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# Observation model for count data

### Use of count data

Longitudinal count data is a special type of longitudinal data that can take only nonnegative integer values {0, 1, 2, …} that come from counting something, e.g., the number of seizures, hemorrhages or lesions in each given time period. In this context, data from individual is the sequence $y_i=(y_{ij},1\leq j \leq n_i)$ where $y_{ij}$ is the number of events observed in the jth time interval $I_{ij}$.
Count data models can also be used for modeling other types of data such as the number of trials required for completing a given task or the number of successes (or failures) during some exercise. Here, $y_{ij}$ is either the number of trials or successes (or failures) for subject i at time $t_{ij}$. For any of these data types we will then model $y_i=(y_{ij},1\leq j \leq n_i)$ as a sequence of random variables that take their values in {0, 1, 2, …}.  If we assume that they are independent, then the model is completely defined by the probability mass functions $\mathbb{P}(y_{ij}=k)$ for $k \geq 0$ and $1 \leq j \leq n_i$. Here, we will consider only parametric distributions for count data.

### Observation model syntax

Considering the observations as a sequence of conditionally independent random variables, the model is completely defined by the probability mass functions $P(y_{ij}=k)$. An observation variable for count data is defined using the type count. Its additional field is:

• P(Y=k): Probability of a given count value k, for the observation named Y. k is a natural number. A transformed probability can be provided instead of a direct one. The transformation can be log, logit, or probit. The bounded variable k supersedes in this scope any predefined variable k.

### Example

In the proposed example, the Poisson distribution is used for defining the distribution of $y_j$:

$y_j \sim \textrm{Poisson}(\lambda_j)$

where the Poisson intensity $\lambda_j$ is function of time $\lambda_j = a+bt_j$. This model is implemented as follows

[LONGITUDINAL]
input = {a,b}

EQUATION:
lambda = a+b*t

DEFINITION:
y = {type=count, P(y=k) = exp(-lambda)*(lambda^k)/factorial(k)}