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• Slide 1
• Slide 1 Stepwise Binary Logistic Regression
• Slide 2
• Slide 2 Stepwise Binary Logistic Regression - 1 Stepwise binary logistic regression is very similar to stepwise multiple regression in terms of its advantages and disadvantages. Stepwise logistic regression is designed to find the most parsimonious set of predictors that are most effective in predicting the dependent variable. Variables are added to the logistic regression equation one at a time, using the statistical criterion of reducing the -2 Log Likelihood error for the included variables. After each variable is entered, each of the included variables are tested to see if the model would be better off the variable were excluded. This does not happen often. The process of adding more variables stops when all of the available variables have been included or when it is not possible to make a statistically significant reduction in -2 Log Likelihood using any of the variables not yet included. Nonmetric variables are added to the logistic regression as a group. It is possible, and often likely, that not all of the individual dummy-coded variables will have a statistically significant individual relationship with the dependent variable. We limit our interpretation to the dummy-coded variables that do have a statistically significant individual relationship.
• Slide 3
• Slide 3 Stepwise Binary Logistic Regression - 2 SPSS provides a table of variables included in the analysis and a table of variables excluded from the analysis. It is possible that none of the variables will be included. It is possible that all of the variables will be included. The order of entry of the variables can be used as a measure of relative importance. Once a variable is included, its interpretation in stepwise logistic regression is the same as it would be using other methods for including variables. The number of cases required for stepwise logistics regression is greater than the number for the other forms. We will use the norm of 20 cases for each independent variable, double the recommendation of Hosmer and Lemeshow.
• Slide 4
• Slide 4 Pros and Cons of Stepwise Logistic Regression Stepwise logistic regression can be used when the goal is to produce a predictive model that is parsimonious and accurate because it excludes variables that do not contribute to explaining differences in the dependent variable. Stepwise logistic regression is less useful for testing hypotheses about statistical relationships. It is widely regarded as atheoretical and its usage is not recommended. Stepwise logistic regression can be useful in finding relationships that have not been tested before. Its findings invite one to speculate on why an unusual relationship makes sense. It is not legitimate to do a stepwise logistic regression and present the results as though one were testing a hypothesis that included the variables found to be significant in the stepwise logistic regression. Using statistical criteria to determine relationships is vulnerable to over-fitting the data set used to develop the model at the expense of generalizability. When stepwise logistic regression is used, some form of validation analysis is a necessity. We will use 75/25% cross-validation.
• Slide 5
• Slide 5 75/25% Cross-validation To do cross validation, we randomly split the data set into a 75% training sample and a 25% validation sample. We will use the training sample to develop the model, and we test its effectiveness on the validation sample to test the applicability of the model to cases not used to develop it. In order to be successful, the follow two questions must be answers affirmatively: Did the stepwise logistic regression of the training sample produce the same subset of predictors produced by the regression model of the full data set? If yes, compare the classification accuracy rate for the 25% validation sample to the classification accuracy rate for the 75% training sample. If the shrinkage (accuracy for the 75% training sample - accuracy for the 25% validation sample) is 2% (0.02) or less, we conclude that validation was successful. Note: shrinkage may be a negative value, indicating that the accuracy rate for the validation sample is larger than the accuracy rate for the training sample. Negative shrinkage (increase in accuracy) is evidence of a successful validation analysis. If the validation is successful, we base our interpretation on the model that included all cases.
• Slide 6
• Slide 6 The Problem in Blackboard The problem statement tells us: the variables included in the analysis to make the assumption that it is not necessary to omit outliers whether each variable should be treated as metric or non-metric the type of dummy coding and reference category for non-metric variables the alpha for both the statistical relationships and for diagnostic tests the random number seed for the validation analysis
• Slide 7
• Slide 7 The Statement about Level of Measurement SPSS Binary Logistic Regression will dummy-code categorical variables for us, provided it is useful to use either the first or last category as the reference category. The first statement in the problem asks about level of measurement. Stepwise binary logistic regression requires that the dependent variable be dichotomous, the metric independent variables be interval level, and the non-metric independent variables be dummy-coded if they are not dichotomous. SPSS Binary Logistic Regression calls non-metric variables categorical.
