Regression for Data Mining

Mgt. 2206 – Introduction to Analytics

Matthew Liberatore

Thomas Coghlan

Learning Objectives

To understand the application of regression analysis in data mining Linear/nonlinear Logistic (Logit)

To understand the key statistical measures of fit

To learn how to run and interpret regression analyses using SAS Enterprise Miner software

Analysis of Association

In business problems interests often go beyond the statistical testing of differences (e.g., female versus male preferences)

Often interested in degree of association between variables.

Regression is one of the techniques that helps uncover those relations.

Linear Regression Analysis

Analysis of the strength of the linear relationship between predictor (independent) variables and outcome (dependent/criterion) variables.

In two dimensions (one predictor, one outcome variable) data can be plotted on a scatter diagram.

EE((yy) = ) = 001 1 (x)(x)

Expected value of y (outcome)

Intercept Term coefficient

Predictorvariable

Estimation Process

Regression ModelRegression Modelyy = = 00 + + 11xx + +

Regression EquationRegression EquationEE((yy) = ) = 00 + + 11xx

Unknown ParametersUnknown Parameters00, , 11

Sample Data:Sample Data:x yx y

xx11 y y11

. .. . . .. . xxnn yynn

bb00 and and bb11

provide estimates ofprovide estimates of00 and and 11

EstimatedEstimatedRegression EquationRegression Equation

Sample StatisticsSample Statistics

bb00, , bb11

0 1y b b x

Simple Linear Regression Equation:Positive Linear RelationshipPositive Linear Relationship

EE((yy): Outcome): Outcome

x : Predictorx : Predictor

Slope Slope 11

is positiveis positive

Regression lineRegression line

InterceptIntercept00

EE((yy): Outcome): Outcome

x: Predictorx: Predictor

Slope Slope 11

is negativeis negative

Regression lineRegression lineInterceptIntercept00

Simple Linear Regression Equation:Negative Linear RelationshipNegative Linear Relationship

EE((yy): Outcome): Outcome

x: Predictorx: Predictor

Simple Linear Regression Equation:No RelationshipNo Relationship

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•

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EE((yy))

xx

Slope Slope 11

is 0is 0

Regression lineRegression lineInterceptIntercept

00

Simple Linear Regression Equation:No RelationshipNo Relationship

EE((yy): Outcome): Outcome

x: Predictorx: Predictor

InterceptIntercept00

Simple Linear Regression Equation:Parabolic RelationshipParabolic Relationship

•••••••••••••••••••••••

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Example

List Variables we have Determine a DV of interest Is there a way to predict DV?

Least Squares Method

Least Squares Criterion: minimize error (distance between actual data & estimated line)

min (y yi i )2min (y yi i )2

where:where:

yyii = = observedobserved value of the dependent variable value of the dependent variable

for the for the iith observationth observation^yyii = = estimatedestimated value of the dependent variable value of the dependent variable

for the for the iith observationth observation

Slope for the Estimated Regression Equation

1 2

( )( )

( )i i

i

x x y yb

x x

1 2

( )( )

( )i i

i

x x y yb

x x

Least Squares Method

yy--Intercept for the Estimated Regression EquationIntercept for the Estimated Regression Equation

Least Squares MethodLeast Squares Method

0 1b y b x 0 1b y b x

where:where:xxii = = value of independent variable for value of independent variable for iithth observationobservation

nn = = total number of observationstotal number of observations

__yy = mean value for dependent variable= mean value for dependent variable

__xx = mean value for independent variable= mean value for independent variable

yyii = = value of dependent variable for value of dependent variable for iithth observationobservation

Least Squares Estimation Procedure

Least Squares Criterion: The sum of the vertical deviations (y axis) of the

points from the line is minimal.

Predicted Line

Actual Data

Example: Kwatts vs. Temp

Temp Kwatts59.2 9,73061.9 9,75055.1 10,18066.2 10,23052.1 10,80069.9 11,16046.8 12,53076.8 13,91079.7 15,11079.3 15,69080.2 17,02083.3 17,880

Is the Relationship Linear?KWatts vs. Temp

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Temp

KW

atts

KWatts

Example ResultsLet X = Temp, Y = Kwatts

Y = 319.04 + 185.27 X

KWatts vs. Temp

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atts KWatts

Forecast

average

Coefficient of Determination

How “strong” is relationship between predictor & outcome? (Fraction of observed variance of outcome variable explained by the predictor variables).

