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no natural response is assumed and so the LCxx value includes the natural response intotal mortality.The choice of model to be fit to a particular study depended on a preliminary inspectionof the data (see Schwarz, 2011). For mortality studies that are very sparse (few animalson test) only simple models are tenable (i.e. without natural responses or random effects)can be fit as more complex models will fail to fit because of a lack of clear effect.For example, consider the plots of fitted models for the EC-RT mortality data found inFigures 1a-1f and summarized in Table 2-EC-RT. The most general model, the Probit,Separate, SNR, RE model (Figure 1a) has a separate dose-response curve for eachhardness levels along with a separate natural response curve for each hardness level. Thismodel has a 10 parameter (a slope, intercept, natural response for each of 3 hardnesslevels plus 1 parameter for the variance of the random effects). The dose-response curvefor hardness level 50 is to the left of the dose-response curve for hardness 100 which inturn is to the left of the dose-response curve for hardness 250 in the range of dosesstudied in this experiment. This ordering is NOT enforced by this model and occurred“naturally” as the data is fairly strong. However, the natural responses do not follow thissame ordering with the natural response at hardness 250 is between that of hardness 50and 100. This model is the most flexible and so has the best fit to the data (largestlikelihood value of -165.6 and an AICc value of 356.3.Figure 1b plots the dose-response curves for the Probit, Separate, CNR, RE model wherethere are three separate dose-response curves for the three hardness levels, but now allthree dose-response curves have the same natural response. This model has 8 parameters(slope and intercept for the 3 dose-response curves plus one parameter for the commonnatural response plus one parameter for the variance of the random effects). This model isless complex than the previous model (fewer parameters), but will fit the data less well(has a lower likelihood value of -165.8). However, the reduction in fit compared to theprevious model is .03 which is a small reduction in fit for a reduction by 2 in the numberof parameters because the three separate natural responses from the model in Figure 1aseems to be too flexible as the three natural responses are not very different.Consequently, the AICc of 350.7 is smaller than the AICc of the previous modelindicating a model with more support from the data.Figure 1c plots the dose-response curves for the Probit, SeparateMono, CNR, RE modelwhere the three dose-response curves are “parallel” on the probit scale (which leads to S-shaped curves on the mortality scale that are shifted left or right). While the doseresponsecurves for this model look very similar to those in Figure 1b (each hardness hasa separate dose-response curve), the fit is not as good (the likelihood for this model (-165.9) is slightly less than the likelihood for the previous model (-165.8)). However, thismodel has fewer parameters (6 in total being the slope and intercept for the first curveplus the variance of the random effects plus the common natural response plus the twoshift for hardness levels 100 and 250). Consequently, the AICc is much improved (345.5for this model vs. 350.7 for the previous model) as the loss in fit (difference in likelihood)
is inconsequential relative to the reduction in complexity. This model has better supportfrom the data than the previous two models.Figure 1d, 1e, and 1f display the fit of models where the LC50, LC25, and LC10 areforced to be the same across all three dose-response curves while the overall doseresponsecurve still varies across hardness levels. Because the natural response is allowedto vary among the hardness levels, the common L50 does NOT occur at total mortality of.50, but rather .50 of the remaining mortality over and above the natural response occursat the same LCxx values. For example, at hardness 50, the natural response is about .18.Consequently, 50% of the remaining mortality (.50(.82)=.41) occurs at a total mortalityof .41+.18=.59 which is at sulphate level just over 1000. Similarly, for hardness 100, thenatural response is about .10 and 50% of the remaining mortality (.50(.90)=.45) leads to atotal mortality of approximately .10+.45=.55 which also occurs at sulphate level just over1000.Counting parameters in these models is not straightforward because of the constraint ofequal LCxx values. For each of these models, the first curve fit to the smallest hardnesslevel has 3 parameters (slope, intercept, natural response). For each successive hardnessconcentration, there is one more parameter for the natural response and only 1 moreparameter to fit a curve that goes though the same LCxx value as the first curve. Addingone more parameter for the variance of the random effects gives a total of 8 parameters.The log-likelihood is worse than the most general model (Figure 1a), but the reduction infit is offset by the reduction in the number of parameters, and so the AICc gives moresupport to these models compared to the most general model.Finally, Figure 1f fits the Probit, Common, CNR, RE model where a single dose-responsecurve is fit for all hardness levels. This model has only 4 parameters but has the worst fitto the data (smallest likelihood of -168.1) but the reduction in fit is again offset by thelarge reduction in the number of parameters required for the fit. The AICc indicates thatthis model has the most support of the models considered in Table 2-EC-RT.Figure 2a-2c illustrates what happens when the SeparateMono model is fit to data that isnot monotonic as the hardness level increases. The fit in shown in Figure 2a shows theapparent mortality at hardness 15 is lower than the mortality at hardness 80. The Probit,Separate, NoRN, NoRe model (and all other models where separate curves are fit) doesnot enforce “protective” effect of hardness. In Figure 2b, the SeparateMono model is fitwhich enforces a “protective” effect of hardness. Consequently, a single curve is drawn.In fact, this model reduces to the Probit, Common, NoNR, NoRE model [This will onlyhappen in cases with two hardness levels.] Table 2-NA-TA-mortality shows that thelikelihood values for the Common and SeparateMono models is the same (implying anindentical fit), but the SeparateMono model has an extra parameter (the effect of hardness80 relative to hardness 15 which happens to be estimated at 0) and so has less supportfrom the data. Because the Separate model does not enforce the “protective” effect ofhardness, it has more support from the data than either of the two other models. In fact,the model with a common LC50 point seems to have the highest support from the data,
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 and 10: fewer mortalities occurred at highe
Page 11: Survival10.90.80.70.60.50.40.30.20.
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

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