Function to compute alpha needed (fixed sample size)

Description

Function to compute two-sided alpha needed to achieve target power given a rate ratio. Useful for computing probability to achieve hurdles.

Usage

alpNeeded(N, bta1, thta, tau, lam0, pow = 0.8, ar = 0.5)

Arguments

Details

This function computes the two-sided alpha alp. Function assumes a rate ratio < 1 is favourable to treatment.

Value

The two-sided alpha level.

Examples

# alpha needed to achieve multiple powers given rate ratio (and other input).
alpNeeded(N = 1000, bta1 = log(0.8), thta = 2, tau = 1, lam0 = 1.1, pow = c( .7, .8))

# alpha needed for many inputs
if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(RR = 0.8) %>%
    crossing(
      thta = c(2, 3, 4),
      lam0 = 1.1,
      pow = c(0.7, 0.8),
      N = c(500, 1000),
      tau = 1
    ) %>%
    mutate(
      alp = alpNeeded(N = N, bta1 = log(RR), thta = thta, tau = tau, lam0 = lam0, pow = pow)
    )
  assumptions %>% data.frame()

}

Function to compute alpha needed (fixed sample size)

Description

Function to compute two-sided alpha needed to achieve target power given a rate ratio. Useful for computing probability to achieve hurdles.

Usage

alpNeeded2(N, bta1, thta, L, pow = 0.8, ar = 0.5)

Arguments

Details

This function computes the two-sided alpha alp. Function assumes a rate ratio < 1 is favourable to treatment.

Value

The two-sided alpha level.

Examples

# alpha needed to achieve multiple powers given rate ratio (and other input).
alpNeeded2(N = 1000, bta1 = log(0.8), thta = 2, L = 1000, pow = c( .7, .8))

# alpha needed for many inputs
if (require("dplyr") & require("tidyr")) {


}

Assurance (expected power) for LWYY

Description

Function to compute assurance given fixed sample size N and follow-up tau, under input assumptions. Assumes a log-normal prior distribution. Here assurance is taken to mean probability to achieve statistical significance (p-value < alp).

Usage

assurance(
  N,
  bta1,
  bta1_sd,
  thta,
  tau,
  lam,
  alp = 0.05,
  ar = 0.5,
  ns = 1000,
  method = c("integration", "montecarlo"),
  frailty.type = c("unblind", "blind"),
  lam.type = c("base", "pool"),
  thtawarning = FALSE
)

Arguments

Details

If working with a blinded estimate of frailty variance (i.e. misspecified model), it is recommended to use frailty.type = "blind" and lam.type = "pool". In which case the frailty variance (i.e. model with treatment effect) is derived using thta and the quantile drawn from the prior distribution of the log-rate ratio. If working with an estimate of frailty variance, should use frailty.type = "unblind" instead.

Function assumes a rate ratio < 1 is favourable to treatment.

Value

The assurance given the input assumptions.

Examples

assurance(N = 500, bta1 = log(0.8), bta1_sd = 1, thta = 2, tau = 1, lam = 1.1, alp = 0.05,
          ns = 100000)


if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
  crossing(
    N = c(500, 1000),
    RR = c(0.6, 0.7, 0.8),
    bta1_sd = 1,
    thta = c(2, 3, 4),
    tau = c(0.8,0.9, 1.0),
    lam0 = c(3, 3.5)
  ) %>%
    mutate(pow = pow(N = N, bta1 = log(RR), thta = thta, tau = tau, lam = lam0, alp = alp)) %>%
    mutate(
      assurance_in_blind = assurance(N = N, bta1 = log(RR), bta1_sd = bta1_sd, thta = thta,
                                     tau = tau, lam = lam0, alp = alp, ns = 1000,
                                     frailty.type = "blind", lam.type = "pool")
    ) %>%
    mutate(
      assurance_mc_blind = assurance(N = N, bta1 = log(RR), bta1_sd = bta1_sd, thta = thta,
                                     tau = tau, lam = lam0, alp = alp, ns = 1000,
                                     method = "monte",
                                     frailty.type = "blind", lam.type = "pool")
    ) %>%
    mutate(
      assurance_in_unblind = assurance(N = N, bta1 = log(RR), bta1_sd = bta1_sd, thta = thta,
                                       tau = tau, lam = lam0, alp = alp, ns = 1000)
    ) %>%
    mutate(
      assurance_mc_unblind = assurance(N = N, bta1 = log(RR), bta1_sd = bta1_sd, thta = thta,
                                       tau = tau, lam = lam0, alp = alp, ns = 1000)
    )

