Fig. 1.

The substitution method (an example with only one group of late participants) Early participants make up the share $\alpha$ of the full population, late participants $\beta$, and non-participants $1-\alpha -\beta$. The proportion with the characteristic of interest is $\overline{y_{\alpha }}$ among early participants and $\overline{y_{\beta }}$ among late participants. The model assumes that non-participants are similar to late participants on average. The estimated population proportion $\overline{y_{\mathit{pop}}}$ is given by $\alpha \overline{y_{\alpha }}+(1-\alpha )\overline{y_{\beta }}$, which lies somewhere in between $\overline{y_{\alpha }}$ and $\overline{y_{\beta }}$ and corresponds to the combined area of the dark and light shaded regions.