Question: Heatmaps and correlation
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gravatar for john herbert
7.7 years ago by
john herbert560
john herbert560 wrote:
Dear bioconductors, >From a google search, I found the following code that confuses me a little. As usual, it is probably something really elementary but reading around does not solve. The code was written by James Mcdonald ( http://www.mail-archive.com/r-help@r-project.org/msg61514.html) and is to compute dendograms based on correlation and plot the results on a heatmap as follows; a <- matrix(rnorm(50), ncol=5) rowv <- as.dendrogram(hclust(as.dist(1-cor(t(a))))) colv <- as.dendrogram(hclust(as.dist(1-cor(a)))) heatmap.2(a, scale="row", Rowv=rowv, Colv=colv) Why the *1*-cor(a)? Orig.cor Adjusted cor 1 0 0.9 0.1 0.8 0.2 0.7 0.3 0.6 0.4 0.5 0.5 0.4 0.6 0.3 0.7 0.2 0.8 0.1 0.9 0 1 -0.1 1.1 -0.2 1.2 -0.3 1.3 -0.4 1.4 -0.5 1.5 -0.6 1.6 -0.7 1.7 -0.8 1.8 -0.9 1.9 -1 2 This removes negative numbers? What is the reason for doing this? [[alternative HTML version deleted]]
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ADD COMMENTlink modified 7.7 years ago • written 7.7 years ago by john herbert560
Answer: Heatmaps and correlation
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gravatar for Kevin Coombes
7.7 years ago by
Kevin Coombes430
United States
Kevin Coombes430 wrote:
This is not really a bioconductor question.... but You need something that behaves like a "distance". By the definition of what you mean by 'distance", two things are close if and only if the distance is near zero. The bigger (more positive) the distance, the further apart things are. And you cannot measure distances as negative; only non-negative values need apply. Using 1-cor, you are taking two things to be close if the correlation is close to 1. And the things that are furthest apart are the ones where the correlation is close to -1. As an exercise, you might want to think about circumstances where the preferred code would be 1 - abs(cor(*)) instead of 1 - cor(*) On 5/11/2011 3:12 PM, john herbert wrote: > Dear bioconductors, > > From a google search, I found the following code that confuses me a little. > As usual, it is probably something really elementary but reading around does > not solve. > > The code was written by James Mcdonald ( > http://www.mail-archive.com/r-help at r-project.org/msg61514.html) and is to > compute dendograms based on correlation and plot the results on a heatmap as > follows; > > a<- matrix(rnorm(50), ncol=5) > rowv<- as.dendrogram(hclust(as.dist(1-cor(t(a))))) > colv<- as.dendrogram(hclust(as.dist(1-cor(a)))) > heatmap.2(a, scale="row", Rowv=rowv, Colv=colv) > > > Why the *1*-cor(a)? > > > Orig.cor Adjusted cor > 1 0 > 0.9 0.1 > 0.8 0.2 > 0.7 0.3 > 0.6 0.4 > 0.5 0.5 > 0.4 0.6 > 0.3 0.7 > 0.2 0.8 > 0.1 0.9 > 0 1 > -0.1 1.1 > -0.2 1.2 > -0.3 1.3 > -0.4 1.4 > -0.5 1.5 > -0.6 1.6 > -0.7 1.7 > -0.8 1.8 > -0.9 1.9 > -1 2 > > > This removes negative numbers? What is the reason for doing this? > > [[alternative HTML version deleted]] > > _______________________________________________ > Bioconductor mailing list > Bioconductor at r-project.org > https://stat.ethz.ch/mailman/listinfo/bioconductor > Search the archives: http://news.gmane.org/gmane.science.biology.informatics.conductor
ADD COMMENTlink written 7.7 years ago by Kevin Coombes430