Relationship Among SST, SSR, SSE

where:where: SST = total sum of squaresSST = total sum of squares SSR = sum of squares due to regressionSSR = sum of squares due to regression SSE = sum of squares due to errorSSE = sum of squares due to error

SST = SSR + SSESST = SSR + SSE

2( )iy y 2( )iy y 2ˆ( )iy y 2ˆ( )iy y 2ˆ( )i iy y 2ˆ( )i iy y

Coefficient of Determination (Coefficient of Determination (rr22))

where:where:

SSR = sum of squares due to regressionSSR = sum of squares due to regression

SST = total sum of squaresSST = total sum of squares

rr22 = SSR/SST = SSR/SST

Kwatts vs. Temp Example

df SSRegression 1 58784708.31Residual 10 38696916.69Total 11 97481625

rr22 = = 0.6030337340.603033734Does the linear regression provide a good fit?

Assumptions About the Error Term

1. The error1. The error is a random variable with mean of zero.is a random variable with mean of zero.

2.2. The variance ofThe variance of , , denoted by denoted by 22, , is the same foris the same for all values of the independent variable.all values of the independent variable.

3.3. The values ofThe values of are independent.are independent.

4.4. The errorThe error is a normally distributed randomis a normally distributed random variable.variable.

Significance Test for Regression

Is the value of Is the value of 11 zero?zero?

Two tests are commonly used:Two tests are commonly used:

tt Test Test andand FF Test Test

Both theBoth the tt test and test and FF test require an estimate oftest require an estimate of the the variance (variance (22) of the error () of the error (..

As in most of our statistical work, we are working withAs in most of our statistical work, we are working witha sample, not the population, so we use a sample, not the population, so we use

mean square error (mean square error (ss22).).

An Estimate of Testing for Significance

210

2 )()ˆ(SSE iiii xbbyyy 210

2 )()ˆ(SSE iiii xbbyyy

where:where:

ss 22 = MSE = SSE/( = MSE = SSE/(n n 2) 2)

Testing for Significance

An Estimate of

2

SSEMSE

n

s2

SSEMSE

n

s

• To estimate To estimate we take the square root of we take the square root of 22..

• The resulting The resulting ss is called the is called the standard error ofstandard error of the estimatethe estimate..

Hypotheses: Coefficient (11) is 0

(no relationship between predictor & outcome)

Calculating t Statistic:

Testing for Significance: t Test

1

1

b

bt

s

1

1

b

bt

s

0 1: 0H 0 1: 0H

0 1: 0H 0 1: 0H 1. Determine if 1. Determine if

2. Specify the level of significance.2. Specify the level of significance.

3. Select the test statistic.3. Select the test statistic.

= .05= .05

4. State the rejection rule.4. State the rejection rule. Reject Reject if if pp-value -value << .05 or | .05 or |t|t| > > 3.182 (with 3 degrees of 3.182 (with 3 degrees of freedom)freedom)

Testing for Significance: Testing for Significance: tt Test Test

1

1

b

bt

s

1

1

b

bt

s

0 1: 0H 0 1: 0H Same Hypotheses: Same Hypotheses:

Different Test Statistic:Different Test Statistic:

Alternative Test: Alternative Test: FF Test Test

FF = MSR/MSE = MSR/MSE

Reject if: Reject if: pp-value-value << or or FF >> FF

Testing for Significance: Testing for Significance: FF TestTest

wherewhere::

FF is based on an is based on an FF distribution withdistribution with

1 degree of freedom in the numerator and1 degree of freedom in the numerator and

nn - 2 degrees of freedom in the denominator- 2 degrees of freedom in the denominator

FF = MSR/MSE = MSR/MSE

0 1: 0H 0 1: 0H

0 1: 0H 0 1: 0H 1. Determine if 1. Determine if

2. Specify the level of significance.2. Specify the level of significance.

3. Select the test statistic. 3. Select the test statistic.

= .05= .05

4. State the rejection rule.4. State the rejection rule. Reject Reject if if pp-value -value << .05 or F > 10.13 .05 or F > 10.13 ((withwith 1 d.f. 1 d.f. in numerator andin numerator and

3 d.f. in denominator)3 d.f. in denominator)

Testing for Significance: Testing for Significance: FF TestTest

FF = MSR/MSE = MSR/MSE

Standard Error of the Estimate

Standard Error of Estimate has properties analogous to those of standard deviation.

