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power

Power and sample size of the two one-sided tests procedure.

The question a bioequivalence study asks after it ran is whether the 90 % interval of the geometric mean ratio lies within the acceptance limits (pkpdutils.stats.bioequivalence); the question it has to answer before it runs is how many subjects that decision needs. Both are the same procedure: power_tost is the probability that the two one-sided tests of Schuirmann (1987) both reject at the assumed ratio and the assumed within-subject variability, and sample_size_tost is the smallest number of subjects which reaches a target power.

The power is exact, not simulated: under the normal model of the log values the two test statistics are a bivariate non-central t pair, whose probability is Owen's Q function (Owen 1965),

\[Q_\nu(t, \delta; a, b) = \frac{1}{\Gamma(\nu/2)\,2^{(\nu-2)/2}} \int_a^b \Phi\!\left(\frac{t x}{\sqrt{\nu}} - \delta\right) x^{\nu-1} e^{-x^2/2}\,dx,\]

evaluated here by numerical integration (scipy.integrate.quad) of the logarithm of the integrand, which is the algorithm of the R package PowerTOST (Labes, Schuetz & Lang). The design enters through the constant \(b_k\) of the standard error and the degrees of freedom, both taken from PowerTOST: a 2x2 crossover has \(b_k = 2\) and \(\nu = n - 2\), two parallel groups \(b_k = 4\) and \(\nu = n - 2\), the four period full replicate (TRTR/RTRT) \(b_k = 1\) and \(\nu = 3n - 4\) and the three period replicate (TRT/RTR) \(b_k = 1.5\) and \(\nu = 2n - 3\), with \(n\) the total number of subjects of the study.

TOSTDesign dataclass 

TOSTDesign(
    name, bk, df_factor, df_offset, step, description
)

A study design of the power calculation.

Attributes:

Name Type Description
name str

the name of the design, as PowerTOST spells it
bk float

the design constant of the standard error, \(\mathrm{se} = \sigma\sqrt{b_k/n}\)
df_factor float

factor of the degrees of freedom, \(\nu = a n + b\)
df_offset float

offset of the degrees of freedom
step int

the step of the sample size search, 2 for a design whose sequences have to carry the same number of subjects
description str

how the design is spelled out in a report

df 

df(n)

The degrees of freedom of a study of n subjects.

Parameters:

Name Type Description Default
n int

the total number of subjects.

 required

Returns:

Type Description
float

The degrees of freedom of the residual.

se 

se(sigma, n)

The standard error of the log ratio of a study of n subjects.

The design is assumed to have two sequences (or two groups) of \(n_1 = \lceil n/2 \rceil\) and \(n_2 = \lfloor n/2 \rfloor\) subjects,

\[\mathrm{se} = \sigma\sqrt{b_{k,ni} \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}, \qquad b_{k,ni} = b_k/4,\]

which is \(\sigma\sqrt{b_k/n}\) for an even n and the standard error of the study with one subject more in one sequence for an odd one. \(b_{k,ni}\) is the unbalanced design constant of PowerTOST (known.designs(): ½, 1, ¼ and ⅜ for the four designs here), which is a quarter of \(b_k\) in every one of them.

Parameters:

Name Type Description Default
sigma float

the standard deviation of the log values.

 required
n int

the total number of subjects.

 required

Returns:

Type Description
float

The standard error.

owens_q 

owens_q(nu, t, delta, a, b)

Owen's Q function, by numerical integration of the log integrand.

\[Q_\nu(t, \delta; a, b) = \frac{1}{\Gamma(\nu/2)\,2^{(\nu-2)/2}} \int_a^b \Phi\!\left(\frac{t x}{\sqrt{\nu}} - \delta\right) x^{\nu-1} e^{-x^2/2}\,dx\]

(Owen 1965), the probability of the bivariate non-central t pair the two one-sided tests form. \(Q_\nu(t, \delta; 0, \infty)\) is the distribution function of the non-central t distribution at t with the non-centrality \(\delta\), which is the regression test of this function. The integrand is evaluated as \(\exp((\nu-1)\ln x - x^2/2 - \ln\Gamma(\nu/2) - \frac{\nu-2}{2}\ln 2)\,\Phi(\cdot)\), so that neither \(x^{\nu-1}\) nor \(\Gamma(\nu/2)\) overflows for the large degrees of freedom of a big study.

