Can the Model Be Improved?

Joint Continuous and Binary Outcomes

A developmental toxicology study looked at 1142 individual live fetuses born to 102 pregnant mice exposed to 0, 0.25, 0.5 or 1.0 unit of a chemical. Table 1 shows the summary data, and indicates malformation rates increasing from 0.31% in the control group to 12.59% in the highest dose group. Live fetuses (individuals) belonging to a single mother constitute a cluster. Switching from unclustered to clustered analysis is as easy as checking one box in ToxTools ö see Figure 1.

Table 1. Summary Data on Malformation Rates

From the ToxTools Model-Fit menu, which features a rich class of dose-response models, the Probit model was chosen for these data (Figure 1). Malformation (Malf) is the outcome variable, Dose the predictor variable, and ClusterID the cluster variable.

Figure 1. Dialog Box for Model Fitting

Table 2 shows the results from fitting the probit model. The results show a highly significant dose effect.

Table 2. Parameter Estimates

Can the Model Be Improved?

With ToxTools, you can easily fit alternative models to your data. Fitting a power model

P(malf)=[a+b*(dose)^g],

resulted in a strong nonlinear fit, reflecting closely the pattern seen in Table 1, where the response rates stayed fairly flat for the lower dose levels and increased markedly at the highest dose (Figure 2).

Figure 2. Fitted Plot for the Power Model

Benchmark Dose Estimation

The excess risk estimates can now be computed from the power model. ToxTools allows us to easily find a BMD and a BMDL (the dose level and associated lower confidence limit linked with a specified level of risk).

Figure 3. Plot of Excess Risk Estimates

Figure 3 shows that, at an excess risk level of 0.1, the BMD is 0.964, the BMDL is 0.931, and the Lower Effective Dose (LED) level is 0.882.

Joint Continuous and Binary Outcomes

The example just cited was for binary data. ToxTools deals just as easily with continuous data. And suppose your analysis must incorporate both binary and continuous outcome data ÷ for example, fetal malformation and fetal weight among live animals in the developmental toxicity context. No problem ÷ with ToxTools you can specify and fit the appropriate multivariate model (Figures 4 and 5).

Figure 4. Developmental Toxicity: Dialog Box

Figure 5. Developmental Toxicity: Fitted Plots

For the above Developmental Toxicity analysis, the risk estimates for the Weight Model, the Marginal Malformation Model and the Multivariate Model in both graphical and tabular form can be obtained through the dialog box shown in Figure 6.

Figure 6. Developmental Toxicity: Risk Estimates