interim.tsd.in: Interim analysis of first stage data of 2-stage 2x2 crossover...
In Detlew/Power2Stage: Power and Sample-Size Distribution of 2-Stage Bioequivalence Studies
interim.tsd.in R Documentation
Interim analysis of first stage data of 2-stage 2x2 crossover designs based on
the Inverse Normal method
Description
Following the design scheme according to power.tsd.in the function
performs the interim analysis of the first stage data.
Usage
interim.tsd.in(alpha, weight, max.comb.test = TRUE, targetpower = 0.8,
GMR1, n1, CV1, df1 = NULL, SEM1 = NULL, theta1, theta2,
GMR, usePE = FALSE, min.n2 = 4, max.n = Inf,
fCpower = targetpower, fCrit = "CI", fClower, fCupper, fCNmax,
ssr.conditional = c("error_power", "error", "no"),
pmethod = c("exact", "nct", "shifted"))
Arguments
alpha
If one element is given, the overall one-sided significance level (not
the adjusted level for stage 1). In this
case the adjusted alpha levels will be calcualted internally. If two
elements are given, the argument refers to the two adjusted one-sided
alpha levels for stage 1 and
stage 2, respectively.
If missing, defaults to 0.05.
weight
Pre-defined weight(s) of stage 1, see
'Details' for more information. Note that using the notation from
Maurer et al, weight corresponds to information fraction, other literature
may refer to sqrt(weight) as being the weight. weight must either
contain one element (in case of max.comb.test = FALSE) or
two elements (in case of max.comb.test = TRUE).
If missing, defaults to 0.5 for max.comb.test = FALSE and to
c(0.5, 0.25) for max.comb.test = TRUE.
max.comb.test
Logical; if TRUE (default) the maximum combination test will be
used, otherwise the standard combination test.
targetpower
Desired (overall) target power to declare BE at the end of the trial.
GMR1
Observed ratio of geometric means (T/R) of stage 1 data (use e.g., 0.95 for 95%).
n1
Sample size of stage 1.
CV1
Observed coefficient of variation of the intra-subject variability of
stage 1 (use e.g., 0.3 for 30%).
df1
Optional; Error degrees of freedom of
stage 1 that can be specified in
addition to n1.
SEM1
Optional; Standard error of the difference of means of
stage 1 that can be specified in
addition to CV1. Must be on additive scale (i.e. usually log-scale).
theta1
Lower bioequivalence limit. Defaults to 0.8.
theta2
Upper bioequivalence limit. Defaults to 1.25.
GMR
Assumed ratio of geometric means (T/R) to be used in power calculation
for stage 1 and sample size re-estimation
for stage 2.
usePE
If TRUE the sample size re-estimation is done with the observed
point estimate (PE) of the treatment difference in
stage 1.
Defaults to FALSE.
Note: The power of stage 1 used for the futility inspection and calculation
of the estimated conditional target power is always calculated with the
planning value GMR.
min.n2
Minimum sample size of stage 2. Defaults to 4.
If the sample size re-estimation step gives a sample size for
stage 2 less than min.n2, then
min.n2 will be used for stage 2.
max.n
Maximum overall sample size stage 1 +
stage 2.
This is not a futility criterion regarding the maximum sample size! If
max.n is set to a finite value and the sample size re-estimation gives a
sample size for stage 2 (n2) such
that n1 + n2 > max.n, then the sample size for
stage 2 will be set to n2 = max.n - n1.
Defaults to Inf, i.e., no constraint on the re-estimated sample size.
fCpower
Threshold for power monitoring step to decide on futility for cases where
BE has not been achieved after
stage 1: If BE has not been
achieved after stage 1 and the power for
stage 1 is greater than or equal to
fCpower, then the study will be considered a failure.
See ‘Details’ for more information on the choice of
fCpower.
fCrit
Futility criterion to use: "No" (no futility criterion regarding
observed point estimate, confidence interval and maximum sample size),
"PE" (observed point estimate of the geometric mean ratio from
stage 1),
"CI" (90% confidence interval of the geometric mean ratio from
stage 1), "Nmax" (overall maximum
sample size); or a combination thereof (concatenate abbreviations).
Defaults to "CI".
fClower
Lower futility limit for the PE or CI of
stage 1.
If the PE or CI is completely outside of fClower
... fCupper the study is to be stopped due to futility (not BE).
May be missing. If "PE" or "CI" is specified within fCrit,
the default will be set to 0.8 for fCrit = "PE" or 0.95 for fCrit = "CI".
If neither "PE" nor "CI" is specified within fCrit, there
will be no futility constraint regarding point estimate or confidence interval
from stage 1 (regardless of any
specification of fClower and/or fCupper).
fCupper
Upper futility limit for the PE or CI of
stage 1.