• Slide 8
• Slide 8 Marking the Statement about Level of Measurement The independent variable "socioeconomic index" [sei] is interval level, satisfying the requirement for independent variables The independent variable "sex" [sex] is dichotomous level, satisfying the requirement for independent variables. The independent variable "respondent's degree of religious fundamentalism" [fund] is ordinal level, which the problem instructs us to dummy-code as a non-metric variable. Mark the check box as a correct statement. The dependent variable "attitude toward abortion when a woman wants one for any reason" [abany] is dichotomous level, satisfying the requirement for the dependent variable. variable. The independent variable "age" [age] is interval level, satisfying the requirement for independent variables. The independent variable "highest year of school completed" [educ] is interval level, satisfying the requirement for independent variables. The independent variable "income" [rincom98] is ordinal level, but the problem calls for treating it as metric by applying the common convention of treating ordinal variables as interval level.
• Slide 9
• Slide 9 The statement about multicollinearity and other numerical problems To check for multicolliearity, we run the binary logistic regression in SPSS and check for outliers. Multicollinearity in the logistic regression solution is detected by examining the standard errors for the b coefficients. A standard error larger than 2.0 indicates numerical problems, such as multicollinearity among the independent variables, cells with a zero count for a dummy-coded independent variable because all of the subjects have the same value for the variable, and 'complete separation' whereby the two groups in the dependent event variable can be perfectly separated by scores on one of the independent variables. Analyses that indicate numerical problems should not be interpreted.
• Slide 10
• Slide 10 Running the Stepwise binary logistic regression Select the Regression | Binary Logistic command from the Analyze menu.
• Slide 11
• Slide 11 Selecting the dependent variable Second, click on the right arrow button to move the dependent variable to the Dependent text box. First, highlight the dependent variable abany in the list of variables.
• Slide 12
• Slide 12 Selecting the independent variables First, move the control independent variables stated in the problem "age" [age], "highest year of school completed" [educ], "income" [rincom98], "socioeconomic index" [sei], "sex" [sex] and "respondent's degree of religious fundamentalism" [fund]) to the Covariates list box.
• Slide 13
• Slide 13 Declare the categorical variables - 1 To indicate that "sex" [sex] and "respondent's degree of religious fundamentalism" [fund] are categorical variables, we click on the Categorical button.
• Slide 14
• Slide 14 Declare the categorical variables - 2 Move the variables sex and fund to the Categorical Covariates list box. SPSS assigns its default method for dummy-coding, Indicator coding, to each variable, placing the name of the coding scheme in parentheses after each variable name.
• Slide 15
• Slide 15 Declare the categorical variables - 3 We will also accept the default of using the last category as the reference category for each variable. Click on the Continue button to close the dialog box. We accept the default of using the Indicator method for dummy-coding variable..
• Slide 16
• Slide 16 Specifying the method for including variables Since the problem calls for a Stepwise binary logistic regression, we select the Forward:LR method for including variables. Forward LR uses likelihood ratio tests to determine which variables are entered in what order.
• Slide 17
• Slide 17 Requesting the output While optional statistical output is available, we do not need to request any optional statistics. Click on the OK button to request the output.
• Slide 18
• Slide 18 Checking for multicollinearity The standard errors for the variables included in the stepwsie procedure were: the standard error for "highest year of school completed" [educ] was.09, the standard error for survey respondents who said they were religiously fundamentalist was.56 and the standard error for survey respondents who said they were religiously moderate was.48.
• Slide 19
• Slide 19 Marking the statement about multicollinearity and other numerical problems Since none of the independent variables in this analysis had a standard error larger than 2.0, we mark the check box to indicate there was no evidence of multicollinearity.
• Slide 20
• Slide 20 The statement about sample size Hosmer and Lemeshow, who wrote the widely used text on logistic regression, suggest that the sample size should be 10 cases for every independent variable. Because stepwise procedures tend to overfit the data at the expense of generalizability, we will double the requirement to 20 cases for every independent variable.
• Slide 21
• Slide 21 The output for sample size The 106 cases available for the analysis did not satisfy the recommended sample size of 140 (7 independent variables times 20 cases per variable), which is based on double the recommended number of 10 cases per independent variable for logistic regression recommended by Hosmer and Lemeshow because of the issue of over-fitting the data when using stepwise methods. The failure to meet the sample size requirement should be mentioned as a limitation to the analysis. The number of independent variables includes 4 metric variables and 3 dummy-coded variables. We find the number of cases included in the analysis in the Case Processing Summary.
