| y | x1 | x2 | w |
|---|---|---|---|
| 10 | 1 | 4 | 1 |
| 12 | 2 | 5 | 1 |
| 13 | 3 | 7 | 2 |
| 16 | 4 | 9 | 2 |
| 18 | 5 | 10 | 3 |
| 22 | 6 | 12 | 3 |
Weighted least squares estimates: β̂ = (Xᵀ W X)⁻¹ Xᵀ W y, where W is diagonal with weights.
Parameter variance: Var(β̂) = σ̂² (Xᵀ W X)⁻¹, with σ̂² = (eᵀ W e)/(n − p).
- Paste rows as y,x1..xk,weight.
- Select how the last column should be treated.
- Choose intercept and confidence level.
- Submit to view coefficients and diagnostics.
- Download CSV or PDF for reporting.
Why weighting changes the fit
Weighted least squares minimizes the weighted sum of squared residuals, Σwᵢ(yᵢ−ŷᵢ)². If high-variance observations receive smaller weights, the fitted line follows the most reliable points. In practice, inverse-variance weights often reduce the influence of noisy tails and stabilize coefficient estimates. This calculator reports both weighted and unweighted R² to highlight how weighting alters explanatory power and how much noise your model is absorbing.
Choosing weights from variance or precision
When measurement variance is known or estimated, the common choice is wᵢ = 1/σᵢ². If your last column is variance, select the variance option; if it is a standard deviation, select the sd option. If you provide direct reliability scores, use weights. Normalizing weights to mean one preserves relative importance while keeping σ̂² and standard errors numerically stable across rescaled inputs.
Interpreting coefficients and uncertainty
Coefficients are computed as β̂ = (XᵀWX)⁻¹XᵀWy. Standard errors come from Var(β̂)=σ̂²(XᵀWX)⁻¹ with σ̂²=(eᵀWe)/(n−p). The table includes t statistics and two‑sided p values based on the t distribution with n−p degrees of freedom. Confidence limits use the selected confidence level and an estimated t* cutoff, helping you compare practical effect sizes across predictors.
Diagnostics that matter for heteroscedastic data
Weighted SSE, weighted RMSE, and the residual plot help you verify that weighting reduced spread across fitted values. If residuals still fan out, consider revising σᵢ estimates or adding predictors that explain the changing variance. AIC and BIC are provided as practical comparison metrics across alternative specifications using the same outcome and dataset size, so you can iterate.
Using the plots to validate assumptions
The Actual vs Predicted chart should cluster near the 45° reference line; systematic curvature suggests missing nonlinear structure. The Residuals vs Predicted chart should be centered around zero with roughly constant vertical spread after weighting. Large residuals at high weights are especially important because they contribute more to the objective and can indicate influential misfit in the most trusted measurements.
Reporting results and exporting outputs
For reporting, cite the weighting rule, whether an intercept was included, and the confidence level used for intervals. Export the coefficients CSV for manuscripts and the predictions CSV for auditing outliers or building further visuals. The PDF download captures the on‑page summary, coefficient table, diagnostics, and plots for quick sharing with collaborators and for consistent archiving in project folders.
FAQs
What problem does weighted least squares solve?
WLS addresses non‑constant error variance by reducing the influence of noisy observations and increasing the influence of precise observations using a positive weight for each row.
When should I use inverse-variance weights?
Use inverse‑variance weights when each observation has a known or estimated variance. Enter variance (or standard deviation) in the last column and select the matching option.
Do I need to normalize weights?
Normalization is optional. It keeps the average weight near one, improving numerical stability and making σ̂² easier to interpret without changing relative importance across rows.
How are p-values and confidence intervals computed?
The calculator uses t statistics from β̂/SE(β̂) with df = n − p. Intervals use the selected confidence level and an estimated t* cutoff.
What do the plots help me check?
Actual vs Predicted highlights bias or nonlinearity. Residuals vs Predicted reveals remaining heteroscedasticity and influential points, especially when large residuals occur at high weights.
How should I format my data?
Provide columns as y, x1..xk, and optionally a final column for weights, variance, or standard deviation. Include a header row if your first line contains names.