Stepwise Multiple Regression

Stepwise Multiple Regression
Differences between stepwise and other methods of multiple regression Sample problem Steps in stepwise multiple regression Homework Problems

Types of multiple regression
Different types of multiple regression are distinguished by the method for entering the independent variables into the analysis. In standard (or simultaneous) multiple regression, all of the independent variables are entered into the analysis at the same. In hierarchical (or sequential) multiple regression, the independent variables are entered in an order prescribed by the analyst. In stepwise (or statistical) multiple regression, the independent variables are entered according to their statistical contribution in explaining the variance in the dependent variable. No matter what method of entry is chosen, a multiple regression that includes the same independent variables and the same dependent variables will produce the same multiple regression equation.

Stepwise multiple regression
Stepwise 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 regression equation one at a time, using the statistical criterion of maximizing the R² of the included variables. 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 improvement in R² using any of the variables not yet included. Since variables will not be added to the regression equation unless they make a statistically significant addition to the analysis, all of the independent variable selected for inclusion will have a statistically significant relationship to the dependent variable.

Differences in statistical outputs
Each time SPSS includes or removes a variable from the analysis, SPSS considers it a new step or model, i.e. there will be one model and result for each variable included in the analysis. 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 regression is the same as it would be using other methods for including regression variables.

Differences in solving stepwise regression problems
The level of significance for the analysis is included in the specifications for the statistical analysis. While we will use 0.05 as the level of significance for our problems, a different level of significance can be entered in the SPSS Options dialog box. The preferred sample size requirement is larger for stepwise regression, i.e. 50 x the number of independent variables. Stepwise procedures are notorious for over-fitting the sample to the detriment of generalizability. Validation analysis is absolutely necessary. If generalizability is compromised, it is permissible to interpret the variables included in the 75% training analysis (though we will not do this in our problems). While multicollinearity for all variable can be examined, it is really only a problem for the variables not included in the analysis. If a variable is included in the stepwise analysis, it will not have a collinear relationship.

A stepwise regression problem
When the problem asks us to identify the best set of predictors, we will do stepwise multiple regression. Multiple regression is feasible if the dependent variable is metric and the independent variables (both predictors and controls) are metric or dichotomous, and the available data is sufficient to satisfy the sample size requirements.

Level of measurement - answer
Stepwise multiple regression requires that the dependent variable be metric and the independent variables be metric or dichotomous. True with caution is the correct answer.

Sample size - question The second question asks about the sample size requirements for multiple regression. To answer this question, we will run the initial or baseline multiple regression to obtain some basic data about the problem and solution.

The baseline regression - 1
After we check for violations of assumptions and outliers, we will make a decision whether we should interpret the model that includes the transformed variables and omits outliers (the revised model), or whether we will interpret the model that uses the untransformed variables and includes all cases including the outliers (the baseline model). In order to make this decision, we run the baseline regression before we examine assumptions and outliers, and record the R² for the baseline model. If using transformations and outliers substantially improves the analysis (a 2% increase in R²), we interpret the revised model. If the increase is smaller, we interpret the baseline model. To run the baseline model, select Regression | Linear… from the Analyze model.

First, move the dependent variable rincom98 to the Dependent text box. Second, move the independent variables hrs1, wkrslf, and prestg80 to the Independent(s) list box. Third, select the method for entering the variables into the analysis from the drop down Method menu. In this example, we select Stepwise to request the best subset of variables.

Click on the Statistics… button to specify the statistics options that we want.

Second, mark the checkboxes for Model Fit, Descriptives, and R squared change. The R squared change statistic will tell us the contribution of each additional variable that the stepwise procedure adds to the analysis. First, mark the checkboxes for Estimates on the Regression Coefficients panel. Fifth, click on the Continue button to close the dialog box. Fourth, mark the the Collinearity diagnostics to get tolerance values for testing multicollinearity. Third, mark the Durbin-Watson statistic on the Residuals panel.

Next, we need to specify the statistical criteria to use for including variables in the analysis. Click on the Options button.

