Sequential-package {Sequential}

Analysis Support, Critical Values, Power, Time to Signal and Sample Size for Sequential Analysis with Poisson and Binomial Data.

Description

Sequential is designed for continuous and group sequential analysis, where statistical hypothesis testing is conducted repeatedly on accumulating data that gradually increases the sample size. This is different from standard statistical analysis, where a single analysis is performed using a fixed sample size. It is possible to analyze either Poisson type data or binomial 0/1 type data. For binomial data, it is possible to incorporate an off-set term to account for variable matching ratios. For Poisson data, the critical value is based on a Wald-type upper boundary, which is flat on the scale of the log-likelihood ratio, and on a predetermined maximum sample size. Alternatively, it is also possible to apply a user defined alpha spending function in order to specify non-flat signaling thresholds. For group sequential analyses, there are functions for pre-specified group sizes and for the situation when the group sizes are not known a priori. It is also possible to perform mixed continuous/group sequential analysis, where, for example, there is at first a big batch of data that arrives in one group, followed by continuous sequential analysis. All results are exact, based on iterative numerical calculations, rather than asymptotic theory or computer simulations. In the Sequential package, there are functions to calculate critical values, statistical power, expected time to signal, and expected sample size at the end of the sequential analyses whether the null hypothesis was rejected or not. For example, for any desired power, relative risk and alpha level, the package can calculate the required upper limit on the sample size (maximum length of surveillance), the critical value needed, and the corresponding expected time to signal when the null hypothesis is rejected.

Details

Package:	Sequential
Type:	Package
Version:	4.3.3
Date:	2024-02-20
License:	GPL 2
LazyLoad:	yes
Index:
Performance.Threshold.Binomial	Power, Expected Time to Signal, Expected
	Sample Size and Alpha Spending for User Defined Threshold
	with Binomial Data.
Performance.Threshold.CondPoisson	Power, Expected Time to Signal, Expected
	Sample Size and Alpha Spending for User Defined Threshold
	with Conditional Poisson Data.
Performance.Threshold.Poisson	Power, Expected Time to Signal, Expected
	Sample Size and Alpha Spending for User Defined Threshold
	with Poisson Data.
Analyze.Binomial	Function to Conduct Group Sequential Analyses for Binomial
	Data When the Goup Sizes are not Known a Priori.
AnalyzeSetUp.Binomial	Function to Set Up the Input Parameters Before Using the
	Analyze.Binomial Function for the First Time.
Analyze.wBinomial	Function for group sequential analyses of multiple
	weighted binomial endpoints.
AnalyzeSetUp.wBinomial	Function to set up input parameters before using the
	Analyze.wBinomial function for the first time.
Analyze.Poisson	Function to Conduct Group Sequential Analyses for Poisson
	Data When the Goup Sizes are not Known a Priori.
AnalyzeSetUp.Poisson	Function to Set Up the Input Parameters Before Using the
	Analyze.Poisson Function for the First Time.
Analyze.CondPoisson	Function to Conduct Group Sequential Analyses for Conditional
	Poisson Data When the Goup Sizes are not Known a Priori.
AnalyzeSetUp.CondPoisson	Function to Set Up the Input Parameters Before Using the
	Analyze.CondPoisson Function for the First Time.
ConfidenceInterval.Binomial	Confidence interval for the relative risk following
	a sequential test with binomial data.
CV.Binomial	Critical Values for Group and Continuous Sequential
	Analysis with Binomial Data.
CV.Poisson	Critical Values for Group and Continuous Sequential
	Analysis with Poisson Data.
CV.CondPoisson	Critical Values for Continuous Sequential CMaxSPRT for
	Limited Information from Historical Cohort Using
	Conditional Poisson Data.
Optimal.Binomial	Optimal alpha spending for continuous and group sequential
	analysis with binomial data.
Performance.Binomial	Power, Expected Time to Signal and Expected Sample Size
	for Group and Continuous Sequential Analysis with
	Binomial Data.
Performance.Poisson	Power, Expected Time to Signal and Expected Sample Size
	for Continuous Sequential Analysis with
	Poisson Data.
Performance.CondPoisson	Power, Expected Time to Signal and Expected Sample
	Size for Continuous Sequential CMaxSPRT Using
	Limited Information from Historical Cohort
	with Conditional Poisson Data.
SampleSize.Binomial	Sample Size Calculation for Continuous Sequential Analysis with
	Binomial Data.
SampleSize.Poisson	Sample Size Calculation for Continuous Sequential Testing with
	Poisson Data.
SampleSize.CondPoisson	Sample Size Calculation for Continuous Sequential CMaxSPRT with
	Poisson Data.
Performance.AlphaSpend.Binomial	Power, Expected Time to Signal, Expected
	Sample Size and Threshold for User Defined Alpha Spending
	with Binomial Data.
Performance.AlphaSpend.CondPoisson	Power, Expected Time to Signal, Expected
	Sample Size and Threshold for User Defined
	Alpha Spending with Conditional Poisson Data.
Performance.AlphaSpend.Poisson	Power, Expected Time to Signal, Expected
	Sample Size and Threshold for User Defined
	Alpha Spending with Poisson Data.

