[R-meta] meta analysis with standard deviation or standard errors
Gerta Ruecker
ruecker @end|ng |rom |mb|@un|-|re|burg@de
Wed Apr 22 15:26:47 CEST 2020
Dear Martin,
I am not sure whether I understand you correctly, but I see the
following cases:
1. You have pre-post changes (differences) with sd (or se) of these
changes -> you can use these for pooling
2. You have pre values and post values and their intra-individual
correlations (not frequently the case) -> you can use the
correlations to calculate the sd/se for the differences (and then
pool as in case 1)
3. You have pre values and post values, but no correlations and no sd
or se for the differences -> you might impute a correlation and
proceed as in case 2
4. You can also mix pre-post changes and post values, but only for mean
differences, not for standardized mean differences, see Cochrane
Handbook
https://training.cochrane.org/handbook/current/chapter-10#section-10-5-2
Best,
Gerta
Am 22.04.2020 um 14:30 schrieb Martin Lobo:
> Dear Gerta, thank tou very mucha for tour time.
>
>
> 1- the MC and SMCC are the methods I found for paired samples on page
> 103 of the metafor manual, I understood that they were equivalent to
> the MD and SMD of the individual samples.
> Manual Link: https://cran.r-project.org/web/packages/metafor/metafor.pdf
>
>
> If I had the pre post standard averages and deviations, only the
> difference with your
> standard deviation, would I no longer need the ri? In that case what
> method do I use
> or what code?
>
> thank you so much
> Martin
> */
> /*
> */Lorenzo Martín Lobo /**/^MTSAC, FACC, FESC /*
> /*Especialista Jerarquizado en Cardiología*/
> /*/*Jefe de Dpto Enf. Cardiovasculares y Cardiometabolismo Hospital
> Militar Campo de Mayo.*/
> */
> /*/*Jefe de Cardiología *//*Hospital Militar Campo de Mayo*/
> */
> /*Ex Jefe de Unidad Coronaria *//*Hospital Militar Campo de Mayo*/
> /
> /*Miembro Titular de la Sociedad Argentina de Cardiología*/
> /
> /
> /*Fellow American College of Cardiology*/
> /
> /
> /*Fellow European Society of Cardiology*/
> /
> /
> /*/*Ex Miembro del Area de Investigación de la SAC*/*/
> /
> /*Ex Director del Consejo de Aterosclerosis y Trombosis de la SAC*/
> /*Miembro Asesor /*del Consejo de Aterosclerosis y Trombosis de la SAC*/*/
> /*/
> /*/
> /*Ex Director del Consejo de Epidemiología y Prevención Cardiovascular
> de la SAC*/
> /*/
> /*/
> /*/
> /*//*/
> /*/
> /*Miembro Asesor del Consejo de Epidemiología y Prevención
> Cardiovascular de la SAC*/
> /*/
> /*/
> /*/
> /*/
> /*Experto en Lipidos de la Sociedad Argentina de Lipidos.*/
> /*Miembro de la Sociedad Argentina de Lipidos.*/
> /
> /*Instructor de ACLS de la American Heart Association*/
> /
>
>
> ------------------------------------------------------------------------
> *De:* Gerta Ruecker <ruecker using imbi.uni-freiburg.de>
> *Enviado:* martes, 21 de abril de 2020 14:44
> *Para:* Martin Lobo <mlobo4370 using hotmail.com>;
> r-sig-meta-analysis using r-project.org <r-sig-meta-analysis using r-project.org>
> *Asunto:* Re: [R-meta] meta analysis with standard deviation or
> standard errors
>
> Dear Martin,
>
>
> Sorry for the delay. The problem is that the mean and sd of pre and
> post do not suffice to know the sd of the pairwise differences, except
> one makes some assumptions about the intraindividual pre-post
> correlation. See the attached R code PrePost.R for illustration.
>
>
> Do you mean by ri the correlation coefficients? If you impute them
> (say, 0.5), you may analyse the pre-post changes, but you should have
> some (external) evidence for using a certain value.
>
>
> I am not sure about each one of your 5 points below, see inline below.
>
>
> Best,
>
> Gerta
>
>
>
> Am 17.04.2020 um 14:22 schrieb Martin Lobo:
>> Thank you very much Gerta.
>>
>> I asked the question to see how I can solve two problems I have.
>>
>> 1- If I want to do an metaanalysis of mean difference analysis
>> (Paired data, pre-post)
>> I have mean and sd pre and post, what methodd i use, MC , SMCC, etc
> What is MC, SMCC? I don't know for what these abbreviations stand.