A biological circumstance? Interesting. Nothing immediately pops to mind. cor 1-cor 1-abs(cor) 1 0 0 0.9 0.1 0.1 0.8 0.2 0.2 0.7 0.3 0.3 0.6 0.4 0.4 0.5 0.5 0.5 0.4 0.6 0.6 0.3 0.7 0.7 0.2 0.8 0.8 0.1 0.9 0.9 0 1 1 -0.1 1.1 0.9 -0.2 1.2 0.8 -0.3 1.3 0.7 -0.4 1.4 0.6 -0.5 1.5 0.5 -0.6 1.6 0.4 -0.7 1.7 0.3 -0.8 1.8 0.2 -0.9 1.9 0.1 -1 2 0 looking at the numbers; anything that is highly correlated ,whether positively or negatively, will be close in distance. the heatmaps look different and obviously the clustering is different. Thanks. . On Wed, May 11, 2011 at 9:19 PM, Kevin R. Coombes <kevin.r.coombes@gmail.com> wrote: > This is not really a bioconductor question.... but > > You need something that behaves like a "distance". > > By the definition of what you mean by 'distance", two things are close if > and only if the distance is near zero. The bigger (more positive) the > distance, the further apart things are. And you cannot measure distances as > negative; only non-negative values need apply. > > Using 1-cor, you are taking two things to be close if the correlation is > close to 1. > And the things that are furthest apart are the ones where the correlation > is close to -1. > > As an exercise, you might want to think about circumstances where the > preferred code would be > 1 - abs(cor(*)) > instead of > 1 - cor(*) > > > On 5/11/2011 3:12 PM, john herbert wrote: > >> Dear bioconductors, >> > From a google search, I found the following code that confuses me a >> little. >> As usual, it is probably something really elementary but reading around >> does >> not solve. >> >> The code was written by James Mcdonald ( >> http://www.mail-archive.com/r-help@r-project.org/msg61514.html) and is to >> compute dendograms based on correlation and plot the results on a heatmap >> as >> follows; >> >> a<- matrix(rnorm(50), ncol=5) >> rowv<- as.dendrogram(hclust(as.dist(1-cor(t(a))))) >> colv<- as.dendrogram(hclust(as.dist(1-cor(a)))) >> heatmap.2(a, scale="row", Rowv=rowv, Colv=colv) >> >> >> Why the *1*-cor(a)? >> >> >> Orig.cor Adjusted cor >> 1 0 >> 0.9 0.1 >> 0.8 0.2 >> 0.7 0.3 >> 0.6 0.4 >> 0.5 0.5 >> 0.4 0.6 >> 0.3 0.7 >> 0.2 0.8 >> 0.1 0.9 >> 0 1 >> -0.1 1.1 >> -0.2 1.2 >> -0.3 1.3 >> -0.4 1.4 >> -0.5 1.5 >> -0.6 1.6 >> -0.7 1.7 >> -0.8 1.8 >> -0.9 1.9 >> -1 2 >> >> >> This removes negative numbers? What is the reason for doing this? >> >> [[alternative HTML version deleted]] >> >> _______________________________________________ >> Bioconductor mailing list >> Bioconductor@r-project.org >> https://stat.ethz.ch/mailman/listinfo/bioconductor >> Search the archives: >> http://news.gmane.org/gmane.science.biology.informatics.conductor >> > [[alternative HTML version deleted]]
ADD REPLYlink written 7.7 years ago by john herbert560
Answer: Heatmaps and correlation
0
gravatar for john herbert
7.7 years ago by
john herbert560
john herbert560 wrote:
Producing some real heatmaps makes me wonder; code; a = data rowv<- as.dendrogram(hclust(as.dist(1-cor(t(a))))) colv<- as.dendrogram(hclust(as.dist(1-cor(a)))) # left map using correlation as a distance heatmap.2(a, scale="row", Rowv=rowv, Colv=colv) x11() # map on the right, using the default settings heatmap.2(a, scale="row") The attached heatmaps don't look that great? The first correlation map seems like it does not cluster correctly. A very basic look makes me think the predominantly yellow/white columns should cluster. the second, the default, looks a little better. This data is Delta CT values, not array data. > sessionInfo() R version 2.13.0 (2011-04-13) Platform: i386-pc-mingw32/i386 (32-bit) locale: [1] LC_COLLATE=English_United Kingdom.1252 