What Does R² Mean?

My regression of daytime sleepiness on depression scores (Investigation 002) produced an R² of 0.158. Two questions immediately came to mind: Is that good? And does it mean daytime sleepiness causes 16% of depression?

The answer to both is no — but unpacking why gets at something important about what R² actually measures.

Starting With Variance

Before R², it helps to think about variance — how spread out your outcome variable is. If you asked me to predict someone’s PHQ-9 depression score without any other information, my best guess would be the average score for everyone in the dataset. Some people would be close to that average; others would be far off. That spread — the total variation in Y — is what we’re trying to explain.

When we build a regression model, we’re hoping to do better than just predicting the average for everyone. We’re saying: knowing something about a person (their daytime sleepiness score, say) should help us predict their depression score more accurately than the average alone.

What R² Actually Measures

R² tells you how much of that spread your model accounts for.

An R² of 0.158 means the model’s predictions are 15.8% closer to the actual values than simply assigning everyone the average. The remaining 84.2% of the variation in depression scores is unexplained — it’s due to other factors, measurement noise, or just the inherent unpredictability of human experience.

So R² is best understood as: how much better are my model’s predictions than a baseline of “everyone gets the average”?

Common Misconceptions

❌ The model is 16% accurate.

❌ Sleepiness causes 16% of depression.

❌ We can predict depression with 16% accuracy.

❌ 84% of people don’t fit the model.

✅ The predictor accounts for 16% of the variability in PHQ-9 scores in this sample.

The accuracy framing is wrong because R² isn’t a measure of prediction accuracy — it’s a measure of explained variance. The causation framing is wrong because R² says nothing about why two variables are related. And “84% of people don’t fit” misreads what residual variance means — it’s not that the model fails for most people, it’s that most of the variation in depression scores isn’t explained by sleepiness alone.

Is 0.158 Good?

It depends entirely on context. In physical sciences, where relationships are tight and measurement is precise, an R² of 0.158 would be poor. In health and behavioral research — where outcomes like depression are shaped by dozens of unmeasured factors — it can be meaningful. There’s no universal threshold for “good.”

What 0.158 tells me here is that daytime sleepiness has a real, detectable relationship with depression scores, but it clearly isn’t the whole story. That’s not a failure of the model; it’s an accurate picture of how complex the phenomenon is.

R² as a Measure of Fit — With Caveats

R² is commonly used to evaluate how well a model fits the data, and that’s reasonable as far as it goes. But it has a critical flaw when used to compare models: it always goes up when you add more variables, even if those variables are noise.

A model with ten predictors will almost always have a higher R² than a model with two, even if eight of those predictors are meaningless. This means chasing a higher R² can lead you to overfit — building a model that fits your training data well but fails to generalize.

For comparing models, better tools exist: adjusted R² (which penalizes for added complexity), AIC/BIC (formal criteria that balance fit against parsimony), and cross-validated error (how well the model predicts on data it hasn’t seen).

R² is a useful summary of how much variation your model accounts for. It’s a poor tool for deciding which model to use.