Code modification for post-hoc power

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Code modification for post-hoc power

R help mailing list-2
Hello everybody,

I am trying to accommodate the R codes provided by Donohue for sample size calculation in the package "longpower" with lmmpower function to estimate the post-hoc power (asked by a reviewer) of a binary GEE model with a three-way interaction (time x condition x continuous predictor) given a fixed sample size. In other words instead of the sample size I would like to estimate the power of my study.

Could anyone please help me to modify these codes as to obtain the power I'm looking for.

I would really appreciate receiving any feedback on this subject.

Yours sincerely,
 
Anne

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Re: Code modification for post-hoc power

R help mailing list-2

> On Aug 26, 2019, at 6:24 AM, CHATTON Anne via R-help <[hidden email]> wrote:
>
> Hello everybody,
>
> I am trying to accommodate the R codes provided by Donohue for sample size calculation in the package "longpower" with lmmpower function to estimate the post-hoc power (asked by a reviewer) of a binary GEE model with a three-way interaction (time x condition x continuous predictor) given a fixed sample size. In other words instead of the sample size I would like to estimate the power of my study.
>
> Could anyone please help me to modify these codes as to obtain the power I'm looking for.
>
> I would really appreciate receiving any feedback on this subject.
>
> Yours sincerely,
>
> Anne


Hi,

Three comments:

1. Don't calculate post hoc power. Do a Google search and you will find a plethora of papers and discussions on why not, including these:

  The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis
  The American Statistician, February 2001, Vol. 55, No. 1
  https://www.vims.edu/people/hoenig_jm/pubs/hoenig2.pdf

  Post Hoc Power: Tables and Commentary
  https://stat.uiowa.edu/sites/stat.uiowa.edu/files/techrep/tr378.pdf

  Observed power, and what to do if your editor asks for post-hoc power analyses
  http://daniellakens.blogspot.com/2014/12/observed-power-and-what-to-do-if-your.html

  Retraction Watch:
  Statisticians clamor for retraction of paper by Harvard researchers they say uses a “nonsense statistic”
  https://retractionwatch.com/2019/06/19/statisticians-clamor-for-retraction-of-paper-by-harvard-researchers-they-say-uses-a-nonsense-statistic/

  PubPeer Comments on the paper cited in the above RW post:
  https://pubpeer.com/publications/4399282A80691D9421B497E8316CF6

  A discussion on Frank's Data Methods forum also related to the same paper cited above:
  "Observed Power" and other "Power" Issues
  https://discourse.datamethods.org/t/observed-power-and-other-power-issues/731/30


2. If you are still compelled (voluntarily or involuntarily), you may want to review the vignette for the longpower package which may have some insights, and/or contact the package maintainer for additional guidance on how to structure the code. See the vignette here:

  https://cran.r-project.org/web/packages/longpower/vignettes/longpower.pdf


3. Don't calculate post hoc power.


Regards,

Marc Schwartz

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Re: Code modification for post-hoc power

Michael Dewey-3
Dear Anne

In addition to Marc's comments if you are forced to do this then,
assuming your package computes sample size from power then just feed it
a range of powers and find the one for which it calculates the sample
size you had. There is a more elegant way to do this using uniroot but
brute force should work.

Michael

On 26/08/2019 13:42, Marc Schwartz via R-help wrote:

>
>> On Aug 26, 2019, at 6:24 AM, CHATTON Anne via R-help <[hidden email]> wrote:
>>
>> Hello everybody,
>>
>> I am trying to accommodate the R codes provided by Donohue for sample size calculation in the package "longpower" with lmmpower function to estimate the post-hoc power (asked by a reviewer) of a binary GEE model with a three-way interaction (time x condition x continuous predictor) given a fixed sample size. In other words instead of the sample size I would like to estimate the power of my study.
>>
>> Could anyone please help me to modify these codes as to obtain the power I'm looking for.
>>
>> I would really appreciate receiving any feedback on this subject.
>>
>> Yours sincerely,
>>
>> Anne
>
>
> Hi,
>
> Three comments:
>
> 1. Don't calculate post hoc power. Do a Google search and you will find a plethora of papers and discussions on why not, including these:
>
>    The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis
>    The American Statistician, February 2001, Vol. 55, No. 1
>    https://www.vims.edu/people/hoenig_jm/pubs/hoenig2.pdf
>
>    Post Hoc Power: Tables and Commentary
>    https://stat.uiowa.edu/sites/stat.uiowa.edu/files/techrep/tr378.pdf
>
>    Observed power, and what to do if your editor asks for post-hoc power analyses
>    http://daniellakens.blogspot.com/2014/12/observed-power-and-what-to-do-if-your.html
>
>    Retraction Watch:
>    Statisticians clamor for retraction of paper by Harvard researchers they say uses a “nonsense statistic”
>    https://retractionwatch.com/2019/06/19/statisticians-clamor-for-retraction-of-paper-by-harvard-researchers-they-say-uses-a-nonsense-statistic/
>
>    PubPeer Comments on the paper cited in the above RW post:
>    https://pubpeer.com/publications/4399282A80691D9421B497E8316CF6
>
>    A discussion on Frank's Data Methods forum also related to the same paper cited above:
>    "Observed Power" and other "Power" Issues
>    https://discourse.datamethods.org/t/observed-power-and-other-power-issues/731/30
>
>
> 2. If you are still compelled (voluntarily or involuntarily), you may want to review the vignette for the longpower package which may have some insights, and/or contact the package maintainer for additional guidance on how to structure the code. See the vignette here:
>
>    https://cran.r-project.org/web/packages/longpower/vignettes/longpower.pdf
>
>
> 3. Don't calculate post hoc power.
>
>
> Regards,
>
> Marc Schwartz
>
> ______________________________________________
> [hidden email] mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
> ---
> This email has been checked for viruses by AVG.
> https://www.avg.com
>
>

--
Michael
http://www.dewey.myzen.co.uk/home.html

______________________________________________
[hidden email] mailing list -- To UNSUBSCRIBE and more, see
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
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Re: Code modification for post-hoc power

Peter Dalgaard-2
That doesn't work. In caricature, post-hoc power is

- I observe a difference of nearly zero
- However, to find a significant difference of that size I'd need 200000 observations
- I only used 100 observations
- Therefore my study is useless and can be discarded

(or: I calculate the probability of Type II error if the true difference is the observed and get 0.999... Therefore, etc.)

Best way out is a confidence interval. Second best (but in principle wrong) is to redo the pre-study power calculation and say that the study was designed to find a difference of delta, which it clearly didn't, so the true difference is probably less than delta.