How “good” is our “fit”?

Interpretation is similar: ~68% of outcomes/predictions within one sest. ~95% of outcomes/predictions within two sest.

Kwatts vs. Temp ExampleANOVA

df SS MS F Significance F

Regression 1 58784708.31 58784708.31 15.19 0.002972726Residual 10 38696916.69 3869691.669

Total 11 97481625

Coefficients Standard Error t Stat P-valueIntercept 319.0414124 3260.412811 0.097853073 0.923982528Temp 185.2702073 47.53479059 3.897570706 0.002972726

Is the regression model statistically significant? Is the coefficient of Temp significant?

Cautions about Interpreting Significance Tests

Statistical significance does not mean Statistical significance does not mean linear linear relationship relationship between between xx and and yy. .

Relationship between Relationship between xx and and yy does not does not meanmean a a cause-and-effect relationshipcause-and-effect relationship is is present between present between xx and and yy..

SAS Enterprise Miner These results can be obtained using Excel or using

a data mining package such as SAS Enterprise Miner 5.3

Using SAS Enterprise Miner requires the following steps: Convert your data (usually in an Excel file) into a SAS data

file Using SAS 9.1 Create a project in Enterprise Miner Within the project:

Create a data source using your SAS data file Create a diagram that includes a data node and a

regression node and a multiplot node for graphs Run the model in the diagram and review the results

Creating a SAS data file from an Excel file: open SAS 9.1. Select File then Import Data

This opens the import wizard. Since the source file is from Excel, click Next. Then click Browse to find the TempKWatts.xls file

Since the data are on sheet1$, click Next. Then enter SASUSER as the Library and TEMPKILOWATTL as the Member. Then click Next

Now click Finish to create your file

Open SAS Enterprise Miner 5.3. Enter the user name and password provided

The Enterprise Window below opens. Select New Project

The Create New Project dialog box appears. Select the General tab, then type the short name of the project, e.g., KWattTemp0. Keep the default path.

In the Startup code tab, enter:libname Ktemps "C:\Documents and Settings\mliberat\My Documents\My SAS Files\9.1\EM_Projects"; This code will be run each time you open the project

The Enterprise Miner application window opens

Right-click on Data Source, opening the wizard. Source is SAS table, so click Next

Browse the SAS libraries to find the SAS table Tempkilowattl found in the SASuser Library (previously created)

Click Next twice. Note that the Table properties shows that we have two variables with 12 observations

The next step controls how Enterprise Miner organizes metadata for the variables in your data. Select advanced, then click next(you can view/change the settings if you click Customize before clicking Next)

Change Role of KWatts to target (outcome variable); change Level of both KWatts and Temp to interval (continuous values); then click Next (Other levels are possible, such as binary). You can click on Explore if you wish to look at some basic stats – we will do this later

Here Role relates to the role of the data set (raw, train, validate, score); raw is fine for our analysis of data, so click Finish

Tempkilowattl now appears under Data Sources in the top left panel called the Project Panel

We need to create a Diagram for our model. Right-click on Diagrams, then enter TempKwatts0 in the dialog box. Now the left panel shows TempKwatts0 as a Diagram, and the right-hand panel is called the Diagram Workspace. Icons can be dragged and dropped onto the Diagram Workspace.

Now add an Input Data Node to the Diagram. From the Data Sources list in the Project Panel drag and drop the Data Source TempKwatts0 onto the Diagram Workspace. Note that when input data node is highlighted, various properties are displayed on the left-hand panel.

If you wish to see the properties of any or all of the variables, highlight the input data node; then on the left hand Properties Panel under Train, click on the box to the right of Variables; in the screen that opens control-click on KWatts and Temp; then click on Explore in the lower right

Frequency distributions for the variables and the raw data are provided. Right-clicking on observations in the lower-left panel will show where they appear in the bar charts. Cancel when finished.

Click on the Explore tab found over the Diagram Workspace, and then drag and drop the Multiplot icon onto the field. Using your cursor, draw a directed arrow from the TempKwattsl icon to the Multiplot icon. With the Multiplot icon highlighted, its properties are found in the left-hand Properties Panel.

