The function `CV.G.Poisson`

calculates exact critical values for group sequential analysis with Poisson data, using a Wald
type upper boundary, which is flat with respect to the likelihood ratio function, and with a pre-specified upper limit on the sample size.

1 2 | ```
CV.G.Poisson(SampleSize,alpha=0.05,GroupSizes,M=1)
``` |

`SampleSize` |
The upper limit on the sample size (length of surveillance) expressed in terms of the expected number of events under the null hypothesis. The "SampleSize" must be greater than 0. There is no default value. |

`M` |
The minimum number of events needed before the null hypothesis can be rejected. The default value is M=1. If there are frequent looks at the data, so that the group sizes are extremely small, a value of M=1 means that even a single event can reject the null hypothesis if it occurs sufficiently early. According to Kulldorff and Silva(2015), a reasonable choice is M=4. |

`alpha` |
The significance level, or the type 1 error probability, which is the probability of rejecting the null hypothesis when it is true. The alpha level must be in the range (0,0.5]. The default value is alpha=0.05. |

`GroupSizes` |
Vector containing the expected number of events under H0 for each test. The values must be positive numbers. The dimension of this vector must be equal to the maximum number of sequential tests. Thus, the sum of the entries in GroupSizes has to be equal to SampleSize. There is no default value. |

For group sequential analysis with Poisson data, `CV.G.Poisson`

calculates the critical value that constitutes the upper boundary used to
determine if the null hypothesis should be rejected. This is done for pre-specified values of the statistical significance level (alpha) and
an upper limit on the sample size, determining the maximum length of surveillance, as well as other parameter settings.
The test is one-sided, so that the null hypothesis is only rejected when there are more events than expected.

For several configurations of SampleSize, Looks and M there is no critical value that gives a probability of Type I error that is exactly equal to alpha.
In such cases, the function `CV.G.Poisson`

returns the largest critical value that will guarantee a
type I error probability that is smaller than alpha, so that the sequential analysis is conservative.

For large values of the maximum SampleSize, such as 200 or more, the computational requirements can be high. To avoid very large computation times, we suggest not using values greater than 1000. Typically, this is not a major restriction. For example, for "RR=1.1" and "alpha=0.01", the statistical power is approximately 1 for a maximum sample size greater than 500.

`cv` |
The critical value for a significance level equal to alpha. The largest conservative value is provided when it is not possible to have a type I error that is exactly equal to alpha. |

Development of the CV.G.Poisson function was funded by:

- Food and Drug Administration, Center for Biologics Evaluation and Research, through the Mini-Sentinel Post-Rapid Immunization Safety Monitoring (PRISM) program (v1.0);

- National Council of Scientific and Technological Development (CNPq), Brazil (v1.0);

- Bank for Development of the Minas Gerais State (BDMG), Brazil (v1.0);

- National Institute of General Medical Sciences, NIH, USA, through grant number R01GM108999 (v2.0.1, v2.0.2).

`Performance.G.Poisson`

: Calculates the statistical power, expected time to signal and expected sample size for group sequential analysis with Poisson data.

`CV.Poisson`

: Calculating critical values for continuous sequential analysis with Poisson data.

`CV.G.Binomial`

: Calculates critical values for group sequential analysis with binomial data.

Ivair Ramos Silva, Ned Lewis, Martin Kulldorff.

Kulldorff M, Silva IR. (2015). Continuous post-market sequential safety surveillance with minimum events to signal. arxiv:1503.01978 [stat.ap].

1 2 3 4 5 6 | ```
## Calculates the critical value for five group sequential looks, at 5, 11,
## 17, 22 and 30 expected events under the null hypothesis, and for a
## statistical signifi-
## cance level of 0.05.
CV.G.Poisson(SampleSize=30,alpha=0.05,GroupSizes= c(5,6,6,5,8))
``` |

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