has 28 inputs that predict a binary target.
Lines 4557 give more information about the model, including the training data set name, target variable name, number of target categories, and most importantly, the number of model parameters.
Model Information
Training Data Set EMWS2.IMPT_TRAIN.VIEW
DMDB Catalog
Target Variable TargetB (Target Gift Flag)
Target Measurement Level Ordinal
Number of Target Categories 2
Error MBernoulli
Link Function Logit
Number of Model Parameters 86
Number of Observations 4843
Based on the introductory material on logistic regression that is presented above, you might expect to have a number of model parameters equal to the number of input variables. This ignores the fact that a single nominal input (for example, DemCluster) can generate scores of model parameters. You can see the number of parameters (or degrees of freedom) that each input contributes to the model, as well as each input’s statistical significance, by viewing lines 117146 of the Output window.
The Type 3 Analysis tests the statistical significance of adding the indicated input to a model that already contains other listed inputs. Roughly speaking, a value near 0 in the Pr > ChiSquare column indicates a significant input; a value near 1 indicates an extraneous input.
Type 3 Analysis of Effects
Wald
Effect DF ChiSquare Pr > ChiSq
DemCluster 53 44.0996 0.8030
DemGender 2 0.0032 0.9984
DemHomeOwner 1 0.0213 0.8840
DemMedHomeValue 1 9.3921 0.0022
DemPctVeterans 1 0.1334 0.7150
GiftAvg36 1 4.5339 0.0332
GiftAvgAll 1 0.8113 0.3677
GiftAvgLast 1 0.0003 0.9874
GiftCnt36 1 0.6848 0.4079
GiftCntAll 1 0.0144 0.9044
GiftCntCard36 1 2.4447 0.1179
GiftCntCardAll 1 0.0073 0.9320
GiftTimeFirst 1 3.8204 0.0506
GiftTimeLast 1 15.8614 <.0001
IMP_DemAge 1 4.7087 0.0300
IMP_DemIncomeGroup 1 11.1804 0.0008
IMP_GiftAvgCard36 1 0.0758 0.7830
M_DemAge 1 0.3366 0.5618
M_DemIncomeGroup 1 0.0480 0.8266
M_GiftAvgCard36 1 5.4579 0.0195
PromCnt12 1 3.4896 0.0618
PromCnt36 1 4.8168 0.0282
PromCntAll 1 0.2440 0.6214
PromCntCard12 1 0.2867 0.5923
PromCntCard36 1 1.1416 0.2853
PromCntCardAll 1 1.6851 0.1942
StatusCat96NK 5 6.5422 0.2570
StatusCatStarAll 1 2.7676 0.0962
The statistical significance measures range from <0.0001 (highly significant) to 0.9997 (highly dubious). Results such as this suggest that certain inputs can be dropped without affecting the predictive prowess of the model.
Restore the Output window to its original size and maximize the Fit Statistics window.
If the decision predictions are of interest, model fit can be judged by misclassification. If estimate predictions are the focus, model fit can be assessed by average square error. There appears to be some discrepancy between the values of these two statistics on the train and validation data. This indicates possible overfit of the model. It can be mitigated by employing an input selection procedure.
The second task that all predictive models should perform is input selection. One way to find the optimal set of inputs for a regression is simply to try every combination. Unfortunately, the number of models to consider using this approach increases exponentially in the number of available inputs. Such an exhaustive search is impractical for realistic prediction problems.
An alternative to the exhaustive search is to restrict the search to a sequence of improving models. While this might not find the single best model, it is commonly used to find models with good predictive performance. The Regression node in SAS Enterprise Miner provides three sequential selection methods.
Forward selection creates a sequence of models of increasing complexity. The sequence starts with the baseline model, a model predicting the overall average target value for all cases. The algorithm searches the set of oneinput models and selects the model that most improves upon the baseline model. It then searches the set of twoinput models that contain the input selected in the previous step and selects the model showing the most significant improvement. By adding a new input to those selected in the previous step, a nested sequence of increasingly complex models is generated. The sequence terminates when no significant improvement can be made.
Improvement is quantified by the usual statistical measure of significance, the pvalue. Adding terms in this nested fashion always increases a model’s overall fit statistic. By calculating the change in the fit statistic and assuming that the change conforms to a chisquared distribution, a significance probability, or pvalue, can be calculated. A large fit statistic change (corresponding to a large chisquared value) is unlikely due to chance. Therefore, a small pvalue indicates a significant improvement. When no pvalue is below a predetermined entry cutoff, the forward selection procedure terminates.
In contrast to forward selection,
backward selection creates a sequence of models of decreasing complexity. The sequence starts with a saturated model, which
is a model that contains all available inputs and, therefore, has the highest
possible fit statistic. Inputs are sequentially removed from the model. At each
step, the input chosen for removal least reduces the overall model fit
statistic. This is equivalent to removing the input with the highest pvalue. The sequence terminates when
all remaining inputs have a
pvalue in excess of the
predetermined stay cutoff.
Stepwise selection combines elements from both the forward and backward selection procedures. The method begins in the same way as the forward procedure, sequentially adding inputs with the smallest p‑value below the entry cutoff. After each input is added, however, the algorithm reevaluates the statistical significance of all included inputs. If the pvalue of any of the included input exceeds a stay cutoff, the input is removed from the model and reentered into the pool of inputs that are available for inclusion in a subsequent step. The process terminates when all inputs available for inclusion in the model have pvalues in excess of the entry cutoff and all inputs already included in the model have pvalues below the stay cutoff.
Implementing a sequential selection method in the regression node requires a minor change to the Regression node settings.
Close the Regression results window.
Select Selection Model Stepwise on the Regression node property sheet.
The Regression node is now configured to use stepwise selection to choose inputs for the model.
Run the Regression node and view the results.
Select the Output tab and scroll to lines 95121.
The stepwise procedure starts with Step 0, an interceptonly regression model. The value of the intercept parameter is chosen so that the model predicts the overall target mean for every case. The parameter estimate and the training data target measurements are combined in an objective function. The objective function is determined by the model form and the error distribution of the target. The value of the objective function for the interceptonly model is compared to the values obtained in subsequent steps for more complex models. A large decrease in the objective function for the more complex model indicates a significantly better model.
Optimization Results
Iterations 0 Function Calls 3
Hessian Calls 1 Active Constraints 0
Objective Function 3356.9116922 Max Abs Gradient Element 7.225331E12
Ridge 0 Actual Over Pred Change 0
Convergence criterion (ABSGCONV=0.00001) satisfied.
Likelihood Ratio Test
for Global Null Hypothesis:
2 Log Likelihood Likelihood
Intercept Intercept & Ratio
Only Covariates ChiSquare DF Pr > ChiSq
6713.823 6713.823 0.0000 0 .
Analysis of Maximum Likelihood Estimates
Standard Wald Standardized
Parameter DF Estimate Error ChiSquare Pr > ChiSq Estimate Exp(Est)
Intercept 1 0.00041 0.0287 0.00 0.9885 1.000
Step 1 adds one input to the interceptonly model. The input and corresponding parameter are chosen to produce the largest decrease in the objective function. To estimate the values of the model parameters, the modeling algorithm makes an initial guess for their values. The initial guess is combined with the training data measurements in the objective function. Based on statistical theory, the objective function is assumed to take its minimum value at the correct estimate for the parameters. The algorithm decides whether changing the values of the initial parameter estimates can decrease the value of objective function. If so, the parameter estimates are changed to decrease the value of the objective function and the process iterates. The algorithm continues iterating until changes in the parameter estimates fail to substantially decrease the value of the objective function.