  assumptions %>% data.frame()

}

Assurance (expected power) for LWYY

Description

Function to compute assurance given fixed sample size N and number of events, under input assumptions. Assumes a log-normal prior distribution. Here assurance is taken to mean probability to achieve statistical significance (p-value < alp).

Usage

assurance2(
  N,
  bta1,
  bta1_sd,
  thta,
  L,
  alp = 0.05,
  ar = 0.5,
  ns = 1000,
  method = c("integration", "montecarlo"),
  frailty.type = c("unblind", "blind"),
  thtawarning = FALSE
)

Arguments

Details

If working with a blinded estimate of frailty variance (i.e. misspecified model), it is recommended to use frailty.type = "blind". In which case the frailty variance (i.e. model with treatment effect) is derived using thta and the quantile drawn from the prior distribution of the log-rate ratio. If working with an estimate of frailty variance, should use frailty.type = "unblind" instead.

Function assumes a rate ratio < 1 is favourable to treatment.

Value

The assurance given the input assumptions.

Examples

assurance2(N = 500, bta1 = log(0.8), bta1_sd = 1, thta = 2, L = 1000, alp = 0.05,
          ns = 100000)


if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
  crossing(
    N = c(500, 1000),
    RR = c(0.6, 0.7, 0.8),
    bta1_sd = 1,
    thta = c(2, 3, 4),
    L = c(500, 1000, 1500)
  ) %>%
    mutate(pow = pow2(N = N, bta1 = log(RR), thta = thta, L = L, alp = alp)) %>%
    mutate(
      assurance_in_blind = assurance2(N = N, bta1 = log(RR), bta1_sd = bta1_sd, thta = thta,
                                     L = L, alp = alp, ns = 1000,
                                     frailty.type = "blind")
    ) %>%
    mutate(
      assurance_mc_blind = assurance2(N = N, bta1 = log(RR), bta1_sd = bta1_sd, thta = thta,
                                     L = L, alp = alp, ns = 1000,
                                     method = "monte",
                                     frailty.type = "blind")
    ) %>%
    mutate(
      assurance_in_unblind = assurance2(N = N, bta1 = log(RR), bta1_sd = bta1_sd, thta = thta,
                                       L = L, alp = alp, ns = 1000)
    ) %>%
    mutate(
      assurance_mc_unblind = assurance2(N = N, bta1 = log(RR), bta1_sd = bta1_sd, thta = thta,
                                       L = L, alp = alp, ns = 1000)
    )

  assumptions %>% data.frame()

}

Function to compute log rate ratio needed (fixed sample size)

Description

Function to compute log rate ratio needed to achieve target power at one-sided Type I control level alp/2. Useful to compute critical value (set pow = 0.5).

Usage

btaNeeded(
  N,
  thta,
  tau,
  lam,
  alp = 0.05,
  pow = 0.8,
  ar = 0.5,
  frailty.type = c("unblind", "blind"),
  lam.type = c("base", "pool"),
  interval = c(log(0.5), log(1))
)

Arguments

Details

This function computes the log rate ratio bta1 as the root of the equation pow(bta1, N, thta_, tau, lam0_, alp, ar) - pow = 0, where thta_ and lam0_ also depend on bta1 if using estimates from blinded analyses. If frailty.type = "blind": thta_ = thtap2thta(thta, bta1); otherwise if frailty = "unblind": thta_ = thta. If lam.type = "pool" then lam0_ = lam * 2 / (1 + exp(bta1)); otherwise if lam.type = "base": lam0_ = lam. Function assumes a rate ratio < 1 is favourable to treatment.

