In this article, we consider the nonparametric modeling of the dose-response function, using locally weighted scatter plot smoothed (LOWESS) regression.
31 Uncertainty intervals are obtained using bootstrap resampling. We also consider parametric linear and nonparametric regression modeling. To identify significant terms in the model, we use parametric tests (analysis of variance) and nonparametric tests that allow the parametric assumptions to be relaxed. The nonparametric tests are performed in an exact setting using permutation methods (SPlus; Mathsoft, Inc., Cambridge, MA).
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In this analysis, we provide a summary of the data and inference results, using the simplest models in an attempt to give a general overview of the various relationships uncovered. In particular, we use a multiple linear regression for the overall treatment response, using amblyopia type as a discrete factor, and age and total dose as covariates. We inspect R 2 values to assess adequacy of fit, and coefficient estimates, and standard errors. Note that all probabilities are two-sided. A stringent criterion is adopted, because of the noise present in the data, and because of the multiple testing of hypotheses in our report: we deem a significance level of 1% to be appropriate in our analyses, but report probabilities exactly.
To assess whether the dose-response differed as a function of amblyopia type we have to capture the difference in an appropriate goodness of fit measure, based on residual sums of squares (RSS), between the best-fitting models. The spline regression (fit using the LOWESS function in a statistical package [SPlus; Mathsoft, Inc.] on a default specification) can be used to obtain a model fit, and hence the RSS, for any data subset. Therefore, let R be the RSS for the whole data set, and let R1, R2, and R3 be the RSS for each of the three data subsets defined by anisometropia, mixed (anisometropia and strabismus), and strabismus, respectively. An appropriate measure of the improvement in fit from fitting amblyopia type as a factor in the regression is T = (R1 + R2 + R3)/R. We use T as a test statistic in a spline-ANOVA type assessment. We compute the exact null distribution of the test statistic using simulation-based methods, in particular, randomization tests, based on randomization of the type label among the dose-response data. We test in the left tail of the randomization distribution only, and compute the simulation probability in the usual way by evaluating the proportion of sampled (permuted) results no larger than the observed test statistic.