The function `SampleSize.Poisson`

obtains the required sample size (length of surveillance) needed to guarantee a desired statistical
power for a pre-specified relative risk, when doing continuous sequential analysis for Poisson data with a Wald type upper boundary, which is flat with
respect to the log-likelihood ratio. It can also be used to approximate the sample size needed when doing group sequential analysis for Poisson data.

1 2 3 | ```
SampleSize.Poisson(alpha=0.05,power=0.9,M=1,D=0,RR=2,
precision=0.000001)
``` |

`alpha` |
The significance level. The default value is alpha=0.05. Must be in the range (0,0.5]. |

`RR` |
The target vector of relative risks to be detected with the requested statistical vector of powers. The default value is RR=2. |

`power` |
The target vector of overall statistical powers to detect an increased relative risk (RR). The default value is power=0.90. |

`M` |
The minimum number of events needed before the null hypothesis can be rejected. It must be a positive integer. A good rule of thumb is to set M=4 (Kulldorff and Silva, 2015). The default value is M=1, which means that even a single event can reject the null hypothesis if it occurs sufficiently early. |

`D` |
The expected number of events under the null hypothesis at the first look at the data. This is used when there is an initial large chunk of data arriving, followed by continuous sequential analysis. The default value is D=0, which is also the best choice. This means that there is no delay in the start of the sequential analyses. If D is very large, the maximum sample size will be set equal to D if a non-sequential analysis provides the desired power. |

`precision` |
The tolerance for the difference between the requested and actual statistical power. Should be very small. The default value is precision=0.000001. |

When using the MaxSPRT and the `CV.Poisson`

function to conduct continuous sequential analysis for Poisson data, the null
hypothesis is rejected when the log likelihood ratio exceeds the pre-determined critical value calculated by `CV.Poisson`

.
The sequential analysis ends without rejecting the null hypothesis when a predetermined upper limit on the sample size is
reached, expressed in terms of the expected number of events under the null hypothesis. For example, the sequential analysis
may end as soon as the sample size is such that there are 50 expected events under the null.

The function `SampleSize.Poisson`

calculates what the upper limit on the sample size (length of surveillance) that is required
for the continuous Poisson based MaxSPRT to achieve the desired statistical power for a pre-specified relative risk RR.
The solution is exact using iterative numerical calculations (Kulldorff et al., (2011).

While designed for continuous sequential analysis, the `SampleSize.Poisson`

function can also be used to approximate the
required upper limit on the sample size that is needed when doing group sequential analysis for Poisson data, using the `CV.G.Poisson function`

.

`SampleSize_by_RR_Power` |
A table containing the main performance measures associated to the required samples sizes, expressed in terms of the expected number of events under the null hypothesis, for each combination of RR and power. |

Development of the `SampleSize.Poisson`

function was funded by:

- 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 (v2.0.1,2.0.2).

`CV.Poisson`

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

`CV.G.Poisson`

: for calculating critical values for group sequential analysis with Poisson data.

`Performance.Poisson`

: for calculating the statistical power, expected time to signal and expected sample size for continuous sequential analysis with Poisson data

`SampleSize.Binomial`

: for calculating the minimum sample size given a target power in continuous sequential analysis with binomial data.

Ivair Ramos Silva, Martin Kulldorff.

Kulldorff M, Davis RL, Kolczak M, Lewis E, Lieu T, Platt R. (2011). A Maximized Sequential Probability Ratio Test for Drug and Safety Surveillance. Sequential Analysis, 30: 58–78. 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 7 8 9 10 11 12 13 14 15 16 17 18 | ```
### Example 1:
## Sample size required to obtain a power of 80%, for a relati-
## ve risk of 3, no delay for starting the surveillance (D=0),
## and when the null hypothesis can be rejected with one event
## (M=1) under an alpha level of 5%.
# result1<- SampleSize.Poisson(alpha=0.05,power=0.8,M=1,D=0,RR=3)
# result1
## Example 2:
## Sample size required to obtain a power of 90%, for a relati-
## ve risk of 2, no delay for starting the surveillance (D=0),
## and when the null hypothesis can be rejected only after 2
## events (M=2) under an alpha level of 10%.
##
# result2<- SampleSize.Poisson(alpha=0.1,power=0.9,M=2,D=0,RR=2)
# result2
``` |

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