Overview

Most of the sequential analysis methods found in the literature are based on asymptotic results. In contrast, this package contains functions for the exact calculation of critical values, statistical power, expected time to signal when the null is rejected and the maximum sample size needed when the null is not rejected. This is done for Poisson and binomial type data with a Wald-type upper boundary, which is flat with respect to the likelihood ratio function, and a predetermined upper limit on the sample size. For a desired statistical power, it is also possible to calculate the latter. The motivation for this package is post-market near real-time drug and vaccine safety surveillance, where the goal is to detect rare but serious safety problems as early as possible, in many cases after only a hand full of adverse events. The package can also be used in other application areas, such as clinical trials.

A tutorial paper by Silva et al. (2021) explains the main features of the package up to version 3.3.1, where sequential test designing and real data analyses are dicussed in more details.

The basis for this package is the Maximized Sequential Probability Ratio Test (MaxSPRT) statistic (Kulldorff et al., 2011), which is a variant of Wald's Sequential Probability Ratio Test (SPRT) (Wald, 1945,47). MaxSPRT uses a composite alternative hypothesis, and upper boundary to reject the null hypothesis when there are more events than expected, no lower boundary, and an upper limit on the sample size at which time the sequential analyses end without rejecting the null. MaxSPRT was developed for post-market vaccine safety surveillance as part of the Vaccine Safety Datalink project run by the Centers for Disease Control and Prevention.

In this package, all critical values, alpha spending strategies, statistical power, expected time to signal and required sample size to achieve a certain power, are obtained exactly to whatever decimal precision desired, using iterative numerical calculations. None of the results are based on asymptotic theory or computer simulations.

Poisson Data

To start, consider continuous sequential analysis for Poisson data. Let C_t be the random variable that counts the number of events up to time t. Suppose that, under the null hypothesis, C_t has a Poisson distribution with mean \mu_t, where \mu_t is a known function reflecting the population at risk. Under the alternative hypothesis, suppose that C_t has a Poisson distribution with mean RR\mu_t, where "RR" is the unknown increased relative risk due to the vaccine. The MaxSPRT statistic defined in terms of the log likelihood ratio is given by:

LLR_t=(\mu_t-c_t)+c_t \log{c_t/\mu_t},

when c_t is at least \mu_t, and LLR_t =0, otherwise. For continuous sequential analysis, the test statistic, LLR_t, is monitored at all times t \in (0,T], where T= SampleSize. SampleSize is defined a priori by the user in order to achieve the desired statistical power, which can be calculated using the SampleSize.Poisson function. The sequential analysis ends, and H_0 is rejected if, and when, LLR_t \geq CV, where CV is calculated using the CV.Poisson function. If \mu_t= SampleSize, the sequential analysis ends without rejecting the null hypothesis. To calculate other important performance metrics, such as the expected time to signal when the null hypothesis is rejected, use the Performance.Poisson, Performance.Threshold.Poisson and Performance.AlphaSpend.Poisson functions. These functions also work for group sequential analysis when the group sizes are fixed a priori. For with fixed and/or unpredictable group sizes, see the function Analyze.Poisson.