Analogous to fClower: Will be set to 1/fClower if missing.
fCNmax
Futility criterion regarding maximum sample size. If the determined sample size
for stage 2 (n2) is such that
n1 + n2 > fCNmax, the study will not continue to
stage 2 and stopped due to futility (not
BE).
If "Nmax" is specified within fCrit and argument fCNmax
is missing, the value will be set to fCNmax = 4*n1. If "Nmax" is
not specified within fCrit, then there will be no futility constraint
regarding maximum sample size (regardless of any specification of fCNmax).
ssr.conditional
Method for sample size re-estimation step: "no" does not use
conditional error rates nor the estimated conditional target power for
the second stage, "error" uses conditional error rates for the
second stage, and "error_power" uses both conditional error rates
and the estimated conditional target power for the second stage.
Defaults to "error_power".
See also ‘Details’.
pmethod
Power calculation method, also to be used in the sample size estimation for
stage 2.
Implemented are "nct" (approximate calculations via non-central
t-distribution, "exact" (exact calculations via Owen’s Q),
and "shifted" (approximate calculation via shifted central t-distribution
like in the paper of Potvin et al.)
In contrast to power.tsd.in the default value here is "exact".
Details
The observed values of stage 1 (e.g. GMR1, n1, CV1) may
be obtained based on the first stage data via the usual ANOVA approach.
The optional arguments df1 and SEM1 require a somewhat
advanced knowledge (provided in the raw output from for example the software
SAS, or may be obtained via emmeans::emmeans).
However, it has the advantage that if there were missing data the exact
degrees of freedom and standard error of the difference can be used,
the former possibly being non-integer valued (e.g. if the
Kenward-Roger method was used).
The weight argument always refers to the first weight of a pair of
weights. For example, in case of max.comb.test = FALSE the standard
combination test requires two weights (w, 1-w) but only the first one, w,
is required as input argument here because the second weight is
automatically specified once the first is given. Similarly for
max.comb.test = TRUE, w and w* need to be specified, which in turn
define the two pairs of weights (w, 1-w) and (w*, 1-w*).
If ssr.conditional = "error_power", the design scheme generally
calculates the estimated conditional target power of the second stage and
uses this value as desired target power in the sample size re-estimation process:
If fCpower > targetpower, then the conditional estimated
target power may be negative. This does not seem sensible. Therefore, for such
cases the desired target power for the sample size re-calculation will be set
to targetpower, i.e. ssr.conditional will be set to "error".
Also, if the futility criterion based on the power of stage 1 is met,
then the conditional estimated target power will be negative. Thus, no further
sample size calculation can be made. To acknowledge that this rule is nonbinding,
for the purpose of calculating n2 the argument ssr.conditional is set
to "error".
Value
Returns an object of class "evaltsd" with all the input arguments and results
as components. As part of the input arguments a component cval is also
presented, containing the critical values for
stage 1 and 2 according to the
input based on alpha, weight and max.comb.test.
The class "evaltsd" has an S3 print method.
The results are in the components:
p11
Observed p-value for first hypothesis.
p12
Observed p-value for second hypothesis.
z1
z statistic value for first null hypothesis.
z2
z statistic value for second null hypothesis.
RCI
Repeated confidence interval for stage 1. Corresponds to the usual CI with level alpha1.
MEUE
If the study stops, the median unbiased point estimate as estimate for the final adjusted
geometric mean ratio after stage 1 (note that the value is identical to GMR1.)
futility
Three dimensional vector with either 0 or 1. The first
component represents futility due to Power of first stage > fCpower,
the second futility due to CI (or PE) outside of
fClower ... fCupper, the third futility due to
n1 + n2 > fCNmax.
Note that the futility rules can be applied in a non-binding manner.
CI90
90% Confidence interval for observed ratio of geometric means
from stage 1. If fCrit != "CI" result will be NULL.
Power Stage 1
Calculated power of stage 1.
stop_s1
Logical, indicating whether to stop after stage 1
(due to BE or due to futility).
stop_fut
Logical, indicating whether study is recommended to be stopped after
stage 1
due to futility.
stop_BE
Logical, indicating whether BE could be concluded after
stage 1 or not (regardless of any
futility criterion).
n2
Required (total) sample size for stage 2
(will be zero if BE has been shown after
stage 1).
alpha_ssr
Only applicable if BE has not been shown after
stage 1. Contains
alpha values for the two hypotheses required for sample size re-calculation.
If ssr.conditional = "no" the result is equal to alpha,
otherwise it contains the conditional error rates for the standard combination
test (in case of max.comb.test = FALSE) or maximum combination test (in
case of max.comb.test = TRUE).
