I am analyzing tumor DNA sequencing data to perform variant calling. I need to find the minimum number of DNA strands (sample size; independent samples, 'depth of coverage') needed to detect mutations occurring at frequencies of 2-5%, with 95% confidence, given a 1% background mutation rate, at power levels of 0.8, 0.9, and 0.99.

Past collaborators did this, using a "cloglog Binomial distribution"(?), and got some of the following results:

frequency: 0.02
power: 0.8
alpha: 0.05
sample size: 1239

frequency: 0.03
power: 0.8
alpha: 0.05
sample size: 423

frequency: 0.04
power: 0.90
alpha: 0.05
sample size: 299

frequency: 0.05
power: 0.99
alpha: 0.05
sample size: 315

I am trying to replicate their analysis in R, to validate and fill in more values, but it seems like I am doing something wrong because I am not getting the same values. Using the pwr library:

pwr.p.test(h = 0.02, 
           sig.level = 0.05, 
           power = 0.80, 
           alternative = "greater")


 proportion power calculation for binomial distribution (arcsine transformation) 

          h = 0.02
          n = 15456.39
  sig.level = 0.05
      power = 0.8
alternative = greater

Here, it is giving me an n of 15456, when the value should be 1239

As per the docs for this package, the 'effect size' is important, so I am wondering if that might be the source of the discrepancy? And I am not sure how the 'coglog Binomial distribution' plays into it, especially since pwr says it uses an 'arcsine transformation' instead.

  • 1
    $\begingroup$ cloglog = "complementary log log" so that the probability is modeled linearly on this scale $\log(-\log(p)) = \beta_0 + \beta_1 X$. It is good for discrete time survival. There's a nice discussion on these types of models in Applied Survival Analysis by Hosmer, Lemeshow, and May 2nd ed or later. $\endgroup$
    – AdamO
    Mar 5, 2018 at 16:28

1 Answer 1


Found the solution, using the binom R package instead. For context:

  • Null Hypothesis: No variant present, background error rate only (1%)

  • Alternative Hypothesis: A variant is present at the given frequency (Variant Allele Frequency; 2%)


VAF <- 0.02
background_seq_error_rate <- 0.01
alpha <- 0.05
conf_level <- 1 - alpha
power <- 0.8

cloglog.sample.size(p.alt = VAF, 
                    p = background_seq_error_rate, 
                    power = power,
                    alpha = alpha)


  p.null p.alt delta alpha power    n phi
1   0.01  0.02  0.01  0.05   0.8 1239   1


  • R version 3.2.3 (2015-12-10)

  • binom_1.1-1


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