Sequential (version 4.5.1)

Performance.Threshold.Poisson: Performance and alpha spending for user-defined signaling threshold for sequential analysis with Poisson data.

Description

The function Performance.Threshold.Poisson calculates power, expected time to signal and alpha spending associated to any user-specified signaling threshold, flat or non-flat, for continuous or group sequential analysis with Poisson data. The user can select the scale for the signaling threshold among MaxSPRT, Pocock, OBrien-Fleming, or Wang-Tsiatis test statistics, all for a pre-specified upper limit on the sample size.

Usage

Performance.Threshold.Poisson(SampleSize,CV.lower="n",CV.upper="n",
CV.events.upper="n",M=1,D=0,GroupSizes="n",Tailed="upper",
Statistic=c("MaxSPRT", "Pocock", "OBrien-Fleming", "Wang-Tsiatis"),
Delta="n",RR)

Value

AlphaSpend

The alpha spending associated to the user-specified threshold.

Performance

A matrix with the following three performance measures for each target RR: statistical power, expected time to signal and expected sample size.

Arguments

SampleSize

The upper limit on the sample size (length of surveillance) expressed in terms of the expected number of events under the null hypothesis. There is no default value.

CV.lower

Signaling threshold for evidence of "RR<1". It is given in the scale of the selected test statistic infomed in "Statistic". There is no default value.

CV.upper

Signaling threshold for evidence of "RR>1". It is given in the scale of the selected test statistic infomed in "Statistic". There is no default value.

CV.events.upper

Signaling threshold for evidence of "RR>1". It is given in the scale of the events. The is no default value unless when one uses "CV.upper" instead.

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.

GroupSizes

Vector with the test-specific expected number of events under the null hypothesis between two looks at the data. Important: Must sums up "SampleSize". There is no default value.

Tailed

Tailed="upper" (default) for H0:RR<=1, and Tailed="lower" for H0:RR>=1 or Tailed="two" for H0:RR=1.

Statistic

The test statistic scale used for "CV.lower" and "CV.upper". There is no default.

Delta

Parameter needed for calculation of Wang-Tsiatis test statistic if this is the option selected in "Statistic". Must be a number in the (0, 0.5] interval. There is no default value.

RR

Vector of relative risks for performance calculation. There is no default value.

Author

Ivair Ramos Silva, Martin Kulldorff.

Acknowledgements

Development of the Performance.Threshold.Poisson function was funded by:
- National Institute of General Medical Sciences, NIH, USA, through grant number R01GM108999 (v2.0,2.0 to 3.1).
- Federal University of Ouro Preto (UFOP), through contract under internal UFOP's resolution CEPE 4600 (v2.0 to 3.1).

Details

For continuous and group sequential analysis with Poisson data, the alpha spending for user-specified thresholds are calculated with Performance.Threshold.Poisson.

The inputs CV.lower, CV.upper, if two-tailed testing (Tailed="two"), both lower and upper signaling thresholds must be informed. If the user desires a constant threshold (critical value), then a single number can be informed for each of these inputs. For time-variable (non-constant) thresholds, the length of CV.lower and CV.upper must coincide with the length of GroupSizes.

For details about the algorithm used to calculate the critical value, see the paper by Silva (2018).

With GroupSizes the user informs the sample size of each subsequent test. Therefore, only positive numbers are accepted in GroupSizes.

The input Statistic specifies the scale selected by the user to inform CV.lower and CV.upper among the classic methods: MaxSPRT (Kulldorf et al., 2011), Pocock (Pocock, 1977), OBrien-Fleming (O'Brien and Fleming, 1979), or Wang-Tsiatis (Jennison and Turnbull, 2000). For Statistic="Wang-Tsiatis", the user has to choose a number in the (0, 0.5] interval for Delta.

For RR the user must specify the target relative risks for calculation of the statistical performance measures to be delivered in the output. It can be a vector of positive number or a single number.

References

Jennison C, Turnbull B. (2000). Group Sequential Methods with Applications to Clinical Trials, London: Chapman and Hall/CRC.

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.

O'Brien PC, Fleming TR. (1979). A multiple testing procedure for clinical trials. Biometrics. 35:549--556.

Pocock SJ. (1977). Group sequential methods in the design and analysis of clinical trials. Biometrika. 64:191--199.

Silva IR. (2018). Type I Error Probability Spending for Post-Market Drug and Vaccine Safety Surveillance With Poisson Data. Methodol Comput Appl Probab, 20(2), 739--750.

Silva IR, Kulldorff M. (2015). Continuous versus Group Sequential Analysis for Vaccine and Drug Safety Surveillance. Biometrics, 71 (3), 851--858.

Silva IR, Maro J, Kulldorff M. (2021). Exact sequential test for clinical trials and post-market drug and vaccine safety surveillance with Poisson and binary data. Statistics in Medicine, DOI: 10.1002/sim.9094.

See Also

Performance.AlphaSpend.Poisson: for calculating performance and signaling threshold for user-specified alpha spending with Poisson data.
CV.Poisson: for calculating Wald-type signaling thresholds for continuous and group sequential analysis with Poisson data.
Analyze.Poisson: for performing sequential analysis with group, continuous or unpredictable sequential fashion for Poisson data.

Examples

## Example 1
#  Performance and alpha spending of group Poisson sequential analysis
#  with a maximum sample size of 90 expected events for two-tailed
#  testing, i.e. H0:RR=1, with irregular group sizes and different
#  and lower and upper thresholds with irregular
#  sample sizes 25, 20, 20, and 25. 
#  The statistical performance is evaluated for four different
#  target RR= 1, 1.2, 2, 3:

# res<- Performance.Threshold.Poisson(SampleSize=90,CV.lower=c(2.5,2.6,2.7,2.8),
# CV.upper=c(3,3.1,3.2,3.3),GroupSizes=c(25,20,20,25),Tailed="two",
# Statistic="MaxSPRT",Delta="n",RR=c(1,1.2,2,3))

## Example 2
#  Suppose that the Analyze.Poisson function was used for an actual analysis.
#  For evaluating the cumulative power after a certain number of subsequent tests,
#  one can enter with the critical values delivered by Analyze.Poisson in the
#  Performance.Threshold.Poisson.
#  For example, suppose that the following thresholds in the scale of the events
#  were printed by Analyze.Poisson for the first three tests:
#     cv.events<- c(2,3,5)
#  which were obtained for the following specific sample sizes:
#     mus<- c(0.05,0.5,1.2)
#  Calculating the cumulative power, the expected time to signal, and 
#  the expected sample size for RR=2:

# res<-Performance.Threshold.Poisson(SampleSize=sum(mus),CV.events.upper=cv.events,
# GroupSizes=mus, Statistic="MaxSPRT",RR=2)


# This returns a power about 30%.