Power and sample size for case-cohort design

Usage

ccsize(n, q, pD, p1, theta, alpha, beta = 0.2, power = TRUE, verbose = FALSE)

Arguments

n: the total number of subjects in the cohort.
q: the sampling fraction of the subcohort.
pD: the proportion of the failures in the full cohort.
p1: proportions of the two groups (p2=1-p1).
theta: log-hazard ratio for two groups.
alpha: type I error – significant level.
beta: type II error.
power: if specified, the power for which sample size is calculated.
verbose: error messages are explicitly printed out.

Value

a value indicating the power or required sample size.

Details

The power of the test is according to $$\Phi\left(Z_\alpha+m^{1/2}\theta\sqrt{\frac{p_1p_2p_D}{q+(1-q)p_D}}\right)$$ where $\alpha$ is the significance level, $\theta$ is the log-hazard ratio for two groups, $p_j$, j=1, 2, are the proportion of the two groups in the population. $m$ is the total number of subjects in the subcohort, $p_D$ is the proportion of the failures in the full cohort, and $q$ is the sampling fraction of the subcohort.

Alternatively, the sample size required for the subcohort is $$m=nBp_D/(n-B(1-p_D))$$ where $B=(Z_{1-\alpha}+Z_\beta)^2/(\theta^2p_1p_2p_D)$, and $n$ is the size of cohort.

When infeaisble configurations are specified, a sample size of -999 is returned.

Note

Programmed for EPIC study. keywords misc

References

Cai J, Zeng D (2004). “Sample size/power calculation for case-cohort studies.” Biometrics, 60(4), 1015-24. doi:10.1111/j.0006-341X.2004.00257.x .

Author

Jing Hua Zhao

Examples

if (FALSE) { # \dontrun{
# Table 1 of Cai & Zeng (2004).
alpha <- 0.05
table1 <- rbind(
  transform(
    within(expand.grid(
      pD = c(0.10, 0.05),
      p1 = c(0.3, 0.5),
      theta = c(0.5, 1.0),
      q = c(0.1, 0.2)
    ), {
      n <- 1000
      power <- mapply(ccsize,
        n = n, q = q, pD = pD, p1 = p1, theta = theta,
        MoreArgs = list(alpha = alpha)
      )
    }),
    power = signif(power, 3)
  ),
  transform(
    within(expand.grid(
      pD = c(0.05, 0.01),
      p1 = c(0.3, 0.5),
      theta = c(0.5, 1.0),
      q = c(0.01, 0.02)
    ), {
      n <- 5000
      power <- mapply(ccsize,
        n = n, q = q, pD = pD, p1 = p1, theta = theta,
        MoreArgs = list(alpha = alpha)
      )
    }),
    power = signif(power, 3)
  )
)

# ARIC study
aric <- within(
  data.frame(
    n = 15792,
    pD = 0.03,
    p1 = 0.25,
    hr = c(1.35, 1.40, 1.45),
    q = c(1463, 722, 468) / 15792
  ), {
    alpha <- 0.05
    beta <- 0.2
    power <- mapply(ccsize,
      n = n, q = q, pD = pD, p1 = p1, theta = log(hr),
      MoreArgs = list(alpha = alpha)
    )
    ssize <- mapply(ccsize,
      n = n, q = q, pD = pD, p1 = p1, theta = log(hr),
      MoreArgs = list(alpha = alpha, beta = beta, power = FALSE)
    )
    power <- signif(power, 3)
  }
)

# EPIC study
epic <- within(
  expand.grid(
    pD = c(0.3, 0.2, 0.1, 0.05),
    p1 = seq(0.1, 0.5, by = 0.1),
    hr = seq(1.1, 1.4, by = 0.1)
  ), {
    n <- 25000
    q <- 0.1
    alpha <- 5e-8
    beta <- 0.2
    ssize <- mapply(ccsize,
      n = n, q = q, pD = pD, p1 = p1, theta = log(hr),
      MoreArgs = list(alpha = alpha, beta = beta, power = FALSE)
    )
  }
)
epic <- subset(epic, !is.na(ssize) & ssize > 0)

# exhaustive search
search <- within(
  expand.grid(
    pD = c(0.3, 0.2, 0.1, 0.05),
    p1 = seq(0.1, 0.5, by = 0.1),
    hr = seq(1.1, 1.4, by = 0.1),
    q  = seq(0.01, 0.5, by = 0.01)
  ), {
    n <- 25000
    alpha <- 5e-8
    power <- mapply(ccsize,
      n = n, q = q, pD = pD, p1 = p1, theta = log(hr),
      MoreArgs = list(alpha = alpha)
    )
    nq <- n * q
  }
)
} # }