- Right-censoring
- Left-censoring
- Interval-censoring/interval data
- Incidence-rate ratios
- Predictions
- Number of events
- Number of events, conditional on censoring
- Probability of a count or range of counts
- Conditional probability of a count or range of counts

Poisson regression is used when the dependent variable is a count from a Poisson process.

Outcomes can be left-censored if they are not observed when they are below a certain level and can be right-censored if are not observed when they are above another level.

Command **cpoisson** fits Poisson regression models on count
data and allows the counts to be left-censored, right-censored, or
both. The censoring can be at constant values, or it can differ across
observations.

An example of a right-censored count outcome is the number of cars in a family, where data might be top-coded at 3 or more.

An example of a left-censored count outcome is the number of cookie boxes sold by Girl Scouts if the first outcome value recorded is 10 or fewer boxes.

Left- and right-censoring combined is also known as interval-censoring.

Distinguish between censored and truncated. With censored outcomes, it
is the outcomes that are not observed even though the observation is in
our data; we observe the other values for the person. In truncated
data, it is the observation that is entirely missing from our data. Stata
has an estimator for truncated Poisson data, see [R] **tpoisson**.

Below we study the number of car accidents a person has during a year. The number recorded is 0, 1, 2, or 3, and 3 means 3 or more accidents. The number is right-censored.

We will model the determinants of accidents as the number of previous accidents, whether the driver is a parent, and the number of traffic tickets the driver received during the previous year.

We type

.cpoisson accidents i.past i.parent i.ntickets, ul(3) irrinitial: log likelihood = -2657.162 rescale: log likelihood = -2657.162 Iteration 0: log likelihood = -2657.162 Iteration 1: log likelihood = -2638.9113 Iteration 2: log likelihood = -2638.7142 Iteration 3: log likelihood = -2638.6901 Iteration 4: log likelihood = -2638.6863 Iteration 5: log likelihood = -2638.6859 Iteration 6: log likelihood = -2638.6858 Iteration 7: log likelihood = -2638.6858 Censored Poisson regression Number of obs = 3,000 Uncensored = 2,840 Limits: Lower = 0 Left-censored = 0 Upper = 3 Right-censored = 160 LR chi2(8) = 1003.81 Log likelihood = -2638.6858 Prob > chi2 = 0.0000

accidents | IRR Std. err. z P>|z| [95% conf. interval] | |

1.past | 2.641695 .1967576 13.04 0.000 2.282884 3.056902 | |

1.parent | .8345776 .0425139 -3.55 0.000 .7552765 .922205 | |

ntickets | ||

1 | 1.994213 .1182697 11.64 0.000 1.775374 2.240027 | |

2 | 3.841546 .2575073 20.08 0.000 3.36859 4.380906 | |

3 | 6.979123 .6090534 22.26 0.000 5.881909 8.281012 | |

4 | 15.97291 2.579757 17.16 0.000 11.63879 21.92099 | |

5 | 66.9069 2547.936 0.11 0.912 2.57e-31 1.74e+34 | |

6 | 58.24981 4426.516 0.05 0.957 1.20e-63 2.82e+66 | |

_cons | .3387513 .015292 -23.98 0.000 .3100673 .3700889 | |

We interpret the model coefficients (or incidence-rate ratios) as if the censoring had not occurred. That is to say, as though we had seen all of the data, uncensored.

We find that past accidents predict more future accidents, that being a parent predicts fewer future accidents, and that the number of tickets generally predicts more future accidents, although having just 1 or 2 tickets has little significance.

Because of the censoring, we do not know which of the people coded as having 3 accidents really had exactly 3 accidents, or which had more.

We can, however, now make predictions of the expected uncensored number of accidents and the probabilities of any specified number of accidents, including values greater than 3.

We wonder, what are the chances anyone had more than 3 accidents in our data? Our data were officially top-coded, but were they practically top-coded? We can obtain each driver's probability of having four or more accidents by typing

. predict fourplus, pr(4,.)

We now have the probability that each driver in our sample had four or more accidents. To get the expected number of drivers who had 4 or more accidents, we simply sum these probabilities

. total fourplusTotal estimation Number of obs = 3,000

Total Std. err. [95% conf. interval] | ||

fourplus | 74.46338 5.614501 63.45472 85.47205 | |

We expect 74.5 drivers in our data had more than 3 accidents, and top-coding almost certainly affected our data.

Almost certainly? We have a standard error above, but the standard
error and confidence interval do not account for the probabilities
having themselves been estimated. If we use **margins** to perform
the computation, it will produce the correct standard error and confidence
interval

. margins, expression(predict(pr(4,.))*3000)Predictive margins Number of obs = 3,000 Model VCE: OIM Expression: predict(pr(4,.))*3000

Delta-method | ||

Margin std. err. z P>|z| [95% conf. interval] | ||

_cons | 74.46338 5.238347 14.22 0.000 64.19641 84.73036 | |

**margins** wants to report a mean, so we had to trick it into giving
us a total by multiplying the probabilities by our sample size of 3000.

With such a small standard error and a lower bound of 43.6 on our confidence interval, we can definitively say, or at least as definitively as any statistician can say, that top-coding affected our data.

Read more about censored Poisson models in *Stata Base Reference Manual*;
see [R] **cpoisson**.