## 'CVcap' [Power / Sample Size]

Hi Mahmoud,

❝ what do mean about 'CVcap' = 0.2142

Quoting my previous post:

If the CVwR >21.42% the additional criterion “must pass 80–125%” becomes increasingly important and power drops.

It’s not explicitly stated in the FDA’s guidances (therefore, we give it in quotes) but can be easily derived with a little algebra. In a nutshell: Limits are scaled based on $$\small{CV_\text{wR}}$$ with the FDA’s regulatory constant $$\small{\theta_0}$$ in the first place. But for any $$\small{CV_\text{wR}>21.42\%}$$ that would result in implied limits $$\small{\left\{L,U\right\}}$$, which are wider than 80.00 – 125.00%. That’s not we want for an NTID.\eqalign{ \Delta&=1/0.9\approx 1.11111\\ \sigma_\text{w0}&=0.10\\ \theta_0&=\frac{\log_e\Delta}{\sigma_\text{w0}}\approx1.053595\\ s_\text{wR}&=\sqrt{\log_e(CV_\text{wR}^2+1)}⁠\\ \left\{L,U\right\}&=100\exp(\mp\theta_0\cdot s_\text{wR}) }Practically the limits are scaled indeed, but if the study would have passed, additionally inclusion within the conventional 80.00 – 125.00% limits is assessed as well.
That’s numerically the same as if scaling would be ‘capped’ at $$\small{CV_\text{wR}=21.42\%}$$.

If algebra is not your thing, try this:

fun <- function(x) {   Delta   <- 1.11111 # aproximate; only the FDA knows why   sigma.0 <- 0.10   theta.0 <- log(Delta) / sigma.0   swR     <- sqrt(log(x^2 + 1))   # the implied upper limit which is ≈1.25   U       <- exp(theta.0 * swR)   return(U - 1.25) } CVcap  <- uniroot(fun, interval = c(0.01, 0.3),                   extendInt = "upX")\$root # numerically find the CV where U ≈1.25 cat(sprintf("\'CVcap\' = %.4g\n", CVcap)) 'CVcap' = 0.2142

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