If the first event occurs sufficiently early, the sequential analysis may end with the null hypothesis rejected after a single event. There is an option to require a minimum number of observed events, c_t= M, before the null can be rejected. Setting M in the range [3,6] is often a good choice (Kulldorff and Silva, 2012). If there is a delay until the sequential analysis starts, but it continuous continuously thereafter, there is an option for that as well, requiring a minimum number \mu_t= D of expected events before the null can be rejected.

With continuous sequential analysis, investigators can repeatedly analyze the data as often as they want, ensuring that the overall probability of falsely rejecting the null hypothesis at any time during the analysis is controlled at the desired nominal significance level (Wald, 1945, 1947). Continuous sequential methods are suitable for real-time or near real-time monitoring. When data is only analyzed intermittently, group sequential methods are used instead (Chin, 2012; Cook and DeMets, 2007; Xia, 2007; Friedman et al., 2010; Ghosh and Sen, 1991; Jennison and Turnbull, 2000; Mukhopadhyay and Silva, 2002; Whitehead, 1997). The data is then analyzed at regular or irregular discrete time intervals after a certain amount of data is accessible. Group sequential statistical methods are commonly used in clinical trials, where a trial may be stopped early due to either efficacy or unexpected adverse events (Jennison and Turnbull, 2000).

The same test statistic, LLR_t, is used for group sequential analyses (Silva and Kulldorff, 2012). The times when LLR_t is evaluated can be defined in several ways, using regular or irregular time intervals that are referenced by calendar period, sample size or some scale involving the distribution of the data. For Poisson data, the group sequential analysis must be conducted with equal size groups, with a constant expected number of adverse events between looks at the accumulating data. In another words, LLR_t is compared against CV whenever \mu_t is a multiple of SampleSize/Looks, where 'Looks' is the total number of looks at the data.

Binomial Data

The MaxSPRT method can also be applied to binomial/Bernoulli data. Let n denote the total number of events that has been observed in a sequential monitoring up to a certain moment in time. Suppose that these n events are categorized as cases and controls. For example, cases may be adverse events happening to a person taking drug A, while controls may be the same adverse event happening to someone in a matched set of individuals taking drug B. As another example, in a self-control sequential analysis, cases may be adverse events happening during the 1-28 days following vaccination, while controls are the same adverse events happening 29-56 days after vaccination.

Let C_t to denote the number of cases among the n events, and assume that C_t follows a binomial distribution with success probability equal to p, where p = 1=(1 + z), and z is the matching ratio between the occurrence of a case and of a control under the null hypothesis. For example, if the probability of having a case (instead of a control) is p = 1=(1 + z) = 0.5, then z=1 (1:1 matching ratio), or, p = 0.25 for z=3 (1:3 matching ratio), etc.

The MaxSPRT statistic (Kulldorff et al., 2011) for a continuous binomial surveillance is:

LR_n=\frac{(c_n/n)^{c_n}\left[(n-c_n)/n\right]^{n-c_n}}{\left[1/(1+z)\right]^{c_n}\left[z/(1+z)\right]^{n-c_n}},

if z c_n/(n-c_n)>1, and LR_n= 1 otherwise.

The monitoring is continued until either there is a signal rejecting the null hypothesis (LR_n > CV) or until n=N, which indicates that the null is not to be rejected. To perform the calculations, use the CV.Binomial, SampleSize.Binomial, Performance.Binomial, Performance.Threshold.Binomial and Performance.AlphaSpend.Binomial functions for continuous and group sequential fashions. For group sequential analysis, the group sizes are fixed a priori. For fixed and/or unpredictable group sizes, see the function Analyze.Binomial.

