Use regression analysis to describe the relationships between a set of independent variables and the dependent variable. Regression analysis produces a regression equation where the coefficients represent the relationship between each independent variable and the dependent variable. You can also use the regression equation to make predictions.
As a statistician, I should probably tell you that I love all statistical analyses equally—like parents with their kids. But, shhh, I have secret! Regression analysis is my favorite because it provides tremendous flexibility, which makes it useful in so many different circumstances.
In this blog post, I explain the capabilities of regression analysis, the types of relationships it can assess, how it controls the variables, and generally why I love it! You’ll learn when you should consider using regression analysis.
Use Regression to Analyze a Wide Variety of Relationships
Regression analysis can handle many things. For example, you can use regression analysis to do the following:
- Model multiple independent variables
- Include continuous and categorical variables
- Use polynomial terms to model curvature
- Assess interaction terms to determine whether the effect of one independent variable depends on the value of another variable
These capabilities are all cool, but they don’t include an almost magical ability. Regression analysis can unscramble very intricate problems where the variables are entangled like spaghetti. For example, imagine you’re a researcher studying any of the following:
- Do socio-economic status and race affect educational achievement?
- Do education and IQ affect earnings?
- Do exercise habits and diet effect weight?
- Are drinking coffee and smoking cigarettes related to mortality risk?
- Does a particular exercise intervention have an impact on bone density that is a distinct effect from other physical activities?
More on the last two examples later!
All these research questions have entwined independent variables that can influence the dependent variables. How do you untangle a web of related variables? Which variables are statistically significant and what role does each one play? Regression comes to the rescue because you can use it for all of these scenarios!
Use Regression Analysis to Control the Independent Variables
As I mentioned, regression analysis describes how the changes in each independent variable are related to changes in the dependent variable. Crucially, regression also statistically controls every variable in your model.
What does controlling for a variable mean?
When you perform regression analysis, you need to isolate the role of each variable. For example, I participated in an exercise intervention study where our goal was to determine whether the intervention increased the subjects’ bone mineral density. We needed to isolate the role of the exercise intervention from everything else that can impact bone mineral density, which ranges from diet to other physical activity.
To accomplish this goal, you must minimize the effect of confounding variables. Regression analysis does this by estimating the effect that changing one independent variable has on the dependent variable while holding all the other independent variables constant. This process allows you to learn the role of each independent variable without worrying about the other variables in the model. Again, you want to isolate the effect of each variable.
How do you control the other variables in regression?
A beautiful aspect of regression analysis is that you hold the other independent variables constant by merely including them in your model! Let’s look at this in action with an example.
A recent study analyzed the effect of coffee consumption on mortality. The first results indicated that higher coffee intake is related to a higher risk of death. However, coffee drinkers frequently smoke, and the researchers did not include smoking in their initial model. After they included smoking in the model, the regression results indicated that coffee intake lowers the risk of mortality while smoking increases it. This model isolates the role of each variable while holding the other variable constant. You can assess the effect of coffee intake while controlling for smoking. Conveniently, you’re also controlling for coffee intake when looking at the effect of smoking.
Note that the study also illustrates how excluding a relevant variable can produce misleading results. Omitting an important variable causes it to be uncontrolled, and it can bias the results for the variables that you do include in the model. This warning is particularly applicable for observational studies where the effects of omitted variables might be unbalanced. On the other hand, the randomization process in a true experiment tends to distribute the effects of these variables equally, which lessens omitted variable bias.
How to Interpret Regression Output
To answer questions using regression analysis, you first need to fit and verify that you have a good model. Then, you look through the regression coefficients and p-values. When you have a low p-value (typically < 0.05), the independent variable is statistically significant. The coefficients represent the average change in the dependent variable given a one-unit change in the independent variable (IV) while controlling the other IVs.
For instance, if your dependent variable is income and your IVs include IQ and education (among other relevant variables), you might see output like this:
The low p-values indicate that both education and IQ are statistically significant. The coefficient for IQ indicates that each additional IQ point increases your income by an average of approximately $4.80 while controlling everything else in the model. Furthermore, an additional unit of education increases average earnings by $24.22 while holding the other variables constant.
I’ve written an entire blog post about how to interpret regression coefficients and their p-values, which I highly recommend.
Obtaining Trustworthy Regression Results
With the vast power of using regression comes great responsibility. Sorry, but that’s the way it must be. To obtain regression results that you can trust, you need to do the following:
- Specify the correct model. As we saw, if you fail to include all the important variables in your model, the results can be biased.
- Check your residual plots. Be sure that your model fits the data adequately.
- Correlation between the independent variables is called multicollinearity. As we saw, some multicollinearity is OK. However, excessive multicollinearity can be a problem.
Using regression analysis gives you the ability to separate the effects of complicated research questions. You can disentangle the spaghetti noodles by modeling and controlling all relevant variables, and then assess the role that each one plays.
There are many different regression analysis procedures. Read my post to determine which type of regression is correct for your data.