> Otherwise, see above.
>>
>> 2- If I only have the mean and standard deviation as I do
> See above.
>>
>> 3 - ri is always necessary or can be imputed in some way
> See also above
>>
>> 4 - without ri the standard deviation of the mean difference can be
>> estimated
> Not without knowing or making assumptions about the correlation, as
> said above.
>>
>> 5 - regarding question 4, both for independent samples and for paired
>> samples
>
> For independent samples it is different, because for differences of
> independent means we have:
>
>
> sd(X + Y) = sqrt(var(X + Y)) = sqrt(var(X) + var(Y)) = sqrt(sd(X)^2 +
> sd(Y)^2)
>
>
> For paired (more general. correlated) variables:
>
>
> sd(X + Y) = sqrt(var(X) + var(Y) - 2Cov(X,Y))
>
>
>>
>>
>>
>> ------------------------------------------------------------------------
>> *De:* Gerta Ruecker <ruecker using imbi.uni-freiburg.de>
>> <mailto:ruecker using imbi.uni-freiburg.de>
>> *Enviado:* viernes, 17 de abril de 2020 08:12
>> *Para:* Martin Lobo <mlobo4370 using hotmail.com>
>> <mailto:mlobo4370 using hotmail.com>; r-sig-meta-analysis using r-project.org
>> <mailto:r-sig-meta-analysis using r-project.org>
>> <r-sig-meta-analysis using r-project.org>
>> <mailto:r-sig-meta-analysis using r-project.org>
>> *Asunto:* Re: [R-meta] meta analysis with standard deviation or
>> standard errors
>>
>> Dear Martin,
>>
>> The answer is no. The standard error is not a measure of dispersion
>> of the data, but a measure of the imprecision of estimation. A large
>> standard error may come from large variability between data, but also
>> from small sample size. The standard error becomes always small if
>> the sample size becomes large (law of large numbers).
>>
>> Best,
>>
>> Gerta
>>
>> Am 17.04.2020 um 13:07 schrieb Martin Lobo:
>>> Hello everyone !
>>>
>>>
>>> I wanted to know if it is possible to use the standard error instead of the standard deviation as a measure of dispersion.
>>> using the MD or SMD method for independent samples.
>>> If this is possible, there would be some difference in the conclusions.
>>>
>>> Thank you so much
>>>
>>>
>>>
>>>
>>> Lorenzo Mart�n Lobo MTSAC, FACC, FESC
>>> Especialista Jerarquizado en Cardiolog�a
>>> Jefe de Dpto Enf. Cardiovasculares y Cardiometabolismo Hospital Militar Campo de Mayo.
>>> Jefe de Cardiolog�a Hospital Militar Campo de Mayo
>>> Ex Jefe de Unidad Coronaria Hospital Militar Campo de Mayo
>>> Miembro Titular de la Sociedad Argentina de Cardiolog�a
>>> Fellow American College of Cardiology
>>> Fellow European Society of Cardiology
>>> Ex Miembro del Area de Investigaci�n de la SAC
>>> Ex Director del Consejo de Aterosclerosis y Trombosis de la SAC
>>> Miembro Asesor del Consejo de Aterosclerosis y Trombosis de la SAC
>>> Ex Director del Consejo de Epidemiolog�a y Prevenci�n Cardiovascular de la SAC
>>>
>>> Miembro Asesor del Consejo de Epidemiolog�a y Prevenci�n Cardiovascular de la SAC
>>>
>>>
>>> Experto en Lipidos de la Sociedad Argentina de Lipidos.
>>> Miembro de la Sociedad Argentina de Lipidos.
