Multiple Regression Summary Calculator

Formulas used

Let y be an n×1 vector and X an n×p design matrix (first column ones if intercept). Ordinary least squares estimates coefficients as β = (X'X)^{-1} X' y. Fitted values are ŷ = Xβ and residuals e = y - ŷ. Residual sum of squares SSE = e'e, total sum of squares SST = ∑(y - \bar{y})^2, and SSR = SST - SSE. With residual variance s^2 = SSE/(n-p), the covariance matrix is Var(β) = s^2 (X'X)^{-1} (or HC1 robust (n/(n-p))(X'X)^{-1} X' diag(e^2) X (X'X)^{-1}). Standard errors are square roots of diagonal terms. t-statistics are t_j = \beta_j / SE(\beta_j) with df = n-p and two-sided p-values from the t distribution. The model F-statistic is F = (SSR/(p-1)) / (SSE/(n-p)) for an intercept model, with p-value from the F distribution. Confidence intervals use \beta_j \pm t_{1-\alpha/2,df} SE. A mean response at new x_0 has variance s^2 x_0' (X'X)^{-1} x_0; prediction interval adds an extra s^2 term.

How to use

1. Place the response in the first column labeled y, predictors in subsequent columns.
2. Tick Treat first row as headers if your data include column names.
3. Choose Include intercept for models with a constant term. Most models require it.
4. Optionally enable robust (HC1) standard errors to lessen heteroskedasticity concerns.
5. Click Run regression to compute coefficients, diagnostics, and intervals.
6. To forecast, enter predictor values in Predict at new values and submit.
7. Use the top-right buttons to download the displayed summary as CSV or PDF.

How to find confidence interval based on summary output of multiple regression in R

You can obtain confidence intervals for coefficients directly with confint(), or compute them manually from the summary() output using the reported standard errors and residual degrees of freedom.

# Fit model
fit <- lm(y ~ x1 + x2 + x3, data = df)

# 95% CIs for coefficients
confint(fit, level = 0.95)

# Mean-response CI and prediction interval at new values
new <- data.frame(x1 = 10, x2 = 5, x3 = 2)
predict(fit, newdata = new, interval = "confidence", level = 0.95)
predict(fit, newdata = new, interval = "prediction", level = 0.95)

s <- summary(fit)
b  <- coef(fit)                               # estimates
se <- coef(s)[, "Std. Error"]                 # standard errors
df <- s$df[2]                                 # residual df (n - p)
tcrit <- qt(0.975, df)                        # two-sided 95%

ci_lo <- b - tcrit * se
ci_hi <- b + tcrit * se
cbind(Estimate = b, `2.5%` = ci_lo, `97.5%` = ci_hi)

# Install once: install.packages(c("sandwich","lmtest"))
library(sandwich); library(lmtest)

Vrob <- vcovHC(fit, type = "HC1")
se_rob <- sqrt(diag(Vrob))
df <- fit$df.residual
tcrit <- qt(0.975, df)

b <- coef(fit)
ci_lo <- b - tcrit * se_rob
ci_hi <- b + tcrit * se_rob
cbind(Estimate = b, `2.5%` = ci_lo, `97.5%` = ci_hi)

# Or: robust tests and CIs via coeftest/coefci
coeftest(fit, vcov = Vrob)
# coefci() from 'lmtest' can build intervals too

1. Rows = variables. Each row is a predictor; the Intercept is the expected response when all predictors are zero.
2. Coef shows the estimated change in y for a one‑unit increase in that predictor, holding others constant. Units are “y‑units per x‑unit.”
3. Std Err is the sampling uncertainty of the coefficient. A large standard error relative to the coefficient means imprecise estimation.
4. t equals Coef ÷ Std Err. As a rough guide with moderate df, absolute t values around 2 or larger often indicate noteworthy effects.
5. p is a two‑sided p‑value testing whether the true coefficient is zero. Values below 0.05 are commonly flagged, but use domain context.
6. 95% CI gives a range of plausible effect sizes. If it excludes 0, the effect is statistically significant at the 5% level.
7. R² / Adjusted R² summarize fit. Adjusted R² penalizes adding weak predictors; prefer it when comparing models with different sizes.
8. F‑statistic & Prob > F test whether at least one slope differs from zero. Small p‑values imply the model explains meaningful variance.
9. RMSE is the typical prediction error in the units of y. Smaller values indicate tighter residuals.
10. ANOVA table: SSR (model), SSE (residual), SST (total). MSR = SSR/df_model, MSE = SSE/df_resid, and F = MSR/MSE.
11. Robust SE (HC1) affect SE, t, p, and CI but not coefficients. Use when heteroskedasticity is a concern.
12. Prediction block reports a mean‑response CI and a wider prediction interval for individual outcomes; intervals use the same t critical value.
13. Good practice: ensure n ≫ p, check multicollinearity, investigate outliers/leverage, and consider transformations or interactions when theory supports them.