Right-click on the Multiplot icon and select Run. After the run is completed select Results from the Run Status window.

Various charts are available as shown below. Descriptive statistics for each variable are given in the lower pane.

Click on the Model tab and drag the Regression icon onto the Model field. Connect the Tempkwattsl icon to the Regression icon. Highlight the Regression icon and on the Property Panel change Regression Type to linear regression.

Run the Regression and select Results. Starting from the upper left and going clockwise, these windows show the fit between target and predicted in percentile terms, the various fit statistics, model output (estimates, F and t stats, R-square), and the two effects (intercept and slope – bars represent size and color represents direction)

For a given percentile, the Target Mean is the actual (or estimated value based on actuals), or what you are trying to predict; the Mean for Predicted is the forecasted values, or the predictions (or estimated values based on forecasts). The results are shown from highest to lowest forecasted values. The distances between the curves shows how well the model predicts the actual data.

A variety of fit statistics are provided. These include SSE, MSE=SSE/(n-2), ASE=SSE/n, RMSE=SQRT(MSE), RASE=SQRT(SSE), FPE = MSE (n+p+1)/n, MAX = largest error in terms of absolute value, where n = no. of observations, p=no. of variables in model (one in our case).Schwartz’s Bayesian Criterion and Akaike’s Information Criterion are used for model selection (comparing one model to another). Schwartz’s adjusts the residual squared error for the number of parameters estimated, while Akaike’s is a relative measure of information lost from fitting the model.

Kwatts vs. Temp Example 2

Another approach to modeling the relationship between Kwatts and Temp is to use a nonlinear regression

This is easily accomplished in Enterprise Miner – highlight the regression node, then in the left hand panel select yes for polynomial terms We use the default of two terms

Is the fit any better???

Multiple RegressionConsider the following data relating family size and income to food expenditures:

family food $ income $ family size1 5.2 28 32 5.1 26 33 5.6 32 24 4.6 24 15 11.3 54 46 8.1 59 27 7.8 44 38 5.8 30 29 5.1 40 110 18 82 611 4.9 42 312 11.8 58 413 5.2 28 114 4.8 20 515 7.9 42 316 6.4 47 117 20 112 618 13.7 85 519 5.1 31 220 2.9 26 2

Multiple Regression We can run this problem in Enterprise Miner using the same

approach followed with the previous example On our model field we have placed the data source called

foodexpenditures, and also both Multiplot and StatExplore found under the Explore tab above the model field

Highlight foodexpenditures, then in the left-hand panel under Training, find variables and click on the box to the right to open up the variables

Change the role of family to rejected (it is just the number of the observation) and change the level of food_ to target, and income_, food_, and fam_size to interval, then click OK

Foodexpenditures Model

Highlight the StatExplore node, right-click to Run, then select Results. Correlations between the input variables and the target are provided, along with basic statistics. The input variables are ordered by the size of the correlations. Now close out the results window and run the regression node and obtain results

Starting from the upper left and going clockwise, these windows show the fit between target and predicted in percentile terms, the various fit statistics, model output (estimates, F and t stats, R-square), and the three effects (intercept and slopes for the two input variables with bars represent size and color represents direction). The model is significant and is a good fit with the data.

What happens in regression analysis when the target variable is binary?

There are many situations when the target variable is binary – some examples: whether a customer will or will not receive credit whether a customer will or will not response to a

promotion Whether a firm will go bankrupt in a year Whether a student will pass an exam!!!

Passing an Exam DataStudent id Outcome Study Hours1 0 32 1 343 0 174 0 65 0 126 1 157 1 268 1 299 0 1410 1 5811 0 212 1 3113 1 2614 0 11

Running a linear regression to predict pass/don’t pass as a function of hours of study provides a model that doesn’t correctly model the data. The data are given in exampassing.xls

Passing an Exam

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hours of s tudy

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Actual

Predicted

The Enterprise Miner results show a poor fit on a percentile basis between predicted and target – another modeling approach is needed.

Similar to linear regression, two main differences Y (outcome or response) is categorical

Yes/No Approve/Reject Responded/Did not respond

Result is expressed as a probability of being in either group.