GMR_ssr
Only applicable if BE has not been shown after
stage 1. Contains the
geometric mean ratio used for sample size re-calculation (accounts for
adaptive planning step).
targetpower_ssr
Only applicable if BE has not been shown after
stage 1. Contains the target power used
for the sample size re-calculation (see also 'Details').
Author(s)
B. Lang
References
König F, Wolfsegger M, Jaki T, Schütz H, Wassmer G.
Adaptive two-stage bioequivalence trials with early stopping and sample size re-estimation.
Vienna: 2014; 35th Annual Conference of the International Society for Clinical Biostatistics. Poster P1.2.88
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.13140/RG.2.1.5190.0967")}.
Patterson SD, Jones B. Bioequivalence and Statistics in Clinical Pharmacology.
Boca Raton: CRC Press; 2nd edition 2017.
Maurer W, Jones B, Chen Y. Controlling the type 1 error rate in two-stage
sequential designs when testing for average bioequivalence.
Stat Med. 2018; 37(10): 1587–1607. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/sim.7614")}.
Wassmer G, Brannath W. Group Sequential and Confirmatory Adaptive Designs
in Clinical Trials.
Springer 2016. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-3-319-32562-0")}.
See Also
power.tsd.in, final.tsd.in
Examples
# Example from Maurer et al.
interim.tsd.in(GMR = 0.95, max.n = 4000,
GMR1 = exp(0.0424), CV1 = 0.3682, n1 = 20)
# Example 2 from Potvin et al.
interim.tsd.in(GMR = 0.95, GMR1 = 1.0876, CV1 = 0.18213, n1 = 12,
fCrit = "No", ssr.conditional = "no")
Detlew/Power2Stage documentation built on Oct. 12, 2023, 9:08 p.m.
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| interim.tsd.in | R Documentation |
Interim analysis of first stage data of 2-stage 2x2 crossover designs based on the Inverse Normal method
Description
Following the design scheme according to power.tsd.in the function
performs the interim analysis of the first stage data.
Usage
interim.tsd.in(alpha, weight, max.comb.test = TRUE, targetpower = 0.8,
GMR1, n1, CV1, df1 = NULL, SEM1 = NULL, theta1, theta2,
GMR, usePE = FALSE, min.n2 = 4, max.n = Inf,
fCpower = targetpower, fCrit = "CI", fClower, fCupper, fCNmax,
ssr.conditional = c("error_power", "error", "no"),
pmethod = c("exact", "nct", "shifted"))
Arguments
alpha |
If one element is given, the overall one-sided significance level (not
the adjusted level for stage 1). In this
case the adjusted alpha levels will be calcualted internally. If two
elements are given, the argument refers to the two adjusted one-sided
alpha levels for stage 1 and
stage 2, respectively. |
weight |
Pre-defined weight(s) of stage 1, see
'Details' for more information. Note that using the notation from
Maurer et al, weight corresponds to information fraction, other literature
may refer to sqrt(weight) as being the weight. |
max.comb.test |
Logical; if |
targetpower |
Desired (overall) target power to declare BE at the end of the trial. |
GMR1 |
Observed ratio of geometric means (T/R) of stage 1 data (use e.g., 0.95 for 95%). |
n1 |
Sample size of stage 1. |
CV1 |
Observed coefficient of variation of the intra-subject variability of stage 1 (use e.g., 0.3 for 30%). |
df1 |
Optional; Error degrees of freedom of
stage 1 that can be specified in
addition to |
SEM1 |
Optional; Standard error of the difference of means of
stage 1 that can be specified in
addition to |
theta1 |
Lower bioequivalence limit. Defaults to 0.8. |
theta2 |
Upper bioequivalence limit. Defaults to 1.25. |
GMR |
Assumed ratio of geometric means (T/R) to be used in power calculation for stage 1 and sample size re-estimation for stage 2. |
usePE |
If |
min.n2 |
Minimum sample size of stage 2. Defaults to 4. |
max.n |
Maximum overall sample size stage 1 +
stage 2. |
fCpower |
Threshold for power monitoring step to decide on futility for cases where
BE has not been achieved after
stage 1: If BE has not been
achieved after stage 1 and the power for
stage 1 is greater than or equal to
|
fCrit |
Futility criterion to use: |
fClower |
Lower futility limit for the PE or CI of
stage 1. |
fCupper |
Upper futility limit for the PE or CI of
stage 1. |
fCNmax |
Futility criterion regarding maximum sample size. If the determined sample size
for stage 2 ( |
ssr.conditional |
Method for sample size re-estimation step: |
pmethod |
Power calculation method, also to be used in the sample size estimation for
stage 2. |
Details
The observed values of stage 1 (e.g. GMR1, n1, CV1) may
be obtained based on the first stage data via the usual ANOVA approach.
