ElMaestro
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2013-08-07 15:18
(4275 d 15:51 ago)

Posting: # 11231
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 Geometric mean of normal dist etc. [General Sta­tis­tics]

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Hi all,

two nasty ones and an easy one:
  1. If X~N(μ, σ2) then what is the geometric mean of X? (note: I am not talking about a log-normal X)
  2. Can you give me some other stats distribution, which is not log-normal, for which we can easily derive random numbers and for which we can do an exact equation for geometric mean and CV?
  3. Can you guess why I ask?
Many thanks.
:-D
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Helmut
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2013-08-07 18:20
(4275 d 12:49 ago)

@ ElMaestro
Posting: # 11240
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 Geometric mean of normal dist etc.

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Hi ElMeastro,

❝ 1. If X~N(μ, σ2) then what is the geometric mean of X?


I would say the general case (for arbitrary u, s²) is not defined since (∏X)1/N will not work if you have at least one negative value of X in the set. Should only always work if you think about a special case of the truncated normal distribution (i.e., X ∈ ℝ>0 or X ]0, +∞] if you prefer).

❝ 2. Can you give me some other stats distribution, which is not log-normal, for which […]



No idea.

❝ 3. Can you guess why I ask?


Yes. :smoke:
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ElMaestro
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Denmark,
2013-08-07 18:28
(4275 d 12:41 ago)

@ Helmut
Posting: # 11242
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 Geometric mean of normal dist etc.

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Hi Hötzi,

❝ Hi ElMeastro,


ElMeastro?? ElMistress? Must be the heat....:-D

❝ ❝ 1. If X~N(μ, σ2) then what is the geometric mean of X?

❝

❝ I would say the general case (for arbitrary u, s²) is not defined since (∏X)1/N will not work if you have at least one negative value of X in the set.


Very good point. One could ask 'then what if I have an even number of negatives' but I will refrain.

Departure from log-normality might be my next playground :ok:....
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Helmut
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2013-08-07 18:37
(4275 d 12:32 ago)

@ ElMaestro
Posting: # 11244
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 Overheated brain

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Hi ElMaestro,

❝ ElMeastro?? ElMistress? Must be the heat....:-D


I’m excused. Today’s forecast is 40 ℃; <30 ℃ not expected before midnight.

[image]

❝ ❝ I would say the general case (for arbitrary u, s²) is not defined since (∏X)1/N will not work if you have at least one negative value of X in the set.

❝

❝ One could ask 'then what if I have an even number of negatives' but I will refrain.


Yes pleezee. See my edit about the truncated normal, BTW.

❝ Departure from log-normality might be my next playground :ok:....


Oh boy!
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Helmut
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2013-08-08 15:10
(4274 d 15:59 ago)

@ ElMaestro
Posting: # 11250
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 Location parameters

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Hi ElMaestro,

what’s the distribution of concentrations (or derived metrics like AUC)? AUC seems to be log-normal (see here). I dunno whether anybody explored concentrations. If we don’t believe in exotic matter (e.g., negative mass) truncation at zero (or LLOQ?) makes sense. Or is the distribution log-normal? Probably (will pool some data time allowing). Note that in PopPK an error model with censored data is often used.
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ElMaestro
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Denmark,
2013-08-08 15:52
(4274 d 15:17 ago)

@ Helmut
Posting: # 11251
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 Location parameters

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Hi Helmut,

❝ what’s the distribution of concentrations (or derived metrics like AUC)? AUC seems to be log-normal (see here). I dunno whether anybody explored concentrations. If we don’t believe in exotic matter (e.g., negative mass) truncation at zero (or LLOQ?) makes sense. Or is the distribution log-normal? Probably (will pool some data time allowing). Note that in PopPK an error model with censored data is often used.


Good input. I have no idea myself. I live in a cage. Whether or not something is relevant to real life is of no particular interest to me :-D:-D:-D

At this point in principle any distribution would suffice for starters as long as:
  1. I can control the geometric mean.
  2. I can control the CV.
  3. The dist. is not log normal.
  4. I can derive random numbers via pts 1+2.
Having said all that I need to air another thought:
We usually say that the error of the linear model is IID and normal with mean zero but actually Chow and Liu's formulation in that regard is not crystal clear. You showed some beautiful plots -can't find the thread right now- which demonstrated that the errors are correlated within subject. I am inclined to think that we do not strictly need IID for the normal linear model. We can relax it a bit:
we only need IID for the difference of test-ref in xovers, and that's why the Potvin/Montague considerations actually work.

Parallel is another matter of course.

Hell, I wish I had studied statistics at the university.





I think the
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Helmut
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2013-08-08 16:35
(4274 d 14:34 ago)

@ ElMaestro
Posting: # 11252
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 Distribution of concentrations

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Hi ElMaestro,

❝ Good input. I have no idea myself. I live in a cage. Whether or not something is relevant to real life is of no particular interest to me :-D:-D:-D


Not unfamiliar to me.

❝ At this point in principle any distribution would suffice for starters as long as:

❝ 1. I can control the geometric mean.

❝ 2. I can control the CV.

❝ 3. The dist. is not log normal.

❝ 4. I can derive random numbers via pts 1+2.


Quick shot about distributions of ~700 concentrations. Drug with quite high between-sub­ject variability in Cmax (~50%). I looked at concentrations in the absorption phase, around Cmax, and in late elimination.

Raw scale

[image]

Log scale

[image]


Lognormal seems to “work” close to Cmax, but otherwise? Look at the crowded first bin.

❝ Having said all that I need to air another thought:

❝ We usually say that the error of the linear model is IID and normal with mean zero but actually Chow and Liu's formulation in that regard is not crystal clear.


Yes.

❝ You showed some beautiful plots -can't find the thread right now- which demonstrated that the errors are correlated within subject.


I know. Will search for it later (have to find the nearest lake now and dive in).

❝ I am inclined to think that we do not strictly need IID for the normal linear model. We can relax it a bit:

❝ we only need IID for the difference of test-ref in xovers, and that's why the Potvin/Montague considerations actually work.


Yep. I remember an article by Carl Metzler where he pointed out that the distribution of metrics is irrelevant. We are only interested in model’s residuals which approximate true error.

❝ Parallel is another matter of course.


Yessir.

❝ Hell, I wish I had studied statistics at the university.


Quoting Ohlbe: Don’t worry, it’s too late.

❝ I think the


What?
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