-pd

> On 26 Aug 2019, at 18:29 , Michael Dewey <[hidden email]> wrote:
>
> Dear Anne
>
> In addition to Marc's comments if you are forced to do this then, assuming your package computes sample size from power then just feed it a range of powers and find the one for which it calculates the sample size you had. There is a more elegant way to do this using uniroot but brute force should work.
>
> Michael
>
> On 26/08/2019 13:42, Marc Schwartz via R-help wrote:
>>> On Aug 26, 2019, at 6:24 AM, CHATTON Anne via R-help <[hidden email]> wrote:
>>>
>>> Hello everybody,
>>>
>>> I am trying to accommodate the R codes provided by Donohue for sample size calculation in the package "longpower" with lmmpower function to estimate the post-hoc power (asked by a reviewer) of a binary GEE model with a three-way interaction (time x condition x continuous predictor) given a fixed sample size. In other words instead of the sample size I would like to estimate the power of my study.
>>>
>>> Could anyone please help me to modify these codes as to obtain the power I'm looking for.
>>>
>>> I would really appreciate receiving any feedback on this subject.
>>>
>>> Yours sincerely,
>>>
>>> Anne
>> Hi,
>> Three comments:
>> 1. Don't calculate post hoc power. Do a Google search and you will find a plethora of papers and discussions on why not, including these:
>>   The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis
>>   The American Statistician, February 2001, Vol. 55, No. 1
>>   https://www.vims.edu/people/hoenig_jm/pubs/hoenig2.pdf
>>   Post Hoc Power: Tables and Commentary
>>   https://stat.uiowa.edu/sites/stat.uiowa.edu/files/techrep/tr378.pdf
>>   Observed power, and what to do if your editor asks for post-hoc power analyses
>>   http://daniellakens.blogspot.com/2014/12/observed-power-and-what-to-do-if-your.html
>>   Retraction Watch:
>>   Statisticians clamor for retraction of paper by Harvard researchers they say uses a “nonsense statistic”
>>   https://retractionwatch.com/2019/06/19/statisticians-clamor-for-retraction-of-paper-by-harvard-researchers-they-say-uses-a-nonsense-statistic/
>>   PubPeer Comments on the paper cited in the above RW post:
>>   https://pubpeer.com/publications/4399282A80691D9421B497E8316CF6
>>   A discussion on Frank's Data Methods forum also related to the same paper cited above:
>>   "Observed Power" and other "Power" Issues
>>   https://discourse.datamethods.org/t/observed-power-and-other-power-issues/731/30
>> 2. If you are still compelled (voluntarily or involuntarily), you may want to review the vignette for the longpower package which may have some insights, and/or contact the package maintainer for additional guidance on how to structure the code. See the vignette here:
>>   https://cran.r-project.org/web/packages/longpower/vignettes/longpower.pdf
>> 3. Don't calculate post hoc power.
>> Regards,
>> Marc Schwartz
>> ______________________________________________
>> [hidden email] mailing list -- To UNSUBSCRIBE and more, see
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>> ---
>> This email has been checked for viruses by AVG.
>> https://www.avg.com
>
> --
> Michael
> http://www.dewey.myzen.co.uk/home.html
>
> ______________________________________________
> [hidden email] mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

--
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Office: A 4.23
Email: [hidden email]  Priv: [hidden email]

______________________________________________
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Re: Code modification for post-hoc power

R help mailing list-2
In reply to this post by Michael Dewey-3
Dear Michael,

Thanks a lot for your suggestion. This is what I am trying to do with R (longpower and gee packages). But I am getting stuck with a confusing error message sent earlier I don't understand.

Best,

Anne

-----Message d'origine-----
De : Michael Dewey [mailto:[hidden email]]
Envoyé : lundi, 26 août 2019 18:29
À : Marc Schwartz <[hidden email]>; CHATTON Anne <[hidden email]>
Cc : R-help <[hidden email]>
Objet : Re: [R] Code modification for post-hoc power

Dear Anne

In addition to Marc's comments if you are forced to do this then, assuming your package computes sample size from power then just feed it a range of powers and find the one for which it calculates the sample size you had. There is a more elegant way to do this using uniroot but brute force should work.

Michael

On 26/08/2019 13:42, Marc Schwartz via R-help wrote:

>
>> On Aug 26, 2019, at 6:24 AM, CHATTON Anne via R-help <[hidden email]> wrote:
>>
>> Hello everybody,
>>
>> I am trying to accommodate the R codes provided by Donohue for sample size calculation in the package "longpower" with lmmpower function to estimate the post-hoc power (asked by a reviewer) of a binary GEE model with a three-way interaction (time x condition x continuous predictor) given a fixed sample size. In other words instead of the sample size I would like to estimate the power of my study.
>>
>> Could anyone please help me to modify these codes as to obtain the power I'm looking for.
>>
>> I would really appreciate receiving any feedback on this subject.
>>
>> Yours sincerely,
>>
>> Anne
>
>
> Hi,
>
> Three comments:
>
> 1. Don't calculate post hoc power. Do a Google search and you will find a plethora of papers and discussions on why not, including these:
>
>    The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis
>    The American Statistician, February 2001, Vol. 55, No. 1
>    https://www.vims.edu/people/hoenig_jm/pubs/hoenig2.pdf
>
>    Post Hoc Power: Tables and Commentary
>    https://stat.uiowa.edu/sites/stat.uiowa.edu/files/techrep/tr378.pdf
>
>    Observed power, and what to do if your editor asks for post-hoc power analyses
>    
> http://daniellakens.blogspot.com/2014/12/observed-power-and-what-to-do
> -if-your.html
>
>    Retraction Watch:
>    Statisticians clamor for retraction of paper by Harvard researchers they say uses a “nonsense statistic”
>    
> https://retractionwatch.com/2019/06/19/statisticians-clamor-for-retrac
> tion-of-paper-by-harvard-researchers-they-say-uses-a-nonsense-statisti
> c/
>
>    PubPeer Comments on the paper cited in the above RW post:
>    https://pubpeer.com/publications/4399282A80691D9421B497E8316CF6
>
>    A discussion on Frank's Data Methods forum also related to the same paper cited above:
>    "Observed Power" and other "Power" Issues
>    
> https://discourse.datamethods.org/t/observed-power-and-other-power-iss
> ues/731/30
>
>
> 2. If you are still compelled (voluntarily or involuntarily), you may want to review the vignette for the longpower package which may have some insights, and/or contact the package maintainer for additional guidance on how to structure the code. See the vignette here:
>
>    
> https://cran.r-project.org/web/packages/longpower/vignettes/longpower.
> pdf
>
>
> 3. Don't calculate post hoc power.
>
>
> Regards,
>
> Marc Schwartz
>
> ______________________________________________
> [hidden email] mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
> ---
> This email has been checked for viruses by AVG.
> https://www.avg.com
>
>

--
Michael
http://www.dewey.myzen.co.uk/home.html
______________________________________________
[hidden email] mailing list -- To UNSUBSCRIBE and more, see
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
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Re: Code modification for post-hoc power

Michael Dewey-3
Dear Anne

Can you resend the eror message which you accidentally sent only to me
please?

Michael

On 27/08/2019 08:02, CHATTON Anne wrote:

> Dear Michael,
>
> Thanks a lot for your suggestion. This is what I am trying to do with R (longpower and gee packages). But I am getting stuck with a confusing error message sent earlier I don't understand.
>
> Best,
>
> Anne
>
> -----Message d'origine-----
> De : Michael Dewey [mailto:[hidden email]]
> Envoyé : lundi, 26 août 2019 18:29
> À : Marc Schwartz <[hidden email]>; CHATTON Anne <[hidden email]>
> Cc : R-help <[hidden email]>
> Objet : Re: [R] Code modification for post-hoc power
>
> Dear Anne
>
> In addition to Marc's comments if you are forced to do this then, assuming your package computes sample size from power then just feed it a range of powers and find the one for which it calculates the sample size you had. There is a more elegant way to do this using uniroot but brute force should work.
>
> Michael
>
> On 26/08/2019 13:42, Marc Schwartz via R-help wrote:
>>
>>> On Aug 26, 2019, at 6:24 AM, CHATTON Anne via R-help <[hidden email]> wrote:
>>>
>>> Hello everybody,
>>>
>>> I am trying to accommodate the R codes provided by Donohue for sample size calculation in the package "longpower" with lmmpower function to estimate the post-hoc power (asked by a reviewer) of a binary GEE model with a three-way interaction (time x condition x continuous predictor) given a fixed sample size. In other words instead of the sample size I would like to estimate the power of my study.
>>>
>>> Could anyone please help me to modify these codes as to obtain the power I'm looking for.
>>>
>>> I would really appreciate receiving any feedback on this subject.
>>>
>>> Yours sincerely,
>>>
>>> Anne
>>
>>
>> Hi,
>>
>> Three comments:
>>
>> 1. Don't calculate post hoc power. Do a Google search and you will find a plethora of papers and discussions on why not, including these:
>>
>>     The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis
>>     The American Statistician, February 2001, Vol. 55, No. 1
>>     https://www.vims.edu/people/hoenig_jm/pubs/hoenig2.pdf
>>
>>     Post Hoc Power: Tables and Commentary
>>     https://stat.uiowa.edu/sites/stat.uiowa.edu/files/techrep/tr378.pdf
>>
>>     Observed power, and what to do if your editor asks for post-hoc power analyses
>>    
>> http://daniellakens.blogspot.com/2014/12/observed-power-and-what-to-do
>> -if-your.html
>>
>>     Retraction Watch:
>>     Statisticians clamor for retraction of paper by Harvard researchers they say uses a “nonsense statistic”
>>    
>> https://retractionwatch.com/2019/06/19/statisticians-clamor-for-retrac
>> tion-of-paper-by-harvard-researchers-they-say-uses-a-nonsense-statisti
>> c/
>>
>>     PubPeer Comments on the paper cited in the above RW post:
>>     https://pubpeer.com/publications/4399282A80691D9421B497E8316CF6
>>
>>     A discussion on Frank's Data Methods forum also related to the same paper cited above:
>>     "Observed Power" and other "Power" Issues
>>    
>> https://discourse.datamethods.org/t/observed-power-and-other-power-iss
>> ues/731/30
>>
>>
>> 2. If you are still compelled (voluntarily or involuntarily), you may want to review the vignette for the longpower package which may have some insights, and/or contact the package maintainer for additional guidance on how to structure the code. See the vignette here:
>>
>>    
>> https://cran.r-project.org/web/packages/longpower/vignettes/longpower.
>> pdf
>>
>>
>> 3. Don't calculate post hoc power.
>>
>>
>> Regards,
>>
>> Marc Schwartz
>>
>> ______________________________________________
>> [hidden email] mailing list -- To UNSUBSCRIBE and more, see
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>>
>> ---
>> This email has been checked for viruses by AVG.
>> https://www.avg.com
>>
>>
>
> --
> Michael
> http://www.dewey.myzen.co.uk/home.html
>