The main assumptions behind the method above are: (i) the monitoring is truly performed in a continuous fashion; (ii) the matching ratio (z) is constant for all of the n events, and (iii) it uses a Wald type rejection boundary that is flat in terms of the likelihood function. Relaxing these assumptions, Fireman et al. (2013) developed exact sequential analysis for group sequential data with varying matching ratios, and for any user specified alpha rejection plan.

Multiple Weighted Binomial Endpoints

When there are multiple different outcomes, one can use weights reflecting practical interpretations, such as an outcome severity in comparison to the others. For example, for two different outcomes, the first with weight w, and the second with weight 1. This way, a single event of the first outcome is equivalent to w outcomes of the second type. This type of analysis is treated by Silva et al (2019). To run the analysis with multiple weighted binary endpoints with fixed and/or unpredictable group sizes, see the function Analyze.wBinomial.

Conditional Poisson data with limited information from historical cohort - CMaxSPRT

In Poisson MaxSPRT, the expected mean \mu_t is assumed to be a known function reflecting the baseline adverse event risk in the absence of the exposure of interest. In practice, it is estimated with historical data and the uncertainty associated with the estimated counts may or may not have a non-negligible impact on the performance of the sequential analysis method. Li and Kulldorff (Li and Kulldorff, 2010) showed in their simulation study that uncertainty in the estimated baseline means can be ignored when the total number of events in the historical data is at least 5 times the specified upper limit T. Otherwise, it is recommended to implement a statistical procedure that takes in account the variation from the historical data, i.e. a procedure that conditionates the likelihood function of the historical Poisson data, here simply called "conditional Poisson data". For this, the Conditional Maximized Sequential Probabilit Ratio Test (CMaxSPRT) to account for variation in both the historical and surveillance cohorts.

Let c and V denote the total number of events and the cumulative person-time in the historical data, let P_k denote the cumulative person-time observed in the surveillance population when the kth event occurred. The CMaxSPRT statistic defined in terms of the log likelihood ratio is given by

U_k=clog(\frac{c(1+P_k/V)}{c+k})+klog(\frac{k(1+P_k/V)}{P_k/V(c+k)}),

when k/c>P_k/V, and U_k=0, otherwise. In the original publication (Li and Kulldorff, 2010), the method was introduced as a continuous sequential analytic approach with the upper limit defined in terms of the maximum number of observed events, i.e., k \leq K, and the critical value calculated via a Monte Carlo approach. A large number of Monte Carlo simulations (e.g., 10 million) might be needed to calculate the critical values with a reasonable precision.

In Silva et al. (2019a), the method was extended i) with another option of defining the surveillance length in terms of the maximum cumulative person-time divided by the total cumulative person-time in the historical cohort, i.e., P_k/V \le T, ii) with an exact calculation of the critical values for both surveillance length definitions, and iii) for group sequential analysis with data updated and analyzed intermittently instead of continuously. The exact critical values are calculated using the interval havling method to solve for the root of a complex, non-linear equation such that the overall Type I error rate is preserved at the nominal level. As K increases, the computing time for the exact critical values increases exponentially.

Silva et al. (2019a) also proposed two approximation methods to calculate the critical values that require substantially less computing time. One approch may overestimate the critical values and thus is referred to as the conservative approach as it may yield lower-than-nominal Type I error rates; the other approach may underestmate the critical values and thus is referred to as the liberal approach as it may yield higher-than-nominal Type I error rates. The recommendation is to use the exact approach when K is small (e.g., 10), use the conservative approach when K is medium or large but c is small, and use the liberal approach when c is medium (e.g., 50) or large. Exact calculations for selected tuning parameters show that the three approaches yield very similar results when K and c are reasonably large.