Value

The log rate ratio.

Examples

# Based on unblinded estimates
btaNeeded(N = 1000, thta = 2, tau = 1, lam = 1.1, alp = c(0.01, 0.05), pow = c(.5))
exp(btaNeeded(N = 1000, thta = 2, tau = 1, lam = 1.1, alp = c(0.01, 0.05), pow = c(.5)))

# Based on blinded estimates
btaNeeded(N = 1000, thta = 2, tau = 1, lam = 0.7, alp = c(0.01, 0.05), pow = c( .5),
  frailty.type = "bl", lam.type = "po")
exp(btaNeeded(N = 1000, thta = 2, tau = 1, lam = 0.7, alp = c(0.01, 0.05), pow = c( .5),
               frailty.type = "bl", lam.type = "po"))

# Based on blinded estimates
if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
    crossing(
      thta = c(2, 3, 4),
      lam = 1.1,
      pow = c(0.5, 0.8),
      N = c(500, 1000),
      tau = 1
    ) %>%
    mutate(
      bta1 = btaNeeded(N = N, thta = thta, tau = tau, lam = lam, alp = alp, pow = pow),
      RR = exp(bta1)
    )
  assumptions %>% data.frame()

}

Function to compute log rate ratio needed (fixed sample size)

Description

Function to compute log rate ratio needed to achieve target power at one-sided Type I control level alp/2. Useful to compute critical value (set pow = 0.5).

Usage

btaNeeded2(
  N,
  thta,
  L,
  alp = 0.05,
  pow = 0.8,
  ar = 0.5,
  frailty.type = c("unblind", "blind"),
  interval = c(log(0.5), log(1))
)

Arguments

Details

This function computes the log rate ratio bta1 as the root of the equation pow2(bta1, N, thta_, L, alp, ar) - pow = 0, where thta_ depends on bta1 if using estimates from blinded analyses. If frailty.type = "blind": thta_ = thtap2thta(thta, bta1); otherwise if frailty = "unblind": thta_ = thta. Function assumes a rate ratio < 1 is favourable to treatment.

Value

The log rate ratio.

Examples

# Based on unblinded estimates

# Based on blinded estimates
btaNeeded2(N = 1000, thta = 2, L = 1000, alp = c(0.01, 0.05), pow = c(.5), frailty.type = "bl")
exp(btaNeeded2(N = 1000, thta = 2, L = 1000, alp = c(0.01, 0.05), pow = c(.5), frailty.type = "bl"))

Function to compute log rate ratio needed

Description

Function to compute log rate ratio needed to achieve target power at one-sided Type I control level alp/2. Useful to compute critical value (set pow = 0.5).

Usage

btaNeeded3(
  thta,
  L,
  tau,
  lam,
  alp = 0.05,
  pow = 0.8,
  ar = 0.5,
  frailty.type = c("unblind", "blind"),
  lam.type = c("base", "pool"),
  interval = c(log(0.5), log(1))
)

Arguments

Details

This function computes the log rate ratio bta1 as the root of the equation eventsNeeded(bta1, thta_, tau, lam0_, alp, pow, ar) - L = 0, where thta_ and lam0_ also depend on bta1 if using estimates from blinded analyses. If frailty.type = "blind": thta_ = thtap2thta(thta, bta1); otherwise if frailty = "unblind": thta_ = thta. If lam.type = "pool" then lam0_ = lam * 2 / (1 + exp(bta1)); otherwise if lam.type = "base": lam0_ = lam. Function assumes a rate ratio < 1 is favourable to treatment.

