Choosing Likelihoods for Binary, Ordinal, and Count PK/PD Data

Let the Response Support Guide the Likelihood

PK/PD analyses often relate concentrations or doses to a response that is not continuous. The response might record whether an adverse event occurred, the severity category of a clinical effect, or the number of events observed during a dosing interval. These outcomes cannot all be modeled as though they were measurements on an unrestricted continuous scale: their possible values and probability structures differ.

A likelihood describes how plausible the observed responses are under specified model parameters. For independent observations conditional on the modeled predictors and parameters, the joint likelihood is the product of the observation-specific probability mass functions. The response support—the set of values the response can take—helps determine which probability mass function is appropriate. A link function is a separate choice: it connects a model predictor, such as concentration, to a valid probability, category distribution, mean, or rate.

Definition: A Bernoulli response takes values 0 and 1, representing absence and presence of an event. An ordinal response takes one value from a finite set of mutually exclusive categories with a meaningful order. A count response is a nonnegative integer representing the number of events in a specified observation window. The likelihood must assign probabilities only to values allowed by the response's support.

For an observation \(Y_i\) and predictors that may include concentration \(C_i\), a model specifies a distribution conditional on those predictors. In the examples below, observations are assumed independent conditional on their predictors and model parameters. This is a substantive assumption, not a consequence of choosing a particular likelihood. Repeated measurements on the same subject, clustering by site, or unmodeled shared factors can violate it.

Binary Outcomes: Bernoulli Probabilities

For a binary outcome \(Y_i\in\{0,1\}\), let \(p_i=P(Y_i=1\mid C_i)\). The Bernoulli probability mass function gives the probability of either possible response in one expression. Across conditionally independent observations, multiplying these contributions gives the likelihood.

P(Y_i=y_i\mid C_i)=p_i^{y_i}(1-p_i)^{1-y_i},\qquad y_i\in\{0,1\}.

The predicted quantity on the response scale is the event probability \(p_i\). A model can relate \(p_i\) to concentration through a link that ensures predictions remain between 0 and 1. The logit link is a familiar choice, but the support alone does not force that link; the link is a modeling decision. The detailed concentration-predictor model is a separate topic.

Worked Example: Likelihood for Three Binary Responses

Suppose a hypothetical PK/PD data set contains three conditionally independent observations. Their modeled event probabilities are 0.20, 0.60, and 0.80, and the observed outcomes are 0, 1, and 1. Calculate the likelihood of these outcomes under the specified probabilities.

The Bernoulli likelihood is

L=\prod_{i=1}^{3}p_i^{y_i}(1-p_i)^{1-y_i}.

Substituting the probabilities and outcomes gives

L=(1-0.20)(0.60)(0.80)=0.80\times0.60\times0.80=0.384.

Thus, under the stated conditional-independence assumption and specified probabilities, the probability of this particular ordered response sequence is 0.384. This is a likelihood value for the specified model; it is not the probability that the model or its parameters are true.

If a record is the number \(Y_i\) of events among a known \(m_i\) independent opportunities, its support is \(0,1,\ldots,m_i\). A binomial likelihood may then be more appropriate than treating the record as one Bernoulli observation. For a binomial observation, the data are event counts out of known denominators, and the predicted quantity is the probability per opportunity. If only a binary indicator—at least one event versus none—is retained, the recorded response instead has Bernoulli support. The data definition matters.

Ordered Categories: Model the Category Distribution

An ordinal response has categories with a meaningful ordering, such as none, mild, moderate, and severe. The distances between categories are not thereby known to be equal. A direct probability description assigns a probability to each category, with all probabilities nonnegative and summing to one. The predicted quantity is therefore a category-probability distribution, not automatically a numerical mean on an equally spaced scale.

P(Y_i=k\mid C_i)=\pi_{ik},\qquad \pi_{ik}\geq 0,\qquad \sum_{k=1}^{K}\pi_{ik}=1.