>>> Instructor de ACLS de la American Heart Association
>>>
>>>
>>> ________________________________
>>> De: R-sig-meta-analysis<r-sig-meta-analysis-bounces using r-project.org> <mailto:r-sig-meta-analysis-bounces using r-project.org> en nombre der-sig-meta-analysis-request using r-project.org <mailto:r-sig-meta-analysis-request using r-project.org> <r-sig-meta-analysis-request using r-project.org> <mailto:r-sig-meta-analysis-request using r-project.org>
>>> Enviado: mi�rcoles, 15 de abril de 2020 07:00
>>> Para:r-sig-meta-analysis using r-project.org <mailto:r-sig-meta-analysis using r-project.org> <r-sig-meta-analysis using r-project.org> <mailto:r-sig-meta-analysis using r-project.org>
>>> Asunto: R-sig-meta-analysis Digest, Vol 35, Issue 8
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>>>
>>> Today's Topics:
>>>
>>> 1. Re: Dear Wolfgang (Viechtbauer, Wolfgang (SP))
>>> 2. Re: Dear Wolfgang (Ju Lee)
>>>
>>> ----------------------------------------------------------------------
>>>
>>> Message: 1
>>> Date: Tue, 14 Apr 2020 20:43:51 +0000
>>> From: "Viechtbauer, Wolfgang (SP)"
>>> <wolfgang.viechtbauer using maastrichtuniversity.nl> <mailto:wolfgang.viechtbauer using maastrichtuniversity.nl>
>>> To: Ju Lee<juhyung2 using stanford.edu> <mailto:juhyung2 using stanford.edu>,
>>> "r-sig-meta-analysis using r-project.org" <mailto:r-sig-meta-analysis using r-project.org>
>>> <r-sig-meta-analysis using r-project.org> <mailto:r-sig-meta-analysis using r-project.org>
>>> Subject: Re: [R-meta] Dear Wolfgang
>>> Message-ID:<b411740819d1411da87d505cdeceb3e6 using UM-MAIL3214.unimaas.nl> <mailto:b411740819d1411da87d505cdeceb3e6 using UM-MAIL3214.unimaas.nl>
>>> Content-Type: text/plain; charset="iso-8859-1"
>>>
>>> Yes, if the effect size measure is the same, one can make such a comparison. Also, there should not be any overlap in the studies included in the two meta-analyses (as otherwise the two estimates are not independent, as assumed by the test). And yes, you don't need sample sizes or tau^2 values or anything else - just the two estimates and their corresponding standard errors. And it doesn't depend on what random effects structure was used in the two meta-analyses -- assuming that the structures used in the two meta-analyses were appropriate for the studies at hand.
>>>
>>> Best,
>>> Wolfgang
>>>
>>>> -----Original Message-----
>>>> From: Ju Lee [mailto:juhyung2 using stanford.edu]
>>>> Sent: Tuesday, 14 April, 2020 18:54
>>>> To: Viechtbauer, Wolfgang (SP);r-sig-meta-analysis using r-project.org <mailto:r-sig-meta-analysis using r-project.org>
>>>> Subject: Re: Dear Wolfgang
>>>>
>>>> Dear Wolfgang,
>>>>
>>>> Thanks for your insights.
>>>> I am reaching out to my colleagues to see how they have made such
>>>> transformation.
>>>>
>>>> In the meantime, based on the information that you have sent, it is possible
>>>> to compare two different meta-analyses if they are using the same effect
>>>> size, say lnRR? and this wald-type test can be performed only with grand
>>>> mean effect sizes and their standard error, without sample sizes or tau
>>>> value, if I understood correctly?
>>>>
>>>> How would this approach be actually applicable to publications that
>>>> seemingly used similar mixed-effect models but there is no guarantee that
>>>> random effect structures are standardized between the two?
>>>>
>>> [[elided Hotmail spam]]
>>>> Best,
>>>> JU
>>>> ________________________________________
>>>> From: Viechtbauer, Wolfgang (SP)
>>>> <wolfgang.viechtbauer using maastrichtuniversity.nl> <mailto:wolfgang.viechtbauer using maastrichtuniversity.nl>
>>>> Sent: Tuesday, April 14, 2020 7:04 AM
>>>> To: Ju Lee<juhyung2 using stanford.edu> <mailto:juhyung2 using stanford.edu>;r-sig-meta-analysis using r-project.org <mailto:r-sig-meta-analysis using r-project.org> <r-
>>>> sig-meta-analysis using r-project.org <mailto:sig-meta-analysis using r-project.org>>
>>>> Subject: RE: Dear Wolfgang
>>>>
>>>> Dear Ju,
>>>>
>>>> In principle, this might be of interest to you:
>>>>
>>>> https://nam01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.metafor-project.org%2Fdoku.php%2Ftips%3Acomp_two_independent_estimates&data=02%7C01%7C%7C7f93a72da7b64707fe6d08d7e12439ac%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637225418004815037&sdata=Tqgh0WpvUo70JTaihNWcZcbVQCQRpbprCYAxGKtlBGY%3D&reserved=0 <https://nam01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.metafor-project.org%2Fdoku.php%2Ftips%3Acomp_two_independent_estimates&data=02%7C01%7C%7Cb7b04d24edb749d80fd808d7e61baa75%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637230879022585452&sdata=a8wDXFJooyKG3NkPWEVrtzN1MGFywU6ha8574C%2F0W2c%3D&reserved=0>
>>>>
>>>> However, a standardized mean difference is given by (m1-m2)/sd, while a
>>>> (log) response ratio is log(m1/m2). I see no sensible way of converting the
>>>> former to the later.
