ElMaestro
★★★

Denmark,
2016-12-26 20:05
(2649 d 00:04 ago)

Posting: # 16878
Views: 7,907
 

 What's your understanding of a standard error? [General Sta­tis­tics]

Hi all,

I can relate to quite a few more or less abstract quantities or objects that we encounter in BE, e.g. critical t-value, or a treatment effect or a density function or a model matrix or....
One of the things I cannot easily relate to (as in: interpret or graphically visualize inside my little head) is the standard error.
Example: the SE is a quantity that is factored onto the cricial t-value to determine the width of a confidence interval, given a set of observations. Yeah right, professor!).
Another example: "The standard error (SE) is the standard deviation of the sampling distribution of a statistic" (from Wikipedia). Yeah right, professor!

So, a question to you experts: How do you interpret the SE (of a mean or of something else)?
I am not asking about a reference to Wikipedia or a stats text which defines it or shows how to calculate it. I've read those:-D. I am solely asking about an interpretation of SE's for dummies.

Thanks if anyone can help.

Pass or fail!
ElMaestro
DavidManteigas
★    

Portugal,
2016-12-27 11:41
(2648 d 08:28 ago)

@ ElMaestro
Posting: # 16879
Views: 7,034
 

 What's your understanding of a standard error?

Hi El Maestro,

Statistical classes define the standard error as the variability of some estimated parameter, such as the mean, odds ratio, proportions, etc. I'm pretty sure this is not what you're looking for, but imo the definition is intuitive especially when you think at the population level. We are working with samples and we know that when we analyse a sample we are only obtaining estimates of the "true" population parameters. If you conduct 3 or 4 bioequivalence studies with the same product under the same conditions you will obtain 3/4 different estimates of the T/R ratio since you used different samples. The standard error is like the "standard deviation" of those estimates. Since we don't conduct lots of studies to estimate a single parameter, we generally obtain the standard error from the standard deviation of the sample. We assume that samples with higher variability will yield parameters with higher variability. The other assumption is regading the sample size. We also assume that higher samples produce estimates with a higher degree of uncertainty regading the population - and therefore with lower standard errors. This is in some way the principal concept of frequentist statistics. The concept of obtaining the variability of the parameter from the variability of the sample. The nicest thing about the standard error is that in general the "parameter distribution" (either a mean, relative risk, odds ratio, etc) will be normal and will let us calculate a "confidence interval" based on the standard error to obtain an estimate of how accurate our estimation process was.

I think that a great way to understand the concept of standard error is to learn the bootstrap techniques. Bootstrapping is in fact using the "frequentist theory" in the world of simulation, including the concept of standar errors to estimate parameters.

Not sure if this helped with your question, since you may already know all of this. I hope the bootstrap may be a better teacher than I am :-D
nobody
nothing

2016-12-27 11:42
(2648 d 08:27 ago)

@ ElMaestro
Posting: # 16880
Views: 6,927
 

 What's your understanding of a standard error?

Hi to stormy Danmark!

Would love to be at the sea now, but sitting here and writing about statistics (*mad*). For me the standard error of the mean is a measure for the difference between the mean of the sample from the real mean of the underlying "population" (Grundgesamtheit) you sample from...

:-)

Kindest regards, nobody
d_labes
★★★

Berlin, Germany,
2016-12-27 12:46
(2648 d 07:23 ago)

@ ElMaestro
Posting: # 16883
Views: 7,032
 

 Standard error for Dummies

Dear ÖbersterGrößterMeister,

❝ So, a question to you experts: How do you interpret the SE (of a mean or of something else)?

❝ ... I am solely asking about an interpretation of SE's for dummies.


There is a Wiley Series "XYZ for Dummies". Has also some books relating to statistics. May be that helps.

For me the Standard Deviation (AKA Standard error SE) is a measure of how variable measurements are.
The formula to calculate is easy: it is the square root of the Variance. So now ask: "What is the Variance?" :cool:

For more dummies explanation see here.

Regards,

Detlew
Ben
★    

2016-12-29 15:32
(2646 d 04:37 ago)

@ d_labes
Posting: # 16906
Views: 6,927
 

 Standard error for Dummies

Dear Maestro,

Not sure if there is a better description that the one from Wiki. For me it is really the best definition and interpretation: it represents the standard deviation of the estimate of a certain statistic/estimator (like the mean).

❝ For me the Standard Deviation (AKA Standard error SE) is a measure of how variable measurements are.


Therefore, imho we should be careful when saying "AKA" here. The standard error is a specific standard deviation, referring not to individual measured values but to an estimator of a certain parameter.