Value

The log rate ratio.

Examples

# Based on unblinded estimates
btaNeeded3(thta = 2, L = 1000, tau = 1, lam = 1.1, alp = c(0.01, 0.05), pow = c(.5))
exp(btaNeeded3(thta = 2, L = 1000, tau = 1, lam = 1.1, alp = c(0.01, 0.05), pow = c(.5)))

# Based on blinded estimates
btaNeeded3(thta = 2, L = 1000, tau = 1, lam = 0.7, alp = c(0.01, 0.05), pow = c( .5),
  frailty.type = "bl", lam.type = "po")
exp(btaNeeded3(thta = 2, L = 1000, tau = 1, lam = 0.7, alp = c(0.01, 0.05), pow = c( .5),
               frailty.type = "bl", lam.type = "po"))

# Based on blinded estimates
if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
    crossing(
      thta = c(2, 3, 4),
      lam = 1.1,
      pow = c(0.5, 0.8),
      L = c(500, 1000),
      tau = 1
    ) %>%
    mutate(
      bta1 = btaNeeded3(thta = thta, L = L, tau = tau, lam = lam, alp = alp, pow = pow),
      RR = exp(bta1)
    )
  assumptions %>% data.frame()

}

Baseline event rate needed (fixed sample size)

Description

Function to compute baseline (control) event rate needed to achieve given power at one-sided Type I control level alp/2.

Usage

erNeeded(
  N,
  bta1,
  thta,
  tau,
  alp = 0.05,
  pow = 0.8,
  ar = 0.5,
  lam0warning = FALSE
)

Arguments

Details

Assumes rate ratio < 1 is favourable to treatment. A negative estimated event rate indicates no event rate is sufficient under the input assumptions.

Value

The baseline event rate needed to achieve target power at one-sided Type I control level alpha/2, given the input assumptions.

Examples

erNeeded(N = 500, bta1 = log(0.6), thta = 2, tau = 1, alp = 0.05, pow = 0.8)
erNeeded(N = 500, bta1 = log(0.6), thta = 3, tau = 1, alp = 0.05, pow = 0.8)


if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
  crossing(
    tau = c(0.8,0.9, 1.0),
    RR = c(0.6, 0.7, 0.8),
    thta = c(2, 3, 4),
    pow = 0.8,
    N = c(500, 1000)
  ) %>%
    mutate(er = erNeeded(N = N, bta1 = log(RR), thta = thta, tau = tau, alp, pow))

  assumptions %>% data.frame()

}

Number of events needed

Description

Function to compute number of events (L) needed to achieve power (pow) at one-sided Type I control level alpha/2 (alp/2).

Usage

eventsNeeded(bta1, thta, tau, lam0, alp = 0.05, pow = 0.8, ar = 0.5)

Arguments

Details

Assumes rate ratio < 1 is favourable to treatment.

Value

The number of events (L) needed.

Examples

eventsNeeded(bta1 = log(0.8), thta = 1, tau = 0.8, lam0 = 3.5, alp = 0.05, pow = 0.8)

if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
    crossing(
      tau = c(0.8,0.9, 1.0),
      RR = c(0.6, 0.7, 0.8),
      lam0 = c(3, 3.5),
      thta = c(2, 3, 4),
      pow = 0.8
    ) %>%
    mutate(
      L = eventsNeeded(bta1 = log(RR), thta = thta, tau = tau,
                       lam0 = lam0, alp = alp, pow = pow)
    )

  assumptions %>% data.frame()

}

Number events needed (fixed sample size)

Description

Function to compute number events (L) needed to achieve given power with fixed sample size N.

Usage

eventsNeeded2(N, bta1, thta, alp = 0.05, pow = 0.8, ar = 0.5, Lmessage = FALSE)

Arguments

Details

Assumes rate ratio < 1 is favourable to treatment. A negative estimated number of events indicates no number of events is sufficient under the input assumptions.

