Adequacy of Interval Estimates of Yield Responses to Ozone Estimated from NCLAN Data

Relative yield losses from O 3 have been estimated by the National Crop Loss Assessment Network (NCLAN) using a nonlinear Weibull response model. The estimated losses were presented in terms of confidence interval estimates based on first‐order linear approximations of variances and normality of estimates. Such interval estimates (Wald estimates) from nonlinear models can be inadequate. Nine Weibull response equations estimated from NCLAN studies were used to assess the adequacy of Wald confidence interval estimates by comparison with interval estimates based on the likelihood ratio test. Three response equations from individual studies were used for illustration of second‐order adjustments to the Wald estimates and the use of measures of parameter‐effects curvature as flags for cases where linear approximations may be inadequate. The Wald interval estimates were clearly inadequate in two of the three individual studies, which had been chosen to represent cases most likely to show inadequacy because of limited coverage of the response curve and/or high variability. In the third study, which had good coverage of the response curve and relatively low variability, the Wald confidence intervals were nearly identical to the likelihood ratio intervals. Measures of parameter effects curvature identified the two cases where the linear approximation was inadequate and showed whether the second‐order adjustments would result in acceptable confidence intervals. Comparisons suggested that Wald confidence interval estimates provided satisfactory approximations for NCLAN response curves that had been computed from data involving more than one experiment, and even for individual experiments where there was adequate coverage of the response curve and moderate variability.