The likelihood for observed categories is the product of the probabilities assigned to the observed category. Because order is meaningful, a model may organize predictions through cumulative probabilities such as \(P(Y_i\leq k\mid C_i)\). A cumulative-link model is one standard way to do this, but choosing that structure imposes assumptions beyond the category support. If the categories have no defensible order, a nominal categorical model is the appropriate comparison rather than an ordinal model.

Worked Example: Likelihood for Ordered Severity Categories

Suppose a hypothetical study records four responses in the ordered categories 1 (none), 2 (mild), and 3 (moderate). At a given predictor setting, the model assigns probabilities 0.20, 0.50, and 0.30, respectively. The observed sequence is 2, 3, 1, 2. Calculate its likelihood and the expected category score if the categories are temporarily coded 1, 2, and 3.

Under conditional independence, the likelihood is the product of the probabilities of the observed categories:

$$ L=(0.50)(0.30)(0.20)(0.50)=0.015. $$

For the stated numerical coding, the expected score is

$$ E(Y)=1(0.20)+2(0.50)+3(0.30)=0.20+1.00+0.90=2.10. $$

The likelihood uses the ordered categories as categories. The expected score of 2.10 is meaningful only if the numerical coding and its interpretation are justified; it does not show that the distance from category 1 to 2 equals the distance from 2 to 3. For ordinal PK/PD reporting, predicted category probabilities or cumulative probabilities are often more directly interpretable.

Counts: Event Numbers, Means, and Observation Windows

A count takes nonnegative integer values. A common starting model is the Poisson distribution, whose mean \(\lambda_i\) is also its variance. If \(\lambda_i\) is the expected count during the observation window, the observation-specific probability is

P(Y_i=y_i\mid \lambda_i)=\frac{e^{-\lambda_i}\lambda_i^{y_i}}{y_i!}, \qquad y_i=0,1,2,\ldots,\quad \lambda_i>0.

The predicted quantity is the conditional expected count \(\lambda_i\), or a rate when the observation window is accounted for. For example, if events accrue at rate \(r_i\) per unit time over exposure duration \(T_i\), a basic mean specification is \(\lambda_i=r_iT_i\). Comparing raw counts across unequal observation durations without accounting for \(T_i\) can confound event frequency with time observed.

Worked Example: Independent Poisson Counts With Different Means

Consider three hypothetical, conditionally independent event counts. Their observed values are 0, 1, and 3, with modeled means 0.5, 1, and 2 events in their respective observation windows. Compute the joint likelihood under independent Poisson distributions.

The likelihood is the product of the three Poisson probability mass functions:

L= \left(\frac{e^{-0.5}(0.5)^0}{0!}\right) \left(\frac{e^{-1}(1)^1}{1!}\right) \left(\frac{e^{-2}(2)^3}{3!}\right).

Since \(0!=1\), \(1!=1\), and \(3!=6\), this becomes

L=e^{-0.5}\times e^{-1}\times\frac{8e^{-2}}{6} =\frac{4}{3}e^{-3.5} \approx 0.04026.

The value 0.04026 is the probability of this particular count vector under the three specified means and conditional independence. The calculation does not establish that the Poisson assumptions are suitable for an actual data set; that requires checking the design and observed variability.

Derivation: Why the Likelihood Follows the Support

For a discrete response, the probability mass function assigns probability to each possible response. If observations are conditionally independent, the joint probability of the observed vector is the product of its observation-specific probabilities. This joint probability, viewed as a function of unknown model parameters with the observed responses held fixed, is the likelihood. For example, let \(Y_i\) be conditionally Poisson with mean \(\lambda_i\). Independence gives

L(\lambda_1,\ldots,\lambda_n) =\prod_{i=1}^{n}\frac{e^{-\lambda_i}\lambda_i^{y_i}}{y_i!}.

