A correlation summarizes how two variables move together, but does not directly say how much average Y changes when X changes by one unit. Regression expresses that question as an equation.

Simple linear regression compresses an average relationship into two coefficients

Yᵢ=β₀+β₁Xᵢ+εᵢ

• β₀, intercept: mean Y when X=0
• β₁, slope: change in mean Y for a one-unit increase in X
• εᵢ, error: individual variation the equation does not explain

In a sample, fit ŷᵢ=b₀+b₁xᵢ. The slope has units of Y units/X units. If X is concentration in mg/mL and Y is viability percentage points, retain those units for scientific meaning.

If X=0 lies outside the observed range, the intercept may be necessary for calculation but have no practical interpretation. Centering X at a scientifically meaningful value makes the intercept the mean response at that observed reference.

Least squares minimizes the sum of squared vertical residuals

A residual is eᵢ=yᵢ−ŷᵢ. Least squares chooses the line with the smallest Σeᵢ². Squaring gives large residuals substantial influence, so always investigate the measurement causes and leverage of outliers.

A positive residual means the observation is above prediction; a negative one is below. Residuals from a suitable linear model should scatter around zero without special structure against fitted values or run order.

R² is an explained proportion, not a quality certificate

R²=1−SSE/SST

R² is the proportion of total observed Y variation explained by the fitted equation within this sample. In simple regression it equals r². A high R² does not guarantee that:

• the linear functional form is right;
• residual variance is constant;
• errors are independent;
• important variables are included;
• the relationship is causal; or
• prediction on new data is good.

Widening the X range can raise R², and narrowing it can lower R². Where response variation is inherently small, even a low R² can accompany a useful average effect.

Four residual patterns point to different problems

1. U or S shape: the linear equation may need curvature or transformation.
2. Funnel: variance changes with the mean level.
3. Run-order trend: drift, autocorrelation, or batch change.
4. Isolated large residual: measurement error, a different population, or an influential point.

Do not select a revision merely by adding increasingly favorable polynomial degrees after viewing residuals. Choose terms using mechanism, design range, and pre-specified candidate models, then check with new data or cross-validation.

Repeated X levels permit separation of pure error from lack of fit. A lack-of-fit test can help assess whether a candidate function has systematic failure beyond repeat error, but non-significance does not make the model true.

Extrapolation is not merely extending the line farther

If observed X ranges from 1–5, predicting at X=20 may encounter different response mechanisms, saturation, toxicity, or physical boundaries. The equation has an algebraic value, but evidence exists inside the design range. A prediction interval is uncertainty conditional on the model being right; it does not justify extrapolation.

In-Silico Lab: make the model fail deliberately

1. At curvature 0, inspect the fitted line and R².
2. Raise curvature to 1.2 and 2, then compare R² and residual structure.
3. Add one outlier and inspect changes in slope and SSE.
4. Change the seed and see how a sample fit estimates the generating model with variation.

JMP interpretation moves from coefficient table to residuals

Read coefficients, SEs, and CIs in Parameter Estimates; R² and RMSE in Summary of Fit; and the model test in Analysis of Variance, then always inspect residual plots and actual by predicted. Add polynomial terms because of scientific candidates and diagnostics—not because a menu makes them easy.

Example result statement

A linear model was fitted to 38 independent batches over 1–5 mg/mL. The slope was 2.4 percentage points/(mg/mL), 95% CI 1.6–3.2, with R²=.48. Residuals versus fitted values showed no marked curvature, but variance increased at high concentration. The slope direction remained in a weighted-model sensitivity analysis. We did not extrapolate this equation beyond the measured range.

Takeaways

• The slope is the average Y change for a one-unit X change.
• Least squares minimizes squared vertical residuals.
• R² is an explained proportion, not a stand-alone fit criterion.
• Residuals reveal curvature, unequal variance, drift, and outliers.
• Choose polynomial terms by mechanism and validation, not post hoc p-value competition.
• Predictive evidence exists within the observed design range.

The next units move beyond observed X–Y relationships to experimental design, where several inputs are deliberately arranged.

Official supplementary resources

• NIST/SEMATECH · How can I tell if a model fits my data?
• NIST/SEMATECH · Check of assumptions

The model and Lab in this article are educational synthetic material.