Logistic Regression

Comparing the Logistic & Linear Regression Models

Logisitic regression

p = Prob(y=1|x) = exp(a+bx)/[1+exp(a+bx)]

1-p =1/[1+exp(a+bx)]

ln [p/(1-p)] = a + bxwhere:

exp or e is the exponential function (e=2.71828…)

ln is the natural logarithm (ln(e) = 1)

p is probability that the event y occurs given x, and can range between 0 and 1

p/(1-p) is the "odds ratio"

ln[p/(1-p)] is the log odds ratio, or "logit"

all other components of the regression model are the same

Odds Ratio Frequently used Related to probability of an event as follows:

Odds Ratio = p/(1-p) Example:

Probability of firm going bankrupt = .25 Odds firm will go bankrupt = .25/(1-.25) = 1/3 or 3 to 1 This is how sports books calculate odds

(e.g., if odds of VU winning a championship are 2:1, probability is 1/3

ln [p/(1-p)] = a + bx means that as x increases by 1, the natural log of the odds ratio increases by b, or the odds ratio increase by a factor of exp(b)

Probability, Odds Ratio, LN of Odds Ratio

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odds

nl(odds)

Running the exam data: Change regression type from linear regression to logistic regressionHighlight the data node; on left-hand panel under Train open variables and change the level of outcome to binary

Results show a much better fit (upper left) and only one misclassification (lower right – a false negative).

The results show that the odds ratio = p(1-p) = exp(-8.4962+0.4949x). For every additional hour of study the odds ratio increases by a factor of exp(0.4949)= 1.640

Understanding Response Rate and LiftTo better understand the top left chart, change cumulative lift to

cumulative % response. The observations are ranked by the predicted probability of response (highest to lowest) for each observation (from the fitted model).

Understanding Response Rate and Lift

Since the first 6 passes were correctly classified, the cumulative % response is 100% through the 40th percentile.

At the 50th percentile the next observation with the highest predicted probability is a non-response, so the cumulative response drops to 6/7 or 85.7%.

The 8th ranked observation, between the 55th and 60th percentile, is a positive response, so the cumulative % response is about 7/8 or 87%. Since there are no more positive responses after the 60th

percentile, the cumulative response rate will drop to 50%. The chart compares how well the cumulative ranked predictions

lead to a match between actual and predicted responses

Understanding Response Rate and Lift Lift calculates the ratio of the actual response rate (passing) of the

top n% of the ranked observations to the overall response rate. Cumulative lift is likewise defined.

At the 50th percentile, the cumulative % response is 88.7%, the cumulative base response is 50%, for a lift of 1.7142.

On the Properties Panel, click on Exported Data to see the predicted probabilities and response for each observation and compare to the actual response.

Logistic regression uses maximum likelihood (and not sum of squared errors) to estimate the model parameters. The results below show that the model is highly significant based on a chi-square test. The Wald chi-square statistic tests whether an effect is significant or not.

Bankruptcy Prediction

To predict bankruptcy a year in advance, you might collect: working capital/total assets (WC/TA) retained earnings/total assets (RE/TA) earnings before interest and taxes/total assets

(EBIT/TA) market value of equity/total debt (MVE/TD) sales/total assets (S/TA)

Bankruptcy Training DataFirm WC/TA RE/TA EBIT/TA MVE/TD S/TA BR/NB1 0.0165 0.1192 0.2035 0.813 1.6702 12 0.1415 0.3868 0.0681 0.5755 1.0579 13 0.5804 0.3331 0.081 0.5755 1.0579 14 0.2304 0.296 0.1225 0.4102 3.0809 15 0.3684 0.3913 0.0524 0.1658 1.1533 16 0.1527 0.3344 0.0783 0.7736 1.5046 17 0.1126 0.3071 0.0839 1.3429 1.5736 18 0.0141 0.2366 0.0905 0.5863 1.4651 19 0.222 0.1797 0.1526 0.3459 1.7237 110 0.2776 0.2567 0.1642 0.2968 1.8904 111 0.2689 0.1729 0.0287 0.1224 0.9277 012 0.2039 -0.0476 0.1263 0.8965 1.0457 013 0.5056 -0.1951 0.2026 0.538 1.9514 014 0.1759 0.1343 0.0946 0.1955 1.9218 015 0.3579 0.1515 0.0812 0.1991 1.4582 016 0.2845 0.2038 0.0171 0.3357 1.3258 017 0.1209 0.2823 -0.0113 0.3157 2.3219 018 0.1254 0.1956 0.0079 0.2073 1.489 019 0.1777 0.0891 0.0695 0.1924 1.6871 020 0.2409 0.166 0.0746 0.2516 1.8524 0

