From Pattern to Prediction

Every regression line makes a prediction for every observation. The residual is the gap between the prediction and reality — how wrong the line is at that specific point. Some residuals are positive (the line underestimated), some negative (it overestimated). They always sum to zero, which means the line is wrong in both directions equally on average. The least squares method chooses the line that makes those residuals as small as possible in total — specifically, that minimizes the sum of their squares.

The slope b₁ tells you how much Y changes for every one-unit increase in X. The intercept b₀ tells you the predicted value of Y when X is zero — which is sometimes meaningful and sometimes an extrapolation beyond your data. Both are derived from the same quantities you already computed for Pearson correlation: the covariance of X and Y, and the variance of X. The slope is just their ratio.

R² is the proportion of variability in Y that is explained by X. An R² of 0.89 means 89% of the variation in FTE counts across hospitals is accounted for by the number of beds alone. The remaining 11% is explained by something else — staffing policies, hospital type, location, patient complexity — variables not in the model. R² doesn't tell you if the relationship is causal. It tells you how much of the story your single predictor is telling.

When one of your variables is time, regression becomes forecasting. The independent variable X is no longer beds or passengers — it's years, months, quarters, coded as integers starting from 1. The slope becomes a trend: how much does the variable grow per time period on average? The intercept becomes the baseline: the predicted value at time zero. Everything else is identical to what you already built. Time series analysis is just regression wearing a watch.

The regression line is built from your data. It knows what happened inside your observed range. It has no knowledge of what happens beyond it. Extrapolation — predicting Y for X values outside the range of your data — assumes the trend continues unchanged indefinitely. Sometimes that's reasonable over a short horizon. Often it isn't. The line doesn't know about recessions, policy changes, pandemics, or anything else that happened after your last data point. The formula keeps calculating. You have to know when to stop trusting it.

The least squares formula will always produce a line. It produces one even if your data is curved, even if one outlier is dragging the slope, even if the residuals show a systematic pattern the line is missing entirely. The formula doesn't warn you. It just calculates. Before trusting a regression line you need to verify three things: that the relationship actually looks linear in a scatter plot, that no single observation is exerting disproportionate influence on the slope, and that the residuals scatter randomly around zero — because systematic residuals mean the line is missing something real.

Test your understanding of the least squares method, regression equations, and extrapolation. These scenarios ask you to build the model and interpret its output correctly.