The Step 1 optimization is summarized in the Output window, lines 121164.
Step 1: Effect GiftCntCard36 entered.
The DMREG Procedure
NewtonRaphson Ridge Optimization
Without Parameter Scaling
Parameter Estimates 2
Optimization Start
Active Constraints 0 Objective Function 3356.9116922
Max Abs Gradient Element
Ratio
Between
Actual
Objective Max Abs and
Function Active Objective Function Gradient Predicted
Iter Restarts Calls Constraints Function Change Element Ridge Change
1 0 2 0 3308 48.5172 2.3587 0 1.014
2 0 3 0 3308 0.0416 0.00805 0 1.002
3 0 4 0 3308 2.979E 6.214E8 0 1.000
Optimization Results
Iterations 3 Function Calls 6
Hessian Calls 4 Active Constraints 0
Objective Function 3308.3529331 Max Abs Gradient Element 6.2144229E8
Ridge 0 Actual Over Pred Change 0.9999995175
Convergence criterion (GCONV=1E6) satisfied.
The output next compares the model fit in step 1 with the model fit in step 0. The objective functions of both models are multiplied by 2 and differenced. The difference is assumed to have a chisquare distribution with 1 degree of freedom. The hypothesis that the two models are identical is tested. A large value for the chisquare statistic makes this hypothesis unlikely.
The hypothesis test is summarized in lines 167173.
Likelihood Ratio Test for Global Null Hypothesis:
2 Log Likelihood Likelihood
Intercept Intercept & Ratio
Only Covariates ChiSquare DF Pr > ChiSq
6713.823 6616.706 97.1175 1 <.0001
Next, the output summarizes an analysis of the statistical significance of individual model effects (lines 176181). For the one input model, this is similar to the global significance test above.
Type 3 Analysis of Effects
Wald
Effect DF ChiSquare Pr > ChiSq
GiftCntCard36 1 93.0507 <.0001
Finally, an analysis of individual parameter estimates is made. The standardized estimates and the odds ratios merit special attention.
Analysis of Maximum Likelihood Estimates
Standard Wald Standardized
Parameter DF Estimate Error ChiSquare Pr > ChiSq Estimate Exp(Est)
Intercept 1 0.3360 0.0450 55.78 <.0001 0.715
GiftCntCard36 1 0.1841 0.0191 93.05 <.0001 0.1596 1.202
Odds Ratio Estimates
Point
Effect Estimate
GiftCntCard36 1.202
The standardized estimates present the effect of the input on the logodds of donation. The values are standardized to be independent of the input’s unit of measure. This provides a means of ranking the importance of inputs in the model.
The odds ratio estimates indicate by what factor the odds of donation increase for each unit change in the associated input. Combined with knowledge of the range of the input, this provides an excellent way to judge the practical (as opposed to the statistical) importance of an input in the model.
The stepwise selection process continues for seven steps. After the eighth step, neither adding nor removing inputs from the model significantly changes the model fit statistic. At this point the Output window provides a summary of the stepwise procedure. The summary shows the step in which each input was added and the statistical significance of each input in the final eightinput model (lines 810825).
NOTE: No (additional) effects met the 0.05 significance level for entry into the model.
Summary of Stepwise Selection
Effect Number Score Wald
Step Entered DF In ChiSquare ChiSquare Pr > ChiSq
1 GiftCntCard36 1 1 95.6966 <.0001
2 GiftTimeLast 1 2 29.9410 <.0001
3 DemMedHomeValue 1 3 25.5086 <.0001
4 GiftTimeFirst 1 4 15.1942 <.0001
5 GiftAvg36 1 5 13.2369 0.0003
6 IMP_DemIncomeGroup 1 6 13.3458 0.0003
7 M_GiftAvgCard36 1 7 10.0412 0.0015
8 IMP_DemAge 1 8 4.2718 0.0387
The default selection criterion selects the model from step 8 as the model with optimal complexity (lines 828830). As the next section shows, this might not be the optimal based on the fit statistic that is appropriate for your analysis objective.
The selected model, based on the CHOOSE=NONE criterion, is the model trained in Step 8. It consists of the following effects:
Intercept DemMedHomeValue GiftAvg36 GiftCntCard36 GiftTimeFirst GiftTimeLast IMP_DemAge IMP_DemIncomeGroup M_GiftAvgCard36
For convenience, the output from step 9 is repeated. An excerpt from the analysis of individual parameter estimates is shown below (lines 857870).
Analysis of Maximum Likelihood Estimates
Standard Wald Standardized
Parameter DF Estimate Error ChiSquare Pr > ChiSq Estimate Exp(Est)
Intercept 1 0.3363 0.2223 2.29 0.1304 0.714
DemMedHomeValue 1 1.349E6 3.177E7 18.03 <.0001 0.0737 1.000
GiftAvg36 1 0.0131 0.00344 14.59 0.0001 0.0712 0.987
GiftCntCard36 1 0.1049 0.0242 18.80 <.0001 0.0910 1.111
GiftTimeFirst 1 0.00305 0.000807 14.28 0.0002 0.0631 1.003
GiftTimeLast 1 0.0376 0.00747 25.32 <.0001 0.0848 0.963
IMP_DemAge 1 0.00440 0.00213 4.27 0.0389 0.0347 1.004
IMP_DemIncomeGroup 1 0.0758 0.0192 15.55 <.0001 0.0680 1.079
M_GiftAvgCard36 0 1 0.1449 0.0462 9.83 0.0017 1.156
The parameter with the largest standardized estimate is GiftCntCard36
The odds ratio estimates show that a unit change in M_GiftAvgCard36 produces the largest change in the donation odds.
Odds Ratio Estimates
Point
Effect Estimate
DemMedHomeValue 1.000
GiftAvg36 0.987
GiftCntCard36 1.111
GiftTimeFirst 1.003
GiftTimeLast 0.963
IMP_DemAge 1.004
IMP_DemIncomeGroup 1.079
M_GiftAvgCard36 0 vs 1 1.336
Restore the Output window and maximize the Fit Statistics window.
Considering the analysis from an estimate prediction perspective, the simpler stepwise selection model, with a lower validation average squared error, is an improvement over the full model (from the previous section). From a decision prediction perspective, however, the larger full model, with a slightly lower misclassification, is preferred.
This somewhat ambiguous result comes from a lack of complexity optimization, by default, in the Regression node. This oversight is handled in the next demonstration.
Regression complexity is optimized by choosing the optimal model in the sequential selection sequence.
The process involves two steps. First, fit statistics are calculated for the models generated in each step of the selection process using both the training and validation data sets.
Then, as with the decision tree in Chapter 4, the simplest model (that is, the one with the fewest inputs) with the optimal fit statistic is selected.
In the same manner as the decision tree, you can tune a regression model to give optimal performance on the validation data. The basic idea involves calculating a fit statistic for each step in the input selection procedure and selecting the step (and corresponding model) with the optimal fit statistic value. To avoid bias, of course, the fit statistic should be calculated on the validation data set.
Select View Model Iteration Plot. The Iteration Plot window opens.
The Iteration Plot window shows (by default) the average squared error from the model selected in each step of the stepwise selection process. Apparently, the smallest average squared error occurs in step 6, rather than in the final model, step 8. If your analysis objective requires estimates predictions, the model from step 6 should provide slightly less biased ones.
