Calculates the boundaries at each interim analysis, i.e. the number of events in the treatment arm that would lead to a stopping of the trial based binomial exact tests, assuming that the events should be equally distributed among both arms. The indicated scenario (and all more extreme) would lead to a rejection of H0 (equal distribution) and a stopping for safety.
Usage
getHarmBound(
nevents,
alpha_test = NULL,
alpha_total = NULL,
power = NULL,
pH0,
maxevents = NULL,
icc = NULL,
pH1 = NULL,
rrH1 = NULL,
orH1 = NULL,
rdH1 = NULL,
r0 = NULL
)Arguments
- nevents
vector with number of events (over both arms) at which an interim analysis is done
- alpha_test
the nominal alpha level to use for each test
- alpha_total
target overall family-wise type I error
- power
target power at the specified alternative
- pH0
proportion of events in the treatment arm under the null hypothesis, typically based on randomization ratio (e.g. 0.5 for a 1:1 randomization)
- maxevents
optional maximum number of events expected for the trial (over both arms), used to calculate the expected number of events
- icc
intraclass correlation if there is more than one event per patient
- pH1
optional alternative, numeric vector, proportion of events in the treatment arm
- rrH1
alternative specification of alternative as risk ratio (treatment / control)
- orH1
alternative specification of alternative as risk ratio (treatment / control). Requires the control proportion (r0).
- rdH1
alternative specification of alternative as risk difference (treatment - control). Requires the control proportion (r0).
- r0
risk in the control arm. Required if the alternative is given as risk difference or odds ratio.
Value
a list with 3 data.frames: bounds, stopprob and opchar. bounds has a row for each interim analysis and columns for number of events (events), number of events in control and treatment arm that would lead to a stop (events_treatment, events_control), and the nominal alpha for each test (alpha_test). stopprob has a row for each interim analysis and columns for number of events (events), the hypothesis (pH), the stopping probability (stop_prob), and the cumulative stopping probability (cum_stop_prob) opchar has a row for each hypothesis (null plus each alternative) and columns for the assumed proportion of events in the treatment arm (p), the cumulative stopping probabilities (cum_stop_prob) and the expected total number of events (expected_events) for the null and each alternative.
Details
If there are several events per patient, the intraclass correlation coefficient has to be given and a beta-binomial framework is used. The overdispersion factor (or design effect) by which the variance exceeds the regular binomial variance is printed. Note that the effective sample size is reduced by that factor.
The rejection region for the binomial or beta-binomial exact tests must be given for either each test (alpha_test), overall (alpha_total, the family-wise error rate) or by the targetted power for the specified alternative.
See also
Available plot methods: plot.harmbound
Examples
getHarmBound(nevents=c(10,50,100), alpha_test=0.025, pH0=0.5)
#> $bounds
#> events events_treatment events_control alpha_test
#> 1 10 9 1 0.025
#> 2 50 33 17 0.025
#> 3 100 61 39 0.025
#>
#> $stopprob
#> $stopprob$`0.5`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.5 H0 0.01074219 0.01074219
#> 2 50 0.5 H0 0.01490030 0.02564249
#> 3 100 0.5 H0 0.01194192 0.03758441
#>
#>
#> $opchar
#> p cum_stop_prob expected_events hyp
#> 1 0.5 0.03758441 98.28819 H0
#>
#> attr(,"class")
#> [1] "harmbound" "list"
#adding an alternative
getHarmBound(nevents=c(10,50,100), alpha_test=0.025, pH0=0.5, pH1=0.6)
#> $bounds
#> events events_treatment events_control alpha_test
#> 1 10 9 1 0.025
#> 2 50 33 17 0.025
#> 3 100 61 39 0.025
#>
#> $stopprob
#> $stopprob$`0.5`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.5 H0 0.01074219 0.01074219
#> 2 50 0.5 H0 0.01490030 0.02564249
#> 3 100 0.5 H0 0.01194192 0.03758441
#>
#> $stopprob$`0.6`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.6 H1 0.0463574 0.0463574
#> 2 50 0.6 H1 0.2098108 0.2561682
#> 3 100 0.6 H1 0.2505214 0.5066895
#>
#>
#> $opchar
#> p cum_stop_prob expected_events hyp
#> 1 0.5 0.03758441 98.28819 H0
#> 2 0.6 0.50668952 85.33730 H1
#>
#> attr(,"class")
#> [1] "harmbound" "list"
#assume that a total of 150 events might occur (for the expected events)
getHarmBound(nevents=c(10,50,100), alpha_test=0.025, pH0=0.5, pH1=0.6, maxevents=150)