First, the default level of significance for entering variables to the regression equation is .05. Since that is the alpha level for our problem we do not need to make any change. The criteria for removing a variable from the analysis is usually set at twice the level for including variables. Second, click on the Continue button to close the dialog box.

Click on the OK button to request the regression output.

R² for the baseline model
The R² of is the benchmark that we will use to evaluate the utility of transformations and the elimination of outliers. Prior to any transformations of variables to satisfy the assumptions of multiple regression or the removal of outliers, the proportion of variance in the dependent variable explained by the independent variables (R²) was 25.7%. In stepwise regression, the relationship will always be significant if any variables are included because the variables can only be included if they contributed to a statistically significant relationship. In stepwise regression, the model number corresponds to the number of variables included in the stepwise analysis. Two variables are included in this problem.

Sample size – evidence and answer
Stepwise multiple regression requires that the minimum ratio of valid cases to independent variables be at least 5 to 1. The ratio of valid cases (145) to number of independent variables (3) was 48.3 to 1, which was equal to or greater than the minimum ratio. The requirement for a minimum ratio of cases to independent variables was satisfied. However, the ratio of 48.3 to 1 did not satisfy the preferred ratio of 50 cases per independent variable. A caution should be added to the interpretation of the analysis and validation analysis should be conducted. True with caution is the correct answer.

Assumption of normality for the dependent variable - question
Having satisfied the level of measurement and sample size requirements, we turn our attention to conformity with three of the assumptions of multiple regression: normality, linearity, and homoscedasticity. First, we will evaluate the assumption of normality for the dependent variable.

Run the script to test normality
First, move the variables to the list boxes based on the role that the variable plays in the analysis and its level of measurement. Second, click on the Assumption of Normality option button to request that SPSS produce the output needed to evaluate the assumption of normality. Fourth, click on the OK button to produce the output. Third, mark the checkboxes for the transformations that we want to test in evaluating the assumption.

Normality of the dependent variable: respondent’s income
The dependent variable "income" [rincom98] satisfied the criteria for a normal distribution. The skewness of the distribution (-0.686) was between -1.0 and +1.0 and the kurtosis of the distribution (-0.253) was between -1.0 and +1.0. True is the correct answer.

Normality of the independent variable: hrs1
Next, we will evaluate the assumption of normality for the independent variable, number of hours worked in the past week.

Normality of the independent variable: number of hours worked in the past week
The independent variable "number of hours worked in the past week" [hrs1] satisfied the criteria for a normal distribution. The skewness of the distribution (-0.324) was between -1.0 and +1.0 and the kurtosis of the distribution (0.935) was between -1.0 and +1.0. True is the correct answer.

Normality of the independent variable: prestg80
Finally, we will evaluate the assumption of normality for the independent variable, The independent variable "occupational prestige score" [prestg80]

Normality of the second independent variable: occupational prestige score
The independent variable "occupational prestige score" [prestg80] satisfied the criteria for a normal distribution. The skewness of the distribution (0.401) was between -1.0 and +1.0 and the kurtosis of the distribution (-0.630) was between -1.0 and +1.0. True is the correct answer.

Assumption of linearity for respondent’s income and number of hours worked last week - question
All of the metric variables included in the analysis satisfied the assumption of normality. Next we will test the relationships for linearity.

Run the script to test linearity
First, click on the Assumption of Linearity option button to request that SPSS produce the output needed to evaluate the assumption of linearity. When the linearity option is selected, a default set of transformations to test is marked. Second, click on the OK button to produce the output.

Linearity test: respondent’s income and number of hours worked last week
True is the correct answer. The correlation between "number of hours worked in the past week" and "income" was statistically significant (r=.337, p<0.001). A linear relationship exists between these variables.

Assumption of linearity for respondent’s income and occupational prestige score - question
All of the metric variables included in the analysis satisfied the assumption of normality. Next we will test the relationships for linearity.

Linearity test: respondent’s income and occupational prestige score
True is the correct answer. The correlation between "occupational prestige score" and "income" was statistically significant (r=.440, p<0.001). A linear relationship exists between these variables.