Best,
Ben.
ElMaestro
★★★

Denmark,
2016-12-29 16:54
(2646 d 03:15 ago)

@ Ben
Posting: # 16907
Views: 6,870
 

 OK

Dear all,

thanks all for your input. I am sorry I am so out of reach of common didactics. The best I could do, and best I still can do:

SD: A measure of how much the observations are scattered.
SE: A measure of how much uncertainty we have on our estimated quantity.

Somehow I thought you guys would have like a mental image of it which you could share. That is how I get my head around things that are hard to grasp. Like e.g. the Hydrogen atom: A football with a pingpong ball flying around it in circles of discrete radia. Other atoms: Still a football of sorts, possibly larger than the previous one, and now with beaucoup de little balls flying here and there. SD in my head easily goes as a dartboard with arrows sitting scattered around the bull's eye. And so forth.
Somehow I am missing such a mental picture for an SE, I hope you see what I mean.
I have the same mental blockade about entropy by the way - people talk of disorder and chaos and information content to describe it, and all that is fancy, but I am only observing a blue screen of death, mentally. Fortunately, entropy hasn't found its way into BE yet :-D

Pass or fail!
ElMaestro
nobody
nothing

2016-12-30 09:52
(2645 d 10:17 ago)

@ ElMaestro
Posting: # 16909
Views: 6,842
 

 OK

...if you want to think something really weird: buy this, built it and sit there for days looking at the result:

http://www.zometool.com/the-hyperdo/

:-D

Kindest regards, nobody
VStus
★    

Poland,
2017-01-17 09:28
(2627 d 10:41 ago)

@ ElMaestro
Posting: # 16961
Views: 6,584
 

 OK

Dear Maestro,

You may want to play with this tool:
https://gallery.shinyapps.io/CLT_mean/

It demonstrates what SE means graphically for different kinds of distributions.

Kind of Statistics for Dummies book can be found here: openintro.org

Best regards, VStus
zizou
★    

Plzeň, Czech Republic,
2016-12-27 20:33
(2647 d 23:36 ago)

(edited by zizou on 2016-12-28 07:15)
@ ElMaestro
Posting: # 16889
Views: 7,073
 

 What's your understanding of a standard error?

Hi ElMaestro,

wikipedia knows more than me and I think there is a good explanation of SE.
The main points to understand (about SD and SE) are:
  • "the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean"
  • "the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean"
With increasing sample size:
  • SD of the sample will be more precise getting closer to the population SD
  • SE of whatever will be closer to zero. This is the property which makes CI narrower with higher sample size.

The calculation of SE of mean (i.e. SD of mean which is different from SD of sample) is on wikipedia (as you mentioned).
I like the proofs:
[image]
(image from wiki)
Easy steps with edit the equation:
var(mean) = 1/N * var(X)
sqrt(var(mean)) = 1/sqrt(N) * sqrt(var(X))
SD(mean) = 1/sqrt(N) * SD(X)
SE of mean = SD(X)/sqrt(N)



In BA/BE 90% CI is calculated from SE of difference T-R for ln-transformed data.
(SE of difference of estimated marginal means T-R)

In the same principles as previous proof (assuming independence and using variance properties) it's possible to derive formula valid for BE in 2x2 crossover design (balanced sequences, if not: mean should be changed somehow for estimated marginal mean in the following):
var(difference T-R) = var(mean_T-mean_R) = var(mean_T) + var((-1)*mean_R) = var(mean_T) + (-1)^2 * var(mean_R) = var(mean_T) + var(mean_R) = 1/N_T * var(X_T) + 1/N_R * var(X_R)
(SE of difference is only square root of variance of difference T-R)

Moreover var(X_T) can be substituted by N_T*SE_T^2 because SE of mean=SD/sqrt(N) which implies SD=sqrt(N)*SE and then var=SD^2=N*SE^2.
So:
SE of difference T-R = SD(difference T-R) = sqrt(var(difference T-R)) = sqrt( 1/N_T * var(X_T) + 1/N_R * var(X_R) ) = sqrt( 1/N_T * N_T*SE_T^2 + 1/N_R * N_R*SE_R^2 ) = sqrt( SE_T^2 + SE_R^2 )
without between steps:
SE of difference T-R = sqrt( SE_T^2 + SE_R^2 )
where SE_T is Standard Error of the mean of ln-data of Test treatment, same for SE_R - Reference treatment.
Nevertheless it is only nice to know (not useful, I guess), GLM gives the result of difference and SE of difference without need of calculation of T and R separately.

Best regards,
zizou
-----
Edit: In second half of this post, for case of unbalanced sequences the mean should be somehow changed for estimated marginal mean (i.e. LSMean in SAS terminology).
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