LC_CTYPE=English_United Kingdom.1252 LC_MONETARY=English_United Kingdom.1252 [4] LC_NUMERIC=C LC_TIME=English_United Kingdom.1252 attached base packages: [1] grid stats graphics grDevices utils datasets methods base other attached packages: [1] HTqPCR_1.6.0 limma_3.8.1 RColorBrewer_1.0-2 Biobase_2.12.1 gplots_2.8.0 caTools_1.11 [7] bitops_1.0-4.1 gdata_2.8.2 gtools_2.6.2 loaded via a namespace (and not attached): [1] affy_1.30.0 affyio_1.20.0 preprocessCore_1.14.0 tools_2.13.0 On Wed, May 11, 2011 at 11:27 PM, john herbert <arraystruggles at="" gmail.com="">wrote: > mmmm, possibly. > Correlation of gene expression = comparing all genes vs. all genes. > > 1) genes A and B, both highly up regulated = high pos correlation, e.g. 0.9 > 2) genes C and D, both lowly down regulated = high pos correlation, e.g. > 0.9 > 3) gene E high up and gene F low down = high neg correlation, e.g. -0.9 > 4) gene G low down and gene H high up = high neg correlation, e.g. -0.9 > > with 1-cor; genes A, B, C and D cluster together. > genes E, F, G and H cluster together but a long way off from ABCD. > > with 1-abs(cor), all genes cluster together as they produce a extreme > correlation, either pos or neg > > To me, 1-cor makes more biological sense, is that agreeable? > > Table. > cor 1-cor 1-abs(cor) > 1 0 0 > 0.9 0.1 0.1 > 0.8 0.2 0.2 > 0.7 0.3 0.3 > 0.6 0.4 0.4 > 0.5 0.5 0.5 > 0.4 0.6 0.6 > 0.3 0.7 0.7 > 0.2 0.8 0.8 > 0.1 0.9 0.9 > 0 1 1 > -0.1 1.1 0.9 > -0.2 1.2 0.8 > -0.3 1.3 0.7 > -0.4 1.4 0.6 > -0.5 1.5 0.5 > -0.6 1.6 0.4 > -0.7 1.7 0.3 > -0.8 1.8 0.2 > -0.9 1.9 0.1 > -1 2 0 > > On Wed, May 11, 2011 at 10:48 PM, James W. MacDonald < > jmacdon at med.umich.edu> wrote: > >> >> >> On 5/11/2011 4:54 PM, john herbert wrote: >> >>> A biological circumstance? Interesting. Nothing immediately pops to mind. >>> >> >> You see the pattern, but think about the underlying biology. If you use >> 1-cor, as you note, two genes that are both highly up-regulated or both >> highly down-regulated will cluster together. >> >> But what if the first gene's product has a negative feedback effect on the >> transcription of the second gene? That implies a relationship that won't be >> captured if you use 1-cor, but will be captured if you use 1-abs(cor). This >> is, I believe, what Kevin was getting at. >> >> Best, >> >> Jim >> >> >> >> >>> cor 1-cor 1-abs(cor) >>> 1 0 0 >>> 0.9 0.1 0.1 >>> 0.8 0.2 0.2 >>> 0.7 0.3 0.3 >>> 0.6 0.4 0.4 >>> 0.5 0.5 0.5 >>> 0.4 0.6 0.6 >>> 0.3 0.7 0.7 >>> 0.2 0.8 0.8 >>> 0.1 0.9 0.9 >>> 0 1 1 >>> -0.1 1.1 0.9 >>> -0.2 1.2 0.8 >>> -0.3 1.3 0.7 >>> -0.4 1.4 0.6 >>> -0.5 1.5 0.5 >>> -0.6 1.6 0.4 >>> -0.7 1.7 0.3 >>> -0.8 1.8 0.2 >>> -0.9 1.9 0.1 >>> -1 2 0 >>> >>> looking at the numbers; anything that is highly correlated ,whether >>> positively or negatively, will be close in distance. the heatmaps look >>> different and obviously the clustering is different. >>> >>> Thanks. . >>> >>> On Wed, May 11, 2011 at 9:19 PM, Kevin R. Coombes< >>> kevin.r.coombes at gmail.com >>> >>>> wrote: >>>> >>> >>> This is not really a bioconductor question.... but >>>> >>>> You need something that behaves like a "distance". >>>> >>>> By the definition of what you mean by 'distance", two things are close >>>> if >>>> and only if the distance is near zero. The bigger (more positive) the >>>> distance, the further apart things are. And you cannot measure distances >>>> as >>>> negative; only non-negative values need apply. >>>> >>>> Using 1-cor, you are taking two things to be close if the