如何近似计算回归方程的预测区间？

Summary of Prediction and Confidence Intervals

1. Difference between Prediction and Confidence Intervals

Prediction intervals estimate the range of individual values for the dependent variable y based on a given value of the independent variable x (x0), reflecting the uncertainty of a single value. Confidence intervals estimate the range for the average value of y, reflecting the uncertainty around the predicted mean. For example, with a regression equation "Workload = 2 * Size + 3", when the size is 10, the mean predicted y value is 23, but actual workload values can vary within a prediction interval. Sample means from multiple samples will fluctuate within a confidence interval.

2. Which Is Wider: Prediction or Confidence Interval?

Prediction intervals (PI) are wider than confidence intervals (CI) because they account for both sampling error and other variabilities, whereas confidence intervals only consider the sampling error. Predicting the average productivity of a company's projects is generally more accurate than predicting the productivity of a single project.

3. How to Approximate Prediction Intervals

3.1 Simple Formula for Prediction Interval

The precise calculation of prediction intervals can be complex; thus, approximations are often used in practice. The upper limit of the prediction interval is the predicted value plus 1.96 times the standard deviation of the residuals, and the lower limit is the predicted value minus 1.96 times the standard deviation of the residuals. This formula assumes that y values for a given x are normally distributed, a hypothesis tested before establishing the regression equation. The 1.96 standard deviations correspond to a 95% confidence level.

3.2 Calculation of Prediction Intervals with Transformed Y

When y is transformed, such as with a logarithmic transformation (lny = ax + b), the prediction interval for lny uses the residual standard deviation of lny. To find the prediction interval for y, exponential transformations are applied: the upper limit is exp(ax + b + 1.96S), and the lower limit is exp(ax + b - 1.96S). Similar principles apply if y undergoes other transformations.