Select Select Chart Misclassification Rate.
The iteration plot shows that the model with the smallest misclassification rate occurs in step 4. If your analysis objective requires decision predictions, the predictions from the step 4 model will be as accurate as the predictions from the final step 8 model.
The selection process stopped at step 8 to limit the amount of time spent running the stepwise selection procedure. In step 8, no more inputs had a chisquared pvalue below 0.05. The value 0.05 is a somewhat arbitrary holdover from the days of statistical tables. With the validation data available to gauge overfitting, it is possible to eliminate this restriction and obtain a richer pool of models to consider.
Close the Results  Regression window.
Select Use Selection Default No from the Regression node Properties panel.
Type in the Entry Significance Level field.
Type in the Stay Significance Level field.
The Entry Significance value enables any input into the model. (The one chosen will have the smallest pvalue.) The Stay Significance value keeps any input in the model with a pvalue less than 0.5. This second choice is somewhat arbitrary. A smaller value can terminate the stepwise selection process earlier, while a larger value can maintain it longer. A Stay Significance of 1.0 forces stepwise to behave in the manner of a forward selection.
Run the Regression node and view the results.
Select View Model Iteration Plot. The Iteration Plot window opens.
The iteration plot shows the smallest average squared errors occurring in steps 15 or 16.
Select Select Chart Misclassification Rate.
The iteration plot shows the smallest validation misclassification rates also occurring near steps 1516.
You can configure the Regression node to select the model with the smallest fit statistic (rather than the final stepwise selection iteration). This method is how SAS Enterprise Miner optimizes complexity for regression models.
Close the Results  Regression window.
If your predictions are decisions, use the following setting:
Select Selection
Criterion Validation
Misclassification. (Equivalently, you can select
Validation Profit / Loss. The
equivalence is demonstrated in Chapter 7.)
If your predictions are estimates (or rankings), use the following setting:
Select Selection Criterion Validation Error.
The continuing demonstration assumes validation error selection criteria.
Run the Regression node and view the results.
Select View Model Iteration Plot.
The vertical blue line shows the model with the optimal validation average squared error (step 17).
Select the Output window and view lines 22562276.
Analysis of Maximum Likelihood Estimates
Standard Wald Standardized
Parameter DF Estimate Error ChiSquare Pr > ChiSq Estimate Exp(Est)
Intercept 1 0.3042 0.2892 1.11 0.2929 0.738
DemMedHomeValue 1 1.352E6 3.183E7 18.05 <.0001 0.0739 1.000
GiftAvg36 1 0.0144 0.00506 8.14 0.0043 0.0782 0.986
GiftAvgAll 1 0.00592 0.00543 1.19 0.2752 0.0326 1.006
GiftCnt36 1 0.0343 0.0305 1.27 0.2606 0.0397 1.035
GiftCntCard36 1 0.0737 0.0414 3.17 0.0751 0.0639 1.076
GiftTimeFirst 1 0.00440 0.00261 2.85 0.0914 0.0911 1.004
GiftTimeLast 1 0.0410 0.00927 19.56 <.0001 0.0925 0.960
IMP_DemAge 1 0.00452 0.00213 4.48 0.0344 0.0356 1.005
IMP_DemIncomeGroup 1 0.0784 0.0193 16.45 <.0001 0.0703 1.082
M_GiftAvgCard36 0 1 0.2776 0.0884 9.86 0.0017 1.320
PromCnt12 1 0.0224 0.0132 2.90 0.0887 0.0607 0.978
PromCnt36 1 0.0236 0.0116 4.13 0.0422 0.1005 1.024
PromCntCard36 1 0.0362 0.0215 2.83 0.0925 0.0909 0.964
PromCntCardAll 1 0.0132 0.0140 0.88 0.3471 0.0620 0.987
StatusCatStarAll 0 1 0.0763 0.0392 3.79 0.0516 0.927
While not all the pvalues are less than 0.05, the model seems to have a better validation average square error (and misclassification) than the model selected using the default Significance Level settings.
In short, there is nothing sacred about 0.05. It is not unreasonable to override the defaults of the Regression node to enable selection from a richer collection of potential models. On the other hand, most of the reduction in the fit statistics occurs during inclusion of the first 10 inputs. If you seek a parsimonious model, it is reasonable to use a smaller value for the Stay Significance parameter
Classical regression analysis makes no assumptions about the distribution of inputs. The only assumption is that the expected value of the target (or some function thereof) is a linear combination of input measurements.
Why should you worry about extreme input distributions?
There are at least two compelling reasons.
First, in most realworld applications, the relationship between expected target value and input value does not increase without bound. Rather, it typically tapers off to some horizontal asymptote. Standard regression models are unable to accommodate such a relationship.
Second, as a point expands from the overall mean of a distribution, the point has more influence, or leverage, on model fit. Models built on inputs with extreme distributions attempt to optimize fit for the most extreme points at the cost of fit for the bulk of the data, usually near the input mean. This can result in an exaggeration or an understating of an input’s association with the target, or both.
The first concern can be addressed by abandoning standard regression models for more flexible modeling methods. Abandoning standard regression models is often done at the cost of model interpretability and, more importantly, failure to address the second concern: leverage.
A simpler and, arguably, more effective approach is to transform offending inputs to less extreme forms and build models on these transformed inputs, which not only reduces the influence of extreme cases, but also creates an asymptotic association between input and target on the original input scale.
Regression models (such as clustering models) are sensitive to extreme or outlying values in the input space. Inputs with highly skewed or highly kurtotic distributions can be selected over inputs that yield better overall predictions. To avoid this problem, analysts often regularize the input distributions using a simple transformation. The benefit of this approach is improved model performance. The cost, of course, is increased difficulty in model interpretation.
The Transform Variables tool enables you to easily apply standard transformations (in addition to the specialized ones seen in Chapter 3) to a set of inputs.
Remove the connection between the Data Partition node and the Impute node.
Select the Modify tab.
Drag a Transform Variables tool into the diagram workspace.
Connect the Data Partition node to the Transform Variables node.
Connect the Transform Variables node to the Impute node.
Adjust the diagram icons for aesthetics.
The Transform Variables node is placed before the Impute node to keep the imputed values at the average (or center of mass) of the model inputs.
Select the Variables . property of the Transform Variables node.
The Variables  Trans window opens.
Select all inputs with Gift in the name.
Select Explore . . The Explore window opens.
The GiftAvg and GiftCnt inputs show some degree of skewness in their distribution. The GiftTime inputs do not. To regularize the skewed distributions, use the log transformation. For these inputs, the order of magnitude of the underlying measure will predict the target rather than the measure itself.
Close the Explore window.
Deselect the two inputs with GiftTime in their names.
Select Method Log for one of the remaining selected inputs. The selected method changes from Default to Log for the GiftAvg and GiftCnt inputs.
Select OK to close the Variables  Trans window.
Run the Transform Variables node and view the results.
Maximize the Output window and examine lines 2030.
