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roman_max ☆ Russia, 2024-02-09 22:08 (419 d 22:52 ago) Posting: # 23860 Views: 4,212	CV in parallel design [🇷 for BE/BA] Post reply
	Hello Helmut! within this code in PowerTOST `100*CVfromCI(alpha=0.05, lower=0.8450985339, upper=1.104975299, n=c(5,5), design="2x2x4")` if design is `"parallel"` it is a CVtotal? Am I right? Edit: I made it a top-level post and changed the subject line and category. [Helmut]

Helmut
★★★

Vienna, Austria,
2024-02-10 01:30
(419 d 19:29 ago)

@ roman_max
Posting: # 23861
Views: 3,567

PowerTOST: CV in different designs

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Hi roman_max,

❝ within this code in PowerTOST

❝ 100*CVfromCI(alpha=0.05, lower=0.8450985339, upper=1.104975299, n=c(5,5), design="2x2x4")

❝ if design is "parallel" it is a CVtotal? Am I right?

Yes, you are.

For the record:

Any replicate design: $\small{CV_\text{intra}}$, assuming homoscedasticity ($\small{s_\text{wT}^2\equiv s_\text{wR}^2}$).
Any crossover and a paired design: $\small{CV_\text{intra}}$.
Parallel design: $\small{CV_\text{total}}$, assuming homoscedasticity ($\small{s_\text{T}^2\equiv s_\text{R}^2}$).

Note that the $\small{s^2}$ components in #1 and #3 cannot be calculated from the confidence interval. You need the raw data.

library(PowerTOST) CI <- c(0.85, 1 / 0.85) n <- 24 designs <- known.designs()[c(1, 13, 3:6, 8:9, 12, 7, 10:11), c(2:3, 9)] res <- data.frame(design = designs$design, name = designs$name, n = rep(n, nrow(designs)), df = designs$df, lower = CI[1], upper = CI[2], CV = NA, type = c("total", rep("intra", nrow(designs) - 1))) for (j in 1:nrow(designs)) { res$CV[j] <- sprintf("%.2f%%", 100 * CI2CV(lower = res$lower[j], upper = res$upper[j], n = n, design = res$design[j])) } print(res, row.names = FALSE, right = FALSE) design name n df lower upper CV type parallel 2 parallel groups 24 n-2 0.85 1.176471 23.50% total paired paired means 24 n-1 0.85 1.176471 33.75% intra 2x2x2 2x2x2 crossover 24 n-2 0.85 1.176471 33.69% intra 3x3 3x3 crossover 24 2*n-4 0.85 1.176471 34.47% intra 3x6x3 3x6x3 crossover 24 2*n-4 0.85 1.176471 34.47% intra 4x4 4x4 crossover 24 3*n-6 0.85 1.176471 34.73% intra 2x2x4 2x2x4 replicate crossover 24 3*n-4 0.85 1.176471 50.60% intra 2x4x4 2x4x4 replicate crossover 24 3*n-4 0.85 1.176471 50.60% intra 2x2x2r Liu's 2x2x2 repeated x-over 24 3*n-2 0.85 1.176471 50.62% intra 2x2x3 2x2x3 replicate crossover 24 2*n-3 0.85 1.176471 40.20% intra 2x3x3 partial replicate (2x3x3) 24 2*n-3 0.85 1.176471 40.20% intra 2x4x2 Balaam's (2x4x2) 24 n-2 0.85 1.176471 16.50% intra

You can calculate $\small{CV_\text{wR}}$ from the upper expanded limit (irrespective of the design and sample size):

U <- 1.4319 cat(sprintf("%.2f%%", 100 * U2CVwR(U)), "\n") 50.00%

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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roman_max ☆ Russia, 2024-02-12 22:27 (416 d 22:33 ago) @ Helmut Posting: # 23862 Views: 3,413	PowerTOST: CV in different designs Post reply
	Hi Helmut! Thanks for the confirmation and "refreshing" info about CVs.

roman_max ☆ Russia, 2024-02-13 20:38 (416 d 00:22 ago) @ Helmut Posting: # 23863 Views: 3,288	PowerTOST: CV in different designs Post reply
	Hi Helmut! One more question in terms of CVintra: what is the algebra for parallel design? In your lectures I could only found for crossover... I expect the same question from our regulatory agency if they cannot find "the formula" in study protocol.

Helmut
★★★

Vienna, Austria,
2024-02-13 21:55
(415 d 23:04 ago)

@ roman_max
Posting: # 23864
Views: 3,318

Formulas for parallel and 2×2×2 designs

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Hi roman_max,

❝ One more question in terms of CVintra: what is the algebra for parallel design?

$\small{CV_\text{intra}}$ is unknown. You mean $\small{CV_\text{total}}$, right?