• Slide 22
• Slide 22 Marking the statement for sample size Since we do not satisfy the sample size requirement, we leave the check box unmarked. We should consider including this as a limitation to the analysis.
• Slide 23
• Slide 23 The stepwise relationship between the dependent and independent variables Three statements in the problem list different combinations of the variables included in the stepwise logistic regression. To determine which is correct, we look at the table of Variables in the Equation for Block 1 in the SPSS output.
• Slide 24
• Slide 24 The output for the stepwise relationship Two independent variables satisfied the statistical criteria for entry into the model. The variable "highest year of school completed" [educ] had the largest individual impact (entered on step 1) on the dependent variable "attitude toward abortion when a woman wants one for any reason" [abany]. The second variable included in the model at step 2 was "respondent's degree of religious fundamentalism" [fund].
• Slide 25
• Slide 25 Marking the statement for stepwise relationship Two independent variables satisfied the statistical criteria for entry into the model. The variable "highest year of school completed" [educ] had the largest individual impact on the dependent variable "attitude toward abortion when a woman wants one for any reason" [abany]. The second variable included in the model was "respondent's degree of religious fundamentalism" [fund]. We mark the first check box in the set of three. Note that in stepwise logistic regression, if any variables are entered, the overall relationship must be significant, since that is the criteria for including variables.
• Slide 26
• Slide 26 The statement about the relationship between education and abortion for any reason Having satisfied the criteria for the stepwise relationship, we examine the findings for individual relationships with the dependent variable. If the overall relationship were not significant, we would not interpret the individual relationships. The first two statements offer alternative interpretations for the relationship between education and abortion for any reason.
• Slide 27
• Slide 27 Output for the relationship between education and abortion for any reason The probability of the Wald statistic for the independent variable "highest year of school completed" [educ] ((1, N = 106) = 5.48, p =.019) was less than or equal to the level of significance of.05. The null hypothesis that the b coefficient for "highest year of school completed" [educ] was equal to zero was rejected. The value of Exp(B) for the variable "highest year of school completed" [educ] was 1.235 which implies an increase in the odds of 23.5% (1.235 - 1.000 =.235). The statement that 'For each unit increase in "highest year of school completed", survey respondents were 23.5% more likely to have thought it should be possible for a woman to obtain a legal abortion if she wants it for any reason' is correct.
• Slide 28
• Slide 28 Marking the statement for relationship between education and abortion for any reason Survey respondents were 23.5% more likely to have thought it should be possible for a woman to obtain a legal abortion if she wants it for any reason, we mark the check box for the second statement.
• Slide 29
• Slide 29 Statement for relationship between fundamentalism and abortion for any reason The next two statements concerns the relationship between the dummy-coded variable for religiously fundamentalist and abortion for any reason.
• Slide 30
• Slide 30 Output for relationship between fundamentalism and abortion for any reason The probability of the Wald statistic for the independent variable survey respondents who said they were religiously fundamentalist ((1, N = 106) = 6.80, p =.009) was less than or equal to the level of significance of.05. The null hypothesis that the b coefficient for survey respondents who said they were religiously fundamentalist was equal to zero was rejected. The value of Exp(B) for the variable survey respondents who said they were religiously fundamentalist was.231 which implies a decrease in the odds of 76.9% (.231 - 1.000 = -.769). The statement that 'Survey respondents who said they were religiously fundamentalist were 76.9% less likely to have thought it should be possible for a woman to obtain a legal abortion if she wants it for any reason compared to those who said they were religiously liberal' is correct.
• Slide 31
• Slide 31 Marking the relationship between fundamentalism and abortion for any reason The statement that 'Survey respondents who said they were religiously fundamentalist were 76.9% less likely to have thought it should be possible for a woman to obtain a legal abortion if she wants it for any reason compared to those who said they were religiously liberal' is correct. The first statement is marked.
• Slide 32
• Slide 32 Statement for relationship between fundamentalism and abortion for any reason The next statement concerns the relationship between the dummy- coded variable for religious moderation and abortion for any reason.