#> $bounds
#> events events_treatment events_control alpha_test
#> 1 10 9 1 0.025
#> 2 50 33 17 0.025
#> 3 100 61 39 0.025
#>
#> $stopprob
#> $stopprob$`0.5`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.5 H0 0.01074219 0.01074219
#> 2 50 0.5 H0 0.01490030 0.02564249
#> 3 100 0.5 H0 0.01194192 0.03758441
#>
#> $stopprob$`0.6`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.6 H1 0.0463574 0.0463574
#> 2 50 0.6 H1 0.2098108 0.2561682
#> 3 100 0.6 H1 0.2505214 0.5066895
#>
#>
#> $opchar
#> p cum_stop_prob expected_events hyp
#> 1 0.5 0.03758441 146.4090 H0
#> 2 0.6 0.50668952 110.0028 H1
#>
#> attr(,"class")
#> [1] "harmbound" "list"
#several alternatives
getHarmBound(nevents=c(10,50,100), alpha_test=0.025, pH0=0.5,
pH1 = seq(0.6,0.8,by=0.05), maxevents=150)
#> $bounds
#> events events_treatment events_control alpha_test
#> 1 10 9 1 0.025
#> 2 50 33 17 0.025
#> 3 100 61 39 0.025
#>
#> $stopprob
#> $stopprob$`0.5`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.5 H0 0.01074219 0.01074219
#> 2 50 0.5 H0 0.01490030 0.02564249
#> 3 100 0.5 H0 0.01194192 0.03758441
#>
#> $stopprob$`0.6`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.6 H1 0.0463574 0.0463574
#> 2 50 0.6 H1 0.2098108 0.2561682
#> 3 100 0.6 H1 0.2505214 0.5066895
#>
#> $stopprob$`0.65`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.65 H1 0.08595444 0.08595444
#> 2 50 0.65 H1 0.43634566 0.52230009
#> 3 100 0.65 H1 0.32449164 0.84679173
#>
#> $stopprob$`0.7`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.7 H1 0.1493083 0.1493083
#> 2 50 0.7 H1 0.6414520 0.7907603
#> 3 100 0.7 H1 0.1910207 0.9817810
#>
#> $stopprob$`0.75`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.75 H1 0.24402523 0.2440252
#> 2 50 0.75 H1 0.70328355 0.9473088
#> 3 100 0.75 H1 0.05210513 0.9994139
#>
#> $stopprob$`0.8`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.8 H1 0.375809638 0.3758096
#> 2 50 0.8 H1 0.618228306 0.9940379
#> 3 100 0.8 H1 0.005959013 0.9999970
#>
#>
#> $opchar
#> p cum_stop_prob expected_events hyp
#> 1 0.50 0.03758441 146.40897 H0
#> 2 0.60 0.50668952 110.00282 H1
#> 3 0.65 0.84679173 78.10723 H1
#> 4 0.70 0.98178099 55.40060 H1
#> 5 0.75 0.99941391 42.90286 H1
#> 6 0.80 0.99999696 35.26587 H1
#>
#> attr(,"class")
#> [1] "harmbound" "list"
#using a risk ratio to specify the alternative
getHarmBound(nevents=c(10,50,100), alpha_test=0.025, pH0=0.5, rrH1=1.5, maxevents=150)
#> $bounds
#> events events_treatment events_control alpha_test
#> 1 10 9 1 0.025
#> 2 50 33 17 0.025
#> 3 100 61 39 0.025
#>
#> $stopprob
#> $stopprob$`0.5`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.5 H0 0.01074219 0.01074219
#> 2 50 0.5 H0 0.01490030 0.02564249
#> 3 100 0.5 H0 0.01194192 0.03758441
#>
#> $stopprob$`0.6`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.6 H1 0.0463574 0.0463574
#> 2 50 0.6 H1 0.2098108 0.2561682
#> 3 100 0.6 H1 0.2505214 0.5066895
#>
#>
#> $opchar
#> p rr cum_stop_prob expected_events hyp
#> 1 0.5 1.0 0.03758441 146.4090 H0
#> 2 0.6 1.5 0.50668952 110.0028 H1
#>
#> attr(,"class")
#> [1] "harmbound" "list"
# define the test so that an family-wise type I error of 5% is achieved
getHarmBound(nevents=c(10,50,100), alpha_total=0.05, pH0=0.5)
#> $bounds
#> events events_treatment events_control alpha_test
#> 1 10 9 1 0.03245429
#> 2 50 33 17 0.03245429
#> 3 100 60 40 0.03245429
#>
#> $stopprob
#> $stopprob$`0.5`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.5 H0 0.01074219 0.01074219
#> 2 50 0.5 H0 0.01490030 0.02564249
#> 3 100 0.5 H0 0.02085634 0.04649882
#>
#>
#> $opchar
#> p cum_stop_prob expected_events hyp
#> 1 0.5 0.04649882 98.28819 H0
#>
#> attr(,"class")
#> [1] "harmbound" "list"
# define the test so that an over power of 80% is achieved
# needs an alternative
getHarmBound(nevents=c(10,50,100), power=0.8, pH0=0.5, pH1=0.6)
#> $bounds
#> events events_treatment events_control alpha_test
#> 1 10 8 2 0.1013193
#> 2 50 31 19 0.1013193
#> 3 100 57 43 0.1013193
#>
#> $stopprob
#> $stopprob$`0.5`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.5 H0 0.05468750 0.0546875
#> 2 50 0.5 H0 0.04648929 0.1011768
#> 3 100 0.5 H0 0.05797765 0.1591544
#>
#> $stopprob$`0.6`
#> events pH hyp stop_prob cum_stop_prob
#> 1 10 0.6 H1 0.1672898 0.1672898
#> 2 50 0.6 H1 0.3260615 0.4933512
#> 3 100 0.6 H1 0.3043610 0.7977123
#>
#>
#> $opchar
#> p cum_stop_prob expected_events hyp
#> 1 0.5 0.1591544 92.75366 H0
#> 2 0.6 0.7977123 68.64085 H1
#>
#> attr(,"class")
#> [1] "harmbound" "list"