Assumption of homogeneity of variance - question
Self-employment is the only dichotomous independent variable in the analysis. We will test if for homogeneity of variance using income as the dependent variable.

Run the script to test homogeneity of variance
First, click on the Assumption of Homogeneity option button to request that SPSS produce the output needed to evaluate the assumption of linearity. When the homogeneity of variance option is selected, a default set of transformations to test is marked. Second, click on the OK button to produce the output.

Assumption of homogeneity of variance
Based on the Levene Test, the variance in "income" [rincom98] is homogeneous for the categories of "self-employment" [wrkslf]. The probability associated with the Levene Statistic (p=0.076) is greater than the level of significance (0.01), so we fail to reject the null hypothesis that the variance is equal across groups, and conclude that the homoscedasticity assumption is satisfied. The homogeneity of variance assumption was satisfied. True is the correct answer.

Detection of outliers - question
In multiple regression, an outlier in the solution can be defined as a case that has a large residual because the equation did a poor job of predicting its value. We will run the baseline regression again and have SPSS compute the standardized residual for each case. Cases with a standardized residual larger than +/- 3.0 will be treated as outliers.

Re-running the baseline regression - 1
Having decided to use the baseline model for the interpretation of this analysis, the SPSS regression output was re-created. To run the baseline model, select Regression | Linear… from the Analyze model.

First, move the dependent variable rincom98 to the Dependent text box. Second, move the independent variables hrs1, wkrslf, and prestg80 to the Independent(s) list box. Third, select the method for entering the variables into the analysis from the drop down Method menu. In this example, we select Stepwise to request the best subset of variables.

Click on the Statistics… button to specify the statistics options that we want.

Second, mark the checkboxes for Model Fit, Descriptives, and R squared change. The R squared change statistic will tell us whether or not the variables added after the controls have a relationship to the dependent variable. First, mark the checkboxes for Estimates on the Regression Coefficients panel. Third, mark the Durbin-Watson statistic on the Residuals panel. Sixth, click on the Continue button to close the dialog box. Fifth, mark the Collinearity diagnostics to get tolerance values for testing multicollinearity. Fourth, mark the checkbox for the Casewise diagnostics, which will be used to identify outliers.

Click on the Save button to save the standardized residuals to the data editor.

Click on the Continue button to close the dialog box. Mark the checkbox for Standardized Residuals so that SPSS saves a new variable in the data editor. We will use this variable to omit outliers in the revised regression model.

Click on the OK button to request the regression output.

Outliers in the analysis
If cases have a standardized residual larger than +/- 3.0, SPSS creates a table titled Casewise Diagnostics, in which it lists the cases and values that results in their being an outlier. If there are no outliers, SPSS does not print the Casewise Diagnostics table. There was no table for this problem. The answer to the question is true. We can verify that all standardized residuals were less than +/- 3.0 by looking the minimum and maximum standardized residuals in the table of Residual Statistics. Both the minimum and maximum fell in the acceptable range. Since there were no outliers, the correct answer is true.

Selecting the model to interpret - question
Since there were no transformations used and there were no outliers, we can use the baseline regression for our interpretation. The correct answer is false.

Assumption of independence of errors - question
We can now check the assumption of independence of errors for the analysis we will interpret.

Assumption of independence of errors: evidence and answer
Multiple regression assumes that the errors are independent and there is no serial correlation. Errors are the residuals or differences between the actual score for a case and the score estimated by the regression equation. No serial correlation implies that the size of the residual for one case has no impact on the size of the residual for the next case. The Durbin-Watson statistic is used to test for the presence of serial correlation among the residuals. The value of the Durbin-Watson statistic ranges from 0 to 4. As a general rule of thumb, the residuals are not correlated if the Durbin-Watson statistic is approximately 2, and an acceptable range is The Durbin-Watson statistic for this problem is which falls within the acceptable range from 1.50 to The analysis satisfies the assumption of independence of errors. True is the correct answer. If the Durbin-Watson statistic was not in the acceptable range, we would add a caution to the findings for a violation of regression assumptions.