Input Output
Input Name Role Level Output Name Level Formula
GiftAvg36 INPUT INTERVAL LOG_GiftAvg36 INTERVAL log(GiftAvg36 + 1)
GiftAvgAll INPUT INTERVAL LOG_GiftAvgAll INTERVAL log(GiftAvgAll + 1)
GiftAvgCard36 INPUT INTERVAL LOG_GiftAvgCard36 INTERVAL log(GiftAvgCard36 + 1)
GiftAvgLast INPUT INTERVAL LOG_GiftAvgLast INTERVAL log(GiftAvgLast + 1)
GiftCnt36 INPUT INTERVAL LOG_GiftCnt36 INTERVAL log(GiftCnt36 + 1)
GiftCntAll INPUT INTERVAL LOG_GiftCntAll INTERVAL log(GiftCntAll + 1)
GiftCntCard36 INPUT INTERVAL LOG_GiftCntCard36 INTERVAL log(GiftCntCard36 + 1)
GiftCntCardAll INPUT INTERVAL LOG_GiftCntCardAll INTERVAL log(GiftCntCardAll + 1)
Notice the Formula column. While a log transformation was specified, the actual transformation used was log(input + 1). This default action of the Transform Variables tool avoids problems with 0values of the underlying inputs.
Close the Transform Variables  Results window.
Run the diagram from the Regression node and view the results.
Examine lines 1653 to 1680 the Output window.
Summary of Stepwise Selection
Effect Number Score Wald
Step Entered Removed DF In ChiSquare ChiSquare Pr > ChiSq
1 LOG_GiftCntCard36 1 1
2 GiftTimeLast 1 2 29.0312 <.0001
3 DemMedHomeValue 1 3 24.8434 <.0001
4 LOG_GiftAvgAll 1 4 28.9692 <.0001
5 IMP_DemIncomeGroup 1 5 13.8240 0.0002
6 GiftTimeFirst 1 6 7.1299 0.0076
7 IMP_DemAge 1 7 4.1407 0.0419
8 LOG_GiftAvgLast 1 8 2.6302 0.1048
9 PromCntCard12 1 9 1.8469 0.1741
10 StatusCatStarAll 1 10 1.8604 0.1726
11 IMP_LOG_GiftAvgCard36 1 11 1.4821 0.2234
LOG_GiftAvgAll 1 10 0.4217 0.5161
13 M_LOG_GiftAvgCard36 1 11 2.0584 0.1514
14 LOG_GiftCnt36 1 12 1.8949 0.1687
15 DemPctVeterans 1 13 0.3578 0.5498
16 DemPctVeterans 1 12 0.3577 0.5498
The selected model, based on the CHOOSE=VERROR criterion, is the model trained in Step 15. It consists of the following effects:
Intercept DemMedHomeValue DemPctVeterans GiftTimeFirst GiftTimeLast IMP_DemAge IMP_DemIncomeGroup IMP_LOG_GiftAvgCard36 LOG_GiftAvgLast LOG_GiftCnt36 LOG_GiftCntCard36 M_LOG_GiftAvgCard36 PromCntCard12 StatusCatStarAll
The stepwise selection process took 16 steps, and the selected model came from step 15. Notice that many of the inputs selected are log transformations of the original.
Lines 1710 to 1750 show more statistics from the selected model.
Analysis of Maximum Likelihood Estimates
Standard Wald Standardized
Parameter DF Estimate Error ChiSquare Pr > ChiSq Estimate Exp(Est)
Intercept 1 0.2908 0.3490 0.69 0.4047 1.338
DemMedHomeValue 1 1.381E6 3.173E7 18.94 <.0001 0.0754 1.000
DemPctVeterans 1 0.00155 0.00260 0.36 0.5498 0.00977 1.002
GiftTimeFirst 1 0.00210 0.000958 4.82 0.0281 0.0436 1.002
GiftTimeLast 1 0.0369 0.00810 20.70 <.0001 0.0831 0.964
IMP_DemAge 1 0.00419 0.00214 3.85 0.0498 0.0330 1.004
IMP_DemIncomeGroup 1 0.0821 0.0193 18.03 <.0001 0.0737 1.086
IMP_LOG_GiftAvgCard36 1 0.1880 0.0985 3.65 0.0562 0.0479 0.829
LOG_GiftAvgLast 1 0.1101 0.0828 1.77 0.1837 0.0334 0.896
LOG_GiftCnt36 1 0.1579 0.1165 1.84 0.1752 0.0417 1.171
LOG_GiftCntCard36 1 0.1983 0.1269 2.44 0.1183 0.0610 1.219
M_LOG_GiftAvgCard36 0 1 0.1111 0.0657 2.86 0.0910 1.117
PromCntCard12 1 0.0418 0.0259 2.60 0.1069 0.0305 0.959
StatusCatStarAll 0 1 0.0529 0.0388 1.86 0.1729 0.948
Odds Ratio Estimates
Point
Effect Estimate
DemMedHomeValue 1.000
DemPctVeterans 1.002
GiftTimeFirst 1.002
GiftTimeLast 0.964
IMP_DemAge 1.004
IMP_DemIncomeGroup 1.086
IMP_LOG_GiftAvgCard36 0.829
LOG_GiftAvgLast 0.896
LOG_GiftCnt36 1.171
LOG_GiftCntCard36 1.219
M_LOG_GiftAvgCard36 0 vs 1 1.249
PromCntCard12 0.959
StatusCatStarAll 0 vs 1 0.900
Select View Model Iteration Plot.
Again, the selected model (based on minimum average squared error) occurs in step 15. The actual value of average squared error for this model is slightly lower than that for the model with the untransformed inputs.
Select Select Chart Misclassification Rate.
The misclassification rate with the transformed input is also lower than that for the untransformed inputs. However, the model with the lowest misclassification rate comes from step 6. If you want to optimize on misclassification rate, you must change this property in the Regression node’s property sheet.
Categorical inputs present another problem for regressions. To represent these nonnumeric inputs in a model, you must convert them to some sort of numeric values. This conversion is most commonly done by creating design variables (or dummy variables), with each design variable representing, roughly, one level of the categorical input. (The total number of design variables required is, in fact, one less than the number of inputs.) A single categorical input can vastly increase a model’s degrees of freedom, which, in turn, increases the chances of a model overfitting.
There are many remedies to this problem. One of the simplest remedies is to use domain knowledge to reduce the number of levels of the categorical input.
The demonstration shows how to use the Replacement tool to facilitate combining input levels.
Remove the connection between the Transform Variables node and the Impute node.
Select the Modify tab.
Drag a Replacement tool into the diagram workspace.
Connect the Transform Variables node to the Replacement node.
Connect the Replacement node to the Impute node.
Select Replacement Editor . from the Replacement node Properties panel. A confirmation dialog box opens.
Select Yes. The Replacement Editor opens.
The Replacement Editor lists all levels of all categorical inputs. You can use the Replacement column to reassign values to any of the levels.
The input with the largest number of levels is DemCluster, which has so many levels that consolidating the levels using the Replacement Editor would be an arduous task. (Another, autonomous method for consolidating the levels of DemCluster (or any categorical input) is presented as a special topic in Chapter 9.)
For this demonstration, combine the levels of another input, StatusCat96NK
Scroll the Replacement Editor to view the levels of StatusCat96NK
The input has six levels, plus a level to represent unknown values (which do not occur in the training data). The levels of StatusCat96NK will be consolidated as follows:
Levels A and S (active and star donors) indicate consistent donors and are grouped into a single level, A.
Levels F and N (firsttime and new donors) indicate new donors and are grouped into a single level, N.
Levels E and L (inactive and lapsing donors) indicate lapsing donors and are grouped into a single level L.
Type A in the Replacement field for StatusCat96NK levels A and S.
Type N in the Replacement field for StatusCat96NK levels F and N.
Type L in the Replacement field for StatusCat96NK levels L and E.
Select OK to close the Replacement Editor.
