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Microarray Analysis of Variance MANAVA

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Microarray Analysis of Variance MANAVA Powered By Docstoc
					   Microarray Analysis of Variance (MANAVA) – the search for a
          general recipe to detect differentiations in gene expression


                                         Robin D.K. Christensen
                                          Copenhagen, July 2001
                     The Royal Veterinary and Agricultural University, Denmark


                                               ABSTRACT


Microarrays is a new class of biotechnologies,            logarithmic mean of the samples: ½[loge(Heart) +
which allow the monitoring of expression levels           loge(IBEC)]. Using the Log-Ratio as response
for thousands of genes simultaneously. The                variable and the Log-Mean as a covariate in a
response from these types of analysis are mostly          linear regression, we adjust (normalize) the
non parametric, which allow no distribution to the        response variable by subtracting the product of the
following statistical analysis. There are many            slope and Log-Mean, by Slides (equals block in a
sources of systematic variation in microarray             block design), in an iterative process. When
experiments, which affect the measured gene               steady state is obtained, the final two way
expression levels. To ensure a valid statistical          ANOVA is expressed as: Ygs -jsfj(xgs) = g + s
analysis we have to adjust for these contributions,       + gs followed by calculation of confidence limits
the process of removing and describing such               to the Log-Ratios ( t1-/2(MSe)). The level of
variations is called normalization. In the present        significance was evaluated by Sidáks formula;
study I have evaluated and analysed the data from         p<0.000023. From these procedures I found 115
a partly visual method, plotting the logarithm of         genes with a significant different degree of
the ratio, between two samples; Heart mRNA                expression, between Heart mRNA and IBEC
(sample of interest) and IBEC mRNA (a                     mRNA.
reference): loge[(Heart/IBEC)]        versus the


                                              1. Introduction

The purpose of this "project-thesis" is to study and practice normalization of a large matrix of data,
from investigation of multiple genes, and to determine which genes that show differences in
expression between an unknown and a "control" sample. In this case I have used a set of data with
2208 genes in a block design, with all the 2208 genes represented in each of the 4 blocks (referred
to as slides), with one channel as the "active" (unknown) sample and the other as a control. The data
is supplied from a study of Scheidl et al. (2001) via Mats Rudemo, Professor in Bio Statistics at the
Royal Danish University of Veterinary- and Agricultural Sciences.
The goal of the study is to define a method by which data (gene material) from microarray
analysis can be analysed and discussed despite the residuals, hopefully without making any Type II




                                                      1
Error misinterpretations (= approving the hypothesis of no differences within any of the genes, on a
false assumption). But, do to the large number of multiple comparisons, a Type I Error
misinterpretation (= rejecting the hypothesis on a false assumption), is probably an even bigger
hazard, on the other hand.
One specific distribution for these types of microarray analysis is not likely to be found, which is
why I can't make any unambiguous distribution assumptions, but via iterative methods I will try to
normalize and reduce the different contributions from e.g. the 4 blocks (slides).




                         2.1 Background concerning Microarray analysis


The relevance for valid statistical/objective analysing tools in molecular biology, when measuring
e.g. DNA sequences, is a subject of increasing interest.
In 1975, Southern developed a technique that made the separation and identification of nucleotides
possible, by separating a biological sample into a mobile and immobile phase. If one of these phases
contains a known DNA sequence, the other can be identified by a relevant interpolation. These
methods can be used in either a qualitative or a quantitative analysis, if a well-known DNA (cDNA)
is available as a standard.


In 1995, Schena et al. developed a technique for quantitative monitoring of gene expression
patterns, by using cDNA “Microarray”, a technique, which typically results in data consisting of
two-dimensional arrays, with genes in one dimension and experimental conditions in the other
dimension (Rudemo, 2001).
The basics of this technique:
The immobile phase consist of single stranded pieces of known DNA ( cDNA), which is attached
to a nylon filter, glass slide or silicon chip. The unknown mobile phase is a mixture of labelled
copies of mRNA. When the hybridisation is over, the label is localised on the spots where the
unknown phase matches the known phase.


It is a big advantage of this method, that an enormous amount of test sequences can be arrayed in a
single experiment. Though the velocity at which results can be obtained with this technique is a
major advantage, there are a few drawbacks: The error of quantitation can reach levels of 30 to




                                                  2
50%, whereas this is 15 to 20% in the traditional methods (e.g. Northern blot or quantitative PCR)
(Dutilh, 1999).
It is important for a potential user of the DNA-chip method, to remember that only known genes
can be analysed using this Microarray technique, because a reference is needed.
Because of the complexity of the topic, and the ease by which a large numbers of genes can be
monitored with this “new” method, a need for many liable databases is inevitable. See e.g.
http://www-binf.bio.uu.nl/~dutilh/gene-networks these kinds of analysis, makes a computational
evaluation of results (conclusions etc.) recommendable. Because of the many sources of systematic
variation in Microarray experiments, normalization is used to “remove” such variation, which is in
analogy to parametric statistics (Gauss’/Normal- distribution), where transformation of the response
variables are preferable, instead of using Non-Parametric statistics. For more information about
normalization see Yang et al. (www.berkeley.EDU/users/terry)
The way the expression level is obtained, is typically by the logarithm of the quotient between the
two intensities in the colours e.g. RED (target-sample) and GREEN (reference), and this is often
finally multiplied by e.g. 2, and given in the “final” matrix of results, without any decimals
(Rudemo, 2001).