For calculating critical values for a Wald type rejection boundary, use the CV.CondPoisson function. For statistical power, expected time to signal, expected time of surveillance, and maximum sample size requirements, use the Performance.CondPoisson, Performance.Threshold.CondPoisson, Performance.AlphaSpend.CondPoisson, and SampleSize.CondPoisson functions. For fixed and/or unpredictable group sizes, see the function Analyze.CondPoisson.

Alpha spending function for unpredictable group sizes

The alpha spending function specifies the cumulative amount, F_{\alpha}(t), of Type I error probability related to each of the possible values of n. Thus, at the end of the monitoring the alpha spending corresponds to a value smaller than or equal to the overall amount of Type I error probability defined for the overall nominal significance level, \alpha.

Denote the single probability of rejecting the null hypothesis at the j-th test by \alpha_j. Then, the alpha spending at test i is given by F_{\alpha}(t_i)=\sum_{j=1}^{i}\alpha_j \leq \alpha.

There is a vast number of proposals for choosing the shape of the alpha spending function. Jennison and Turnbull (2000) present a rich discussion about this topic. They dedicated a special attention to the alpha spending of the form: F_{\alpha}(t)=\alpha t^{\rho}, where \rho>0, and t represents a fraction of the maximum length of surveillance.

A new aproach for alpha spending selection is the optimal solution, introduced by Silva and Kulldorff (2020). The optimal solution is based on exact calculations through linear programing, and it is operated through the function Optimal.Binomial.

For MaxSPRT with Poisson data, Silva (2018b) presents a rule of thumb for a near-optimal alpha spending shape. Likewise, Silva et al. (2019b) discuss the choice of alpha spending shapes for CMaxSPRT.

To run continuous or group sequential analysis with an user defined alpha spending function, and/or, when the group sizes are not known a prior, Analyze.Binomial, Analyze.wBinomial (for multiple weighted binomial endpoints), Analyze.Poisson, and Analyze.CondPoisson should be used for binomial and Poisson, and conditional Poisson data, respectively. These functions work differently than the other functions mentioned above. Those other functions are designed to be used before the start of the sequential analysis, in order to determine what the maximum sample size and critical value should be. Once the sequential analysis is under way, the test statistic is then calculated using a hand calculator or an excel spread sheet, and compared with the critical value. The functions Analyze.Binomial, Analyze.Poisson, and Analyze.CondPoisson work very differently, in that they are run at each look at the accumulating data, whenever a new group of data arrives, and it is meant to perform the test itself, i.e., there is no need to use hand calculators or excel spread sheets or any other auxiliar code. The results and conclusions, including a descriptive table and illustrative graphics, are automatically provided after running Analyze.Binomial, Analyze.wBinomial, Analyze.Poisson, or Analyze.CondPoisson.

Important: before using these functions, though, it is necessary to first run the functions AnalyzeSetup.Binomial, AnalyzeSetup.wBinomial, AnalyzeSetup.Poisson, or AnalyzeSetup.CondPoisson once in order to set everything up for the sequential analysis.

Comparison with Other R Packages for Sequential Analysis

The R Sequential package is designed for sequential analysis where statistical hypothesis testing is performed using gradually accumulating data. It is not designed for quality control problems, where a process is monitored over time to detect an emerging problem due to a sudden increase in the excess risk. Although the methods for sequential analysis and quality control may seem similar, as they both analyze gradually accumulating data, they are actually very different in both their purpose and design. Under the sequential hypothesis testing approach, the objective is to quickly determine if there is some intrinsic excess risk, with the assumption that this risk does not change over time. For example, we may want to know if drug A is better than drug B, and there is no reason to believe that the behavior of the drugs change over time. In the quality control setting, the objective is instead to detect a possible change in a stochastic process that may occur in the future, and to detect that change as soon as possible after it occurs. For example, the heart of a hospital patient is beating as it should, but if there is a sudden deterioration, the alarm should sound as soon as possible without generating a lot of false alarms. This package is only meant for sequential analysis of the former type, and it should not be used for quality control type problems. For quality control type analyses, there are other R packages available, such as graphicsQC, IQCC, MetaQC, MSQC, qcc, and qcr.