Multicollinearity - question
The final condition that can have an impact on our interpretation is multicollinearity.

Multicollinearity – evidence and answer
Multicollinearity occurs when one independent variable is so strongly Since multicollinearity will result in a variable not being included in the analysis, out examination of tolerances focuses on the table of excluded variables. The tolerance values for all of the independent variables are larger than 0.10: "number of hours worked in the past week" [hrs1] (.954), "self-employment" [wrkslf] (.979) and "occupational prestige score" [prestg80] (.954). Multicollinearity is not a problem in this regression analysis. True is the correct answer.

Overall relationship between dependent variable and independent variables - question
The first finding we want to confirm concerns the overall relationship between the dependent variable and one or more of the independent variables.

Overall relationship between dependent variable and independent variables – evidence and answer 1
Stepwise multiple regression was performed to identify the best predictors of the dependent variable "income" [rincom98] among the independent variables "number of hours worked in the past week" [hrs1], "self-employment" [wrkslf], and "occupational prestige score" [prestg80]. Based on the results in the ANOVA table (F(2, 142) = , p<0.001), there was an overall relationship between the dependent variable "income" [rincom98] and one or more of independent variables. Since the probability of the F statistic (p<0.001) was less than or equal to the level of significance (0.05), the null hypothesis that the Multiple R for all independent variables was equal to 0 was rejected. The purpose of the analysis, to identify a relationship between some of independent variables and the dependent variable, was supported.

Overall relationship between dependent variable and independent variables – evidence and answer 2
The Multiple R for the relationship between the independent variables included in the analysis and the dependent variable was 0.507, which would be characterized as moderate using the rule of thumb that a correlation less than or equal to 0.20 is characterized as very weak; greater than 0.20 and less than or equal to 0.40 is weak; greater than 0.40 and less than or equal to 0.60 is moderate; greater than 0.60 and less than or equal to 0.80 is strong; and greater than 0.80 is very strong. The relationship between the independent variables and the dependent variable was correctly characterized as moderate. True with caution is the correct answer. Caution in interpreting the relationship should be exercised because of inclusion of ordinal variables; and cases to variables ratio less than 50:1.

Best subset of predictors - question
The next finding concerns the list of independent variables that are statistically significant.

Best subset of predictors – evidence and answer
The best predictors of scores for the dependent variable "income" [rincom98] were "occupational prestige score" [prestg80]; and "number of hours worked in the past week" [hrs1]. The variable "number of hours worked in the past week" [hrs1] was not included in the list of predictors in the question, so false is the correct answer.

Relationship of the first independent variable and the dependent variable - question
In the stepwise regression problems, we will focus on the entry order of the independent variables and the interpretation of individual relationships of independent variables on the dependent variable.

Relationship of the first independent variable and the dependent variable – evidence and answer 1
In the table of variables entered and removed, "number of hours worked in the past week" [hrs1] was added to the regression equation in model 2. In the table of variables entered and removed, "number of hours worked in the past week" [hrs1] was added to the regression equation in model 2. The increase in R Square as a result of including this variable was .077 which was statistically significant, F(1, 142) = , p<0.001.

Relationship of the first independent variable and the dependent variable – evidence and answer 2
The b coefficient for the relationship between the dependent variable "income" [rincom98] and the independent variable "number of hours worked in the past week" [hrs1]. was .118, which implies a direct relationship because the sign of the coefficient is positive. Higher numeric values for the independent variable "number of hours worked in the past week" [hrs1] are associated with higher numeric values for the dependent variable "income" [rincom98]. The statement in the problem that "survey respondents who worked longer hours in the past week had higher incomes" is correct. True with caution is the correct answer. Caution in interpreting the relationship should be exercised because of cases to variables ratio less than 50:1; and an ordinal variable treated as metric.