>>>>
>>>> Best,
>>>> Wolfgang
>>>>
>>>>> -----Original Message-----
>>>>> From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-
>>>> project.org]
>>>>> On Behalf Of Ju Lee
>>>>> Sent: Monday, 13 April, 2020 22:47
>>>>> To:r-sig-meta-analysis using r-project.org <mailto:r-sig-meta-analysis using r-project.org>
>>>>> Subject: [R-meta] Dear Wolfgang
>>>>>
>>>>> Dear Wolfgang,
>>>>>
>>>>> I hope you are doing well.
>>>>>
>>>>> My research group is currently working on a project where they are trying
>>>> to
>>>>> compare effect sizes generated from their current mixed-effect meta-
>>>> analysis
>>>>> with effect sizes (based on similar response variables) calculated in other
>>>>> meta-analysis publications.
>>>>>
>>>>> We are currently using log response ratio and are trying to make some
>>>>> statement or analysis to compare our grand mean effect sizes with other
>>>>> studies. In more details, we are examining how herbivorous animal control
>>>>> plant growth in degraded environment. Now, there is already a meta-analysis
>>>>> out there that has examined this (in comparable manner) in natural
>>>>> environment as opposed to our study.
>>>>>
>>>>> My colleagues want to know if there is a way to make some type of
>>>> comparison
>>>>> (ex. whether responses are stronger in degraded vs. natural environemnts)
>>>>> between two effect sizes from these different studies using statistical
>>>>> approaches.
>>>>> So far what they have from other meta-analysis publication is grand mean
>>>>> hedges'd and var which they transformed to lnRR and var in hopes to compare
>>>>> with our lnRR effect sizes.
>>>>>
>>>>> My view is that this is not possible unless we can have their actual raw
>>>>> dataset and run a whole new model combining with our original raw dataset.
>>>>> But I wanted to reach out to you and the community if there is an
>>>>> alternative approaches to compare mean effect sizes among different meta-
>>>>> analysis which are assumed to have used similar approaches in study
>>>>> selection and models (another issue being different random effect
>>>> structures
>>>>> used in different meta-analysis which may not be very apparent from method
>>>>> description).
>>>>>
>>> [[elided Hotmail spam]]
>>>>> Best,
>>>>> JU
>>> ------------------------------
>>>
>>> Message: 2
>>> Date: Wed, 15 Apr 2020 05:33:16 +0000
>>> From: Ju Lee<juhyung2 using stanford.edu> <mailto:juhyung2 using stanford.edu>
>>> To: "Viechtbauer, Wolfgang (SP)"
>>> <wolfgang.viechtbauer using maastrichtuniversity.nl> <mailto:wolfgang.viechtbauer using maastrichtuniversity.nl>,
>>> "r-sig-meta-analysis using r-project.org" <mailto:r-sig-meta-analysis using r-project.org>
>>> <r-sig-meta-analysis using r-project.org> <mailto:r-sig-meta-analysis using r-project.org>
>>> Subject: Re: [R-meta] Dear Wolfgang
>>> Message-ID:
>>> <BYAPR02MB5559407370455A06F0B047A8F7DB0 using BYAPR02MB5559.namprd02.prod.outlook.com> <mailto:BYAPR02MB5559407370455A06F0B047A8F7DB0 using BYAPR02MB5559.namprd02.prod.outlook.com>
>>>
>>> Content-Type: text/plain; charset="utf-8"
>>>
>>> Dear Wolfgang,
>>>
>>> [[elided Hotmail spam]]
>>> I am not sure how my colleagues have transformed hedges' d to lnRR, based on what sources, but I will reach out again once I have more details. I, too, have not known if there is a way to make such effect size transformation.
>>>
>>> Thank you very much!