correlation is >>>> close to 1. >>>> And the things that are furthest apart are the ones where the >>>> correlation >>>> is close to -1. >>>> >>>> As an exercise, you might want to think about circumstances where the >>>> preferred code would be >>>> 1 - abs(cor(*)) >>>> instead of >>>> 1 - cor(*) >>>> >>>> >>>> On 5/11/2011 3:12 PM, john herbert wrote: >>>> >>>> Dear bioconductors, >>>>> >>>>>> From a google search, I found the following code that confuses me a >>>>>> >>>>> little. >>>>> As usual, it is probably something really elementary but reading around >>>>> does >>>>> not solve. >>>>> >>>>> The code was written by James Mcdonald ( >>>>> http://www.mail-archive.com/r-help at r-project.org/msg61514.html) and is >>>>> to >>>>> compute dendograms based on correlation and plot the results on a >>>>> heatmap >>>>> as >>>>> follows; >>>>> >>>>> a<- matrix(rnorm(50), ncol=5) >>>>> rowv<- as.dendrogram(hclust(as.dist(1-cor(t(a))))) >>>>> colv<- as.dendrogram(hclust(as.dist(1-cor(a)))) >>>>> heatmap.2(a, scale="row", Rowv=rowv, Colv=colv) >>>>> >>>>> >>>>> Why the *1*-cor(a)? >>>>> >>>>> >>>>> Orig.cor Adjusted cor >>>>> 1 0 >>>>> 0.9 0.1 >>>>> 0.8 0.2 >>>>> 0.7 0.3 >>>>> 0.6 0.4 >>>>> 0.5 0.5 >>>>> 0.4 0.6 >>>>> 0.3 0.7 >>>>> 0.2 0.8 >>>>> 0.1 0.9 >>>>> 0 1 >>>>> -0.1 1.1 >>>>> -0.2 1.2 >>>>> -0.3 1.3 >>>>> -0.4 1.4 >>>>> -0.5 1.5 >>>>> -0.6 1.6 >>>>> -0.7 1.7 >>>>> -0.8 1.8 >>>>> -0.9 1.9 >>>>> -1 2 >>>>> >>>>> >>>>> This removes negative numbers? What is the reason for doing this? >>>>> >>>>> [[alternative HTML version deleted]] >>>>> >>>>> _______________________________________________ >>>>> Bioconductor mailing list >>>>> Bioconductor at r-project.org >>>>> https://stat.ethz.ch/mailman/listinfo/bioconductor >>>>> Search the archives: >>>>> http://news.gmane.org/gmane.science.biology.informatics.conductor >>>>> >>>>> >>>> >>> [[alternative HTML version deleted]] >>> >>> _______________________________________________ >>> Bioconductor mailing list >>> Bioconductor at r-project.org >>> https://stat.ethz.ch/mailman/listinfo/bioconductor >>> Search the archives: >>> http://news.gmane.org/gmane.science.biology.informatics.conductor >>> >> >> -- >> James W. MacDonald, M.S. >> Biostatistician >> Douglas Lab >> University of Michigan >> Department of Human Genetics >> 5912 Buhl >> 1241 E. Catherine St. >> Ann Arbor MI 48109-5618 >> 734-615-7826 >> ********************************************************** >> Electronic Mail is not secure, may not be read every day, and should not >> be used for urgent or sensitive issues >> > >
ADD COMMENTlink written 7.7 years ago by john herbert560
Answer: Heatmaps and correlation
0
gravatar for john herbert
7.7 years ago by
john herbert560
john herbert560 wrote:
double mmmmmm, thanks for all interesting points of discussion. It is some very interesting points and very educational. My heatmaps, don't look very informative and judging naively by colour, they don't seem to cluster that well with the two methods I tried. I have a copy of the bioconductor case studies that has a good section on distances etc, I will try some of those too. PAM and Kmeans etc. On Wed, May 11, 2011 at 11:52 PM, john herbert <arraystruggles@gmail.com>wrote: > Producing some real heatmaps makes me wonder; > code; > > a = data > rowv<- as.dendrogram(hclust(as.dist(1-cor(t(a))))) > colv<- as.dendrogram(hclust(as.dist(1-cor(a)))) > # left map using correlation as a distance > heatmap.2(a, scale="row", Rowv=rowv, Colv=colv) > > x11() > # map on