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• Slide 33 Output for relationship between fundamentalism and abortion for any reason The probability of the Wald statistic for the independent variable survey respondents who said they were religiously moderate ((1, N = 106) = 2.87, p =.090) was greater than the level of significance of.05. The null hypothesis that the b coefficient for survey respondents who said they were religiously moderate was equal to zero was not rejected. Survey respondents who said they were religiously moderate does not have an impact on the odds that survey respondents have thought it should be possible for a woman to obtain a legal abortion if she wants it for any reason. The analysis does not support the relationship that 'Survey respondents who said they were religiously moderate were 56.0% less likely to have thought it should be possible for a woman to obtain a legal abortion if she wants it for any reason compared to those who said they were religiously liberal.
• Slide 34
• Slide 34 Marking the relationship between fundamentalism and abortion for any reason Since the relationship was not statistically significant, we do not mark the check box for the statement.
• Slide 35
• Slide 35 Statement for relationship between socioeconomic index and abortion for any reason The next statement concerns the relationship between the metric variable socioeconomic index and abortion for any reason.
• Slide 36
• Slide 36 Output for relationship between socioeconomic index and abortion for any reason The independent variable "socioeconomic index" [sei] was not included among the statistically significant predictors and should not be intepreted. The statement that "For each unit increase in "socioeconomic index", survey respondents were 10.5% more likely to have thought it should be possible for a woman to obtain a legal abortion if she wants it for any reason" is not correct.
• Slide 37
• Slide 37 Marking the relationship between socioeconomic index and abortion for any reason Since the relationship was not statistically significant, the statement is marked.
• Slide 38
• Slide 38 Statement about the usefulness of the model based on classification accuracy The final statement concerns the usefulness of the logistic regression model. The independent variables could be characterized as useful predictors distinguishing survey respondents who use a computer from survey respondents who not use a computer if the classification accuracy rate was substantially higher than the accuracy attainable by chance alone. Operationally, the classification accuracy rate should be 25% or more higher than the proportional by chance accuracy rate.
• Slide 39
• Slide 39 Computing proportional by-chance accuracy rate The proportional by chance accuracy rate was computed by calculating the proportion of cases for each group based on the number of cases in each group in the classification table at Step 0, and then squaring and summing the proportion of cases in each group (.509 +.491 =.500). The proportion in the largest group is 50.9% or.509. The proportion in the other group is 1.0 0.509 =.491. At Block 0 with no independent variables in the model, all of the cases are predicted to be members of the modal group, 0=NO in this example.
• Slide 40
• Slide 40 Output for the usefulness of the model based on classification accuracy To be characterized as a useful model, the accuracy rate should be 25% higher than the by chance accuracy rate. The by chance accuracy criteria is computed by multiplying the by chance accurate rate of.500 times 1.25, or 1.25 x.500 =.625 (62.5%).. The classification accuracy rate computed by SPSS was 67.9% which was greater than or equal to the proportional by chance accuracy criteria of 62.5% (1.25 x 50.0% = 62.5%). The criteria for classification accuracy is satisfied.
• Slide 41
• Slide 41 Marking the statement for usefulness of the model Since the criteria for classification accuracy was satisfied, the check box is marked.
• Slide 42
• Slide 42 Statement about Cross-validation The final statement concerns the generalizability of our findings to the larger population. To answer this question, we will do a 75/25% cross-validation. The findings from our analysis are generalizable to the extent that they are applicable to cases not included in the analysis. Since we cannot collect new cases, we will divide our sample into two subsets, using one subset to create the model and test the findings on the second subset of cases which were not included in the analysis that created the model.
• Slide 43
• Slide 43 Creating the Training Sample and the Validation Sample - 1 The 75/25% cross-validation requires that we randomly divide the cases for this analysis into two parts: 75% of the cases will be used to run the stepwise logistic regression (the training sample), which will be tested for accuracy on the remaining 25% of the cases (the validation sample). To set the seed for the random number generator, select Random Number Generator from the Transform menu. NOTE: you must use the random number seed that is stated in the problem in order to produce the same results that I found. Any other seed will generate a different random sequence that can produce results that are very different from mine.
• Slide 44
• Slide 44 Creating the Training Sample and the Validation Sample - 2 Third, type the seed number provided in the problem directions: 981982. First, mark the check for Set Starting Point. Second, select the option button for a Fixed Value. Fourth, click on the OK button to complete the action. NOTE: SPSS does not provide any feedback that the seed has been set or changed. If you are in doubt, you can reopen the dialog box and see what it indicates.