>>> Best wishes,
>>> JU
>>> ________________________________
>>> From: Viechtbauer, Wolfgang (SP)<wolfgang.viechtbauer using maastrichtuniversity.nl> <mailto:wolfgang.viechtbauer using maastrichtuniversity.nl>
>>> Sent: Tuesday, April 14, 2020 1:43 PM
>>> To: Ju Lee<juhyung2 using stanford.edu> <mailto:juhyung2 using stanford.edu>;r-sig-meta-analysis using r-project.org <mailto:r-sig-meta-analysis using r-project.org> <r-sig-meta-analysis using r-project.org> <mailto:r-sig-meta-analysis using r-project.org>
>>> Subject: RE: Dear Wolfgang
>>>
>>> Yes, if the effect size measure is the same, one can make such a comparison. Also, there should not be any overlap in the studies included in the two meta-analyses (as otherwise the two estimates are not independent, as assumed by the test). And yes, you don't need sample sizes or tau^2 values or anything else - just the two estimates and their corresponding standard errors. And it doesn't depend on what random effects structure was used in the two meta-analyses -- assuming that the structures used in the two meta-analyses were appropriate for the studies at hand.
>>>
>>> Best,
>>> Wolfgang
>>>
>>>> -----Original Message-----
>>>> From: Ju Lee [mailto:juhyung2 using stanford.edu]
>>>> Sent: Tuesday, 14 April, 2020 18:54
>>>> To: Viechtbauer, Wolfgang (SP);r-sig-meta-analysis using r-project.org <mailto:r-sig-meta-analysis using r-project.org>
>>>> Subject: Re: Dear Wolfgang
>>>>
>>>> Dear Wolfgang,
>>>>
>>>> Thanks for your insights.
>>>> I am reaching out to my colleagues to see how they have made such
>>>> transformation.
>>>>
>>>> In the meantime, based on the information that you have sent, it is possible
>>>> to compare two different meta-analyses if they are using the same effect
>>>> size, say lnRR? and this wald-type test can be performed only with grand
>>>> mean effect sizes and their standard error, without sample sizes or tau
>>>> value, if I understood correctly?
>>>>
>>>> How would this approach be actually applicable to publications that
>>>> seemingly used similar mixed-effect models but there is no guarantee that
>>>> random effect structures are standardized between the two?
>>>>
>>> [[elided Hotmail spam]]
>>>> Best,
>>>> JU
>>>> ________________________________________
>>>> From: Viechtbauer, Wolfgang (SP)
>>>> <wolfgang.viechtbauer using maastrichtuniversity.nl> <mailto:wolfgang.viechtbauer using maastrichtuniversity.nl>
>>>> Sent: Tuesday, April 14, 2020 7:04 AM
>>>> To: Ju Lee<juhyung2 using stanford.edu> <mailto:juhyung2 using stanford.edu>;r-sig-meta-analysis using r-project.org <mailto:r-sig-meta-analysis using r-project.org> <r-
>>>> sig-meta-analysis using r-project.org <mailto:sig-meta-analysis using r-project.org>>
>>>> Subject: RE: Dear Wolfgang
>>>>
>>>> Dear Ju,
>>>>
>>>> In principle, this might be of interest to you:
>>>>
>>>> https://nam01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.metafor-project.org%2Fdoku.php%2Ftips%3Acomp_two_independent_estimates&data=02%7C01%7C%7C7f93a72da7b64707fe6d08d7e12439ac%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637225418004815037&sdata=Tqgh0WpvUo70JTaihNWcZcbVQCQRpbprCYAxGKtlBGY%3D&reserved=0 <https://nam01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.metafor-project.org%2Fdoku.php%2Ftips%3Acomp_two_independent_estimates&data=02%7C01%7C%7Cb7b04d24edb749d80fd808d7e61baa75%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637230879022595432&sdata=PMzH%2FelqtN5EAhFmK6pqcyq%2FLNGyHonwyBdWtal57Mo%3D&reserved=0>
>>>>
>>>> However, a standardized mean difference is given by (m1-m2)/sd, while a
>>>> (log) response ratio is log(m1/m2). I see no sensible way of converting the
>>>> former to the later.
>>>>
>>>> Best,
>>>> Wolfgang
>>>>
>>>>> -----Original Message-----
>>>>> From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-
>>>> project.org]
>>>>> On Behalf Of Ju Lee
>>>>> Sent: Monday, 13 April, 2020 22:47
>>>>> To:r-sig-meta-analysis using r-project.org <mailto:r-sig-meta-analysis using r-project.org>
>>>>> Subject: [R-meta] Dear Wolfgang
>>>>>
>>>>> Dear Wolfgang,
>>>>>
>>>>> I hope you are doing well.
