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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.

Intercept Term

Predictor

variable

coefficient

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.

E(y) = b0 + b1 (x)

x y

x1 y1

. .

. .

xnyn

Estimated

Regression Equation

Sample Statistics

b0, b1

Estimation ProcessRegression Model

y = b0 + b1x +e

Regression Equation

E(y) = b0 + b1x

Unknown Parameters

b0, b1

b0 and b1

provide estimates of

b0 and b1

Simple Linear Regression Equation:Positive Linear Relationship

E(y): Outcome

Regression line

Intercept

b0

Slope b1

is positive

x : Predictor

Simple Linear Regression Equation:Negative Linear Relationship

E(y): Outcome

Regression line

Intercept

b0

Slope b1

is negative

x: Predictor

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Simple Linear Regression Equation:No Relationship

E(y): Outcome

x: Predictor

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Simple Linear Regression Equation:Parabolic Relationship

E(y): Outcome

Intercept

b0

x: Predictor

Example

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

yi = observed value of the dependent variable

for the ith observation

^

yi = estimated value of the dependent variable

for the ith observation

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

Least Squares Method

- Slope for the Estimated Regression Equation

_

x= mean value for independent variable

y= mean value for dependent variable

Least Squares Method

- y-Intercept for the Estimated Regression Equation

where:

xi = value of independent variable for ith

observation

yi = value of dependent variable for ith

observation

n = total number of observations

Actual Data

Least Squares Estimation Procedure- Least Squares Criterion:

The sum of the vertical deviations (y axis) of the points from the line is minimal.

Example: Kwatts vs. Temp

Temp Kwatts

59.2 9,730

61.9 9,750

55.1 10,180

66.2 10,230

52.1 10,800

69.9 11,160

46.8 12,530

76.8 13,910

79.7 15,110

79.3 15,690

80.2 17,020

83.3 17,880

where:

SST = total sum of squares

SSR = sum of squares due to regression

SSE = sum of squares due to error

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

Coefficient of Determination (r2)

r2 = SSR/SST

where:

SSR = sum of squares due to regression

SST = total sum of squares

Kwatts vs. Temp Example

df SS

Regression 1 58784708.31

Residual 10 38696916.69

Total 11 97481625

r2 = 0.603033734

Does the linear regression provide a good fit?

Assumptions About the Error Term e

1. The erroris a random variable with mean of zero.

2.The variance of , denoted by 2, is the same for

all values of the independent variable.

3.The values of are independent.

4.The erroris a normally distributed random

variable.

Significance Test for Regression

Is the value of b1zero?

Two tests are commonly used:

F Test

t Test

and

Both thettest and Ftest require an estimate of the

variance (s2) of the error (e).

As in most of our statistical work, we are working with

a sample, not the population, so we use

mean square error (s2).

Testing for Significance

- An Estimate of s

- To estimate swe take the square root of s 2.

- The resulting sis called the standard error of
- the estimate.

Testing for Significance: t Test

- Hypotheses: Coefficient (b1) is 0

(no relationship between predictor & outcome)

- Calculating t Statistic:

Testing for Significance: t Test

1. Determine if

2. Specify the level of significance.

a = .05

3. Select the test statistic.

4. State the rejection rule.

Reject if

p-value < .05 or |t| > 3.182 (with 3 degrees of freedom)

Testing for Significance: FTest

- Reject if: p-value<a or F>F

F = MSR/MSE

where:

Fis based on an Fdistribution with

1 degree of freedom in the numerator and

n- 2 degrees of freedom in the denominator

Testing for Significance: FTest

1. Determine if

2. Specify the level of significance.

a = .05

3. Select the test statistic.

F = MSR/MSE

4. State the rejection rule.

Reject if

p-value < .05 or F > 10.13

(with1 d.f. in numerator and

3 d.f. in denominator)

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 Example

ANOVA

df SS MS F Significance F

Regression 1 58784708.31 58784708.31 15.19 0.002972726

Residual 10 38696916.69 3869691.669

Total 11 97481625

Coefficients Standard Error t Stat P-value

Intercept 319.0414124 3260.412811 0.097853073 0.923982528

Temp 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 linear relationship between x and y.
- Relationship between x and ydoes not mean a cause-and-effect relationship is present between x and y.

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

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

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

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 Regression

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

family food $ income $ family size

1 5.2 28 3

2 5.1 26 3

3 5.6 32 2

4 4.6 24 1

5 11.3 54 4

6 8.1 59 2

7 7.8 44 3

8 5.8 30 2

9 5.1 40 1

10 18 82 6

11 4.9 42 3

12 11.8 58 4

13 5.2 28 1

14 4.8 20 5

15 7.9 42 3

16 6.4 47 1

17 20 112 6

18 13.7 85 5

19 5.1 31 2

20 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

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 Data

Student id Outcome Study Hours

1 0 3

2 1 34

3 0 17

4 0 6

5 0 12

6 1 15

7 1 26

8 1 29

9 0 14

10 1 58

11 0 2

12 1 31

13 1 26

14 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

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

Logistic Regression

- 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.

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 + bx

where:

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)

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 Lift

To 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 Data

Firm WC/TA RE/TA EBIT/TA MVE/TD S/TA BR/NB

1 0.0165 0.1192 0.2035 0.813 1.6702 1

2 0.1415 0.3868 0.0681 0.5755 1.0579 1

3 0.5804 0.3331 0.081 0.5755 1.0579 1

4 0.2304 0.296 0.1225 0.4102 3.0809 1

5 0.3684 0.3913 0.0524 0.1658 1.1533 1

6 0.1527 0.3344 0.0783 0.7736 1.5046 1

7 0.1126 0.3071 0.0839 1.3429 1.5736 1

8 0.0141 0.2366 0.0905 0.5863 1.4651 1

9 0.222 0.1797 0.1526 0.3459 1.7237 1

10 0.2776 0.2567 0.1642 0.2968 1.8904 1

11 0.2689 0.1729 0.0287 0.1224 0.9277 0

12 0.2039 -0.0476 0.1263 0.8965 1.0457 0

13 0.5056 -0.1951 0.2026 0.538 1.9514 0

14 0.1759 0.1343 0.0946 0.1955 1.9218 0

15 0.3579 0.1515 0.0812 0.1991 1.4582 0

16 0.2845 0.2038 0.0171 0.3357 1.3258 0

17 0.1209 0.2823 -0.0113 0.3157 2.3219 0

18 0.1254 0.1956 0.0079 0.2073 1.489 0

19 0.1777 0.0891 0.0695 0.1924 1.6871 0

20 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 Data

Firm WC/TA RE/TA EBIT/TA MVE/TDS/TA

A 0.1759 0.1343 0.09560.19551.9218

B 0.3732 0.3483 -0.0013 0.3483 1.8223

C 0.1725 0.3238 0.104 0.8847 0.5576

D 0.163 0.3555 0.0110.373 2.8307

E 0.1904 0.2011 0.1329 0.558 1.6623

F 0.1123 0.2288 0.01 0.1884 2.7186

G 0.0732 0.3526 0.0587 0.2349 1.7432

H 0.2653 0.2683 0.0235 0.5118 1.835

I 0.107 0.0787 0.0433 0.1083 1.2051

J 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

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