Relationship of the second independent variable and the dependent variable - question

Relationship of the second independent variable and the dependent variable – evidence and answer
The independent variable "self-employment" [wrkslf] was not included in the regression equation. It did not increase the percentage of variance explained in the dependent variable by an amount large enough to be statistically significant. False is the correct answer.

Relationship of the third independent variable and the dependent variable - question

Relationship of the third independent variable and the dependent variable – evidence and answer 1
In the table of variables entered and removed, "occupational prestige score" [prestg80] was added to the regression equation in model 1. The increase in R Square as a result of including this variable was .180 which was statistically significant, F(1, 143) = , p<0.001.

Relationship of the third independent variable and the dependent variable – evidence and answer 2
The b coefficient for the relationship between the dependent variable "income" [rincom98] and the independent variable "occupational prestige score" [prestg80]. was .135, which implies a direct relationship because the sign of the coefficient is positive. Higher numeric values for the independent variable "occupational prestige score" [prestg80] are associated with higher numeric values for the dependent variable "income" [rincom98]. The statement in the problem that "survey respondents who had more prestigious occupations had lower incomes" is incorrect. The direction of the relationship is stated incorrectly. False is the correct answer.

Validation analysis - question
The problem states the random number seed to use in the validation analysis.

Validation analysis: set the random number seed
Validate the results of your regression analysis by conducting a 75/25% cross-validation, using as the random number seed. To set the random number seed, select the Random Number Seed… command from the Transform menu.

Set the random number seed
First, click on the Set seed to option button to activate the text box. Second, type in the random seed stated in the problem. Third, click on the OK button to complete the dialog box. Note that SPSS does not provide you with any feedback about the change.

Validation analysis: compute the split variable
To enter the formula for the variable that will split the sample in two parts, click on the Compute… command.

The formula for the split variable
First, type the name for the new variable, split, into the Target Variable text box. Second, the formula for the value of split is shown in the text box. The uniform(1) function generates a random decimal number between 0 and 1. The random number is compared to the value 0.75. If the random number is less than or equal to 0.75, the value of the formula will be 1, the SPSS numeric equivalent to true. If the random number is larger than 0.75, the formula will return a 0, the SPSS numeric equivalent to false. Third, click on the OK button to complete the dialog box.

The split variable in the data editor
In the data editor, the split variable shows a random pattern of zero’s and one’s. To select the cases for the training sample, we select the cases where split = 1.

Repeat the regression for the validation
To repeat the multiple regression analysis for the validation sample, select Regression | Linear from the Analyze tool button.

Using "split" as the selection variable
First, scroll down the list of variables and highlight the variable split. Second, click on the right arrow button to move the split variable to the Selection Variable text box.

Setting the value of split to select cases
When the variable named split is moved to the Selection Variable text box, SPSS adds "=?" after the name to prompt up to enter a specific value for split. Click on the Rule… button to enter a value for split.

Completing the value selection
First, type the value for the training sample, 1, into the Value text box. Second, click on the Continue button to complete the value entry.

Requesting output for the validation analysis
Click on the OK button to request the output. When the value entry dialog box is closed, SPSS adds the value we entered after the equal sign. This specification now tells SPSS to include in the analysis only those cases that have a value of 1 for the split variable.

Validation – Overall Relationship
The validation analysis requires that the regression model for the 75% training sample replicate the pattern of statistical significance found for the full data set. In the analysis of the 75% training sample, the relationship between the set of independent variables and the dependent variable was statistically significant, F(2, 105) = , p<0.001, as was the overall relationship in the analysis of the full data set, F(2, 142) = , p<0.001.

Validation - Relationship of Individual Independent Variables to Dependent Variable
In stepwise multiple regression, the pattern of individual relationships between the dependent variable and the independent variables will be the same if the same variables are selected as predictors for the analysis using the full data set and the analysis using the 75% training sample. In this analysis, the same two variables entered into the regression model: "occupational prestige score" [prestg80]; and "number of hours worked in the past week" [hrs1].

