## Confuse-a-Cat [General Statistics]

Hi ElMaestro,

» I am sorry that I am once again not able to tell what I am looking for .

No,

» I am not talking about simulations and not about transformations either.

OK.

» Two functions, generally, and the key word is really

I think:

» We can think of f=mean(x) and g=median(x). I guess we can easily do a mental picture of plotting means versus medians, often seeing the strong relationship. Visually appealing.

Go back to my script and

» Independence?

No. Both are measures of the

» OK then let us say f=range(x) and g=sine function of the range (x).

» Or an F statistics with a variance in both f=numerator and g=denominator in an unbalanced anova.

Oh dear!

» Or Cmax and AUCt (which I guess are correlated and dependent(?), …

Correlated, yes. Highly sometimes. In some cases not so much (recall that one). Why? Duno. Though correlation ≠ causation.

Actually there are hidden (confounding) variables – the entire PK stuff – which drives the apparent correlation. So are they dependent even with a high correlation? I would say no. Both depend on the underlying PK.

» … but the example is not great in my perspective since the two functions are not applied to a random sample but to a time series).

Yep.

» There is no end to the possible examples.

Already the ones you mentioned gave me headaches.

» Without debating to much about the specific cases, how do we generally approach it to define two (outcomes of) functions as being independent?

See above. Maybe I’m completely wrong.

» Which mathematical/algebraic/statistical/whatever properties of functions render them mutually independent? When I understand it, I think or hope I will understand the nature of independence.

» For inspiration: Are estimates of any two statistical moments independent? If yes, why? Is it only the first and second? Why? Is it generally so? Why? Etc. I am looking for the

Sorry, again.

» I am sorry that I am once again not able to tell what I am looking for .

No,

*I’m*sorry that I’m not able to comprehend your message. As you know, walnut-sized brain.» I am not talking about simulations and not about transformations either.

OK.

» Two functions, generally, and the key word is really

**generally**, when are they (or their results) to be considered independent?I think:

- The results of f(x) and g(x) should have a very (very!) low correlation.

- Both should convey different information about the properties of x.

- What else?

» We can think of f=mean(x) and g=median(x). I guess we can easily do a mental picture of plotting means versus medians, often seeing the strong relationship. Visually appealing.

Go back to my script and

`med.spl <- aggregate(xs[, 2], list(sample = xs$sample), median)`

plot(mean.spl$x, med.spl$x)

cor(mean.spl$x, med.spl$x)

» Independence?

No. Both are measures of the

*location*of the data. IMHO, that would*not*be suitable to construct a test as stated at the end of this post.» OK then let us say f=range(x) and g=sine function of the range (x).

» Or an F statistics with a variance in both f=numerator and g=denominator in an unbalanced anova.

Oh dear!

» Or Cmax and AUCt (which I guess are correlated and dependent(?), …

Correlated, yes. Highly sometimes. In some cases not so much (recall that one). Why? Duno. Though correlation ≠ causation.

Actually there are hidden (confounding) variables – the entire PK stuff – which drives the apparent correlation. So are they dependent even with a high correlation? I would say no. Both depend on the underlying PK.

» … but the example is not great in my perspective since the two functions are not applied to a random sample but to a time series).

Yep.

» There is no end to the possible examples.

Already the ones you mentioned gave me headaches.

» Without debating to much about the specific cases, how do we generally approach it to define two (outcomes of) functions as being independent?

See above. Maybe I’m completely wrong.

» Which mathematical/algebraic/statistical/whatever properties of functions render them mutually independent? When I understand it, I think or hope I will understand the nature of independence.

» For inspiration: Are estimates of any two statistical moments independent? If yes, why? Is it only the first and second? Why? Is it generally so? Why? Etc. I am looking for the

**general**clarity.Sorry, again.

—

Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮

Science Quotes

*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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### Complete thread:

- Statistical independence, what is it? I mean really, what is it?? ElMaestro 2020-06-27 21:35 [General Statistics]
- Die don’t remember the last roll. Really. Helmut 2020-06-28 13:35
- Die don’t remember the last roll. Really. ElMaestro 2020-06-28 14:45
- Die don’t remember the last roll. Really. Helmut 2020-06-28 15:36
- Still none the wiser ElMaestro 2020-06-28 18:20
- You’ve lost me now. Helmut 2020-06-28 21:55
- Worded differently ElMaestro 2020-06-29 08:30
- Still not sure what you are aiming at… Helmut 2020-06-29 16:46
- Still not sure what you are aiming at… ElMaestro 2020-06-30 00:55
- Confuse-a-Cat Helmut 2020-06-30 11:33
- Confuse-a-Cat ElMaestro 2020-06-30 13:07
- Confuse-a-CatHelmut 2020-06-30 14:27
- pseudorandom and linear independence mittyri 2020-07-01 00:04

- Confuse-a-Cat ElMaestro 2020-06-30 13:07

- Confuse-a-Cat Helmut 2020-06-30 11:33

- Still not sure what you are aiming at… ElMaestro 2020-06-30 00:55

- Still not sure what you are aiming at… Helmut 2020-06-29 16:46

- Worded differently ElMaestro 2020-06-29 08:30

- You’ve lost me now. Helmut 2020-06-28 21:55

- Still none the wiser ElMaestro 2020-06-28 18:20

- Die don’t remember the last roll. Really. Helmut 2020-06-28 15:36

- Die don’t remember the last roll. Really. ElMaestro 2020-06-28 14:45
- Statistical independence, what is it? I mean really, what is it?? martin 2020-07-01 08:40
- Statistical independence, what is it? I mean really, what is it?? ElMaestro 2020-07-01 09:42
- Statistical independence, what is it? I mean really, what is it?? martin 2020-07-01 10:07

- Statistical independence, what is it? I mean really, what is it?? ElMaestro 2020-07-01 09:42

- Die don’t remember the last roll. Really. Helmut 2020-06-28 13:35