In a number of ways, the R Sequential package differs from other R packages for sequential analyses. Historically, most sequential analysis has been conducted using asymptotic statistical theory, and that is also what is used in the gsDesign, ldbounds, PwrGSD, seqDesign, seqmon, and sglr R packages. In contrast, the R Sequential package is based on exact results, using iterative numerical calculations, rather than using asymptotic theory or computer simulations.

With this package, it is only possible to analyze binomial/Bernoulli, Poisson, or conditional Poisson data. For other probability distributions, such as normal or exponential data, other R packages should be consulted, such as GroupSeq or SPRT. Moreover, all functions in this package uses a one-sided upper bound to reject the null hypothesis, while the analyses end without rejecting the null when an upper limit on the sample size is reached. For two sided sequential analysis, or other types of rejection boundaries, other R packages must be used, such as e.g. ldbounds and Binseqtest. Finally, in this package, there are functions for both continuous and group sequential analysis, and it is also possible to analyze situations where some of the data arrives continuously while other parts of the data arrives in groups. Most other R packages are exclusively designed for group sequential analysis, but there are some that also do continuous sequential analysis, such as Binseqtest and SPRT, but Binseqtest is only for binomial data type, and SPRT is for simple alternative hypothis, while Sequential can be used for binomial and Poisson data and is meant to composite alternative hypothesis. The present package offers the possibility to calculate the expected time to signal through the Performance.Poisson, Performance.G.Poisson, Performance.Binomial, Performance.G.Binomial, and Performance.CondPoisson functions, which is not offered by the other packages cited above. Another important advantage of the Sequential package is the possibility of eliciting, through exact calculations, the minimum sample size needed to accomplish with target statistical powers through the functions SampleSize.Poisson, SampleSize.CondPoisson, and SampleSize.Binomial.

Acknowledgements

Development of the R Sequential package has been funded and supported by:
- Food and Drug Administration, USA, through the Mini-Sentinel Project (v1.0,1.1,2.0).
- National Institute of General Medical Sciences, NIH, USA, through grant number R01GM108999 (v2.0 to 3.1).
- Federal University of Ouro Preto (UFOP), through contract under internal UFOP's resolution CEPE 4600 (v2.0 to 3.1).
- National Council of Scientific and Technological Development (CNPq), Brazil, process number 301391/2019-0. (v3.1 to v4.3).
- National Council of Scientific and Technological Development (CNPq), Brazil, process number 302882/2022-7. (v4.3.1).
- Support Foundation to Minas Gerais State Research-Fapemig, Brazil, grant numbers PQ-00787-21 and RED-00133-21. (v3.1 to v4.3).
- Bank for Development of the Minas Gerais State (BDMG), Brazil (v1.0).

Feedback from users is greatly appreciated. Very valuable suggestions concerning the R Sequential package have been received from various individuals, including:
- Ron Berman, University of California Berkeley.
- Claudia Coronel-Moreno, Harvard Pilgrim Health Care Institute.
- Bruce Fireman, Kaiser Permanente Northern California.
- Josh Gagne, Harvard Medical School and Brigham and Women's Hospital.
- Ned Lewis, Kaiser Permanente Northern California.
- Judith Maro, Harvard Medical School and Harvard Pilgrim Health Care Institute.
- Azadeh Shoaibi, Food and Drug Administration.
- Katherine Yih, Harvard Medical School and Harvard Pilgrim Health Care Institute.
- Jie Tang, Clinical biostatistics, Janssen R and D US, Johnson and Johnson LLC.
- Tuomo A. Nieminen, The National Institute for Health and Welfare (THL), Finland.
- Andreia Leite, Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine.
- Laura Hou, Harvard Pilgrim Health Care Institute, Boston, USA.
- Abdurrahman Abdurrob, Division of Pharmacoepidemiology and Pharmacoeconomics, Department of Medicine, Brigham and Women's Hospital and Harvard Medical School.
- Kirk Snyder, Information Management Services, Inc.
- Joselito M. Montalban, Active Living, Population and Public Health Branch, Winnipeg, Manitoba, Canada.
- Sarah Spruin and Vaccine Safety Surveillance, Public Health Agency of Canada.
- Lola Adewale, Stateam LLC, New Jersey, United States.
- Lian Lin, Ph.D., Director, Safety Statistics, Stats and Programming, Moderna.