Run the Replacement node and view the results.
The Total Replacement Counts window shows the number of replacements that occurr in the training and validation data.
Select View Model Replaced Levels. The Replaced Levels window opens.
The replaced level values are consistent with expectations.
Close the Results window.
Run the Regression node and view the results.
View lines 23102340 of the Output window.
Summary of Stepwise Selection
Effect Number Score Wald
Step Entered Removed DF In ChiSquare ChiSquare Pr > ChiSq
1 LOG_GiftCntCard36 1 1
2 GiftTimeLast 1 2 29.0312 <.0001
3 DemMedHomeValue 1 3 24.8434 <.0001
4 LOG_GiftAvgAll 1 4 28.9692 <.0001
5 IMP_DemIncomeGroup 1 5 13.8240 0.0002
6 GiftTimeFirst 1 6 7.1299 0.0076
7 IMP_DemAge 1 7 4.1407 0.0419
8 REPL_StatusCat96NK 2 8 5.2197 0.0735
9 IMP_LOG_GiftAvgCard36 1 9 2.4956 0.1142
10 M_LOG_GiftAvgCard36 1 10 1.6207 0.2030
11 StatusCatStarAll 1 11 1.9076 0.1672
12 LOG_GiftCnt36 1 12 1.5708 0.2101
13 LOG_GiftAvgLast 1 13 1.4316 0.2315
14 LOG_GiftAvgAll 1 12 0.2082 0.6482
15 PromCntCardAll 1 13 0.6704 0.4129
16 PromCnt36 1 14 1.0417 0.3074
17 PromCnt12 1 15 0.9800 0.3222
18 PromCntAll 1 16 1.0446 0.3068
19 M_DemAge 1 17 0.4815 0.4877
20 LOG_GiftCntAll 1 18 0.3887 0.5330
21 LOG_GiftCntAll 1 17 0.3886 0.5330
The REPL_StatusCat96NK input (created from the original StatusCat96NK input) is included in the stepwise selection process. The threelevel input is represented by two degrees of freedom.
View lines 24222423 of the Output window.
REPL_StatusCat96NK A vs N 0.822
REPL_StatusCat96NK L vs N 1.273
Based on the odds ratios, active donors in the 96NK campaign are less likely than new donors to contribute in the 97NK campaign. On the other hand, lapsing donors in the 96NK campaign are more likely than new donors to contribute in the 97NK campaign.
Select View Model Iteration Plot.
The selected model from step 19 has, again, a slightly smaller average squared error than previous models.
The Regression tool assumes (by default) a linear and additive association between the inputs and the logit of the target. If the true association is more complicated, such an assumption might result in biased predictions. For decisions and rankings, this bias can (in some cases) be unimportant. For estimates, this bias will appear as a higher value for the validation average squared error fit statistic.
In the dot color problem, the (standard logistic regression) assumption that the concentration of yellow dots increases toward the upperright corner of the unit square seems to be suspect.
When minimizing prediction bias is important, you can increase the flexibility of a regression model by adding polynomial combinations of the model inputs. This enables predictions to better match the true input/target association. It also increases the chances of overfitting while simultaneously reducing the interpretability of the predictions. Therefore, polynomial regression must be approached with some care.
In SAS Enterprise Miner, adding polynomial terms can be done selectively or autonomously.
This demonstration shows how to selectively add polynomial regression terms.
You can modify the existing Regression node or add a new Regression node. If you add a new node, you must configure the Polynomial Regression node to perform the same tasks as the original. An alternative is to make a copy of the existing node.
Rightclick the Regression node and select Copy from the menu.
Rightclick the diagram workspace and select Paste from the menu. A new Regression node is added with the label Regression (2) to distinguish it from the existing one.
Select the Regression (2) node. The properties are identical to the existing node.
Rename the new regression node Polynomial Regression
Connect the Polynomial Regression node to the Impute node.
The Term Editor enables you to add specific polynomial terms to the regression model.
Select Term Editor . from the Polynomial Regression Properties panel. The Terms dialog box opens.
Suppose that you suspect an interaction between home value and time since last gift. (Perhaps a recent change in property values affected the donation patterns.)
Select DemMedHomeValue in the Variables panel of the Terms dialog box.
Select the Add button, _{}. The DemMedHomeValue input is added to the Term panel.
Repeat for GiftTimeLast
Select Save. An interaction between the selected inputs is now available for consideration by the regression node.
Similarly, suppose that you suspect a parabolashaped relationship between donation and donor age. (Donation odds increase with age, peak, and then decline with age.)
Select IMP_DemAge
Select the Add button, _{}. The IMP_DemAge input is added to the Term panel.
Select the Add button, _{} again. Another IMP_DemAge input is added to the Term panel.
Select Save. A quadratic age term is available for consideration by the model.
Select OK to close the Terms dialog box.
To use the terms you defined in the Terms dialog box, you must enable the User Terms option in the Regression node.
Select User Terms Yes in the Polynomial Regression Properties panel.
Run the Polynomial Regression node and view the results.
Scroll the Output window to lines 25152540.
Summary of Stepwise Selection
Effect Number Score Wald
Step Entered Removed DF In ChiSquare ChiSquare Pr > ChiSq
1 LOG_GiftCntCard36 1 1
2 GiftTimeLast 1 2 29.0312 <.0001
3 DemMedHomeValue*GiftTimeLast 1 3 26.3762 <.0001
4 LOG_GiftAvgAll 1 4 29.2016 <.0001
5 IMP_DemIncomeGroup 1 5 13.8523 0.0002
6 GiftTimeFirst 1 6 7.1518 0.0075
7 IMP_DemAge 1 7 4.0914 0.0431
8 IMP_DemAge*IMP_DemAge 1 8 12.2214 0.0005
9 M_DemAge 1 9 4.1796 0.0409
10 REPL_StatusCat96NK 2 10 5.7905 0.0553
11 LOG_GiftAvgLast 1 11 2.3447 0.1257
12 StatusCatStarAll 1 12 1.8360 0.1754
13 LOG_GiftCnt36 1 13 1.3030 0.2537
14 M_LOG_GiftAvgCard36 1 14 1.6761 0.1954
15 IMP_LOG_GiftAvgCard36 1 15 1.7173 0.1900
16 LOG_GiftAvgAll 1 14 0.3704 0.5428
17 PromCntCardAll 1 15 1.0003 0.3172
18 PromCntAll 1 16 1.2101 0.2713
19 PromCnt12 1 17 0.6678 0.4138
20 PromCnt36 1 18 1.3816 0.2398
21 LOG_GiftCntAll 1 19 0.4570 0.4990
22 LOG_GiftAvgAll 1 20 0.3898 0.5324
23 LOG_GiftAvgAll 1 19 0.3897 0.5325
The stepwise selection summary shows the polynomial terms added in steps 3 and 8.
View the Iteration Plot window.
While present in the selected model, the interaction terms appear to have negligible effect on the model performance (at least for average squared error).
This begs the question: How do you know which nonlinear terms to include in a model?
Unfortunately, there is no simple solution to this question in SAS Enterprise Miner, short of trying all of them
SAS Enterprise Miner has the ability to add every polynomial combination of inputs to a regression model. Obviously, this feature must be used with some care, because the number of polynomial input combinations increases rapidly with input count.
For instance, the
The trick is to only consider polynomial expressions of those inputs found to be linearly related to the logit of the target. That is, start with those inputs selected by the original regression model. A simple modification to your analysis diagram makes this possible.
