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Stepwise Binary Logistic Regression

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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.

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.

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.

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.

The Problem in Blackboard

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

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 Regressioncalls non-metric variables “categorical.”

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 Statement about Level of MeasurementMarking the Statement about Level of Measurement measurement.

- 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.

- 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 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.

Running the Stepwise binary logistic regression problems

Select the Regression | Binary Logistic… command from the Analyze menu.

Selecting the dependent variable problems

First, highlight the dependent variable abany in the list of variables.

Second, click on the right arrow button to move the dependent variable to the Dependent text box.

Selecting the independent variables problems

- 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.

Declare the categorical variables - 1 problems

To indicate that "sex" [sex] and "respondent's degree of religious fundamentalism" [fund] are categorical variables, we click on the Categorical button.

Declare the categorical variables - 2 problems

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.

Declare the categorical variables - 3 problems

We accept the default of using the Indicator method for dummy-coding variable..

Click on the Continue button to close the dialog box.

We will also accept the default of using the last category as the reference category for each variable.

Specifying the method for including variables problems

Since the problem calls for a Stepwise binary logistic regression, we select the Forward:LRmethod for including variables.

Forward LR uses likelihood ratio tests to determine which variables are entered in what order.

Requesting the output problems

Click on the OK button to request the output.

While optional statistical output is available, we do not need to request any optional statistics.

Checking for multicollinearity problems

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.

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.

The statement about sample size numerical problems

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.

The output for sample size numerical problems

We find the number of cases included in the analysis in the Case Processing Summary.

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.

Marking the statement for sample size numerical problems

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.

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.

The output for the stepwise relationship independent variables

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].

Marking the statement for stepwise relationship independent variables

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.

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.

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] (χ²(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.

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.

Statement for relationship between abortion for any reasonfundamentalism 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.

Output for relationship between abortion for any reasonfundamentalism and abortion for any reason

The probability of the Wald statistic for the independent variable survey respondents who said they were religiously fundamentalist (χ²(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.

Marking the relationship between abortion for any reasonfundamentalism 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.

Statement for relationship between abortion for any reasonfundamentalism and abortion for any reason

The next statement concerns the relationship between the dummy-coded variable for religious moderation and abortion for any reason.

Output for relationship between abortion for any reasonfundamentalism and abortion for any reason

The probability of the Wald statistic for the independent variable survey respondents who said they were religiously moderate (χ²(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‘.

Marking the relationship between abortion for any reasonfundamentalism and abortion for any reason

Since the relationship was not statistically significant, we do not mark the check box for the statement.

Statement for relationship between abortion for any reasonsocioeconomic index and abortion for any reason

The next statement concerns the relationship between the metric variable socioeconomic index and abortion for any reason.

Output for relationship between abortion for any reasonsocioeconomic 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.

Marking the relationship between abortion for any reasonsocioeconomic index and abortion for any reason

Since the relationship was not statistically significant, the statement is marked.

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.

Computing proportional by-chance accuracy rate classification accuracy

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.

The proportion in the largest group is 50.9%% or .509. The proportion in the other group is 1.0 – 0.509 = .491.

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 (.509² + .491² = .500).

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.

Marking the statement for usefulness of the model classification accuracy

Since the criteria for classification accuracy was satisfied, the check box is marked.

Statement about Cross-validation classification accuracy

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.

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.

Creating the Training Sample and the Validation Sample - 1 classification accuracy

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.

Creating the Training Sample and the Validation Sample - 2 classification accuracy

First, mark the check for Set Starting Point.

Fourth, click on the OK button to complete the action.

Second, select the option button for a Fixed Value.

Third, type the seed number provided in the problem directions: 981982.

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.

Creating the Training Sample and the Validation Sample - 3 classification accuracy

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.

Creating the Training Sample and the Validation Sample - 4 classification accuracy

Type the formula as shown in the Numeric Expression text box.