the right, using the default settings > heatmap.2(a, scale="row") > > The attached heatmaps don't look that great? The first correlation map > seems like it does not cluster correctly. A very basic look makes me think > the predominantly yellow/white columns should cluster. > > the second, the default, looks a little better. > > This data is Delta CT values, not array data. > > > sessionInfo() > R version 2.13.0 (2011-04-13) > Platform: i386-pc-mingw32/i386 (32-bit) > > locale: > [1] LC_COLLATE=English_United Kingdom.1252 LC_CTYPE=English_United > Kingdom.1252 LC_MONETARY=English_United Kingdom.1252 > [4] LC_NUMERIC=C LC_TIME=English_United > Kingdom.1252 > > attached base packages: > [1] grid stats graphics grDevices utils datasets methods > base > > other attached packages: > [1] HTqPCR_1.6.0 limma_3.8.1 RColorBrewer_1.0-2 Biobase_2.12.1 > gplots_2.8.0 caTools_1.11 > [7] bitops_1.0-4.1 gdata_2.8.2 gtools_2.6.2 > > loaded via a namespace (and not attached): > [1] affy_1.30.0 affyio_1.20.0 preprocessCore_1.14.0 > tools_2.13.0 > > > > > On Wed, May 11, 2011 at 11:27 PM, john herbert <arraystruggles@gmail.com>wrote: > >> mmmm, possibly. >> Correlation of gene expression = comparing all genes vs. all genes. >> >> 1) genes A and B, both highly up regulated = high pos correlation, e.g. >> 0.9 >> 2) genes C and D, both lowly down regulated = high pos correlation, e.g. >> 0.9 >> 3) gene E high up and gene F low down = high neg correlation, e.g. -0.9 >> 4) gene G low down and gene H high up = high neg correlation, e.g. -0.9 >> >> with 1-cor; genes A, B, C and D cluster together. >> genes E, F, G and H cluster together but a long way off from ABCD. >> >> with 1-abs(cor), all genes cluster together as they produce a extreme >> correlation, either pos or neg >> >> To me, 1-cor makes more biological sense, is that agreeable? >> >> Table. >> cor 1-cor 1-abs(cor) >> 1 0 0 >> 0.9 0.1 0.1 >> 0.8 0.2 0.2 >> 0.7 0.3 0.3 >> 0.6 0.4 0.4 >> 0.5 0.5 0.5 >> 0.4 0.6 0.6 >> 0.3 0.7 0.7 >> 0.2 0.8 0.8 >> 0.1 0.9 0.9 >> 0 1 1 >> -0.1 1.1 0.9 >> -0.2 1.2 0.8 >> -0.3 1.3 0.7 >> -0.4 1.4 0.6 >> -0.5 1.5 0.5 >> -0.6 1.6 0.4 >> -0.7 1.7 0.3 >> -0.8 1.8 0.2 >> -0.9 1.9 0.1 >> -1 2 0 >> >> On Wed, May 11, 2011 at 10:48 PM, James W. MacDonald < >> jmacdon@med.umich.edu> wrote: >> >>> >>> >>> On 5/11/2011 4:54 PM, john herbert wrote: >>> >>>> A biological circumstance? Interesting. Nothing immediately pops to >>>> mind. >>>> >>> >>> You see the pattern, but think about the underlying biology. If you use >>> 1-cor, as you note, two genes that are both highly up-regulated or both >>> highly down-regulated will cluster together. >>> >>> But what if the first gene's product has a negative feedback effect on >>> the transcription of the second gene? That implies a relationship that won't >>> be captured if you use 1-cor, but will be captured if you use 1-abs(cor). >>> This is, I believe, what Kevin was getting at. >>> >>> Best, >>> >>> Jim >>> >>> >>> >>> >>>> cor 1-cor 1-abs(cor) >>>> 1 0 0 >>>> 0.9 0.1 0.1 >>>> 0.8 0.2 0.2 >>>> 0.7 0.3 0.3 >>>> 0.6 0.4 0.4 >>>> 0.5 0.5 0.5 >>>> 0.4 0.6 0.6 >>>> 0.3 0.7 0.7 >>>> 0.2 0.8 0.8 >>>> 0.1 0.9 0.9 >>>> 0 1 1 >>>> -0.1 1.1 0.9 >>>> -0.2 1.2 0.8 >>>> -0.3 1.3 0.7 >>>> -0.4 1.4 0.6 >>>> -0.5 1.5 0.5 >>>> -0.6 1.6 0.4 >>>> -0.7 1.7 0.3 >>>> -0.8 1.8 0.2 >>>> -0.9 1.9 0.1 >>>> -1 2 0 >>>> >>>> looking at the numbers; anything that is highly correlated ,whether >>>> positively or negatively, will be close in distance. the heatmaps look >>>> different