Delete the connection between the Impute node and the Polynomial Regression node.
Connect the Regression node to the Polynomial Regression node.
Rightclick the Polynomial Regression node and select Update from the menu. A Status dialog box informs you of the completion of the update process.
Select OK to close the Status dialog box.
Select Variables . from the Polynomial Regression Properties panel.
All inputs not selected in the original regression’s stepwise procedure are rejected.
Select OK to close the Variables dialog box.
Select Polynomial Terms Yes in the Polynomial Regression Properties panel. This adds all quadratic combinations of the interval inputs.
You can increase this to a higher order polynomial by changing the Polynomial Degree field.
Select User Terms No. The terms added in the last demonstration will be included by specifying the Polynomial Terms option.
Run the Polynomial Regression node and view the results. (This might take a few minutes to complete on slower computers.)
Scroll to the bottom of the Output window.
NOTE: File view has been truncated.
Refer to C:Documents and SettingsusernameMy DocumentsMy SAS Files9.1EM_ProjectsMy ProjectWorkspacesEMWS2Reg2EMOUTPUT.out
on this server for entire file contents.
The amount of output generated by the selection history exceeds the 10,000line, clienttransfer limit of SAS Enterprise Miner. To view the complete output listing, you must open the referenced file.
Doing so reveals a 121step and 18,000line input selection freeforall. The selection summary is shown below.
Summary of Stepwise Selection
Effect Number Score Wald
Step Entered Removed DF In ChiSquare ChiSquare Pr > ChiSq
1 IMP_DemAge*LOG_GiftCntCard36 1 1 112.3900 <.0001
2 GiftTimeLast*LOG_GiftAvgLast 1 2 50.4645 <.0001
3 DemMedHomeValue*IMP_DemIncomeGroup 1 3 35.6614 <.0001
4 GiftTimeFirst*IMP_DemIncomeGroup 1 4 15.9535 <.0001
5 DemMedHomeValue*GiftTimeLast 1 5 5.1609 0.0231
6 DemMedHomeValue*IMP_DemIncomeGroup 1 4 0.0257 0.8726
7 IMP_DemAge*PromCntCardAll 1 5 4.5051 0.0338
8 GiftTimeFirst*LOG_GiftCnt36 1 6 4.6421 0.0312
9 GiftTimeLast*LOG_GiftCntCard36 1 7 8.0742 0.0045
10 GiftTimeLast*IMP_DemIncomeGroup 1 8 3.7680 0.0522
11 StatusCatStarAll 1 9 4.0508 0.0442
12 M_LOG_GiftAvgCard36 1 10 4.9476 0.0261
13 IMP_DemIncomeGroup*IMP_DemIncomeGroup 1 11 3.3268 0.0682
14 IMP_DemIncomeGroup*LOG_GiftAvgLast 1 12 5.6563 0.0174
15 GiftTimeLast 1 13 6.6747 0.0098
16 GiftTimeLast*IMP_DemIncomeGroup 1 12 0.0375 0.8465
17 LOG_GiftCntCard36 1 13 5.4244 0.0199
18 IMP_DemAge*LOG_GiftAvgLast 1 14 5.0260 0.0250
19 IMP_DemAge*IMP_DemAge 1 15 5.7455 0.0165
20 IMP_DemAge 1 16 5.2865 0.0215
21 M_DemAge 1 17 8.9336 0.0028
22 LOG_GiftAvgLast*PromCntAll 1 18 2.6281 0.1050
23 LOG_GiftCnt36*LOG_GiftCnt36 1 19 2.5730 0.1087
24 GiftTimeFirst*IMP_DemIncomeGroup 1 18 0.0289 0.8651
25 GiftTimeFirst*PromCntAll 1 19 3.6941 0.0546
26 IMP_DemAge*LOG_GiftAvgLast 1 18 0.3456 0.5566
27 IMP_DemAge*LOG_GiftCntCard36 1 17 0.2592 0.6107
28 GiftTimeLast*PromCntAll 1 18 2.3912 0.1220
29 GiftTimeFirst*IMP_DemAge 1 19 1.9879 0.1586
30 LOG_GiftCnt36*PromCntCardAll 1 20 4.8338 0.0279
31 REPL_StatusCat96NK 2 21 3.4798 0.1755
32 IMP_DemAge*IMP_DemIncomeGroup 1 22 1.1337 0.2870
33 IMP_DemAge*IMP_LOG_GiftAvgCard36 1 23 1.0241 0.3116
34 GiftTimeLast*IMP_LOG_GiftAvgCard36 1 24 3.8889 0.0486
35 IMP_LOG_GiftAvgCard36 1 25 3.6415 0.0564
36 IMP_DemAge*IMP_LOG_GiftAvgCard36 1 24 0.2008 0.6541
37 IMP_LOG_GiftAvgCard36*LOG_GiftAvgLast 1 25 5.6166 0.0178
38 IMP_LOG_GiftAvgCard36*LOG_GiftCntCard36 1 26 2.8395 0.0920
39 LOG_GiftAvgLast 1 27 1.4026 0.2363
40 LOG_GiftAvgLast*LOG_GiftCntCard36 1 28 1.0598 0.3033
41 M_LOG_GiftAvgCard36 1 27 0.2257 0.6348
42 GiftTimeLast*PromCnt12 1 28 0.9554 0.3284
43 IMP_LOG_GiftAvgCard36*PromCnt36 1 29 1.1580 0.2819
44 GiftTimeFirst*PromCnt36 1 30 1.2396 0.2655
45 GiftTimeFirst*PromCntAll 1 29 0.0060 0.9380
(Continued on the next page.)