Parameters:

Name Type Description Default
nu float

degrees of freedom, positive.

 required
t float

the argument of the normal distribution function.

 required
delta float

the non-centrality.

 required
a float

lower bound of the integral, non-negative.

 required
b float

upper bound of the integral.

 required

Returns:

Type Description
float

The value of the integral, 0 for an empty interval.

Raises:

Type Description
ValueError

if nu is not positive or a is negative.

power_tost 

power_tost(
    *,
    cv,
    n,
    gmr=0.95,
    limits=(0.8, 1.25),
    design="2x2",
    alpha=0.05,
)

The exact power of the two one-sided tests procedure.

With \(\sigma = \sqrt{\ln(1 + \mathrm{CV}^2)}\) the standard deviation of the log values, \(\mathrm{se} = \sigma\sqrt{b_k/n}\) the standard error of the log ratio of the design, \(\Delta = \ln \mathrm{GMR}\) and the log limits \(\ln\theta_L\), \(\ln\theta_U\), the two non-centralities are

\[\delta_1 = \frac{\Delta - \ln\theta_L}{\mathrm{se}}, \qquad \delta_2 = \frac{\Delta - \ln\theta_U}{\mathrm{se}},\]

and with \(t = t_{1-\alpha,\nu}\) and \(R = (\delta_1 - \delta_2)\sqrt{\nu} / (2t)\) the power is the difference of two Owen's Q values,

\[1 - \beta = Q_\nu(-t, \delta_2; 0, R) - Q_\nu(t, \delta_1; 0, R),\]

the exact probability that both one-sided tests reject (Owen 1965; the algorithm of PowerTOST::power.TOST). An odd n is split into the two sequences (or groups) of \(\lceil n/2 \rceil\) and \(\lfloor n/2 \rfloor\) subjects the study would have, which widens the standard error a little against the balanced formula, as PowerTOST does for a dropout.

Parameters:

Name Type Description Default
cv float

the within-subject coefficient of variation as a fraction (the total CV for a parallel design), 0.3 for 30 %.

 required
n int

the total number of subjects of the study.

 required
gmr float

the geometric mean ratio the study is powered for; 0.95 is the usual assumption of a 5 % difference of the formulations.

 0.95
limits tuple[float, float]

the acceptance limits of the ratio.

 (0.8, 1.25)
design DesignName | str

the design, one of DESIGNS.

 '2x2'
alpha float

the level of each one-sided test, 0.05 for a 90 % interval.

 0.05

Returns:

Type Description
float

The power, a probability.

Raises:

Type Description
ValueError

for an unknown design, a negative cv, a non-positive gmr, reversed limits, an alpha outside (0, 0.5), or an n which leaves no degree of freedom.

sample_size_tost 

sample_size_tost(
    *,
    cv,
    gmr=0.95,
    target_power=0.8,
    design="2x2",
    alpha=0.05,
    limits=(0.8, 1.25),
    max_n=100000,
)

The smallest number of subjects which reaches a target power.

The search walks the sample size upwards from a lower bound derived from the normal approximation and returns the first size whose power_tost reaches target_power. A crossover is searched in steps of two, so that the sequences carry the same number of subjects, and starts at four; a parallel design is searched in steps of one and starts at three. The number returned is the total number of subjects of the study, not the number per sequence or per group.

Parameters:

Name Type Description Default
cv float

the within-subject coefficient of variation as a fraction (the total CV for a parallel design).

 required
gmr float

the geometric mean ratio the study is powered for.

 0.95
target_power float

the power to reach, 0.8 or 0.9 in practice.

 0.8
design DesignName | str

the design, one of DESIGNS.

 '2x2'
alpha float

the level of each one-sided test.

 0.05
limits tuple[float, float]

the acceptance limits of the ratio.

 (0.8, 1.25)
max_n int

the largest sample size the search looks at.

 100000

Returns:

Type Description
int

The total number of subjects.

Raises:

Type Description
ValueError

for an unknown design, a target_power outside (0, 1), a gmr outside the limits (no sample size reaches any power then), or as power_tost; also when max_n subjects do not reach the power.