and obviously the clustering is different. >>>> >>>> Thanks. . >>>> >>>> On Wed, May 11, 2011 at 9:19 PM, Kevin R. Coombes< >>>> kevin.r.coombes@gmail.com >>>> >>>>> wrote: >>>>> >>>> >>>> This is not really a bioconductor question.... but >>>>> >>>>> You need something that behaves like a "distance". >>>>> >>>>> By the definition of what you mean by 'distance", two things are close >>>>> if >>>>> and only if the distance is near zero. The bigger (more positive) the >>>>> distance, the further apart things are. And you cannot measure >>>>> distances as >>>>> negative; only non-negative values need apply. >>>>> >>>>> Using 1-cor, you are taking two things to be close if the correlation >>>>> is >>>>> close to 1. >>>>> And the things that are furthest apart are the ones where the >>>>> correlation >>>>> is close to -1. >>>>> >>>>> As an exercise, you might want to think about circumstances where the >>>>> preferred code would be >>>>> 1 - abs(cor(*)) >>>>> instead of >>>>> 1 - cor(*) >>>>> >>>>> >>>>> On 5/11/2011 3:12 PM, john herbert wrote: >>>>> >>>>> Dear bioconductors, >>>>>> >>>>>>> From a google search, I found the following code that confuses me a >>>>>>> >>>>>> little. >>>>>> As usual, it is probably something really elementary but reading >>>>>> around >>>>>> does >>>>>> not solve. >>>>>> >>>>>> The code was written by James Mcdonald ( >>>>>> http://www.mail-archive.com/r-help@r-project.org/msg61514.html) and >>>>>> is to >>>>>> compute dendograms based on correlation and plot the results on a >>>>>> heatmap >>>>>> as >>>>>> follows; >>>>>> >>>>>> a<- matrix(rnorm(50), ncol=5) >>>>>> rowv<- as.dendrogram(hclust(as.dist(1-cor(t(a))))) >>>>>> colv<- as.dendrogram(hclust(as.dist(1-cor(a)))) >>>>>> heatmap.2(a, scale="row", Rowv=rowv, Colv=colv) >>>>>> >>>>>> >>>>>> Why the *1*-cor(a)? >>>>>> >>>>>> >>>>>> Orig.cor Adjusted cor >>>>>> 1 0 >>>>>> 0.9 0.1 >>>>>> 0.8 0.2 >>>>>> 0.7 0.3 >>>>>> 0.6 0.4 >>>>>> 0.5 0.5 >>>>>> 0.4 0.6 >>>>>> 0.3 0.7 >>>>>> 0.2 0.8 >>>>>> 0.1 0.9 >>>>>> 0 1 >>>>>> -0.1 1.1 >>>>>> -0.2 1.2 >>>>>> -0.3 1.3 >>>>>> -0.4 1.4 >>>>>> -0.5 1.5 >>>>>> -0.6 1.6 >>>>>> -0.7 1.7 >>>>>> -0.8 1.8 >>>>>> -0.9 1.9 >>>>>> -1 2 >>>>>> >>>>>> >>>>>> This removes negative numbers? What is the reason for doing this? >>>>>> >>>>>> [[alternative HTML version deleted]] >>>>>> >>>>>> _______________________________________________ >>>>>> Bioconductor mailing list >>>>>> Bioconductor@r-project.org >>>>>> https://stat.ethz.ch/mailman/listinfo/bioconductor >>>>>> Search the archives: >>>>>> http://news.gmane.org/gmane.science.biology.informatics.conductor >>>>>> >>>>>> >>>>> >>>> [[alternative HTML version deleted]] >>>> >>>> _______________________________________________ >>>> Bioconductor mailing list >>>> Bioconductor@r-project.org >>>> https://stat.ethz.ch/mailman/listinfo/bioconductor >>>> Search the archives: >>>> http://news.gmane.org/gmane.science.biology.informatics.conductor >>>> >>> >>> -- >>> James W. MacDonald, M.S. >>> Biostatistician >>> Douglas Lab >>> University of Michigan >>> Department of Human Genetics >>> 5912 Buhl >>> 1241 E. Catherine St. >>> Ann Arbor MI 48109-5618 >>> 734-615-7826 >>> ********************************************************** >>> Electronic Mail is not secure, may not be read every day, and should not >>> be used for urgent or sensitive issues >>> >> >> > [[alternative HTML version deleted]]
ADD COMMENTlink written 7.7 years ago by john herbert560
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