Bankruptcy Example

Using the BankruptTrain.xls data create a SAS data file called bankrupt BR_NB: role is target and level is binary Firm: role is rejected and level is nominal (it is

simply the firm number) Remaining five financial ratio variables: role is input

and level is interval

Create a diagram named bankrupt1. Drag and drop the data node onto the model. Highlight the data node and on the left hand panel under variables click on the box to its right to see the variables data

From the Explore tab drag and drop the StatExplore node onto the diagram and link it to the bankrupt node. Highlight the StatExplore node, right-click and run it, and obtain results. On top, correlations between the five input variables and the target are shown via bars ordered from largest to smallest. Below the mean variable score for bankrupt vs. non-bankrupt observations is shown.

From the Model tab drag and drop the regression node onto the diagram and connect it to the bankrupt node. Highlight the regression node and run, and obtain the results

The results show that the model fits the data very well with highly significant overall chi square statistic, low error values, and 0 misclassifications. Cumulative lift shows that for the top 50% of observations that are bankrupt, they are twice as likely to be classified as bankrupt.

Scoring

Once you have specified a model you might wish to apply it to new data whose outcome is unknown -- make predictions

This can be easily accomplished in Enterprise Miner using scoring

Convert the data set BankruptScore.xls to a SAS file called bankruptscore. The role of this data is score.

Bankruptcy Scoring DataFirm WC/TA RE/TA EBIT/TA MVE/TD S/TAA 0.1759 0.1343 0.0956 0.1955 1.9218B 0.3732 0.3483 -0.0013 0.3483 1.8223C 0.1725 0.3238 0.104 0.8847 0.5576D 0.163 0.3555 0.011 0.373 2.8307E 0.1904 0.2011 0.1329 0.558 1.6623F 0.1123 0.2288 0.01 0.1884 2.7186G 0.0732 0.3526 0.0587 0.2349 1.7432H 0.2653 0.2683 0.0235 0.5118 1.835I 0.107 0.0787 0.0433 0.1083 1.2051J 0.2921 0.239 0.9673 0.3402 0.9277

Drag and drop the bankruptscore data node to the bankrupt1 diagram. From the Assess tab, drag and drop the Score node into the diagram. Link the regression and bankruptscore nodes together and connect them to the Score node.

Run the Score node and obtain the Results. Of the 10 firms, 6 are predicted to become bankrupt.

For details about the individual predictions, highlight the Score node and on the left-hand panel click on the square to the right of Exported Data. Then in the box that appears click on the row whose Port entry is Score. Then click on Explore.

The lower portion of the output is shown below. The predictions are given, along with the probabilities of the firm becoming bankrupt or not.

Regression Using Selection Models

When there are a number of possible input variables, procedures are available to sort through them and include those that have a certain level of statistical significance

SAS Enterprise Miner 5.3 offers three selection methods: Backward Forward Stepwise

Regression Using Selection Models Backward: training begins with all candidate effects

in the model and removes effects until the stay significance level or the stop criterion is met

Forward: training begins with no candidate effects in the model and adds effects until the entry significance level or the stop criterion is met.

Stepwise: training begins as in the forward model but may remove effects already in the model. This continues until the stay significance level or the stop criterion is met

Note that the default significance levels (p values) values are 0.05 and no stop criteria (such as maximum number of steps in the regression) are set

Regression Using Selection Models – Bankruptcy Model

To select stepwise regression

for the bankruptcy model, highlight

the regression node and in the

properties panel under

Selection Model choose

Stepwise. The default significance

level of 0.05 is used

Regression Using Selection Models – Bankruptcy Model

Interestingly, the Training Model only uses RE/TA as a predictor There are 3 misclassifications (.15 rate) in this set vs. 0 in

the original model The results are very different: the original model with

all 5 input variables predicted bankruptcy for G, E, C, and J, while the stepwise model predicted B, C, D, F, G, H, and J would become bankrupt.

Changing the significance levels to 0.1 (to make it easier for input variables to enter/leave the stepwise model) produces the same results