Value

The number of events (L) needed to achieve target power at one-sided Type I control level alpha/2, given the input assumptions.

Examples

eventsNeeded2(N = 500, bta1 = log(0.8), thta = 2, alp = 0.05, pow = 0.8)
eventsNeeded2(N = 500, bta1 = log(0.8), thta = 3, alp = 0.05, pow = 0.8)

if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
  crossing(
    RR = c(0.6, 0.7, 0.8),
    thta = c(2, 3, 4),
    pow = 0.8,
    N = c(500, 1000)
  ) %>%
    mutate(L = eventsNeeded2(N = N, bta1 = log(RR), thta = thta, alp, pow))

  assumptions %>% data.frame()

}

Function to compute sample size needed to achieve target power

Description

Computes sample size needed to achieve target power at one-sided Type I control level alp/2.

Usage

nNeeded(bta1, thta, tau, lam0, alp = 0.05, pow = 0.8, ar = 0.5)

Arguments

Value

The sample size required given the input assumptions to achieve target power.

Examples

nNeeded(bta1 = log(0.8), thta = 1, tau = 0.8, lam0 = 3.5, alp = 0.05, pow = 0.8)

if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
  crossing(
    tau = c(0.8,0.9, 1.0),
    RR = c(0.6, 0.7, 0.8),
    lam0 = c(3, 3.5),
    thta = c(2, 3, 4),
    pow = 0.8
  ) %>%
    mutate(N = nNeeded(bta1 = log(RR), thta = thta, tau = tau, lam0 = lam0, alp = alp, pow = pow))

  assumptions %>% data.frame()

}

Function to compute sample size needed to achieve target power

Description

Computes sample size needed to achieve target power at one-sided Type I control level alp/2. A negative estimated sample size indicates no sample size is sufficient under the input assumptions.

Usage

nNeeded2(bta1, thta, L, alp = 0.05, pow = 0.8, ar = 0.5)

Arguments

Value

The sample size required given the input assumptions to achieve target power.

Examples

nNeeded2(bta1 = log(0.8), thta = 1, L = 1000, alp = 0.05, pow = 0.8)

if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
  crossing(
    L = c(500, 1000, 1500),
    RR = c(0.6, 0.7, 0.8),
    thta = c(2, 3, 4),
    pow = 0.8
  ) %>%
    mutate(N = nNeeded2(bta1 = log(RR), thta = thta, L = L, alp = alp, pow = pow))

  assumptions %>% data.frame()

}

Power for LWYY (fixed sample size)

Description

Function to compute power at one-sided Type I control level alp/2.

Usage

pow(N, bta1, thta, tau, lam0, alp = 0.05, ar = 0.5)

Arguments

Details

Note: Approximation breaks down in no event scenario. For example, pow(N=1000, bta1 = log(1), thta = 1, lam0 = 0, tau = 1, alp = 0.05) returns a power of 0.025

Value

The power given the input assumptions.

Examples

pow(N = 500, bta1 = log(0.8), thta = 2, tau = 1, lam0 = 1.1, alp = 0.05)
pow(N = 500, bta1 = log(0.8), thta = 3, tau = 1, lam0 = 1.1, alp = 0.05)

if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
  crossing(
    tau = c(0.8,0.9, 1.0),
    RR = c(0.6, 0.7, 0.8),
    lam0 = c(3, 3.5),
    thta = c(2, 3, 4),
    N = c(500, 1000)
  ) %>%
    mutate(pow = pow(N = N, bta1 = log(RR), thta = thta, tau = tau, lam0 = lam0, alp = alp))

  assumptions %>% data.frame()

}

Power for LWYY (fixed sample size)

Description

Function to compute power at one-sided Type I control level alp/2.

Usage

pow2(N, bta1, thta, L, alp = 0.05, ar = 0.5)

Arguments

Value

The power given the input assumptions.