• Slide 45
• Slide 45 Creating the Training Sample and the Validation Sample - 3 We will create a variable that will contain the information about whether a case is in the training sample or the validation sample. We will name this variable split and use a value of 1 to indicate the training sample and a value of 0 to indicate the validation sample. To create the new variable, select Compute from the Transform menu.
• Slide 46
• Slide 46 Creating the Training Sample and the Validation Sample - 4 Type the name of the new variable, split, in the Target Variable text box. Type the formula as shown in the Numeric Expression text box. Click on the OK button to create the variable. The formula uses the SPSS UNIFORM function to create a uniform distribution of decimal numbers between 0 and 1. If the generated number for a case is less than or equal to 0.75, the statement in the text box is True and the split variable will be assigned a 1 for that case. If the generated number is larger than 0.75, the statement is false and the case will be assigned a 0 for split.
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• Slide 47 Creating the Training Sample and the Validation Sample - 5 If we scroll the data editor window to the right, we see the split variable in a new column.
• Slide 48
• Slide 48 Creating the Training Sample and the Validation Sample - 6 If we created a frequency distribution for the split variable, we see that the breakdown is approximately, not exactly, correct. This is a consequence of generating random numbers you have no control over the sequence that it generates beyond setting an initial seed. Though I have done it to create specific results for homework problems, it is not acceptable to run repeated series of random numbers until one gets a sequence that has desirable properties.
• Slide 49
• Slide 49 An Additional Task before Running the Stepwise Logistic Regression on Training Sample Before we run the regression on the training sample, we need an additional step that will enable us to compare the accuracy of the model for the training sample to the accuracy of the model for the validation sample, using the R 2 for each as our measure of accuracy. We need to exclude from the analysis cases that are missing data for any of the variables that we have designated as candidates for inclusion. If we dont specifically do this, SPSS may include different cases in predicting values for the dependent variable than it does in determining which variables to include in the model. In model building, SPSS does listwise exclusion of missing data and omits any cases that have missing data for any variable. In predicting scores on the dependent variable, it excludes cases that are missing data for only the variables included in the stepwise model. Thus, when selecting variables, SPSS assumes that only respondents who answer all questions are valid cases; in predicting scores, it assumes that failing to answer a question on a variable that is not included has no importance in the analysis.
• Slide 50
• Slide 50 Selecting Cases with Valid Data for All Variables in the Analysis - 1 To include only those cases that have valid data for all variables in the analysis, choose the Select Cases command from the Data menu.
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• Slide 51 Selecting Cases with Valid Data for All Variables in the Analysis - 2 First, mark the option button for If condition is satisfied. Second, click on the If button to add the condition.
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• Slide 52 Selecting Cases with Valid Data for All Variables in the Analysis - 3 Type NMISS(abany,age,educ,sex,rincom98,fund,sei) = 0 in the condition textbox. In the parentheses, we type the names of the dependent variable and all of the independent variables. The SPSS NMISS function counts the number of variables in the list that have missing data. Telling SPSS to include cases for which this calculation results in 0 indicates that the case was not missing data for any of the variables.
• Slide 53
• Slide 53 Selecting Cases with Valid Data for All Variables in the Analysis - 4 Click on the Continue button to close the dialog box.
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• Slide 54 Selecting Cases with Valid Data for All Variables in the Analysis - 5 Click on the OK button to execute the command.
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• Slide 55 Selecting Cases with Valid Data for All Variables in the Analysis - 6 The excluded cases have a slash through the case number.
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• Slide 56 Run the Stepwise Logistic Regression on the Training Sample - 1 To run the logistic regression, select Regression > Binary Logisitic from the Analyze menu.
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• Slide 57 Run the Stepwise Logistic Regression on the Training Sample - 2 Move the dependent variable: "attitude toward abortion when a woman wants one for any reason" [abany] to the Dependent text box. Move the control independent variables stated in the problem "age" [age], "highest year of school completed" [educ], "sex" [sex] and "respondent's degree of religious fundamentalism" [fund]) "income" [rincom98], "socioeconomic index" [sei], to the Covariates list box.
• Slide 58
• Slide 58 Run the Stepwise Logistic Regression on the Training Sample - 3 To indicate that "sex" [sex] and "respondent's degree of religious fundamentalism" [fund] are categorical variables, we click on the Categorical button.
• Slide 59
• Slide 59 Run the Stepwise Logistic Regression on the Training Sample - 4 Move the variables sex and fund to the Categorical Covariates list box. Click on the Continue button to close the dialog box.