Version History of the R Sequential Package

Version 1.1, February 2013
Exact sequential analysis for Poisson data:
- Exact continuous sequential analysis.
- Exact group sequential analysis with pre-defined and constant groups sizes.
- Wald type rejection boundary.
- Statistical power, expected time to signal and sample size calculations.
- User guide.

Version 1.2, January 2014
- Improved code structure and efficiency.
- More extensive user guide.

Version 2.0, June 2015
Exact sequential analysis for binomial data:
- Continuous sequential analysis.
- Group sequential analysis with pre-defined group sizes.
- Group sequential analysis with unpredictable group sizes, not specified a priori.
- Fixed or variable binomial probabilities (matching ratios).
- User specified alpha spending function.
- Statistical power, expected time to signal and sample size calculations.
- Updated user guide.

Version 2.0.1, June 2015
- Correction of bugs in CV.Poisson function.
- Updated user guide.

Version 2.0.2, Octuber 2015
- Improved user guide.

Version 2.1, May 2016
Exact sequential analysis for Poisson data:
- Group sequential analysis with unpredictable group sizes, not specified a priori.
- User specified alpha spending function.
- Mixed group-continuous sequential analysis.
- Statistical power, expected time to signal and sample size calculations for non-constant groups sizes.
Other:
- Directory address parameter in AnalyzeSetUp functions.
- Probability parameter in binomial functions.
- Updated user guide.

Version 2.1.1, June 2016
- Correction of bugs in Poisson functions.
- Updated user guide.

Version 2.2, July 2016
- Critical Value, Performance, and SampleSize calculations for CMaxSPRT with Poisson data.
- Updated user guide.

Version 2.2.1, September 2016
- Correction of bugs in CV.Poisson and CV.G.Poisson functions.
- Updated user guide.

Version 2.3, Dec 2016
- Correction of bugs in the SampleSize.Binomial function.
- Improvement of SampleSize functions for considering vectors for the input parameters R and power.
- Inclusion of the new functions AnalyzeSetUp.CondPoisson and Analyze.CondPoisson.
- Updated user guide.

Version 2.3.1, Feb 2017
- Correction of bugs in Analyze.Binomial and AnalyzeSetUp.Poisson functions.
- Adjustment on the relative risk estimation method for Analyze.Binomial function.
- Updated user guide.

Version 2.3.2, Aug 2017
- Correction of bugs in Analyze.Binomial.
- Updated user guide.

Version 3.0, Jan 2019
- Functions Optimal.Binomial and Analyze.wBinomial.
- Updated user guide.

Version 3.0.1, Feb 2019
- Updated user guide.

Version 3.1, Sept 2019
- New Functions Performance.Threshold.Binomial, Performance.Threshold.CondPoisson,
Performance.Threshold.Poisson, Performance.AlphaSpend.Binomial,
Performance.AlphaSpend.CondPoisson, Performance.AlphaSpend.Poisson.
- Implemented two-tailed testing for the functions: CV.Poisson, Performance.Poisson,
SampleSize.Poisson, Analyze.wBinomial, Optimal.Binomial,
Performance.Threshold.Binomial, Performance.AlphaSpend.Binomial,
Performance.Threshold.Poisson.
- Updated user guide.
- Concatenation of functions for continuous and group fashions for critical values and performance.

Version 3.2, Nov 2020
- Function Analyze.Binomial improved for fixed-width and fixed-accuracy confidence intervals,
- Function Performance.AlphaSpend.Binomial improved for new inputs having lower and upper alpha spending in two tailed testing,
- Correction of bugs in Analyze.Poisson and CV.Poisson functions,
- Updated user guide.

