[R-meta] paired meta-analyzes with different n for loss of patients to follow-up

Enviado: mi?rcoles, 22 de abril de 2020 11:16
Para: Martin Lobo <mlobo4370 at hotmail.com>; r-sig-meta-analysis at r-project.org <r-sig-meta-analysis at r-project.org>
Asunto: Re: [R-meta] meta analysis with standard deviation or standard errors


For paired samples (changes, differences) I would preferably use MD.


Gerta


Am 22.04.2020 um 16:02 schrieb Martin Lobo:
THANK YOU GERTA !!!

Ok

For individual samples I use MD or SMD, depending on whether the measurements are on the same scale or not.

for paired samples, should i use MC or SMCC?

Regard




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: Gerta Ruecker <ruecker at imbi.uni-freiburg.de><mailto:ruecker at imbi.uni-freiburg.de>
Enviado: mi?rcoles, 22 de abril de 2020 10:26
Para: Martin Lobo <mlobo4370 at hotmail.com><mailto:mlobo4370 at hotmail.com>; r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at r-project.org> <r-sig-meta-analysis at r-project.org><mailto:r-sig-meta-analysis at r-project.org>
Asunto: Re: [R-meta] meta analysis with standard deviation or standard errors


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<https://eur05.safelinks.protection.outlook.com/?url=https%3A%2F%2Ftraining.cochrane.org%2Fhandbook%2Fcurrent%2Fchapter-10%23section-10-5-2&data=02%7C01%7C%7C50bd0edfa39a4d34702f08d7e6c7caeb%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637231618285603045&sdata=VYSIicza9TQ%2Fuj5zZEwtd19GAO1Qn0Z4RhOyTSzztig%3D&reserved=0>

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<https://eur05.safelinks.protection.outlook.com/?url=https%3A%2F%2Fcran.r-project.org%2Fweb%2Fpackages%2Fmetafor%2Fmetafor.pdf&data=02%7C01%7C%7C50bd0edfa39a4d34702f08d7e6c7caeb%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637231618285613037&sdata=Us9VTOVTOQLicqIv1ZvHqzKXRHXCBsbmgzNEwF4XnOM%3D&reserved=0>


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 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: Gerta Ruecker <ruecker at imbi.uni-freiburg.de><mailto:ruecker at imbi.uni-freiburg.de>
Enviado: martes, 21 de abril de 2020 14:44
Para: Martin Lobo <mlobo4370 at hotmail.com><mailto:mlobo4370 at hotmail.com>; r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at r-project.org> <r-sig-meta-analysis at r-project.org><mailto:r-sig-meta-analysis at 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 at imbi.uni-freiburg.de><mailto:ruecker at imbi.uni-freiburg.de>
Enviado: viernes, 17 de abril de 2020 08:12
Para: Martin Lobo <mlobo4370 at hotmail.com><mailto:mlobo4370 at hotmail.com>; r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at r-project.org> <r-sig-meta-analysis at r-project.org><mailto:r-sig-meta-analysis at 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 at r-project.org><mailto:r-sig-meta-analysis-bounces at r-project.org> en nombre de r-sig-meta-analysis-request at r-project.org<mailto:r-sig-meta-analysis-request at r-project.org> <r-sig-meta-analysis-request at r-project.org><mailto:r-sig-meta-analysis-request at r-project.org>
Enviado: mi?rcoles, 15 de abril de 2020 07:00
Para: r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at r-project.org> <r-sig-meta-analysis at r-project.org><mailto:r-sig-meta-analysis at 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 at maastrichtuniversity.nl><mailto:wolfgang.viechtbauer at maastrichtuniversity.nl>
To: Ju Lee <juhyung2 at stanford.edu><mailto:juhyung2 at stanford.edu>,
        "r-sig-meta-analysis at r-project.org"<mailto:r-sig-meta-analysis at r-project.org>
        <r-sig-meta-analysis at r-project.org><mailto:r-sig-meta-analysis at r-project.org>
Subject: Re: [R-meta] Dear Wolfgang
Message-ID: <b411740819d1411da87d505cdeceb3e6 at UM-MAIL3214.unimaas.nl><mailto:b411740819d1411da87d505cdeceb3e6 at 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 at stanford.edu]
Sent: Tuesday, 14 April, 2020 18:54
To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at 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 at maastrichtuniversity.nl><mailto:wolfgang.viechtbauer at maastrichtuniversity.nl>
Sent: Tuesday, April 14, 2020 7:04 AM
To: Ju Lee <juhyung2 at stanford.edu><mailto:juhyung2 at stanford.edu>; r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at r-project.org> <r-
sig-meta-analysis at r-project.org<mailto:sig-meta-analysis at 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&amp;data=02%7C01%7C%7C7f93a72da7b64707fe6d08d7e12439ac%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637225418004815037&amp;sdata=Tqgh0WpvUo70JTaihNWcZcbVQCQRpbprCYAxGKtlBGY%3D&amp;reserved=0<https://eur05.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.metafor-project.org%2Fdoku.php%2Ftips%3Acomp_two_independent_estimates&data=02%7C01%7C%7C50bd0edfa39a4d34702f08d7e6c7caeb%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637231618285623034&sdata=XsMRHy8lut9U%2FXh%2BsIqcZ%2BvghN2%2FQrsxNXGnv6bJa5Y%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 at r-


project.org]


