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Stepwise Regression
Stepwise regression is a method of regression modeling in which the choice of predictive variables is carried out by an automatic procedure.
Learning Objective

Appraise and criticize stepwise regression approaches that automatically choose predictive variables.
Key Points

Forward selection involves starting with no variables in the model, testing the addition of each variable using a chosen model comparison criterion, adding the variable (if any) that improves the model the most, and repeating this process until none improves the model.

Backward elimination involves starting with all candidate variables, testing the deletion of each variable using a chosen model comparison criterion, deleting the variable that improves the model the most by being deleted, and repeating this process until no further improvement is possible.

Bidirectional elimination is a combination of forward selection and backward elimination, testing at each step for variables to be included or excluded.

One of the main issues with stepwise regression is that it searches a large space of possible models. Hence it is prone to overfitting the data.
Terms

Bonferroni point
how significant the best spurious variable should be based on chance alone

Akaike information criterion
a measure of the relative quality of a statistical model, for a given set of data, that deals with the tradeoff between the complexity of the model and the goodness of fit of the model

Bayesian information criterion
a criterion for model selection among a finite set of models that is based, in part, on the likelihood function
Full Text
Stepwise regression is a method of regression modeling in which the choice of predictive variables is carried out by an automatic procedure. Usually, this takes the form of a sequence of Ftests; however, other techniques are possible, such as ttests, adjusted Rsquare, Akaike information criterion, Bayesian information criterion, Mallows's C_{p}, or false discovery rate. The frequent practice of fitting the final selected model, followed by reporting estimates and confidence intervals without adjusting them to take the model building process into account, has led to calls to stop using stepwise model building altogether  or to at least make sure model uncertainty is correctly reflected.
Main Approaches
 Forward selection involves starting with no variables in the model, testing the addition of each variable using a chosen model comparison criterion, adding the variable (if any) that improves the model the most, and repeating this process until none improves the model.
 Backward elimination involves starting with all candidate variables, testing the deletion of each variable using a chosen model comparison criterion, deleting the variable (if any) that improves the model the most by being deleted, and repeating this process until no further improvement is possible.
 Bidirectional elimination, a combination of the above, tests at each step for variables to be included or excluded.
Another approach is to use an algorithm that provides an automatic procedure for statistical model selection in cases where there is a large number of potential explanatory variables and no underlying theory on which to base the model selection. This is a variation on forward selection, in which a new variable is added at each stage in the process, and a test is made to check if some variables can be deleted without appreciably increasing the residual sum of squares (RSS).
Selection Criterion
One of the main issues with stepwise regression is that it searches a large space of possible models. Hence it is prone to overfitting the data. In other words, stepwise regression will often fit much better insample than it does on new outofsample data. This problem can be mitigated if the criterion for adding (or deleting) a variable is stiff enough. The key line in the sand is at what can be thought of as the Bonferroni point: namely how significant the best spurious variable should be based on chance alone. Unfortunately, this means that many variables which actually carry signal will not be included.
Model Accuracy
A way to test for errors in models created by stepwise regression is to not rely on the model's Fstatistic, significance, or multipler, but instead assess the model against a set of data that was not used to create the model. This is often done by building a model based on a sample of the dataset available (e.g., 70%) and use the remaining 30% of the dataset to assess the accuracy of the model. Accuracy is often measured as the standard error between the predicted value and the actual value in the holdout sample. This method is particularly valuable when data is collected in different settings.
Criticism
Stepwise regression procedures are used in data mining, but are controversial. Several points of criticism have been made:
 The tests themselves are biased, since they are based on the same data.
 When estimating the degrees of freedom, the number of the candidate independent variables from the best fit selected is smaller than the total number of final model variables, causing the fit to appear better than it is when adjusting the r^{2} value for the number of degrees of freedom. It is important to consider how many degrees of freedom have been used in the entire model, not just count the number of independent variables in the resulting fit.
 Models that are created may be toosmall than the real models in the data.
Key Term Reference
 Accuracy
 Appears in this related concepts: Bias, Deploying Evidence, and Variations in Accuracy
 confidence interval
 Appears in this related concepts: Variation and Prediction Intervals, Estimating a Population Proportion, and When to Use These Tests
 data mining
 Appears in this related concepts: Applications of Statistics, Exploratory Data Analysis (EDA), and Analyze Data
 datum
 Appears in this related concepts: Change of Scale, Comparing Nested Models, and Controlling for a Variable
 degrees of freedom
 Appears in this related concepts: Specific Heat and Heat Capacity, Structure of the ChiSquared Test, and Inelastic Collisions in One Dimension
 error
 Appears in this related concepts: The Year the Polls Elected Dewey, Estimation, and Precise Definition of a Limit
 independent
 Appears in this related concepts: Probability Histograms, Regression Analysis for Forecast Improvement, and Party Identification
 independent variable
 Appears in this related concepts: Graphing Functions, Formulating the Hypothesis, and Experimental Research
 line
 Appears in this related concepts: Plotting Lines, Line, and Varieties of Line
 mean
 Appears in this related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, The Mean Value Theorem, Rolle's Theorem, and Monotonicity, and Understanding Statistics
 regression
 Appears in this related concepts: Making a Box Model, Standard Error, and Coefficient of Determination
 residual
 Appears in this related concepts: Plotting the Residuals, Models with Both Quantitative and Qualitative Variables, and Degrees of Freedom
 residuals
 Appears in this related concepts: The Correction Factor, Two Regression Lines, and Inferences of Correlation and Regression
 sample
 Appears in this related concepts: Sampling, What Is a Confidence Interval?, and Sampling
 spurious variable
 Appears in this related concepts: Experimental Design and Some Pitfalls: Estimability, Multicollinearity, and Extrapolation
 standard error
 Appears in this related concepts: Estimating the Accuracy of an Average, Calculations for the tTest: One Sample, and Chance Error and Bias
 ttest
 Appears in this related concepts: Assumptions, tTest for One Sample, and One, Two, or More Groups?
 variable
 Appears in this related concepts: Related Rates, Fundamentals of Statistics, and Math Review
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Cite This Source
Source: Boundless. “Stepwise Regression.” Boundless Statistics. Boundless, 14 Nov. 2014. Retrieved 21 May. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/correlationandregression11/multipleregression49/stepwiseregression2382694/