❝ In your lectures I could only found for crossover... I expect the same question from our regulatory agency if they cannot find "the formula" in study protocol. :confused:

I was never asked for by any agency since I derived the formulas in the late 80s… The algebra is simple. The CI depends on the point estimate, the t-value for given $\alpha$ / degrees of freedom, and the variance. Only the last is unknown. Work backwards. However, to satisfy your curiosity:
$$\eqalign{
PE&=\sqrt{CL_\text{lower}\times CL_\text{upper}}\\
s_1&=\frac{\log_e\,PE-\log_e\,CL_\text{lower}}{t_{1-\alpha,\,n_1+n_2-2}\;\sqrt{f\left(\tfrac{1}{n_1}+\tfrac{1}{n_2}\right)}}\\
s_2&=\frac{\log_e\,CL_\text{upper}-\log_e\,PE}{t_{1-\alpha,\,n_1+n_2-2}\;\sqrt{f\left(\tfrac{1}{n_1}+\tfrac{1}{n_2}\right)}}\\
CV&=\sqrt{\exp\left[\left(\frac{s_1+s_2}{2}\right)^2\right]-1}
}$$The only difference of the formula for the parallel design to the one for the 2×2×2 design is the factor $f$ in the denominator of the $s$ calculations. For the parallel design it is $1$, whereas for the 2×2×2 design it is $\small{{^1}/_{2}}$. Note that $s_1\sim s_2$. The formulas above are algebraic transformations of those in my lectures and are implemented as such in the function CVfromCI() / CI2CV() of PowerTOST.
If you are interested in the source code, type CVfromCI in the console. In the function, the user can not only specify the Confidence Limits, but also one of them together with the Point Estimate. If $\left|s_1-s_2\right|\,/\,\frac{s_1+s_2}{2}>0.1$ (i.e., $s_1$ and $s_2$ are more than 10% apart), a warning is issued prompting the user to check their input. The code is more complicated than the one below because it handles all implemented designs and contains input checking.

The example of my previous post in Base 🇷:

CI <- setNames(c(0.85, 1 / 0.85), c("lower", "upper")) n <- c(12, 12) f <- setNames(c(1, 0.5), c("parallel", "2x2x2")) PE <- sqrt(CI[["lower"]] * CI[["upper"]]) s1 <- (log(PE) - log(CI[["lower"]])) / (qt(1 - 0.05, sum(n) - 2) * sqrt(f * sum(1 / n))) s2 <- (log(CI[["upper"]]) - log(PE)) / (qt(1 - 0.05, sum(n) - 2) * sqrt(f * sum(1 / n))) CV <- sqrt(exp(((s1 + s2) / 2)^2) - 1) print(100 * CV, digits = 4) parallel 2x2x2 23.50 33.69

To see $f$ of the implemented designs:

designs <- PowerTOST::known.designs()[c(1, 13, 3:6, 8:9, 12, 7, 10:11), c(9, 2, 7)] names(designs)[3] <- " f" print(designs, row.names = FALSE, right = FALSE) name design f 2 parallel groups parallel 1/1 paired means paired 2/1 2x2x2 crossover 2x2x2 1/2 3x3 crossover 3x3 2/9 3x6x3 crossover 3x6x3 1/18 4x4 crossover 4x4 1/8 2x2x4 replicate crossover 2x2x4 1/4 2x4x4 replicate crossover 2x4x4 1/16 Liu's 2x2x2 repeated x-over 2x2x2r 1/4 2x2x3 replicate crossover 2x2x3 3/8 partial replicate (2x3x3) 2x3x3 1/6 Balaam's (2x4x2) 2x4x2 1/2

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Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
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roman_max ☆ Russia, 2024-02-15 17:42 (414 d 03:17 ago) @ Helmut Posting: # 23865 Views: 3,140	Formulas for parallel and 2×2×2 designs Post reply
	Hi Helmut! Thank you for the reply and useful explanations :) Especially for our regulator as well )) ❝ ❝ One more question in terms of CVintra: what is the algebra for parallel design? ❝ $\small{CV_\text{intra}}$ is unknown. You mean $\small{CV_\text{total}}$, right? Of course, I meant CVtotal.

fethiye ☆ Colchester, 2024-02-23 08:11 (406 d 12:48 ago) @ Helmut Posting: # 23874 Views: 3,000	PowerTOST: CV in different designs Post reply
	Thanks for information Edit: Full quote removed. Please delete everything from the text of the original poster which is not necessary in understanding your answer; see also this post #5! [Helmut]

CV in parallel design [🇷 for BE/BA]

PowerTOST: CV in different designs

PowerTOST: CV in different designs

PowerTOST: CV in different designs

Formulas for parallel and 2×2×2 designs

Formulas for parallel and 2×2×2 designs

PowerTOST: CV in different designs