On Behalf Of Ju Lee
Sent: Monday, 13 April, 2020 22:47
To: r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at 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 at stanford.edu><mailto:juhyung2 at stanford.edu>
To: "Viechtbauer, Wolfgang (SP)"
        <wolfgang.viechtbauer at maastrichtuniversity.nl><mailto:wolfgang.viechtbauer at maastrichtuniversity.nl>,
        "r-sig-meta-analysis at r-project.org"<mailto:r-sig-meta-analysis at r-project.org>
        <r-sig-meta-analysis at r-project.org><mailto:r-sig-meta-analysis at r-project.org>
Subject: Re: [R-meta] Dear Wolfgang
Message-ID:
        <BYAPR02MB5559407370455A06F0B047A8F7DB0 at BYAPR02MB5559.namprd02.prod.outlook.com><mailto:BYAPR02MB5559407370455A06F0B047A8F7DB0 at 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 at maastrichtuniversity.nl><mailto:wolfgang.viechtbauer at maastrichtuniversity.nl>
Sent: Tuesday, April 14, 2020 1:43 PM
To: Ju Lee <juhyung2 at stanford.edu><mailto:juhyung2 at stanford.edu>; r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at r-project.org> <r-sig-meta-analysis at r-project.org><mailto:r-sig-meta-analysis at 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 at stanford.edu]
Sent: Tuesday, 14 April, 2020 18:54
To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at 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 at maastrichtuniversity.nl><mailto:wolfgang.viechtbauer at maastrichtuniversity.nl>
Sent: Tuesday, April 14, 2020 7:04 AM
To: Ju Lee <juhyung2 at stanford.edu><mailto:juhyung2 at stanford.edu>; r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at r-project.org> <r-
sig-meta-analysis at r-project.org<mailto:sig-meta-analysis at 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&amp;data=02%7C01%7C%7C7f93a72da7b64707fe6d08d7e12439ac%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637225418004815037&amp;sdata=Tqgh0WpvUo70JTaihNWcZcbVQCQRpbprCYAxGKtlBGY%3D&amp;reserved=0<https://eur05.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.metafor-project.org%2Fdoku.php%2Ftips%3Acomp_two_independent_estimates&data=02%7C01%7C%7C50bd0edfa39a4d34702f08d7e6c7caeb%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637231618285623034&sdata=XsMRHy8lut9U%2FXh%2BsIqcZ%2BvghN2%2FQrsxNXGnv6bJa5Y%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 at r-


project.org]


On Behalf Of Ju Lee
Sent: Monday, 13 April, 2020 22:47
To: r-sig-meta-analysis at r-project.org<mailto:r-sig-meta-analysis at 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


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--

Dr. rer. nat. Gerta R?cker, 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 at imbi.uni-freiburg.de<mailto:ruecker at imbi.uni-freiburg.de>
Homepage: https://www.imbi.uni-freiburg.de/persons/ruecker/person_view<https://eur05.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.imbi.uni-freiburg.de%2Fpersons%2Fruecker%2Fperson_view&data=02%7C01%7C%7C50bd0edfa39a4d34702f08d7e6c7caeb%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637231618285643023&sdata=unn63E9aAWHY7dFlimdVsbN5AsXlryiYsifTVPliwb4%3D&reserved=0>


--

Dr. rer. nat. Gerta R?cker, 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 at imbi.uni-freiburg.de<mailto:ruecker at imbi.uni-freiburg.de>
Homepage: https://www.imbi.uni-freiburg.de/persons/ruecker/person_view<https://eur05.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.imbi.uni-freiburg.de%2Fpersons%2Fruecker%2Fperson_view&data=02%7C01%7C%7C50bd0edfa39a4d34702f08d7e6c7caeb%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637231618285643023&sdata=unn63E9aAWHY7dFlimdVsbN5AsXlryiYsifTVPliwb4%3D&reserved=0>


--

Dr. rer. nat. Gerta R?cker, 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 at imbi.uni-freiburg.de<mailto:ruecker at imbi.uni-freiburg.de>
Homepage: https://www.imbi.uni-freiburg.de/persons/ruecker/person_view<https://eur05.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.imbi.uni-freiburg.de%2Fpersons%2Fruecker%2Fperson_view&data=02%7C01%7C%7C50bd0edfa39a4d34702f08d7e6c7caeb%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637231618285643023&sdata=unn63E9aAWHY7dFlimdVsbN5AsXlryiYsifTVPliwb4%3D&reserved=0>


--

Dr. rer. nat. Gerta R?cker, 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 at imbi.uni-freiburg.de<mailto:ruecker at imbi.uni-freiburg.de>
Homepage: https://www.imbi.uni-freiburg.de/persons/ruecker/person_view<https://eur05.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.imbi.uni-freiburg.de%2Fpersons%2Fruecker%2Fperson_view&data=02%7C01%7C%7C50bd0edfa39a4d34702f08d7e6c7caeb%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637231618285653020&sdata=raYwUS8%2FSRV1x6fzwZe%2B%2FmCFMzghHsoOUPuiVsZK2H0%3D&reserved=0>