Examples

pow2(N = 500, bta1 = log(0.8), thta = 2, L = 1000, alp = 0.05)
pow2(N = 500, bta1 = log(0.8), thta = 3, L = 1000, alp = 0.05)

if (require("dplyr") & require("tidyr")) {

  assumptions = tibble(alp = 0.05) %>%
  crossing(
    thta = c(1),
    RR = c(0.6, 0.7, 0.8),
    L = c(1000, 1500, 2000),
    N = c(500, 1000)
  ) %>%
    mutate(pow = pow2(N = N, bta1 = log(RR), thta = thta, L = L, alp = alp))

  assumptions %>% data.frame()

}

Power calculations for one-sided two-sample test of LWYY model parameter.

Description

Function to compute power at one-sided Type I control level alp/2 or determine parameters to target given power to test the hypotheses H0: RR = 1 vs HA: RR < 1 for the LWYY (Andersen-Gill with robust standard errors) model.

Usage

power.lwyy.test(
  N = NULL,
  RR = NULL,
  bta1 = NULL,
  thta,
  L = NULL,
  tau = NULL,
  lam = NULL,
  alp = 0.05,
  pow = NULL,
  ar = 0.5,
  frailty.type = c("unblind", "blind"),
  lam.type = c("base", "pool")
)

Arguments

Details

An object of class "power.lwyytest" has the following components:

• method: Description of the test.

• N: Total sample size. N = Nc + Nt, where Nc = ar * N is the number on control and Nt = (1-ar) * N is the number on treatment.

• RR, RRCV: Rate ratio powered for and critical value for rate ratio.

• bta1, bta1CV: The log of RR, RRCV.

• thta: The variance of the frailty parameter.

• thtap: If frailty.type = "blind", the input variance from a blinded source.

• L: Number of events. Note that all else being equal, an increase (decrease) in N requires a decrease (increase) in L.

• tau, lam0: The mean follow-up time and control event rate. Note if tau = 1, then lam0 may be interpreted as mean cumulative intensity on control.

• lamp: If lam.type = "pool" and argument lam is not null, the input rate from a blinded source.

• alp: alp/2 is the one-sided Type I control level of the test H0: RR = 1 vs HA: RR < 1.

• pow: Power of the test.

• ar: Allocation to control ratio Nc/N.

• note: Set of key notes about the assessment.

Value

An object of class "power.lwyytest", a list of arguments, including those computed, with method and note elements.

Examples

x = power.lwyy.test(N = 1000, RR = 0.8, thta = 1, L = 1000, tau = 0.9, alp = 0.05, ar = 0.5)
print(x)

x = power.lwyy.test(N = 1000, RR = 0.8, thta = 1, tau = 0.9, lam = 1.23, alp = 0.05, ar = 0.5)
print(x)

x = power.lwyy.test(N = 1000, RR = 0.8, thta = 1, tau = NULL, lam = 1.23, alp = 0.05, ar = 0.5)
print(x, digits = 3)

Obtain theta from pooled theta (model without treatment).

Description

Function transforms estimate of blinded pooled theta (from misspecified model without adjusting for treatment) to estimate of theta for model adjusting for treatment.

Usage

thtap2thta(bta1, thtap, ar = 0.5, thtawarning = FALSE)

Arguments

Details

This function assumes a recurrent event distribution with exponential baseline function and frailty parameter distribution with mean 1 and variance \theta, and derives from expectations of the variance under misspecification.

Specifically, the following relationship between the frailty variance for the misspecified model (\theta_p) and correctly specified model (\theta) is used

\theta_p = Var(Z \cdot \exp{\beta}) = c^2 - 2pc - 2(1-p)\exp{\beta}c + (p + (1-p)\exp{2\beta})(\theta + 1)

, where c = E(Z\exp{X\beta}) and p = ar.

Value

An estimate of the variance of the frailty parameter when the model includes treatment.

Examples

thtap2thta(2, log(c(0.8, 0.7, 0.6)))