Overfitting a model is a condition where a statistical model begins to describe the random error in the data rather than the relationships between variables. This problem occurs when the model is too complex. In regression analysis, overfitting can produce misleading R-squared values, regression coefficients, and p-values. In this post, I explain how overfitting models is a problem and how you can identify and avoid it.
Overfit regression models have too many terms for the number of observations. When this occurs, the regression coefficients represent the noise rather than the genuine relationships in the population.
That’s problematic by itself. However, there is another problem. Each sample has its own unique quirks. Consequently, a regression model that becomes tailor-made to fit the random quirks of one sample is unlikely to fit the random quirks of another sample. Thus, overfitting a regression model reduces its generalizability outside the original dataset.
Taking the above in combination, an overfit regression model describes the noise, and it’s not applicable outside the sample. That’s not very helpful, right? I’d really like these problems to sink in because overfitting often occurs when analysts chase a high R-squared. In fact, inflated R-squared values are a symptom of overfit models! Despite the misleading results, it can be difficult for analysts to give up that nice high R-squared value.
When choosing a regression model, our goal is to approximate the true model for the whole population. If we accomplish this goal, our model should fit most random samples drawn from that population. In other words, our results are more generalizable—we can expect that the model will fit other samples.
Graphical Illustration of Overfitting Regression Models
The image below illustrates an overfit model. The green line represents the true relationship between the variables. The random error inherent in the data causes the data points to fall randomly around the green fit line. The red line represents an overfit model. This model is too complex, and it attempts to explain the random error present in the data.
The example above is very clear. However, it’s not always that obvious. Below, the fitted line plot shows an overfit model. In the graph, it appears that the model explains a good proportion of the dependent variable variance. Unfortunately, this is an overfit model, and I’ll show you how to detect it shortly.
If you have more than two independent variables, it’s not possible to graph them in this manner, which makes it harder to detect.
How Overfitting a Model Causes these Problems
Let’s go back to the basics of inferential statistics to understand how overfitting models causes problems. You use inferential statistics to draw conclusions about a population from a random sample. An important consideration is that the sample size limits the quantity and quality of the conclusions you can draw about a population. The more you need to learn, the larger the sample must be.
This concept is fairly intuitive. Suppose we have a total sample size of 20 and we need to estimate one population mean using a 1-sample t-test. We’ll probably obtain a good estimate. However, if we want to use a 2-sample t-test to estimate the means of two populations, it’s not as good because we have only ten observations to estimate each mean. If we want to estimate three or more means using one-way ANOVA, it becomes pretty bad.
As the number of observations per estimate decreases (20, 10, 6.7, etc.), the estimates become more erratic. Furthermore, a new sample is unlikely to replicate the inconsistent estimates produced by the smaller sample sizes.
In short, the quality of the estimates deteriorates as you draw more conclusions from a sample. This idea is directly related to the degrees of freedom in the analysis. To learn more about this concept, read my post: Degrees of Freedom in Statistics.
Applying These Concepts to Overfitting Regression Models
Overfitting a regression model is similar to the example above. The problems occur when you try to estimate too many parameters from the sample. Each term in the model forces the regression analysis to estimate a parameter using a fixed sample size. Therefore, the size of your sample restricts the number of terms that you can safely add to the model before you obtain erratic estimates.
Similar to the example with the means, you need a sufficient number of observations for each term in the regression model to help ensure trustworthy results. Statisticians have conducted simulation studies* which indicate you should have at least 10-15 observations for each term in a linear model. The number of terms in a model is the sum of all the independent variables, their interactions, and polynomial terms to model curvature.
For instance, if the regression model has two independent variables and their interaction term, you have three terms and need 30-45 observations. Although, if the model has multicollinearity or if the effect size is small, you might need more observations.
To obtain reliable results, you need a sample size that is large enough to handle the model complexity that your study requires. If your study calls for a complex model, you must collect a relatively large sample size. If the sample is too small, you can’t dependably fit a model that approaches the true model for your independent variable. In that case, the results can be misleading.
How to Detect Overfit Models
As I discussed earlier, generalizability suffers in an overfit model. Consequently, you can detect overfitting by determining whether your model fits new data as well as it fits the data used to estimate the model. In statistics, we call this cross-validation, and it often involves partitioning your data.
However, for linear regression, there is an excellent accelerated cross-validation method called predicted R-squared. This method doesn’t require you to collect a separate sample or partition your data, and you can obtain the cross-validated results as you fit the model. Statistical software calculates predicted R-squared using the following automated procedure:
- It removes a data point from the dataset.
- Calculates the regression equation.
- Evaluates how well the model predicts the missing observation.
- And, repeats this for all data points in the dataset.
Predicted R-squared has several cool features. First, you can just include it in the output as you fit the model without any extra steps on your part. Second, it’s easy to interpret. You simply compare predicted R-squared to the regular R-squared and see if there is a big difference.
If there is a large discrepancy between the two values, your model doesn’t predict new observations as well as it fits the original dataset. The results are not generalizable, and there’s a good chance you’re overfitting the model.
For the fitted line plot above, the model produces a predicted R-squared (not shown) of 0%, which reveals the overfitting. For more information, read my post about how to interpret predicted R-squared, which also covers the model in the fitted line plot in more detail.
How to Avoid Overfitting Models
To avoid overfitting a regression model, you should draw a random sample that is large enough to handle all of the terms that you expect to include in your model. This process requires that you investigate similar studies before you collect data. The goal is to identify relevant variables and terms that you are likely to include in your own model. After you get a sense of the typical complexity of models in your study area, you’ll be able to estimate a good sample size.
To read about an analysis I performed where I had to be extremely careful to avoid overfit models, read Understanding Historians’ Rankings of U.S. Presidents using Regression Models.
For more information about successful regression modeling, read my post: Five Regression Analysis Tips to Avoid Common Mistakes.
If you’re learning regression and like the approach I use in my blog, check out my eBook!
Babyak, MA., What You See May Not Be What You Get: A Brief, Nontechnical Introduction to Overfitting in Regression-Type Models, Psychosomatic Medicine 66:411-421 (2004).