--
Michael
http://www.dewey.myzen.co.uk/home.html

______________________________________________
[hidden email] mailing list -- To UNSUBSCRIBE and more, see
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
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Re: Code modification for post-hoc power

Michael Dewey-3
Anne sent me off-line the error message.
============ error message starts here ==========
This error message:
"Beginning Cgee S-function, @(#) geeformula.q 4.13 98/01/27 Error in
`[.default`(xj, i) : type 'closure' d'indice incorrect"

appears after the following R codes:

library(gee)
attach(geefile) # stored in some spss directory
geereg <- gee(outcome ~ trial + group + trial*group + trial*group*beckce
+ trial*group*beckref, id = subject, data = geefile, na.omit, tol =
0.001, maxiter = 25, family = binomial, corstr = "exchangeable", silent
= TRUE)

The following command would have allowed me to modify the power so as to
obtain the sample size I have:
lmmpower(geereg, pct.change = 0.10, t = seq(0,6,3), power = 0.80)

================ end ===============

I am not an expert on R programming but in my experience that error has
meant that I was including as a parameter something which was not of the
type which the program expected. I would suggest as a first step not
using attach() at all and instead using data=geefile in the call to gee
or investigating the with() command if it does not allow a data parameter.

On 27/08/2019 12:25, Michael Dewey wrote:

> Dear Anne
>
> Can you resend the eror message which you accidentally sent only to me
> please?
>
> Michael
>
> On 27/08/2019 08:02, CHATTON Anne wrote:
>> Dear Michael,
>>
>> Thanks a lot for your suggestion. This is what I am trying to do with
>> R (longpower and gee packages). But I am getting stuck with a
>> confusing error message sent earlier I don't understand.
>>
>> Best,
>>
>> Anne
>>
>> -----Message d'origine-----
>> De : Michael Dewey [mailto:[hidden email]]
>> Envoyé : lundi, 26 août 2019 18:29
>> À : Marc Schwartz <[hidden email]>; CHATTON Anne
>> <[hidden email]>
>> Cc : R-help <[hidden email]>
>> Objet : Re: [R] Code modification for post-hoc power
>>
>> Dear Anne
>>
>> In addition to Marc's comments if you are forced to do this then,
>> assuming your package computes sample size from power then just feed
>> it a range of powers and find the one for which it calculates the
>> sample size you had. There is a more elegant way to do this using
>> uniroot but brute force should work.
>>
>> Michael
>>
>> On 26/08/2019 13:42, Marc Schwartz via R-help wrote:
>>>
>>>> On Aug 26, 2019, at 6:24 AM, CHATTON Anne via R-help
>>>> <[hidden email]> wrote:
>>>>
>>>> Hello everybody,
>>>>
>>>> I am trying to accommodate the R codes provided by Donohue for
>>>> sample size calculation in the package "longpower" with lmmpower
>>>> function to estimate the post-hoc power (asked by a reviewer) of a
>>>> binary GEE model with a three-way interaction (time x condition x
>>>> continuous predictor) given a fixed sample size. In other words
>>>> instead of the sample size I would like to estimate the power of my
>>>> study.
>>>>
>>>> Could anyone please help me to modify these codes as to obtain the
>>>> power I'm looking for.
>>>>
>>>> I would really appreciate receiving any feedback on this subject.
>>>>
>>>> Yours sincerely,
>>>>
>>>> Anne
>>>
>>>
>>> Hi,
>>>
>>> Three comments:
>>>
>>> 1. Don't calculate post hoc power. Do a Google search and you will
>>> find a plethora of papers and discussions on why not, including these:
>>>
>>>     The Abuse of Power: The Pervasive Fallacy of Power Calculations
>>> for Data Analysis
>>>     The American Statistician, February 2001, Vol. 55, No. 1
>>>     https://www.vims.edu/people/hoenig_jm/pubs/hoenig2.pdf
>>>
>>>     Post Hoc Power: Tables and Commentary
>>>     https://stat.uiowa.edu/sites/stat.uiowa.edu/files/techrep/tr378.pdf
>>>
>>>     Observed power, and what to do if your editor asks for post-hoc
>>> power analyses
>>> http://daniellakens.blogspot.com/2014/12/observed-power-and-what-to-do
>>> -if-your.html
>>>
>>>     Retraction Watch:
>>>     Statisticians clamor for retraction of paper by Harvard
>>> researchers they say uses a “nonsense statistic”
>>> https://retractionwatch.com/2019/06/19/statisticians-clamor-for-retrac
>>> tion-of-paper-by-harvard-researchers-they-say-uses-a-nonsense-statisti
>>> c/
>>>
>>>     PubPeer Comments on the paper cited in the above RW post:
>>>     https://pubpeer.com/publications/4399282A80691D9421B497E8316CF6
>>>
>>>     A discussion on Frank's Data Methods forum also related to the
>>> same paper cited above:
>>>     "Observed Power" and other "Power" Issues
>>> https://discourse.datamethods.org/t/observed-power-and-other-power-iss
>>> ues/731/30
>>>
>>>
>>> 2. If you are still compelled (voluntarily or involuntarily), you may
>>> want to review the vignette for the longpower package which may have
>>> some insights, and/or contact the package maintainer for additional
>>> guidance on how to structure the code. See the vignette here:
>>>
>>> https://cran.r-project.org/web/packages/longpower/vignettes/longpower.
>>> pdf
>>>
>>>
>>> 3. Don't calculate post hoc power.
>>>
>>>
>>> Regards,
>>>
>>> Marc Schwartz
>>>
>>> ______________________________________________
>>> [hidden email] mailing list -- To UNSUBSCRIBE and more, see
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guide
>>> http://www.R-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>> ---
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>>>
>>>
>>
>> --
>> Michael
>> http://www.dewey.myzen.co.uk/home.html
>>
>

--
Michael
http://www.dewey.myzen.co.uk/home.html

______________________________________________
[hidden email] mailing list -- To UNSUBSCRIBE and more, see
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.