Taking logarithms converts the product into a sum because \(\log(ab)=\log(a)+\log(b)\). Therefore,

\ell =\log L =\sum_{i=1}^{n}\left[-\lambda_i+y_i\log(\lambda_i)-\log(y_i!)\right].

This derivation illustrates the general pattern: select a probability model whose support includes the observed response values, then multiply its probabilities under the stated dependence assumptions. The likelihood does not itself determine the predictor structure. A concentration-response model must additionally specify how the predictor determines a valid Bernoulli probability, ordinal category distribution, or count mean.

Assumptions, Diagnostics, and Alternatives

The likelihood choice should reflect both how the response was collected and how its observations are related. Check that the response is coded with the correct unit, that all relevant denominators or observation windows are known, and that the modeled distribution is plausible conditional on predictors. A good fit of the mean relationship does not guarantee a suitable distribution for residual variation.

Observed response	Typical likelihood starting point	Predicted quantity	Important check
One event indicator, 0 or 1	Bernoulli	Event probability	Verify the outcome is truly binary, not an event count or count out of a denominator.
Ordered severity category	Ordinal categorical model	Category or cumulative probabilities	Verify category order; assess whether the chosen ordinal structure is defensible.
Nonnegative event count	Poisson as a starting point	Expected count or rate	Check exposure time and whether observed variation is compatible with the mean-variance assumption.

The Poisson model assumes conditional variance equal to conditional mean. If the data show extra variability, a negative-binomial model is one alternative; if there are more zeros than the chosen model can explain, first investigate exposure, covariates, dependence, and data collection before considering a model with a separate zero-generating process. For counts bounded by a known number of opportunities, a binomial model may better reflect the finite support. Repeated outcomes may require a model that represents within-subject dependence rather than a product of independent contributions.

For ordinal data, treating category labels as a continuous outcome assumes more than ordering, while treating them as nominal discards the order. The appropriate alternative depends on the scientific meaning and the assumptions of the model, not simply which approach is easiest to fit. For binary data, aggregating outcomes into proportions without their denominators loses information: a proportion of one-half from 2 observations is not as precise as one-half from 200 observations.

Support mismatch: A Gaussian model can predict impossible values, such as a probability below zero or a negative event count. Use a response distribution that respects the observed support.
Wrong unit of analysis: A binary indicator for any event and a count of how many events occurred are different responses, even if both came from the same event records.
Ignoring denominators or time: Event counts from unequal opportunities or unequal observation periods are not directly comparable without representing those differences.
Assuming independence by default: Repeated measures from one subject can remain correlated after conditioning on concentration. A product likelihood needs conditional independence or an explicit dependence model.
Confusing likelihood and link: A link maps a predictor to a valid parameter of a response distribution; it does not replace the response distribution itself.

Key takeaway: First identify exactly what was recorded and its support. Then choose a likelihood consistent with that support and the sampling design; only afterward specify how concentration predicts the relevant probability, category distribution, mean, or rate.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

A study records whether each participant experienced nausea during a visit. What likelihood is a natural starting point, and what is the predicted response-scale quantity?
A response has four ordered severity categories. Why is its expected numerical score not necessarily a complete description of the prediction?
Counts are recorded over observation periods that differ in duration. What feature should be represented when modeling expected counts?
A count model has a conditional mean of 2.5 events. What variance does a Poisson model imply at that mean?
Why is a proportion of events insufficient to describe binomial precision if the denominator is omitted?

Solutions. (1) Bernoulli is a natural starting point for a single binary event indicator; the response-scale prediction is the event probability. (2) The category spacings may not be equal, so category probabilities or cumulative probabilities preserve the categorical nature and ordering more directly. (3) Represent the observation-window duration or exposure, for example through an expected count equal to a rate times duration. (4) The Poisson variance equals its mean, so the implied conditional variance is 2.5 events squared. (5) The same proportion can arise from different numbers of opportunities, and the larger denominator generally provides more information; retain the event count and its denominator when using a binomial likelihood.