Version 3.2.1, Dec 2020
- Correction of bugs in the Analyze.Poisson function,
- Updated user guide.

Version 3.3, Feb 2021
- Optimized Analyze.Poisson function for running three times faster than earlier versions.
- Updated user guide.

Version 3.3.1, Feb 2021
- Adjusted Analyze.Binomial function for a flexible H0:R<=R0 under the null hypothesis.
- Updated user guide.

Version 3.3.2, Aug 2021
- Adjusted functions: Analyze.Poisson, AnalyzeSetUp.Poisson, Performance.AlphaSpend.Poisson, and SampleSize.Poisson for the surveillance of COVID-19 vaccination.
- Updated user guide.

Version 3.3.3, Aug 2021
- Adjusted Performance.AlphaSpend.Poisson function.
- Updated user guide.

Version 3.4, Sept 2021
- Adjusted Analyze.Poisson function.
- Updated user guide.

Version 4.0, Sept 2021
- New function ConfidenceInterval.Binomial, adjusted Analyze.wBinomial and new parameter R0 for Analyze.Poisson and Performance.AlphaSpend.Poisson.
- Updated user guide.

Version 4.1, Oct 2021
- Adjusted SampleSize.Poisson.
- Updated user guide.

Version 4.2, Feb 2022
- Adjusted AnalyzeSetUp.Poisson.
- Updated user guide.

Version 4.3, April 2022
- Adjusted Analyze.Binomial for bugs on the relative risk estimate under variable matching ratio.
- Updated user guide.

Version 4.3.1, Octuber 2023
- Updated reference for the bounded-length confidence interval following sequential analysis with binary data.

Version 4.3.2, Dec 2023
- Correction of bugs in the Analyze.CondPoisson function.

Version 4.3.3, Feb 2024
- Updated user guide.

Author(s)

Ivair Ramos Silva, Martin Kulldorff.
Maintainer: Ivair Ramos Silva <[e-mail]>

References

Chin R. (2012), Adaptive and Flexible Clinical Trials, Boca Raton, FL: Chapman and Hall/CRC.

Cook TD, DeMets DL. (2007), Introduction to Statistical Methods for Clinical Trials: Chapman and Hall/CRC Texts in Statistical Science.

Fireman B, et al. (2013), Exact sequential analysis for binomial data with timevarying probabilities. Manuscript in Preparation.

Friedman LM, Furberg CD, DeMets D. (2010), Fundamentals of Clinical Trials, 4th ed.: Springer.

Ghosh BK, Sen PK. (1991), Handbook of Sequential Analysis, New York: MARCEL DEKKER, Inc.

Ghosh M, Mukhopadhyay N, Sen PK. (2011), Sequential Estimation: Wiley.

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.

Kulldorff M, Silva IR. (2015), Continuous Post-market Sequential Safety Surveillance with Minimum Events to Signal. REVSTAT Statistical Journal, 15(3): 373–394.

Li L, Kulldorff M. (2010), A conditional maximized sequential probability ratio test for Pharmacovigilance. Statistics in Medicine, 29 (2), 284–295.

Mukhopadhyay N, Silva BM. (2002), Sequential Methods and Their Applications, 1th ed.: Chapman and Hall/CRC.

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Examples

## Critical value for continuous sequential analyses for Poisson Data.
## Maximum sample size = 10, alpha = 0.05 and minimum number of events = 3:

cvt<- CV.Poisson(SampleSize=10,D=0,M=3,alpha=0.05)

## Statistical power and the expected time to signal for relative risk RR=2:

result<- Performance.Poisson(SampleSize=10,D=0,M=3,cv=cvt,RR=2)

# And if you type:
result

# Then you will see the following:
#          Power ESignalTime ESampleSize
#     [1,] 0.7329625    4.071636    5.654732