>>>>>
>>>>> My research group is currently working on a project where they are trying
>>>> to
>>>>> compare effect sizes generated from their current mixed-effect meta-
>>>> analysis
>>>>> with effect sizes (based on similar response variables) calculated in other
>>>>> meta-analysis publications.
>>>>>
>>>>> We are currently using log response ratio and are trying to make some
>>>>> statement or analysis to compare our grand mean effect sizes with other
>>>>> studies. In more details, we are examining how herbivorous animal control
>>>>> plant growth in degraded environment. Now, there is already a meta-analysis
>>>>> out there that has examined this (in comparable manner) in natural
>>>>> environment as opposed to our study.
>>>>>
>>>>> My colleagues want to know if there is a way to make some type of
>>>> comparison
>>>>> (ex. whether responses are stronger in degraded vs. natural environemnts)
>>>>> between two effect sizes from these different studies using statistical
>>>>> approaches.
>>>>> So far what they have from other meta-analysis publication is grand mean
>>>>> hedges'd and var which they transformed to lnRR and var in hopes to compare
>>>>> with our lnRR effect sizes.
>>>>>
>>>>> My view is that this is not possible unless we can have their actual raw
>>>>> dataset and run a whole new model combining with our original raw dataset.
>>>>> But I wanted to reach out to you and the community if there is an
>>>>> alternative approaches to compare mean effect sizes among different meta-
>>>>> analysis which are assumed to have used similar approaches in study
>>>>> selection and models (another issue being different random effect
>>>> structures
>>>>> used in different meta-analysis which may not be very apparent from method
>>>>> description).
>>>>>
>>> [[elided Hotmail spam]]
>>>>> Best,
>>>>> JU
>>> [[alternative HTML version deleted]]
>>>
>>>
>>>
>>>
>>> ------------------------------
>>>
>>> Subject: Digest Footer
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>>>
>>>
>>> ------------------------------
>>>
>>> End of R-sig-meta-analysis Digest, Vol 35, Issue 8
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>> --
>>
>> Dr. rer. nat. Gerta Rücker, Dipl.-Math.
>>
>> Institute of Medical Biometry and Statistics,
>> Faculty of Medicine and Medical Center - University of Freiburg
>>
>> Stefan-Meier-Str. 26, D-79104 Freiburg, Germany
>>
>> Phone: +49/761/203-6673
>> Fax: +49/761/203-6680
>> Mail:ruecker using imbi.uni-freiburg.de <mailto:ruecker using imbi.uni-freiburg.de>
>> Homepage:https://www.imbi.uni-freiburg.de/persons/ruecker/person_view <https://nam01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.imbi.uni-freiburg.de%2Fpersons%2Fruecker%2Fperson_view&data=02%7C01%7C%7Cb7b04d24edb749d80fd808d7e61baa75%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637230879022625386&sdata=iWguskBjyqRZpTn%2BI4uaATKnkPT1Wlwfc2NTKlU9CNs%3D&reserved=0>
> --
>
> Dr. rer. nat. Gerta Rücker, Dipl.-Math.
>
> Institute of Medical Biometry and Statistics,
> Faculty of Medicine and Medical Center - University of Freiburg
>
> Stefan-Meier-Str. 26, D-79104 Freiburg, Germany
>
> Phone: +49/761/203-6673
> Fax: +49/761/203-6680
> Mail:ruecker using imbi.uni-freiburg.de <mailto:ruecker using imbi.uni-freiburg.de>
> Homepage:https://www.imbi.uni-freiburg.de/persons/ruecker/person_view <https://nam01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.imbi.uni-freiburg.de%2Fpersons%2Fruecker%2Fperson_view&data=02%7C01%7C%7Cb7b04d24edb749d80fd808d7e61baa75%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637230879022640361&sdata=Zvxi1tsRfeepkG9%2BzqXGpijxl%2BJFKfqqGrstldcZHgc%3D&reserved=0>
--
Dr. rer. nat. Gerta Rücker, Dipl.-Math.
Institute of Medical Biometry and Statistics,
Faculty of Medicine and Medical Center - University of Freiburg
Stefan-Meier-Str. 26, D-79104 Freiburg, Germany
Phone: +49/761/203-6673
Fax: +49/761/203-6680
Mail: ruecker using imbi.uni-freiburg.de
Homepage: https://www.imbi.uni-freiburg.de/persons/ruecker/person_view
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