46 IMP_LOG_GiftAvgCard36*LOG_GiftCntCard36 1 28 0.4424 0.5060
47 IMP_LOG_GiftAvgCard36*IMP_LOG_GiftAvgCard36 1 29 1.0828 0.2981
48 LOG_GiftCnt36*PromCntAll 1 30 1.0412 0.3075
49 IMP_DemAge*PromCntCardAll 1 29 0.3676 0.5443
50 LOG_GiftAvgLast*PromCntCardAll 1 30 2.0774 0.1495
51 LOG_GiftCnt36*PromCntCardAll 1 29 0.3967 0.5288
52 PromCnt36*PromCntCardAll 1 30 1.2143 0.2705
53 GiftTimeLast*PromCntAll 1 29 0.4198 0.5170
54 LOG_GiftAvgLast*PromCnt36 1 30 0.9148 0.3388
55 PromCntAll 1 31 1.0177 0.3131
56 IMP_DemIncomeGroup*LOG_GiftCntCard36 1 32 0.8736 0.3500
57 IMP_DemIncomeGroup*LOG_GiftCnt36 1 33 1.3926 0.2380
58 GiftTimeLast*LOG_GiftCnt36 1 34 1.9062 0.1674
59 DemMedHomeValue*GiftTimeFirst 1 35 0.8606 0.3536
60 DemMedHomeValue*PromCnt12 1 36 1.7609 0.1845
61 DemMedHomeValue*DemMedHomeValue 1 37 1.6746 0.1956
62 IMP_LOG_GiftAvgCard36*PromCnt12 1 38 1.0289 0.3104
63 LOG_GiftAvgLast*PromCnt12 1 39 1.1348 0.2867
64 LOG_GiftCnt36*PromCnt12 1 40 0.8804 0.3481
65 PromCntAll 1 39 0.1284 0.7201
66 IMP_DemAge*PromCntAll 1 40 0.9530 0.3290
67 IMP_DemAge*PromCnt36 1 41 0.7239 0.3949
68 IMP_LOG_GiftAvgCard36*PromCntAll 1 42 0.7922 0.3734
69 GiftTimeFirst*IMP_LOG_GiftAvgCard36 1 43 1.1685 0.2797
70 IMP_DemIncomeGroup*PromCnt36 1 44 0.8058 0.3694
71 IMP_DemIncomeGroup*PromCntCardAll 1 45 1.7767 0.1826
72 IMP_DemIncomeGroup*PromCnt12 1 46 3.3905 0.0656
73 LOG_GiftAvgLast*PromCnt12 1 45 0.1157 0.7337
74 IMP_DemAge*IMP_DemIncomeGroup 1 44 0.3121 0.5764
75 IMP_DemIncomeGroup*PromCntAll 1 45 1.5760 0.2093
76 DemMedHomeValue*IMP_DemAge 1 46 0.7067 0.4005
77 DemMedHomeValue 1 47 0.8369 0.3603
78 GiftTimeLast*PromCntCardAll 1 48 0.6305 0.4272
79 GiftTimeLast*PromCnt36 1 49 1.3586 0.2438
80 IMP_DemAge*IMP_LOG_GiftAvgCard36 1 50 0.5447 0.4605
81 IMP_DemAge*LOG_GiftAvgLast 1 51 1.8152 0.1779
82 IMP_LOG_GiftAvgCard36*PromCntCardAll 1 52 0.5547 0.4564
83 PromCnt12*PromCnt12 1 53 0.4997 0.4796
84 PromCnt36*PromCntCardAll 1 52 0.2620 0.6088
85 GiftTimeFirst*IMP_LOG_GiftAvgCard36 1 51 0.3309 0.5651
86 GiftTimeFirst*GiftTimeLast 1 52 0.9405 0.3322
87 PromCnt12 1 53 0.6663 0.4143
88 PromCnt36 1 54 1.1938 0.2746
89 IMP_LOG_GiftAvgCard36*PromCnt12 1 53 0.1271 0.7215
90 LOG_GiftCnt36*LOG_GiftCntCard36 1 54 0.9120 0.3396
91 LOG_GiftCntCard36 1 53 0.1168 0.7326
92 LOG_GiftCntCard36*LOG_GiftCntCard36 1 54 1.2887 0.2563
93 LOG_GiftCntCard36*PromCnt36 1 55 0.8773 0.3489
94 PromCntCardAll 1 56 0.9834 0.3214
95 GiftTimeLast*PromCnt36 1 55 0.3743 0.5407
96 GiftTimeLast*PromCnt12 1 54 0.0612 0.8046
97 PromCnt36*PromCnt36 1 55 1.0367 0.3086
98 LOG_GiftAvgLast*LOG_GiftCnt36 1 56 0.7673 0.3811
99 GiftTimeLast*LOG_GiftCnt36 1 55 0.3944 0.5300
100 LOG_GiftCntCard36*PromCntCardAll 1 56 0.8257 0.3635
(Continued on the next page.)
101 LOG_GiftCntCard36*PromCntAll 1 57 2.3132 0.1283
102 GiftTimeFirst*PromCnt36 1 56 0.3471 0.5558
103 GiftTimeFirst*LOG_GiftCntCard36 1 57 1.6283 0.2019
104 LOG_GiftCntCard36*PromCnt36 1 56 0.0007 0.9792
105 LOG_GiftCnt36*PromCntAll 1 55 0.2986 0.5848
106 LOG_GiftCntCard36*PromCnt12 1 56 2.1837 0.1395
107 LOG_GiftCnt36*PromCnt12 1 55 0.1815 0.6701
108 GiftTimeLast 1 54 0.4189 0.5175
109 LOG_GiftCntCard36*PromCnt36 1 55 0.9461 0.3307
110 DemMedHomeValue*PromCnt36 1 56 0.5590 0.4547
111 DemMedHomeValue*PromCnt12 1 55 0.4496 0.5025
112 DemMedHomeValue*PromCntCardAll 1 56 0.8854 0.3467
DemMedHomeValue*GiftTimeFirst 1 55 0.0000 0.9972
114 GiftTimeLast*IMP_DemIncomeGroup 1 56 0.6227 0.4300
115 PromCnt12*PromCntAll 1 57 0.5536 0.4568
116 PromCnt12*PromCntCardAll 1 58 0.5737 0.4488
117 GiftTimeFirst 1 59 0.4895 0.4842
118 GiftTimeFirst*GiftTimeLast 1 58 0.1034 0.7478
119 GiftTimeFirst*LOG_GiftAvgLast 1 59 1.4562 0.2275
120 GiftTimeLast 1 60 0.3704 0.5428
121 GiftTimeLast 1 59 0.3701 0.5430
Examine the iteration plot.
Surprisingly, the selected model comes from step 12 and involves only a few regression terms.
Analysis of Maximum Likelihood Estimates
Standard Wald Standardized
Parameter DF Estimate Error ChiSquare Pr > ChiSq Estimate Exp(Est)
Intercept 1 0.0164 0.1395 0.01 0.9063 1.017
StatusCatStarAll 0 1 0.0761 0.0378 4.05 0.0442 0.927
DemMedHomeValue*GiftTimeLast 1 7.885E8 1.694E8 21.67 <.0001 1.000
GiftTimeFirst*IMP_DemIncomeGroup 1 0.000345 0.000324 1.13 0.2872 1.000
GiftTimeFirst*LOG_GiftCnt36 1 0.00309 0.000920 11.26 0.0008 1.003
GiftTimeLast*IMP_DemIncomeGroup 1 0.00327 0.00157 4.34 0.0371 1.003
GiftTimeLast*LOG_GiftAvgLast 1 0.0136 0.00265 26.42 <.0001 0.986
GiftTimeLast*LOG_GiftCntCard36 1 0.0239 0.00695 11.86 0.0006 0.976
IMP_DemAge*LOG_GiftCntCard36 1 0.0114 0.00202 32.02 <.0001 1.012
IMP_DemAge*PromCntCardAll 1 0.00032 0.000092 12.23 0.0005 1.000
Unfortunately, all but one of the selected terms is an interaction.
While this model is the best predictor of PVA97NK response found up to this time, it is nearly impossible to interpret. This is an important factor to remember when considering polynomial regression models.
Predictive Modeling Using Regression
Return to the Chapter 4 Organics diagram in the Exercises project. Explore the ORGANICS data source.
In preparation for regression, is any missing values imputation needed? If yes, should you do this imputation before generating the decision tree models? Why or why not?
Add an Impute node to the diagram and connect it to the Data Partition node.
Change Default Input Method to Tree for both class and interval variables. Create missing value indicator variables. Replace missing values for GENDER with U for unknown.
Add a Regression node to the diagram and connect it to the Impute node.
Choose the stepwise selection and average squared error as the selection criterion.
Run the Regression node and view the results. Which variables are included in the final model? Which variables are important in this model?
In preparation for regression, are any transformations of the data warranted? Why or why not?
Add a Transform Variables node to the diagram and connect it to the Impute node.
The variable AFFL appears to be skewed to the
right. Use a square root transformation for AFFL. The variables
Run the Transform Variables node. Explore the exported training data.
Did the transformation of AFFL appear to result in a less
skewed distribution? What transformation was chosen for the variables
Add another Regression node to the diagram and connect it to the Transform Variables node.
