Probit model

Probit model A mathematical model of dosage and response in which the dependent variable (response) is a probit number that is related through a statistical function directly to a probability. 

Probit models have been found generally useful to describe the effects of incident outcome cases on people or property for more complex risk analyses. At the other end of the sc e, the estimation of a distance within which the population would be exposed to a concentration of ERPG-2 or higher may be sufficient to describe the impact of a simple risk analysis. 

The Log-Probit Model. The log-probit model has been utilized widely in the risk assessment literature, although it has no physiological justification. It was first proposed by Mantel and Bryan, and has been found to provide a good fit with a considerable amount of empirical data (10). The model rests on the assumption that the susceptibility of a population or organisms to a carcinogen has a lognormal distribution with respect to dose, i.e., the logarithm of the dose will produce a positive response if normally distributed. The functional form of the model is ... 

The most widely-accepted dose response model at the present time is the multi-stage model, which has great flexibility in curve-fitting, and also has a strong physiological justification. Although it is difficult to implement, there are already computer codes in existence that estimate the model parameters (13). The two most widely-used models, until recently, were the one-hit model and the log-probit model. They are both easy to implement, and represent opposite extremes in terms of shape - the former represents the linear non-threshold assumption, whereas the latter has a steep threshold-like curvature. In numerous applications with different substances it has been found that these three... 

One approach is to use the probit models developed in chapter 2. These models are also capable of including the effects resulting from transient changes in toxic concentrations. Unfor-... 

Fig. 8.6 Estimated risk of liver cancer, P(d), in relation to dose of aflatoxin, d, as determined with different dose-incidence models. The models for the different curves. are as follows OH. one-hit model MS, multi-stage model W, Weibull model MH, multihit model MB, Mantel-Bryan (log-probit model) (from Krewski and Van Ryzin, 1981).

Another model, widely used in the past, is the Mantel-Bryan probit model (Mantel et al., 1975). This can be derived by assuming that the dose-response relationship for each individual has a threshold, and that the thresholds for different individuals in the population are distributed log-normally. This model gives a lower risk at low doses than does any power law and, therefore, a lower risk than the multistage or proportional models (Figure 8.5). Moreover, when backgroimd is included, Crump et al., (1976) and Pfeto (1977) have shown that it... 

Probit model. This assumes a lognormal distribution for tolerance in the exposed population. 

The cancer risk values, which these models generate, are of course very different. For example, for the chemical chlordane, the lifetime risk for one cancer death in one million people ranges from exposures of0.03 pg/L of drinking water for the one-hit model, 0.07 pg/L from the linearized multistage model to 50 pg/L for the probit model. 

A fully parametric model/estimator provides consistent, efficient, and comparatively precise results. The semiparametric model/estimator, by comparison, is relatively less precise in general terms. But, the payoff to this imprecision is that the semiparametric formulation is more likely to be robust to failures of the assumptions of the parametric model. Consider, for example, the binary probit model of Chapter 21, which makes a strong assumption of normality and homoscedasticity. If the assumptions are coirect, the probit estimator is the most efficient use of the data. However, if the normality assumption or the homoscedasticity assumption are incorrect, then the probit estimator becomes inconsistent in an unknown fashion. Lewbel s semiparametric estimator for the binary choice model, in contrast, is not very precise in comparison to the probit model. But, it will remain consistent if the normality assumption is violated, and it is even robust to certain kinds of heteroscedasticity. 

The probit model produces a set of marginal effects, as discussed in the text. These cannot be computed for the Klein and Spady estimator. 

These are the fit measures for the probabilities computed for the Klein and Spady model. The probit model... 

Consider the probit model analyzed in Section 17.8. The model states that for given vector of independent variables,... 

Notice that in spite of the quite different coefficients, these are identical to the results for the probit model. Remember that we originally estimated the probabilities, not the parameters, and these were independent of the distribution. Then, the Hessian is computed in the same manner as for the probit model using hy = Fi/l-Fy) instead of X0Xx in each cell. The asymptotic covariance matrix is the inverse of... 

Estimate a probit model, and test the hypothesis that X is not influential in determining the probability that Y equals one. 

We are interested in the ordered probit model. Our data consist of 250 observations, of which the... 

The town of Eleven is contemplating initiating a recycling program but wishes to achieve a 95% rate of participation. Using a probit model for your analysis,... 

In the panel data models estimated in Example 21.5.1, neither the logit nor the probit model provides a framework for applying a Hausman test to determine whether fixed or random effects is preferred. Explain. (Hint Unlike our application in the linear model, the incidental parameters problem persists here.) Look at the two cases. Neither case has an estimator which is consistent in both cases. In both cases, the unconditional fixed effects effects estimator is inconsistent, so the rest of the analysis falls apart. This is the incidental parameters problem at work. Note that the fixed effects estimator is inconsistent because in both models, the estimator of the constant terms is a function of 1/T. Certainly in both cases, if the fixed effects model is appropriate, then the random effects estimator is inconsistent, whereas if the random effects model is appropriate, the maximum likelihood random effects estimator is both consistent and efficient. Thus, in this instance, the random effects satisfies the requirements of the test. In fact, there does exist a consistent estimator for the logit model with fixed effects - see the text. However, this estimator must be based on a restricted sample observations with the sum of the ys equal to zero or T muust be discarded, so the mechanics of the Hausman test are problematic. This does not fall into the template of computations for the Hausman test. 

The one-hit and linearized multi-stage models usually will predict the highest response rates and the probit model the lowest (Paustenbach, 1989a 1989b). 

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