• Slide 60
• Slide 60 Run the Stepwise Logistic Regression on the Training Sample - 5 Since the problem calls for a Stepwise binary logistic regression, we select the Forward:LR method for including variables. Forward LR uses likelihood ratio tests to determine which variables are entered in what order.
• Slide 61
• Slide 61 Run the Stepwise Logistic Regression on the Training Sample - 6 First, highlight the split variable. To select the training sample, we move the split variable to the Selection Variable text box. Second, click on the right arrow button to the left of the Selection Variable text box..
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• Slide 62 Run the Stepwise Logistic Regression on the Training Sample - 7 Click on the Rule button to specify the value that we want split to use to select cases.
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• Slide 63 Run the Stepwise Logistic Regression on the Training Sample - 7 First, type 1 in the Value text box. Recall that this is the value of split indicating training cases. Second, click on the Continue button to close the dialog box.
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• Slide 64 Run the Stepwise Logistic Regression on the Training Sample - 8 Click on the OK button to produce the output.
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• Slide 65 Validating the Model - 1 If the number of steps were different, the validation would fail. The stepwise binary logistic regression of the training sample resulted in the same number of steps as the full sample model (2).
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• Slide 66 Validating the Model - 2 If the variables included were different, the validation would fail. The same variables were selected in the stepwise logistic regression of the training sample that were selected in the stepwise logistic regression of the full sample "highest year of school completed" [educ], "respondent's degree of religious fundamentalism" [fund].
• Slide 67
• Slide 67 Validating the Model - 3 Third, we compare the accuracy of the model for the validation sample to the accuracy of the model for the training sample. The classification accuracy rate for the model using the training sample was 67.9%, compared to 72.7% for the validation sample. The classification accuracy for the validation sample was actually larger than the classification accuracy for the training sample, implying a better fit than obtained for the training sample. This supports a conclusion that the logistic regression model based on this analysis would be effective in predicting scores for cases other than those included in the sample.
• Slide 68
• Slide 68 Marking the Check Box for the Cross-validation Statement The validation analysis supported the generalizability of the findings of the analysis to the population represented by the sample in the data set. We mark the check box for the validation.
• Slide 69
• Slide 69 Stepwise Binary Logistic Regression: Level of Measurement No Ordinal level variable treated as metric? Yes Level of measurement ok? Consider limitation in discussion of findings Mark check box for level of measurement Do not mark check box for level of measurement Mark: Inappropriate application of the statistic Stop
• Slide 70
• Slide 70 Stepwise Binary Logistic Regression: Multicollinearity and Sample Size No Yes Multicollinearity/Numeric al Problems (S. E. > 2.0) Stop Yes No Adequate Sample Size (Number of IVs x 20) Consider limitation in discussion of findings Mark check box for no multicollinearity Do not mark check box for no multicollinearity Mark check box for sample size Do not mark check box for sample size Run Stepwise Binary Logistic Regression, Assuming that it is not necessary to remove any outliers
• Slide 71
• Slide 71 Logic Diagram for Solving Homework Problems: Stepwise Relationship 1+ variables entered in model? No Yes Stop (no significant predictors) Note: model will be statistically significant if any variables entered Do not mark check box for correct subset Yes Parsimonious subset of variables correctly identified? No Mark check box for correct subset
• Slide 72
• Slide 72 Stepwise Binary Logistic Regression: Individual Relationships Yes Individual relationship (Wald Sig )? No Mark check box for individual relationship Correct interpretation of direction and strength of relationship? Yes Do not mark check box for individual relationship No Additional individual Relationships to interpret? Yes No For each of the variables included by the stepwise procedure.
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• Slide 73 Stepwise Binary Logistic Regression: Classification Accuracy Yes Classification accuracy = or > 1.25 x by chance accuracy rate Do not mark check box for classification accuracy No Mark check box for classification accuracy Stop (the model does meet criteria for usefulness)
• Slide 74
• Slide 74 Stepwise Binary Logistic Regression: Cross-validation Create split variable using specified seed Select cases with no missing values for all variables Run stepwise logistic regression on training sample Same variables entered in full model? Yes Do not mark check box for supporting validation No Shrinkage for accuracy rate < or = 2%? Yes Mark check box for supporting validation No Do not mark check box for supporting validation