Type the name of the new variable, split, in the Target Variable text box.

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.

Click on the OK button to create the variable.

Creating the Training Sample and the Validation Sample - 5 classification accuracy

If we scroll the data editor window to the right, we see the split variable in a new column.

Creating the Training Sample and the Validation Sample - 6 classification accuracy

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.

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 R2 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 don’t 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.

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.

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.

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.

Selecting Cases with Valid Data for All Variables in the Analysis - 4

Click on the Continue button to close the dialog box.

Selecting Cases with Valid Data for All Variables in the Analysis - 5

Click on the OK button to execute the command.

Selecting Cases with Valid Data for All Variables in the Analysis - 6

The excluded cases have a slash through the case number.

Run the Stepwise Logistic Regression on the Training Sample - 1

To run the logistic regression, select Regression > Binary Logisitic from the Analyze menu.

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.

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.

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.

Run the Stepwise Logistic Regression on the Training Sample - 5

Since the problem calls for a Stepwise binary logistic regression, we select the Forward:LRmethod for including variables.

Forward LR uses likelihood ratio tests to determine which variables are entered in what order.

Run the Stepwise Logistic Regression on the Training Sample - 6

To select the training sample, we move the split variable to the Selection Variable text box.

First, highlight the split variable.

Second, click on the right arrow button to the left of the Selection Variable text box..

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.

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.

Run the Stepwise Logistic Regression on the Training Sample - 8

Click on the OK button to produce the output.

Validating the Model - 1 - 8

The stepwise binary logistic regression of the training sample resulted in the same number of steps as the full sample model (2).

If the number of steps were different, the validation would fail.

Validating the Model - 2 - 8

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].

If the variables included were different, the validation would fail.

Validating the Model - 3 - 8

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.

Marking the Check Box for the Cross-validation Statement - 8

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.

No - 8

Yes

Yes

No

Stepwise Binary Logistic Regression: Level of MeasurementLevel of

measurement ok?

Do not mark check box

for level of measurement

Mark: Inappropriate application of the statistic

Stop

Mark check box

for level of measurement

Ordinal level variable

treated as metric?

Consider limitation in discussion of findings

Yes - 8

No

No

Yes

Stepwise Binary Logistic Regression: Multicollinearity and Sample SizeRun Stepwise Binary Logistic Regression,

Assuming that it is not necessary to remove any outliers

Multicollinearity/Numerical Problems (S. E. > 2.0)

Do not mark check box

for no multicollinearity

Stop

Mark check box

for no multicollinearity

Adequate Sample Size

(Number of IV’s x 20)

Do not mark check box

for sample size

Consider limitation in discussion of findings

Mark check box

for sample size

No - 8

No

Yes

Yes

Logic Diagram for Solving Homework Problems: Stepwise Relationship1+ variables entered

in model?

Stop (no significant predictors)

Note: model will be statistically significant if any variables entered

Parsimonious subset of variables correctly identified?

Do not mark check box for correct subset

Mark check box

for correct subset

Yes - 8

Yes

No

No

Stepwise Binary Logistic Regression: Individual RelationshipsFor each of the variables included by the stepwise procedure.

Individual relationship

(Wald Sig ≤ α)?

No

Correct interpretation of direction and strength of relationship?

Do not mark check box for individual relationship

Mark check box

for individual relationship

Additional individual

Relationships to interpret?

Yes

Yes - 8

No

Stepwise Binary Logistic Regression: Classification AccuracyClassification accuracy = or > 1.25 x by chance

accuracy rate

Do not mark check box for classification accuracy

Stop (the model does meet criteria for usefulness)

Mark check box for classification accuracy

Yes - 8

Yes

No

No

Stepwise Binary Logistic Regression: Cross-validationCreate 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?

Do not mark check box for supporting validation

Shrinkage for accuracy rate < or = 2%?

Do not mark check box for supporting validation

Mark check box for supporting validation

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