Sequential (version 4.5.1)

CV.Binomial: Calculates exact critical values for group and continuous sequential analysis with binomial data.

Description

The function CV.Binomial obtains critical values for the group continuous sequential MaxSPRT test with binomial data, using a Wald-type upper boundary, which is flat with respect to the likelihood ratio function, and an pre-specified upper limit on the sample size.

Usage

CV.Binomial(N,alpha=0.05,M=1,z="n",p="n",GroupSizes=1,Tailed="upper")

Value

cv

The critical value for a significance level equal to alpha. The largest conservative value is provided when it is not possible to have an Type I error exactly equal to alpha.

Type_I_Error

The exact Type I error probability given cv. Always less than or equal to alpha.

Arguments

N

The upper limit on the sample size (length of surveillance) expressed in terms of the total number of events (cases plus controls). "N" must be a positive integer. 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", "alpha=0.01" and "z=1", the statistical power is approximately 1 for "N>500". There is no default value.

alpha

The significance level. The "alpha" level must be in the range (0,0.5]. The default value is "alpha=0.05".

M

The minimum number of events needed before the null hypothesis can be rejected. "M" must be a positive integer, and the default value is "M=1".

z

For a matched case-control analysis, z is the number of controls matched to each case under the null hypothesis. There is no default value.

p

The probability of having a case under the null hypothesis. There is no default value.

GroupSizes

Vector with the number of events (cases+controls) between two consecutive looks (tests) at the data, i.e, the group sizes. The length of this vector is equal to the maximum number of tests. The entries do not have to be the same, but they must sum up "N". If the group sizes is an integer instead of a vector, then that integer is the group size for all looks at the data, and the number of looks is "N/GroupSizes". The default is GroupSizes=1 for continuous sequential analysis.

Tailed

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

Author

Ivair Ramos Silva, Ned Lewis, Ron Berman, Martin Kulldorff.

Acknowledgements

Development of the CV.Binomial function was funded by:
- Food and Drug Administration, Center for Drug Evaluation and Research, through the Mini-Sentinel Project; base version, documentation;
- National Institute of General Medical Sciences, NIH, USA, through grant number R01GM108999; code revisions, increased computational speed, improved documentation.

We thank Ron Berman, University of California, Berkeley, for a key suggestion to speed up the calculations, and Bruce Fireman for helpful discussions.

Details

For the continuous and group binomial MaxSPRT, CV.Binomial calculates the upper boundary used to determine if the null hypothesis is to be rejected at each analysis. This is done for pre-specified values of the statistical significance level (alpha) and an upper limit on the sample size equal to N.

The input z represents the number of controls matched to each case. For example, if there are 3 controls matched to each case, "z=3". In a self-control analysis, z is the ratio of the control interval to the risk interval. For example, if the risk interval is 2 days long and the control interval is 7 days long, z=7/2. In terms of p, the binomial probability under the null hypothesis, p=1/(1+z), or equivalently, z=1/p-1. The parameter z must be a positive number.

Alternatively, instead of z the user can specify p directly. Note that only one of these inputs, z or p, has to be specified, but if both are entered the code will only work if z and p are such that p=1/(1+z). Otherwise, an error message will appear to remind that such condition must be complied.

For details about the algorithm used to calculate the critical value, see the paper by Kulldorff et al. (2011).

For some configurations of N and alpha and GroupSizes, there is no critical value that gives a Type I error probability that is exactly equal to the requested "alpha". This is because of the discrete nature of binomial data. In such situations, CV.Binomial returns the greatest critical value that guarantees a Type I error probability smaller than "alpha". Thus the critical value for the binomial sequential analysis is conservative in this sense.

References

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.

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

See Also

Analyze.Binomial: for performing sequential analysis with group, continuous or unpredictable sequential fashion.

Examples

# Example 1:
## Critical value for continuous binomial sequential analysis with
#  a maximum sample size of 20 events, requiring at 
#  least 3 events to reject the null, and with a significance level of 0.05:

CV.Binomial(N=20,alpha=0.05,M=3,z=1.1)

# Example 2:
## Critical value for five-group sequential analysis with
#  a maximum sample size of 25 events, requiring at 
#  least 1 event to reject the null, and with a significance level of 0.05:
result<- CV.Binomial(N=25,alpha=0.05,M=1,z=7/2,GroupSizes=5)
# if you type:
result
# then you will get the following output:
# [[1]]
# [1] 1.9852

# [[2]]
# [1] 0.04775995

# Example 3:
## Critical value for four-group sequential analysis with
#  a maximum sample size of 50 events, requiring at 
#  least 1 event to reject the null, and with a significance level of 0.05:
result<- CV.Binomial(N=50,alpha=0.05,M=1,z=7/2,GroupSizes=c(10,10,15,15))
cv<- as.numeric(result[1])
# if you type:
cv
# then you will get the following output:
# [1] 1.99202