--- title: R-squared vs P-value in Regression | RGM® url: https://realgrowthmatters.com/learn/measurement/r-squared-vs-p-value-regression/ updated: 2026-06-10 source_html: https://realgrowthmatters.com/learn/measurement/r-squared-vs-p-value-regression/ --- # R-squared vs P-value in Regression — Which Statistic Answers Which Question R-squared and p-value answer different questions in regression. R² (and adjusted R²) measures how much of the variation in the outcome is explained by the model — model fit. P-value measures whether a specific coefficient is significantly different from zero — coefficient significance. Marketers confuse these constantly. A high R² with insignificant coefficients means the model fits but you don't know why; significant coefficients with low R² means the predictors matter but most variation is unexplained. Regression output displays both R-squared and p-values, and most marketers learn to nod at them without internalizing what each one actually measures. The confusion is costly — using R² to justify a model that has no significant coefficients, or dismissing a model with significant coefficients because R² seems low. The two statistics answer separate questions. ## R-squared — how much variation does the model explain? R² is the proportion of variance in the dependent variable that the model explains. R² = 1 - (SS\_residual / SS\_total). Ranges from 0 (model explains nothing) to 1 (model explains everything). **Interpretation**: R² = 0.65 means the model explains 65% of the variation in the outcome. The other 35% is unexplained — random variation, omitted variables, or noise. **What R² doesn't tell you**: whether the model is correct, whether individual predictors matter, whether the residuals are well-behaved, whether the model will generalize. **Adjusted R²**: penalizes R² for adding more predictors. Use Adjusted R² when comparing models with different numbers of predictors. Plain R² always increases when you add predictors, even useless ones. ## P-value — is this coefficient significantly different from zero? For each predictor's coefficient, the p-value tests the null hypothesis: 'This coefficient is zero (the predictor has no effect)'. A small p-value (typically <0.05) rejects the null — the coefficient is significantly different from zero. **Interpretation**: p-value = 0.03 on a coefficient means there's a 3% chance of seeing this coefficient (or larger) by random chance if the predictor truly has no effect. **What p-value doesn't tell you**: the size of the effect, the practical importance, whether the coefficient is causally meaningful, whether the model is correct. **Multiple testing correction**: when testing many coefficients simultaneously, p-values inflate. Use Bonferroni or false-discovery-rate adjustments for high-dimensional regressions. ## The four combinations operators see - **High R² + significant coefficients** — model fits well and you know which predictors matter. Best case. - **High R² + insignificant coefficients** — model fits but you can't attribute the fit to specific predictors. Often caused by multicollinearity (correlated predictors). Action: refit with ridge regression or remove correlated predictors. - **Low R² + significant coefficients** — predictors matter but most variation is unexplained. Common in marketing data where outcomes have huge noise. Action: trust the directional finding, don't over-claim the magnitude. - **Low R² + insignificant coefficients** — model fits poorly and predictors don't matter. Time to rethink the model. ## When R² is misleading R² can be high even when the model is wrong: - **Overfitting** — too many predictors fit the training noise. Test on held-out data; the held-out R² is typically much lower. - **Time series with trend** — both outcome and predictors trend over time, producing high R² with no real relationship (spurious regression). Check for stationarity; difference if needed. - **Influential outliers** — a single observation can dominate R²; check Cook's distance. - **Functional form mismatch** — linear regression on nonlinear relationship may have low R² that doesn't reflect the underlying relationship. - **Heteroscedastic residuals** — variance differs across the range of predictors; R² is still computed but standard errors are wrong. ## When p-value is misleading P-values have well-known pathologies: - **P-hacking** — running many tests, reporting only those with p<0.05; vastly inflates false positive rate - **Multiple comparisons** — testing 20 coefficients, expect 1 false positive at α=0.05 by chance; adjust with Bonferroni - **Large sample sizes** — with millions of observations, every coefficient becomes 'significant' even if practically meaningless. Look at effect sizes, not just p-values. - **Confounded coefficients** — significant p-value on a confounded predictor is not a causal claim - **Multicollinearity** — correlated predictors give unstable coefficients with inflated standard errors; p-values become unreliable #### RGM Experts Say The operating rule we use: R² tells you whether the model is useful for prediction. P-values tell you whether to trust the individual coefficients for explanation. MMM models often have R² of 0.7–0.9 with mixed coefficient significance — that means the overall fit is good but you should treat individual channel coefficients with skepticism. Use both statistics, but never use either one alone. ## Other regression diagnostics that matter - **Residual plots** — check for non-random patterns (curvature, fan shapes, outliers) - **VIF (Variance Inflation Factor)** — measures multicollinearity; VIF > 5–10 indicates problematic correlation - **Durbin-Watson statistic** — tests for autocorrelation in time-series residuals; should be near 2 - **AIC / BIC** — penalized model fit measures; lower is better; useful for model comparison - **F-statistic and overall p-value** — tests whether the model as a whole has explanatory power - **Confidence intervals on coefficients** — more informative than p-values for effect-size interpretation - **Cross-validation R²** — out-of-sample R² is the honest measure of model quality ## Related guides - See [regression analysis for marketing](/learn/measurement/regression-analysis-for-marketing/) - See [statistical analysis](/learn/measurement/statistical-analysis-for-marketers/) - See [MMM Ultimate Guide](/learn/measurement/marketing-mix-modeling-guide/) ## Sources 1. [1]Wasserman, All of Statistics; James, Witten, Hastie, Tibshirani, An Introduction to Statistical Learning; Gelman and Hill, Data Analysis Using Regression ### Related guides - [Regression analysis](/learn/measurement/regression-analysis-for-marketing/) - [Statistical analysis](/learn/measurement/statistical-analysis-for-marketers/) - [MMM Ultimate Guide](/learn/measurement/marketing-mix-modeling-guide/)