Validation - Comparison of Training Sample and Validation Sample
The total proportion of variance explained in the model using the training sample was 27.8% (.527²), compared to 70.7% (.841²) for the validation sample. The value of R² for the validation sample was actually larger than the value of R² for the training sample, implying a better fit than obtained for the training sample. This supports a conclusion that the regression model would be effective in predicting scores for cases other than those included in the sample. The validation analysis supported the generalizability of the findings of the analysis to the population represented by the sample in the data set. The answer to the question is true.

Steps in complete stepwise regression analysis
The following flow charts depict the process for solving the complete regression problem and determining the answer to each of the questions encountered in the complete analysis. Text in italics (e.g. True, False, True with caution, Incorrect application of a statistic) represent the answers to each specific question. Many of the steps in stepwise regression analysis are identical to the steps in standard regression analysis. Steps that are different are identified with a magenta background, with the specifics of the difference underlined.

Complete stepwise multiple regression analysis: level of measurement
Question: do variables included in the analysis satisfy the level of measurement requirements? Is the dependent variable metric and the independent variables metric or dichotomous? No Incorrect application of a statistic Examine all independent variables – controls as well as predictors Yes Ordinal variables included in the relationship? No Yes True True with caution

Complete stepwise multiple regression analysis: sample size
Question: Number of variables and cases satisfy sample size requirements? Include both controls and predictors, in the count of independent variables Compute the baseline regression in SPSS Ratio of cases to independent variables at least 5 to 1? No Inappropriate application of a statistic Yes Yes Ratio of cases to independent variables at preferred sample size of at least 50 to 1? No True with caution Yes True

Complete stepwise multiple regression analysis: assumption of normality
Question: each metric variable satisfies the assumption of normality? Test the dependent variable and independent variables The variable satisfies criteria for a normal distribution? No False Yes Log, square root, or inverse transformation satisfies normality? Use untransformed variable in analysis, add caution to interpretation for violation of normality No True If more than one transformation satisfies normality, use one with smallest skew Yes Use transformation in revised model, no caution needed

Complete stepwise multiple regression analysis: assumption of linearity
Question: relationship between dependent variable and metric independent variable satisfies assumption of linearity? If independent variable was transformed to satisfy normality, skip check for linearity. If dependent variable was transformed for normality, use transformed dependent variable in the test for linearity. If more than one transformation satisfies linearity, use one with largest r Probability of correlation (r) for relationship with any transformation of IV <= level of significance? Probability of Pearson correlation (r) <= level of significance? No No Yes Yes Weak relationship. No caution needed Use transformation in revised model True

Complete stepwise multiple regression analysis: assumption of homogeneity of variance
Question: variance in dependent variable is uniform across the categories of a dichotomous independent variable? If dependent variable was transformed for normality, substitute transformed dependent variable in the test for the assumption of homogeneity of variance Probability of Levene statistic <= level of significance? Yes False No Do not test transformations of dependent variable, add caution to interpretation for violation of homoscedasticity True

Complete stepwise multiple regression analysis: detecting outliers
Question: After incorporating any transformations, no outliers were detected in the regression analysis. If any variables were transformed for normality or linearity, substitute transformed variables in the regression for the detection of outliers. Is the standardized residual for any case greater than +/-3.00? Yes False No Remove outliers and run revised regression again. True

Complete stepwise multiple regression analysis: picking regression model for interpretation
Question: interpretation based on model that includes transformation of variables and removes outliers? R² for revised regression greater than R² for baseline regression by 2% or more? Yes No Pick revised regression with transformations and omitting outliers for interpretation Pick baseline regression with untransformed variables and all cases for interpretation True False

Complete stepwise multiple regression analysis: assumption of independence of errors
Question: serial correlation of errors is not a problem in this regression analysis? Residuals are independent, Durbin-Watson between 1.5 and 2.5? No False NOTE: caution for violation of assumption of independence of errors Yes True

Complete stepwise multiple regression analysis: multicollinearity
Question: Multicollinearity is not a problem in this regression analysis? Tolerance for all IV’s greater than 0.10, indicating no multicollinearity? No False NOTE: halt the analysis if it is not okay to simply exclude the variable from the analysis. Yes True

