What are the benefits and drawbacks of using stepwise methods for variable selection in multiple regression?

There are multiple independent(X) variables that influence the dependent variable(Y). Through the multiple regression equation, the relative importance of each X in determining the Y is established. In this context, a distinguishing feature of multiple linear regression from simple linear regression is that the relationships between each independent variable and the dependent variable are determined simultaneously. For example, if we look at the reason for the increase in per capita income as an increase in GDP, taking only one effect from this may not sufficiently explain the dependent variable. Therefore, considering all factors that determine GDP as independent variables allows the model to make better predictions.

The last sentence should really be elaborated on, since it potentially implies that testing the hypothesis about the effect of a variable in a multiple regression is directly comparable to adding another variable to the model, while the use of a quasi or other experimental methodology is really essential for determining causality. Additionally, "You can use multiple regression to compare the fit of different models" doesn't really make any sense, since you can of course have many different types of multiple regression models.

Multiple regression provides advantages such as understanding complex relationships, determining individual variable effects, and controlling for confounding variables. By simultaneously testing the effects of multiple variables, this statistical method enhances predictive accuracy and strengthens the robustness of the model, while also accommodating non-linear relationships. It is worth noting the robustness of the model, as simultaneously observing the effects of variables can make it sensitive to outliers and extreme values.

Including multitude of independent variables on one hand improves the explanation of the dependent variable but may sometimes also lead to other problems. For eg R2 is generally used as a measure to test the robustness of the linear models. Higher the value of R2, more descriptive is the model. However, value of R2 is directly proportional to the number of independent variables and may inflate the value of it causing to settle on a less accurate model. It also leads to the problem of multicollinearity and autocorrelation. So it’s essential to first look at the correlation between the independent variables and causality on dependent variable to only include the ones in the model that actually explain the variance in the dependent variable.

Researchers should be cautious when using multiple regression due to challenges such as the problem of changing variance, non-normal distribution of errors, the concept of multicollinearity, the potential selection of an inappropriate regression model, and the issue of overfitting. This situation can affect the reliability of the regression model, making the results misleading, and to address this issue, researchers should consider error transformations or different regression methods. Multicollinearity can complicate the stability and accuracy of regression coefficients. Special regression techniques that address variable selection or multicollinearity can be employed to tackle this problem.

There is also the risk of collider bias...https://catalogofbias.org/biases/collider-bias/ If exposure and outcome both act on the same variable, but are not directly associated, then controlling for that variable can produce a distorted estimate of the association between exposure and outcome Whether independence is a problem depends on your goal. Including correlated variables can cause increased uncertainty in parameter estimates (variance inflation factor) but the model predictive power can still be good.

The stepwise method is an important fundamental statistical technique in building multiple regression models. It involves both forward selection and backward elimination approaches. The goal is to observe the individual contributions of variables one by one. When a variable enters the model, it may not necessarily remain in it. The effects with other variables are examined, aiming to create the most optimum model. In this way, the complexity of the model can be reduced, saving it from unnecessary variables.

Stepwise methods comprise a class of methods in which at each step explanatory variables are compared to either add or remove one of them. The underlying principle is as follows:? The stepwise method starts with some variables in the model (can be none or all of them) and gradually adjusts the number of variables. ? It stops when no further improvement is obtained based upon a certain statistical criterion. Several criteria are available. Stepwise procedures are automated in most software packages.

Using stepwise methods for variable selection in multiple regression can save time and effort, particularly with many potential predictors. These methods automate the process of testing and comparing models, selecting the best variable combination based on objective criteria. They can reveal new or unexpected relationships between variables and eliminate irrelevant or redundant ones. In my experience, this approach was invaluable during a large-scale data analysis project, where manually testing models would have been impractical. However, it’s important to be aware of the limitations, such as the potential for overfitting and the reliance on the chosen criteria for selection.

One of the advantages is the automatic process of variable selection. It relies on statistical criteria at each step, leading to more effective variable selection. At each step, the most significant variable is added. The addition of the variable is evaluated based on whether it increases the model's significance. Forward selection continues, adding a variable at each step until a predefined criterion (such as AIC, BIC, etc.) is met. Additionally, it controls the complexity of the model. This is crucial to prevent the model from overfitting.

Literature I've seen suggests that if you find stepwise regression necessary, you should: * Use backward regression, so all candidate variables have a chance to contribute * Prune based on minimizing AIC or BIC rather than statistical significance * Prune interaction terms before pruning main terms

Stepwise variable selection although saves time in variable selection when the list of causal variables is huge and manual selection isn’t possible, but it comes with its drawbacks. The variable selected may not at the end make a theoretical sense and the variable selection criteria may overemphasize model parsimony while compromising predictability. There’s a trade off between the descriptive and predictive value of the model. So it’s advisable to do follow pruning approach while checking how R2 changes with decreasing AIC and BIC scores.

Overfitting is the condition where a model fits the training data too closely, leading to a tendency to perform poorly on new, unseen data. However, in this process, adding or removing each variable can cause the model to overfit the training data. In other words, while the application of stepwise regression aims to reduce model complexity, at times, this process may increase the risk of overfitting. This situation may arise from the model developing very specific responses to the training data. The stability of the model may decrease due to the addition and removal of variables. Factors such as selection criteria and the starting point of the model can influence the results and should therefore be carefully chosen.

Advice: Use an all subset models approach unless computationally infeasible. All stepwise methodologies, on average, fare weakly relative to this more computationally intensive regression examination. The main failing of any form of stepwise regression model fitting is that significance levels are not simultaneously corrected for data structure and the number and nature of models searched. Expect substantial overfitting to result from such an exercise based on the reported F values. Models that suddenly get good as terms are added after seemingly going nowhere for a while require suspicion that a data structure artifact or modeling artifact is occurring rather than a good model. Some stepwise algorithms are much better than others.

Stepwise methods were developed before powerful computers were available. The algorithms are looking for a good solution without checking all possible models. The risk is to find only a local optimum, i.e. not the overall best model. With more computing power, makes more sense to look at a much larger number of possible models (best subset methods). I strongly recommend against stepwise selection methods as they are prone to bias. Much better using best subsets method. Introduce domain knowledge into the variable selection process is key.

The predictability and insights gleaned is highly dependent on population sampled and the defined environment. As one who has used these tools in both science and business, care needs to be taken to be inclusive of all factors when interpreting results. Like in the article, the ability to describe the results in a concise yet impactful way is paramount to data analysis and visualization.