rme/_subfiles/count-regression/_sec_pois-reg_intro.qmd at 96c7ad7feb9a53563348cde1731339d5ba1365d2 · d-morrison/rme · GitHub

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{{< include latex-macros/macros.qmd >}}

::: notes
This chapter presents models for
[count data](data.qmd#sec-count-vars) outcomes.
With covariates,
the event rate $\lambda$
becomes a function of the covariate vector
$\vX = \tp{(X_1, \dots, X_p)} \in \reals^p$.
Typically, count data models use
a $\logf{}$ link function,
and thus an $\exp{}$ inverse-link function.
Specifically, the model relates the expected outcome count
to the event rate and linear predictor as:
:::

$$
\ba
\Expp[Y \mid \vX = \vx, T = t] &= \mu(\vx,t) \\
\mu(\vx,t) &= \lambda(\vx)\cdot t \\
\lambda(\vx) &= \exp{\eta(\vx)} \\
\eta(\vx) &= \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p
\ea
$${#eq-mean-poisson}

::: notes
The term $T=t$ represents the [exposure magnitude](probability.qmd#def-exposure)
(such as person-years or observation time)
and plays a structural role in scaling rates to expected counts.
:::

---

:::{#exr-analogy-pois-offset-binom}
Where have we seen a relationship like
$$\mu = \lambda \cdot t$$
before?
:::

---

:::{#sol-analogy-pois-offset-binom}
The relationship
$$\mu = \lambda \cdot t$$
in count regression models
is analogous to the relationship
$$\mu = n \pi$$
in Binomial models.
:::

---

::: notes
We can also express the exposure magnitude $t$
directly as a component of the linear predictor:
:::

$$
\ba
\logf{\Expp[Y \mid \vX = \vx, T = t]}
&= \logf{\mu(\vx, t)} \\
&\quad \text{(definition of the conditional mean)} \\
&= \logf{\lambda(\vx) \cdot t} \\
&\quad \text{(substituting the rate relationship)} \\
&= \logf{\lambda(\vx)} + \logf{t} \\
&\quad \text{(logarithmic product rule)} \\
&= \logf{\exp{\eta(\vx)}} + \logf{t} \\
&\quad \text{(substituting the rate function)} \\
&= \eta(\vx) + \logf{t} \\
&\quad \text{(by inverse relationship of log and exp)} \\
&= (\beta_0 + \beta_1 x_1 + \dots + \beta_p x_p) + \logf{t} \\
&\quad \text{(expanding the linear predictor } \eta(\vx)\text{)}
\ea
$$

::: notes
In contrast with the other covariates (represented by $\vX$),
$t$ enters this expression with a $\log{}$ transformation
and without an estimated $\beta$ coefficient;
in other words, $\logf{t}$ is an [offset term](probability.qmd#def-offset).
:::

---

:::{#exr-eq-mean-poisson}
What are the units of $\mu$ in @eq-mean-poisson?
:::

---

:::{#sol-eq-mean-poisson}

$\mu$ is the expected value of $Y$.
Because $Y$ represents a count of events,
$\mu$ is expressed in units of event counts;
for example:

- 3.1 cyclones,
- 10.23 ER visits,
- 15.01 infections.

:::

---

:::{#exr-eq-rate-poisson}
What are the units of $\lambda$ in @eq-mean-poisson?
:::

---

:::{#sol-eq-rate-poisson}
$\lambda = \mu/t$,
so $\lambda$ is an event rate per unit of exposure $t$.
For example:

- 3.1 cyclones *per year*,
- 2.023 ER visits per 10 person-years,
- 15.01 infections per 1000 person-years at risk.

:::