Sample Size Calculations in Simple Linear Regression: A New Approach

The problem tackled is the determination of sample size for a given level and power in the context of a simple linear regression model. The standard approach deals with planned experiments in which the predictor X is observed for a number n of times and the corresponding observations on the response variable Y are to be drawn. The statistic that is used is built on the least squares' estimator of the slope parameter. Its conditional distribution given the data on the predictor X is utilized for sample size calculations. This is problematic. The sample size n is already presaged and the data on X is fixed. In unplanned experiments, in which both X and Y are to be sampled simultaneously, we do not have data on the predictor X yet. This conundrum has been discussed in several papers and books with no solution proposed. We overcome the problem by determining the exact unconditional distribution of the test statistic in the unplanned case. We have provided tables of critical values for given levels of significance following the exact distribution. In addition, we show that the distribution of the test statistic depends only on the effect size, which is defined precisely in the paper.

Automated summary

This article addresses the issue of determining the sample size for a given level and power in a simple linear regression model. The standard approach deals with planned experiments, where both the predictor X and the response variable Y are observed. The statistic used relies on the least squares' estimator, but it is problematic because it is based on the conditional distribution given the data on X. For unplanned experiments where X and Y are sampled simultaneously, we don't have data on X yet. To solve this, the article determines the exact unconditional distribution of the test statistic in unplanned cases.