Bioequivalence and Bioavailability Forum • Geometric mean ≈ MLE of median

Geometric mean ≈ MLE of median [Regulatives / Guidelines]

posted by Helmut – Vienna, Austria, 2024-11-14 10:54 (229 d 11:50 ago) – Posting: # 24275
Views: 13,014

Hi BEQool,

❝ ❝ Well, the geometric mean is the maximum likelihood estimator of the median.

❝ Is geometric mean always equal to median when we have "perfect" log-normal distribution of data set (very large sample)?

Yes. Try a simple simulation:

set.seed(20241114) mue <- 1 CV <- 0.2 n <- 1e8 x <- rlnorm(n = n, meanlog = log(mue), sdlog = sqrt(log(CV^2 + 1))) # instead of prod(x)^(1/n) we use exp(mean(log(x))) because with # very large n we will underrun the numeric resolution of the machine y <- data.frame(estimator = c("arithmetic mean", "geometric mean", "median"), estimate = c(mean(x), exp(mean(log(x))), median(x))) y$RE <- sprintf("%+.5f%%", 100 * (y$estimate - mue) / mue) print(y, row.names = FALSE) estimator estimate RE arithmetic mean 1.019806 +1.98057% geometric mean 1.000005 +0.00046% median 1.000012 +0.00115%

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Complete thread:

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        Postmenopausal women Dr_Dan 2016-02-15 17:45
        
        Baseline correction examples BEQool 2024-11-11 07:34
        
        Baseline correction GeoMean and Median mittyri 2024-11-11 11:20
        
        Geometric mean? Helmut 2024-11-11 14:53
        
        Geometric mean? BEQool 2024-11-14 08:19
        
        Geometric mean ≈ MLE of medianHelmut 2024-11-14 09:54
        
        Geometric mean ≈ MLE of median Mahmoud 2024-11-15 11:26
        
        Bias? Helmut 2024-11-16 08:43
        
        Bias? Mahmoud 2024-11-16 11:12
    - Baseline; another example Helmut 2015-09-18 17:16
      - Baseline; another example nobody 2016-02-12 09:10
        
        Baseline; another example Helmut 2016-02-12 13:37
        
        Baseline; another example nobody 2016-02-12 14:08
        
        Slowly going OT: BE study simulations Helmut 2016-02-12 15:47
        
        Slowly going OT: BE study simulations nobody 2016-02-12 16:15
        
        Religious debate Helmut 2016-02-12 17:04
        
        Religious debate nobody 2016-02-12 17:26
        
        Religious debate ElMaestro 2016-02-13 12:53
        
        Religious debate nobody 2016-02-14 15:23
        
        Slowly going OT: BE study simulations Ohlbe 2016-02-13 13:10
        
        Slowly going OT: BE study simulations Helmut 2016-02-13 15:02
        
        Slowly going OT: BE study simulations lizhao 2016-02-25 17:31
        
        Business as usual Helmut 2016-02-25 17:44
        
        Business as usual lizhao 2016-02-25 17:59