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for control doses of sulphate, where ! ijis a latent random effect that comes from anormal distribution with mean 0 and variance! 2 , i.e. adding extra variation in themortality rate at a specified dose. So even if the expected mortality rate at a particulardose is 30%, the random effect (applied at the batch level) could vary this higher orlower. This model can also be fit using maximum likelihood (e.g. Proc Nlmixed in SAS).Estimates from the fitted model automatically incorporate the effects of the excessrandom variation in their standard errors.The primary goal of this paper is to investigate the effect of hardness levels on the doseresponsecurve. We accomplish this by fitting two (or more) models to the combined datafrom the three hardness levels. In first model (the Separate response model), a separateprobit curve is fit to each hardness level. So, if the basic probit model is used with 3hardness levels, this model will require 6 parameters (an intercept and a slope for eachhardness level). This can be done in a single model fit rather than (the equivalent)running three separate models (one for each dose). In the second model (the Commonresponse) model, the data are pooled over all hardness levels and single Probit model isfit. This model has 2 parameters. In some cases, additional models were run whereindividual Probit curves were fit (one for each hardness level), but the curve wereconstrained to have a common LC10, LC25, or LC50 values (e.g. refer to Jeske et al.,2009). For these models, penalized maximum likelihood was used where the penalty iscomputed as the difference among the LCxx values from the individual curves. Thispenalty declines to zero as the common LCxx value is achieved across all the curves.The Separate response model is very general. Each hardness level has its own doseresponsecurve and these curves do not have to have the same shape. Consequently, it ispossible that the dose-response curve for lower hardness levels give rise to higherestimated mortalities than the dose-response curve for a higher hardness value. Anintermediate model between that of the separate curves for each hardness level and acommon curve for all hardness levels, is the Monotonic-Separate response curve wherethe probit curves are “parallel” at different hardness levels and increasing hardness isalways “protective”, i.e. higher hardness values does not lead to an increase in mortalityat any sulphate dose. More formally,Dead ij! Binomial(BatchSize ij, p i)( ( ) +$ hardness )p i= NR + (1! NR)" # 0+ # ilog D ijwhere ! hardnessis the “shift” in the curve due to different hardness’s and constraints areplaced on these parameters to ensure that the dose-response curve never decreases ashardness increases. A schematic of the results from such a model is:
Survival10.90.80.70.60.50.40.30.20.101 2 3 4 5 6 7 8 9log(Sulphate)Notice that as hardness increases (the three lines from left to right), the mortalitydecreases at any sulphate level, i.e. hardness is “protective”. The shape of the threecurves is identical – all that happens is that the S-shape is shifted to the right as hardnessincreases.The suite of potential probit models fit is described by a 3 part “code”. First, is themodeling of the effects of hardness as either a separate model for each hardness(Separate), or a model with a separate curve for each hardness but a common LCxx(CommonLCxx), or a common model for each hardness (Common), or a model with ashifted-to-the-right (“protective”) dose-response curves as hardness increases(SeparateMono). Next, does the model assume no natural response (NoNR), a commonnatural response over all hardness levels (CNR), or a separate natural response for eachhardness level (SNR). Finally, does the model include a random effect (RE) or excluderandom effects (NoRE) to account for overdispersion. For example, a Probit modelidentified as CommonLC10, NoNR, NoRE corresponds to fitting the model withseparate curves for each hardness level but a common LC10, no natural responses, and norandom effects.Other possible choices for quantal responses are logistic, Gompertz, log-logistic, etc, butthese lead to very similar dose-response curves especially for estimating BMDs in the .10to .90 range (Ritz, 2010) and so were not fit.Many of the analyses from EC and NA of the individual studies used isotonic regression(see next section) applied to the observed mortalities. This method would be applicable ifthere is evidence of a structural lack of fit in the Probit model (i.e. the response is notlinear on the probit scale) but no large lack of fit was detected in any o the studies. Theisotonic model treats a natural response as simply another set of data values. In thesecases, the LCxx values from isotonic regression are not directly comparable to those fromthe maximum likelihood Probit approach with a natural response. In the isotonic method,
Page 1 and 2: An assessment of the effect of hard
Page 3 and 4: 2. Studies used. There are two gr
Page 5 and 6: number of free parameters in the mo
Page 7 and 8: isotonicity constraint can be appli
Page 9: fewer mortalities occurred at highe
Page 13 and 14: is inconsequential relative to the
Page 18 and 19: The function fit isA(1+ E • X)Y =
Page 20 and 21: In some cases, model averaged estim
Page 22 and 23: experimental studies of quantal res
Page 24 and 25: Figure 1b. The Probit, Separate, CN
Page 26: Figure 1d. The Probit, Common LC50,
Page 29 and 30: Figure 1g. The Probit, Common, CNR,
Page 31 and 32: Figure 2b. The Probit, SeparateMono
Page 33 and 34: Table 1. Summary of sampling protoc
Page 35 and 36: Table 2-EC-CH-mortality. Summary of
Page 37 and 38: Table 2-EC-FH-weight. Summary of AI
Page 39 and 40: Table 2-EC-HY-weight. Summary of AI
Page 42 and 43: Table EC-MY-mortality. Summary of A
Page 44 and 45: Table 2-NA-AL-cell. Summary of AIC
Page 46 and 47: Table 2-NA-DA-repro. Summary of mod
Page 48 and 49: Table 2-NA-FH-weight. Summary of AI
Page 51: Table 2-NA-TA-weight. Summary of mo
Page 54 and 55: Table 3. Summary of model averaged
Page 56 and 57: Table 3. Summary of model averaged

Summary - People.stat.sfu.ca - Simon Fraser University

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