The optional arguments df1 and SEM1 require a somewhat
advanced knowledge (provided in the raw output from for example the software
SAS, or may be obtained via emmeans::emmeans).
However, it has the advantage that if there were missing data the exact
degrees of freedom and standard error of the difference can be used,
the former possibly being non-integer valued (e.g. if the
Kenward-Roger method was used).
The weight argument always refers to the first weight of a pair of
weights. For example, in case of max.comb.test = FALSE the standard
combination test requires two weights (w, 1-w) but only the first one, w,
is required as input argument here because the second weight is
automatically specified once the first is given. Similarly for
max.comb.test = TRUE, w and w* need to be specified, which in turn
define the two pairs of weights (w, 1-w) and (w*, 1-w*).
If ssr.conditional = "error_power", the design scheme generally
calculates the estimated conditional target power of the second stage and
uses this value as desired target power in the sample size re-estimation process:
If fCpower > targetpower, then the conditional estimated
target power may be negative. This does not seem sensible. Therefore, for such
cases the desired target power for the sample size re-calculation will be set
to targetpower, i.e. ssr.conditional will be set to "error".
Also, if the futility criterion based on the power of stage 1 is met,
then the conditional estimated target power will be negative. Thus, no further
sample size calculation can be made. To acknowledge that this rule is nonbinding,
for the purpose of calculating n2 the argument ssr.conditional is set
to "error".
Value
Returns an object of class "evaltsd" with all the input arguments and results
as components. As part of the input arguments a component cval is also
presented, containing the critical values for
stage 1 and 2 according to the
input based on alpha, weight and max.comb.test.
The class "evaltsd" has an S3 print method.
The results are in the components:
p11 |
Observed p-value for first hypothesis. |
p12 |
Observed p-value for second hypothesis. |
z1 |
z statistic value for first null hypothesis. |
z2 |
z statistic value for second null hypothesis. |
RCI |
Repeated confidence interval for stage 1. Corresponds to the usual CI with level alpha1. |
MEUE |
If the study stops, the median unbiased point estimate as estimate for the final adjusted
geometric mean ratio after stage 1 (note that the value is identical to |
futility |
Three dimensional vector with either 0 or 1. The first
component represents futility due to Power of first stage > |
CI90 |
90% Confidence interval for observed ratio of geometric means
from stage 1. If |
Power Stage 1 |
Calculated power of stage 1. |
stop_s1 |
Logical, indicating whether to stop after stage 1 (due to BE or due to futility). |
stop_fut |
Logical, indicating whether study is recommended to be stopped after stage 1 due to futility. |
stop_BE |
Logical, indicating whether BE could be concluded after stage 1 or not (regardless of any futility criterion). |
n2 |
Required (total) sample size for stage 2 (will be zero if BE has been shown after stage 1). |
alpha_ssr |
Only applicable if BE has not been shown after
stage 1. Contains
alpha values for the two hypotheses required for sample size re-calculation.
If |
GMR_ssr |
Only applicable if BE has not been shown after stage 1. Contains the geometric mean ratio used for sample size re-calculation (accounts for adaptive planning step). |
targetpower_ssr |
Only applicable if BE has not been shown after stage 1. Contains the target power used for the sample size re-calculation (see also 'Details'). |
Author(s)
B. Lang
References
König F, Wolfsegger M, Jaki T, Schütz H, Wassmer G.
Adaptive two-stage bioequivalence trials with early stopping and sample size re-estimation.
Vienna: 2014; 35th Annual Conference of the International Society for Clinical Biostatistics. Poster P1.2.88
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.13140/RG.2.1.5190.0967")}.
Patterson SD, Jones B. Bioequivalence and Statistics in Clinical Pharmacology.
Boca Raton: CRC Press; 2nd edition 2017.
Maurer W, Jones B, Chen Y. Controlling the type 1 error rate in two-stage
sequential designs when testing for average bioequivalence.
Stat Med. 2018; 37(10): 1587–1607. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/sim.7614")}.
Wassmer G, Brannath W. Group Sequential and Confirmatory Adaptive Designs
in Clinical Trials.
Springer 2016. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-3-319-32562-0")}.
See Also
power.tsd.in, final.tsd.in
Examples
# Example from Maurer et al.
interim.tsd.in(GMR = 0.95, max.n = 4000,
GMR1 = exp(0.0424), CV1 = 0.3682, n1 = 20)
# Example 2 from Potvin et al.
interim.tsd.in(GMR = 0.95, GMR1 = 1.0876, CV1 = 0.18213, n1 = 12,
fCrit = "No", ssr.conditional = "no")
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