Choose the stepwise selection method and average squared error selection criterion.
Run this new Regression node and view the results. Which variables are included in the final model? Which variables are important in the model?
How do the validation average squared errors of the two regression models compare? How do the regression models compare to the tree models?
Regression models are a prolific and useful way to create predictions. New cases are scored using a prediction formula. Inputs are selected via a sequential selection process. Model complexity is controlled by fit statistics calculated on validation data.
To use regression models, there are several issues with which to contend that go beyond the predictive modeling essentials. First, a mechanism for handling missing input values must be included in the model development process. Second, methods for handling extreme or outlying predictions should be included. Third, the levelcount of a categorical should be reduced to avoid overfitting. Finally, the model complexity might need to be increased beyond what is provided by standard regression methods. One approach to this is polynomial regression. Polynomial regression models can be fit by hand with specific interactions in mind. They also can be fit autonomously by selecting polynomial terms from a list of all polynomial candidates.
Predictive Modeling Using Regression
Rightclick the ORGANICS data source in the Project Panel and select Explore.
Examine the ORGANICS data table to reveal several inputs with missing values. A more precise way to determine the extent of missing values in this data set is to use the StatExplore node.
Connect a StatExplore node to the ORGANICS data source in the diagram workspace.
Select the StatExplore node in the diagram and examine the Properties panel.
Change the option to Hide Rejected Variables to No and change the option for Interval Variables to Yes using the menus.
Run the StatExplore node and examine the Output window in the Results.
Class Variable Summary Statistics
(maximum 500 observations printed)
Variable Role Numcat NMiss Mode
CLASS INPUT 4 0 Silver
GENDER INPUT 4 2512 F
NGROUP INPUT 8 674 C
REGION INPUT 6 465 South East
TV_
ORGYN TARGET 2 0 0
The variable GENDER has a relatively large number of missing values.
Interval Variable Summary Statistics
(maximum 500 variables printed)
Variable ROLE Mean StdDev Non Missing Missing
AFFL INPUT 9 3 21138 1085
LTIME INPUT 7 5 21942 281
The variables AFFL and
Imputation of missing values should be done prior to generating a regression model. However, imputation is not necessary before generating a decision tree model because a decision tree can use missing values in the same way as any other data value in the data set.
Close the StatExplore node results and return to the diagram workspace.
After you add an Impute node, the diagram should appear as shown below.
Choose the imputation methods.
Select the Impute node in the diagram.
To use tree imputation as the default method of imputation, change the Default Input Method for both class and interval variables to Tree using the menus.
To create missing value indicator variables, change the Indicator Variable property to Unique and the Indicator Variable Role to Input using the menus in the Properties panel.
To replace missing values for the variable GENDER with U, first change Default Character Value to U. The Properties panel should appear as shown.
Select Variables . . Change the value in the Method column for the variable GENDER to Constant as shown below.
Select OK to confirm the change.
Add a Regression node to the diagram as shown below.
Choose the stepwise selection and average squared error as the selection criterion.
Select the Regression node in the diagram.
Select Selection Model Stepwise.
Select Selection Criterion Validation Error.
Run the Regression node and view the results. The Output window shows which variables are in the final model and which variables are important.
Line 658 lists the inputs included in the final model:
Intercept IMP_AFFL IMP_
Important inputs can be ascertained by Wald ChiSquare, standardized estimates, and odds ratio estimates.
Analysis of Maximum Likelihood Estimates
Standard Wald Standardized
Parameter DF Estimate Error ChiSquare Pr > ChiSq Estimate Exp(Est)
Intercept 1 0.7674 0.1336 33.01 <.0001 0.464
IMP_AFFL 1 0.2513 0.00721 1215.16 <.0001 0.4637 1.286
IMP_
IMP_GENDER F 1 0.9688 0.0480 407.57 <.0001 2.635
IMP_GENDER M 1 0.0485 0.0528 0.84 0.3584 1.050
M_AFFL 0 1 0.1276 0.0464 7.56 0.0060 0.880
M_
M_GENDER 0 1 0.2324 0.0767 9.18 0.0024 0.793
Odds Ratio Estimates
Point
Effect Estimate
IMP_AFFL 1.286
IMP_
IMP_GENDER F vs U 7.287
IMP_GENDER M vs U 2.903
M_AFFL 0 vs 1 0.775
M_
M_GENDER 0 vs 1 0.628
Important inputs include IMP_AFFL IMP_
Close the regression results and return to the diagram workspace.
Explore the input distributions.
Select the ORGANICS node and select Variables . in the Properties panel.
Select all interval input variables.
Select Explore . .
The variables AFFL
Connect a Transform Variables node as shown below.
Transform inputs.
Select Variables . in the Transform Variables node’s Properties panel.
In the Method column for the
variable
Select OK to confirm the changes.
Run the Transform Variables node. Do not view the results. Instead, select Exported Data . from the Transform Variables node’s Properties panel. The Exported Data  Transform Variables window opens.
Select the Train data set and select Explore . .
In the Explore window, select Actions Plot . .
Select Histogram as the type of chart and then select Next >.
Change the role of SQRT_
Select Finish.
The transformed variable appears to be less skewed than the original variable.
To view the transformations for the other variables,
close the graph and examine the Transformations window in the results. The
transformation chosen for the variable
Transformations
(Maximum 500 observations printed)
Input Input Output
Name Role Level Output Name Level Formula
BILL INPUT INTERVAL SQRT_BILL INTERVAL Sqrt(_SCALEVAR_)
IMP_AFFL INPUT INTERVAL SQRT_IMP_AFFL INTERVAL Sqrt(IMP_AFFL + 1)
IMP_LTIME INPUT INTERVAL
Close the results and return to the diagram workspace.
Add a second Regression node to the diagram as shown below.
Choose the stepwise selection and average squared error as the selection criterion.
Select the Regression node in the diagram.
Select Selection Model Stepwise.
Select Selection Criterion Validation Error.
Run the Regression node and view the results. The Output window shows which variables are in the final model and which variables are important.
Line 563 lists the inputs included in the final model:
Intercept IMP_
Important inputs can be ascertained by Wald ChiSquare, standardized estimates, and odds ratio estimates.
Analysis of Maximum Likelihood Estimates
Standard Wald Standardized
Parameter DF Estimate Error ChiSquare Pr > ChiSq Estimate Exp(Est)
Intercept 1 4.1258 0.2170 361.43 <.0001 0.016
IMP_
IMP_GENDER F 1 0.8995 0.0416 468.16 <.0001 2.458
IMP_GENDER M 1 0.00905 0.0496 0.03 0.8551 0.991
SQRT_IMP_AFFL 1 1.5685 0.0571 755.15 <.0001 0.4545 4.799
Odds Ratio Estimates
Point
Effect Estimate
IMP_
IMP_GENDER F vs U 5.990
IMP_GENDER M vs U 2.414
SQRT_IMP_AFFL 4.799
Important inputs include SQRT_IMP_AFFL IMP_
Model fit comparisons.
Examine the Fit Statistics window.
The validation average squared error for this model equals 0.1376.
Open the Results window for the original regression and examine the Fit Statistics window.
The validation average squared error for the original regression model equals 0.1365, or slightly lower. In this case, transforming the inputs does not improve model fit.
Open the Results window for the original tree model and examine the Fit Statistics window.
The average squared error of the tree model is smaller. It appears that the regression models are less adept at modeling the association between inputs and target.
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