Top panel: Estimated <i>σ</i> and <i>ρ</i> as a function of p^j for 20 areas in Greenland <i>et. al</i> [12], with loess smooth overlayed.

<p>Bottom panel: Solid line represents OC curve for <i>n</i> = 60, <i>d</i> = 50 for <i>p</i>_<i>l</i> = .75, <i>p</i>_<i>u</i> = .9 design when <i>σ</i> and <i>ρ</i> are fixed at the mean value of </p><p></p><p></p><p></p><p><mi>σ</mi><mo>^</mo></p><p></p><p></p><p></p> (left) and <p></p><p></p><p></p><p><mi>ρ</mi><mo>^</mo></p><p></p><p></p><p></p> (right). Dashed line represents OC curve when <i>σ</i> and <i>ρ</i> vary over <i>p</i> according to the predicted loess smooth of <p></p><p></p><p></p><p><mi>σ</mi><mo>^</mo></p><p></p><p></p><p></p> (left) and <p></p><p></p><p></p><p><mi>ρ</mi><mo>^</mo></p><p></p><p></p><p></p> (right).<p></p

Comparing distributional assumptions between the methods when <i>σ</i> = .1 and <i>ρ</i> = .1.

<p>For the 55–70% and 75–90% couplets, the risks for the 6 × 10 design with <i>d</i> = 38 and <i>d</i> = 50, respectively, are shown. For the 90–95% couplet, the risks for the 33 × 6 design with <i>d</i> = 13 are shown. We compare the binomial-scaled (Pezzoli), beta (Hedt), and quasi-binomial (Hund) models.</p><p>Comparing distributional assumptions between the methods when <i>σ</i> = .1 and <i>ρ</i> = .1.</p

Comparing the Pezzoli, Hund, and Hedt designs for <i>ρ</i> = .1, <i>α</i> = <i>β</i> = .1, and <i>m</i> = 10.

<p>Comparing the Pezzoli, Hund, and Hedt designs for <i>ρ</i> = .1, <i>α</i> = <i>β</i> = .1, and <i>m</i> = 10.</p

Comparing the Pezzoli, Hund, and Hedt designs for <i>σ</i> = .1, <i>α</i> = <i>β</i> = .1, and <i>m</i> = 10.

<p>Comparing the Pezzoli, Hund, and Hedt designs for <i>σ</i> = .1, <i>α</i> = <i>β</i> = .1, and <i>m</i> = 10.</p

Top panel: Estimated <i>σ</i> and <i>ρ</i> as a function of p^j for 37 areas in Minetti <i>et. al</i> [27], with loess smooth overlayed.

<p>Bottom panel: Solid line represents OC curve for <i>n</i> = 60, <i>d</i> = 50 for <i>p</i>_<i>l</i> = .75, <i>p</i>_<i>u</i> = .9 design when <i>σ</i> and <i>ρ</i> are fixed at the mean value of </p><p></p><p></p><p></p><p><mi>σ</mi><mo>^</mo></p><p></p><p></p><p></p> (left) and <p></p><p></p><p></p><p><mi>ρ</mi><mo>^</mo></p><p></p><p></p><p></p> (right). Dashed line represents OC curve when <i>σ</i> and <i>ρ</i> vary over <i>p</i> according to the predicted loess smooth of <p></p><p></p><p></p><p><mi>σ</mi><mo>^</mo></p><p></p><p></p><p></p> (left) and <p></p><p></p><p></p><p><mi>ρ</mi><mo>^</mo></p><p></p><p></p><p></p> (right).<p></p

Shape of the beta and binomial-scaled distributions with standard deviation <i>σ</i> = .1 and means <i>p</i> = .5, <i>p</i> = .7, and <i>p</i> = .9.

<p>Shape of the beta and binomial-scaled distributions with standard deviation <i>σ</i> = .1 and means <i>p</i> = .5, <i>p</i> = .7, and <i>p</i> = .9.</p

Differences between the 3 methods.

<p>Differences between the 3 methods.</p

Plot of beta distributions with fixed mean coverages and varying standard deviation and/or intraclass correlation.

<p>The dotted lines represent the mean coverages <i>p</i>_<i>l</i> = .75 and <i>p</i>_<i>u</i> = .9, which are the same in all 3 panels. Left: standard deviation is fixed for both distributions at <i>σ</i> = .1; middle: intraclass correlation is fixed at <i>ρ</i>_<i>l</i> = .05 (note: <i>ρ</i>_<i>l</i> = .1²/(.75*.25)); right: intraclass correlation is fixed at <i>ρ</i>_<i>u</i> = .11 (note: <i>ρ</i>_<i>u</i> = .1²/(.9*.1)).</p

Appendix A. Measured data and generated data for four variables used in the model.

Measured data and generated data for four variables used in the model

Appendix B. Means and standard deviations used in the generation of pH data.

Means and standard deviations used in the generation of pH data