# Stepwise Regression with Diagnostic Checking

Stepwise regression is used to build models by adding or removing predictor variables. This is usually done using a series of T-tests or F-test. The test statistics of the estimated coefficients determine the variables to be added or removed. Stepwise regression has its benefits, but the researcher should be skillful. It should be performed by analysts who are well-versed with statistical testing. Stepwise regression is not as efficient as most regression models. For this reason, the models created with stepwise regression should be taken with a grain of salt. These models require a keen eye to detect whether they make sense or not.

How does Stepwise Regression Work?

Just like the name suggests, stepwise regression is a procedure that selects variables in a step-by-step manner. It uses stat statistical significance to add or remove independent variables one at a time. In this procedure, either the most significant variable is added or the least significant variable is removed. Stepwise regression produces a single regression model when the algorithm ends and does not consider all possible models.

It is possible to control the specifics of the stepwise procedure. For example, you can choose to remove the variables, add the variables, or do both. Additionally, the significant level for excluding and including independent variables can be set.

A Software can perform stepwise regression in two ways. Our online tutors have discussed these methods below.

• Use all the available predictor variables to start the test

This is also known as the backward method. It involves deleting one variable at a time as the regression model progresses. You should use this method if you have a number of predictor variables and you want to eliminate a few. The variable with the lowest “F-to-remove” statistic is deleted from the model at each step. Here is the step used to calculate the “F-to-remove” statistic:

1. Calculate a t-statistic for the estimated coefficient of each variable in the model
2. Create the “F-to-remove” statistic by squaring the t-statistic
• The forward method (Starting the test with no predictor variables)

It involves adding the variables one at a time as the regression model progresses. You should use this method if you have a large set of predictor variables. You can create the “F-to-add” statistic using the same steps mentioned above. The only difference is that the system will calculate the statistic for each variable not in the model. The forward method adds to the model the variable with the highest “F-to-add” statistic.

• Stepwise regression is capable of managing large amounts of potential predictor variables. This makes it easy to fine-tune the model to choose the best predictor variables from the available options.
• Compared to other automatic model-selection methods, the stepwise regression is faster
• We can get valuable information about the quality of the predictor variables by watching the order in which variables are removed or added.

Several analysts and statisticians agree that stepwise regression is marred with many problems and should not be used. Some of the issues associated with it include:

• It has several potential variables but little data to meaningfully estimate coefficients.
• Only one of two predictor variables will make it to the model if they are highly correlated.
• In most cases, the R-squared values are high.
• As the model progresses, the adjusted r-squared values might dip sharply from a high. You should identify the variables that were added or removed and adjust the model if this happens.
• Confidence intervals and predicted values are too narrow
• The P-values in the model do not have the correct meaning
• Coefficients for other variables are too high and the regression coefficients are biased
• Another major issue with stepwise regression is collinearity. The program can dump predictor variables into the model if there is excessive collinearity.
• Some variables which are deemed important like the dummy variables may be removed from the model. However, these variables can be manually added back.

Alternatives to Stepwise Regression

• Partial Least Squares

The standard data reduction technique that you should use when you have too many variables is the principal components analysis (PCA). It reduces the number of IVs by using the largest eigen values of X’X

• LASSO

The LASSO is probably one of the best-known shrinkage methods in ridge regression.

• LAR (Least Angle Regression)

The LAR was developed by Hastie, Efron, Johnstone, and Tibshirani (2004). This method centers all the variables and scales the covariates. LAR initially sets all the parameters to zero and then based on correlations with the current residuals, adds the parameters.

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