Complete stepwise multiple regression analysis: overall relationship
Question: Finding about overall relationship between dependent variable and independent variables. Probability of F test of regression for last model <= level of significance? No False Yes Strength of relationship for included variables interpreted correctly? No False Yes Small sample, ordinal variables, or violation of assumption in the relationship? Yes True with caution No True

Complete stepwise multiple regression analysis: subset of best predictors
Question: Finding about list of best subset of predictors? Listed variables match variables in table of variables entered/removed. No False Yes Small sample, ordinal variables, or violation of assumption in the relationship? Yes True with caution No True

Complete stepwise multiple regression analysis: individual relationships - 1
Question: Finding about individual relationship between independent variable and dependent variable. Order of entry into regression equation stated correctly? No False Yes Significance of R2 change for variable <= level of significance? No False Yes Yes

Complete stepwise multiple regression analysis: individual relationships - 2
Direction of relationship between included variables and DV interpreted oorrectly? No False Yes Yes Small sample, ordinal variables, or violation of assumption in the relationship? Yes True with caution No True

Complete stepwise multiple regression analysis: validation analysis - 1
Question: The validation analysis supports the generalizability of the findings? Set the random seed and randomly split the sample into 75% training sample and 25% validation sample. Probability of ANOVA test for training sample <= level of significance? No False Yes Yes

Complete stepwise multiple regression analysis: validation analysis - 2
Same variables entered into regression equation in training sample? No False Yes Shrinkage in R² (R² for training sample - R² for validation sample) < 2%? No False Yes True

Homework Problems Multiple Regression – Stepwise Problems - 1
The stepwise regression homework problems parallel the complete standard regression problems and the complete hierarchical problems. The only assumption made is the problems is that there is no problem with missing data. The complete stepwise multiple regression will include: Testing assumptions of normality and linearity Testing for outliers Determining whether to use transformations or exclude outliers, Testing for independence of errors, Checking for multicollinearty, and Validating the generalizability of the analysis.

The statement of the stepwise regression problem identifies the dependent variable and the independent variables from which we will extract a parsimonious subset.

The findings, which must all be correct for a problem to be true, include: an ordered listing of the included independent variables an interpretive statement about each of the independent variables. a statement about the strength of the overall relationship.

The first prerequisite for a problem is the satisfaction of the level of measurement and minimum sample size requirements. Failing to satisfy either of these requirement results in an inappropriate application of a statistic.

The assumption of normality requires that each metric variable be tested. If the variable is not normal, transformations should be examined to see if we can improve the distribution of the variable. If transformations are unsuccessful, a caution is added to any true findings.

The assumption of linearity is examined for any metric independent variables that were not transformed for the assumption of normality.

After incorporating any transformations, we look for outliers using standard residuals as the criterion.

We compare the results of the regression without transformations and exclusion of outliers to the model with transformations and excluding outliers to determine whether we will base our interpretation on the baseline or the revised analysis.

We test for the assumption of independence of errors and the presence of multicollinearity. If we violate the assumption of independence, we attach a caution to our findings. If there is a mutlicollinearity problem, we halt the analysis, since we may be reporting erroneous findings.

In stepwise regression, we interpret the R² for the overall relationship at the step or model when the last statistically significant variable was entered.

The primary purpose of stepwise regression is to identify the best subset of predictors and the order in which variables were included in the regression equation. The order tells us the relative importance of the predictors, i.e. best predictor, second best, …

The relationships between predictor independent variables and the dependent variable stated in the problem must be statistically significant, and worded correctly for the direction of the relationship. The interpretation of individual predictors is the same for standard, hierarchical, and stepwise regression.

We use a 75-25% validation strategy to support the generalizability of our findings. The validation must support: the significance of the overall relationship, the inclusion of the same variables in the validation model that were included in the full model, though not necessarily in the same order, and the shrinkage in R² for the validation sample must not be more than 2% less than the training sample.

Cautions are added as limitations to the analysis, if needed.

Stepwise Multiple Regression

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