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# 14.5 Probability of quantal effects

Normally the dose-response function for a quantal effect represents the probability of the effect occurring as a function of the intake of the relevant contaminant or nutrient. Generally the function is continuous, e.g. probit, and can be summarised by 2 or more parameters (e.g. intercept and slope) relating the probability of effect to an appropriate measure of intake.

Equations (1) and (3) for quantal effects are readily applicable to chronic effects, or to acute effects that may not recur for the same individual within the same year. The direct health loss method is not well suited for dealing with recurrent effects: possible work-arounds for these are outlined earlier in section 9.

Effects which occur in future generations, e.g. the children of mothers exposed to a substance, also require modifications to the calculations (see section 10).

Dose-response relationships may include covariates, e.g. the dose-response relationships can be different for different population groups, e.g. sex or age classes. If so, this needs to be taken into account in the calculations. The Qalibra software allows the use of discrete covariates (e.g. sex); continuous covariates have to be represented by discretising them (e.g. age must be divided into age classes). The number of classes is not specifically limited: the spacing and number of classes should be chosen by the user so as to adequately represent the influence of each covariate.

A key issue concerning the representation of dose-responses is that the directly attributable health loss calculation requires the probability of effect to be expressed as the
*
probability of onset
per year
for
specified ages or age groups
*
. Dose-response epidemiological studies may report dose-response relationships in this form (e.g. relative risk per year, as a function of age) but animal studies generally estimate lifetime probabilities.

When dose-responses are expressed as lifetime probabilities, it will be necessary to convert them to annual probabilities, which will often vary with age. This requires data or assumptions on age of onset of effect. Age of onset may be available from epidemiological or intervention studies of humans. Age of onset is rarely determined in animal studies, and even when it is, there would be substantial uncertainty if it is assumed the same relative age would apply for humans. In some cases, the age of onset may be clear, e.g. for effects manifested in offspring at birth. For other cases, a possible approach is to assume age of onset when the effect is caused by the dietary change follows the general distribution for the same or similar diseases in the human population. However it must be taken into account that such assumptions are likely to involve considerable uncertainty, as it is very likely that the two distributions will differ (e.g. if part of the incidence of the disease is due to causes other than the contaminant or nutrient being assessed, with different ages of onset). When the appropriate treatment for age of onset is very uncertain, it will be prudent to try different assumptions to explore the sensitivity of the assessment to this choice.

Equations (1) and (3) also require the probability of effect to be expressed in absolute not relative terms. When the dose response is relative, e.g. a relative risk from an epidemiological study, it will be necessary to convert it to absolute probability of effects by combining the relative risk with data on baseline effect probabilities: this may be available from medical statistics for the population under assessment, or for another population which is considered similar.

Equations (1) and (3) also require that the probability of effect must relate to humans. If the dose-response relationship comes from animal studies, it will be necessary either to assume the human dose-response is the same, or to apply a suitable form of extrapolation (e.g. apply an adjustment consistent with the inter-species extrapolation component of uncertainty factors used in risk assessment).

Generally, dose response relationships are continuous, and need to be specified by a mathematical equation containing one or more parameters. The original plan for the Qalibra software was to provide a drop-down menu of commonly-used dose-response models, but it was considered desirable to provide more flexibility to allow use of any dose-response relationship the user considered appropriate. This is achieved by requiring the user to input the dose-response in discretised form, i.e. as a series of paired values for dose and response. The Qalibra software then forms the continuous relationship by linear interpolation between the successive pairs of values. Users need to check that the number and spacing of the pairs of values is sufficient for linear extrapolation between them to represent the continuous relationship adequately (this can be checked by repeating the assessment with more or less points to see if it alters the outcome).

The Qalibra software also provides the option of specifying the dose-response relationship for probability of effect using a single value. Qalibra treats this value as a threshold dose, below which the probability of the effect is zero, and above which it is one. This allows the user to carry out simplified assessments, e.g. using NOAEL or LOAEL values as thresholds, which may be useful for exploratory or screening purposes. Note that Qalibra offers this option only for increasing relationships for probability of effect; if the probability of effect decreases with increasing dose, then this must be specified as a series of pairs of values (see previous paragraph).

It will be clear from the issues discussed above that deriving appropriate dose-response relationships for use in risk-benefit assessment requires specialised expertise in toxicology, epidemiology & modelling, and involves substantial uncertainties. The impact of all identified uncertainties should be considered when interpreting the results of the assessment (see section 15.1).