In the present thesis I have used the results from a study by Scheidl et al. (2001), in which they
presented and evaluated an experimental procedure for ‘global gene expression analysis of slender
embryonic structures using laser microbeam microdissection and laser pressure catapulting’.
As described earlier, the purpose of this study haven’t been to examine the problem in a biological
sense, but to investigate the possibilities to determine which genes that are expressed differently,
from a large gene sample (2208 different genes) one “unknown” (the sample of interest) compared
to the “known” (the sample used as a reference).
In the following: The sample of interest is referred to as Heart (from mRNA) and the “reference” is
referred to as IBEC (also descended from mRNA).




                                                   3
 2.2 How do we normalize and analyse our response’ variables, to obtain a “true” picture of
                      the differences between the genes that are expressed


In Microarray experiments there are many sources of systematic variation, which cannot be,
neglected because of the huge possibility of making either a result without proper meaning, or
extracting a false conclusion, due to a Type I Error. The process of reducing (and possibly
removing) these variations is termed normalization.
We describe a within/between -slide normalization approach, which makes a stepwise reduction of
the response variable. When given data that needs normalization, I speculate a method for the
identifying of single differentially expressed genes, by using a two way analysis of variance
(ANOVA), from which we adjust (normalize) our response variables by using linear regression to
calculate the coefficient of slope from an assumed linear distribution. These two steps can then be
used repeatable, to adjust and normalize the response variables by iteration.
The approach of using proper normalization should make it possible to consider and analyse data by
univariate testing for each gene (Dudoit et al., 2000), indicating the use of methods known from e.g.

the Gauss’ distribution N(0, 2) - assuming varians homogeneity and E{ij}= 0.


A model for Microarray data with two treatments:
Suppose that we have Microarray data from S slides, denoted s = 1,….., S. For each slide there are
two treatments t = 1, 2
Let                        Zgts , g = 1,…….., G,       t = 1, 2      s = 1,……., S


Denote the observed intensity value for the spot corresponding to gene g and treatment t in slide s,
where G is the number of genes. We assume here that each spot corresponds to one gene.
Put
                           Ygs = loge(Zg1s) - loge(Zg2s)                          (referred to as M)
And
                           Xgs = ½ (loge(Zg1s) + loge(Zg2s))                      (referred to as A)


We will regard the following model:
                                               J

(Model A )                 Ygs = g + s +    js fj(xgs)   + gs



                                                   4
where gs, g = 1,…….., G s = 1,…….., S are independent with E(gs) = 0 and var(gs) = gs2
We assume that:
                                            S

                                            s       =0


The functions fj, j = 1, …….., J, are assumed to be known and to satisfy:
                                            S   G

                                            fj(xgs)      =0                       j = 1, ………, J

As an example we could let fj(x) be a polynomial of degree:             j

In the following we assume j = 1  linear, and (maybe an extreme case) that we have only one
variance.

From the linear regression (fj(xgs)) we want a slope (s) for each of the slides (s = 1, …, S) by which
we can normalize by subtracting this estimate from Ygs 
                                    J

(Model A’ )                 Ygs -   js fj(xgs)    = g + s + gs



Because of the way all the response variables are calculated (Ygs = loge(Zg1s) - loge(Zg2s)), a
significant difference from 0, could (‘would’) be an indication of a difference in the expression of
the Geneg
After a suitable number of iterations, we calculate confidence limits to every each of the genes
(1,…..,2208) on the basis of their means, and the MSe


We want to test the hypothesis              H0 : loge[ZgHeart/ZgIBEC ]  0


                           - CL1-/2 < Yg  t1-/2 (MSe) < CL1-/2


Because of the large amount of comparisons between Heart (mRNA) and IBEC (mRNA), it is
necessary to adjust the p-values (because of the multiple testing), to reduce the risk of making a
Type I Error.




                                                       5
There are two simple and slightly different methods to adjust the p-values:
- The Bonferroni adjustment:                   Padjusted= kPcalculated
- The Sidák adjustment:                        Padjusted= 1 – (1 – Pcalculated)k
                                               Where the letter k = the number of genes tested.


I have chosen to use Sidák’s adjusted P-value, for multiple comparisons.




              3. Results from the statistical analysis of the Microarray data, using SAS


After sorting of data from the study by Scheidl et al. (2001), I made a program in SAS to display the
outputs of all the 4 slides (249, 286, 287 and 346) without any adjustments. I made scatter plots of:
                M = loge(Zg1s) - loge(Zg2s) versus              A = ½ (loge(Zg1s) + loge(Zg2s))
- note that g indicates the number of one specific gene (g = 1, ….., 2208) and the treatment 1 or 2 (1
= Heart mRNA & 2 = IBEC mRNA). As shown in figure 1 there is four scatter plots, one for each
slides


Fig. 1 Scatter plots of M versus A, for unadjusted data. The figure show (from left to the right): s = 249, 286,
287 and 346. (n = 2208 Genes).




To define the estimates of the Log-Ratio’s from a two way ANOVA (as mentioned in section 2.2) I
made the following SAS-editor. Because of the huge amounts of data (if build as a Matrix) I chose
to use an ANOVA statement, which is a valid method when the design is balanced and complete,
instead of the more obvious ‘Generalised Linear model’ (proc GLM).


                /*The following is the SAS program used:*/


data data1 ;
input Slide       Gene      MeanHe       MeanIB       RatioHI       M       A      ;
cards;

249      39       295.91      495.01    0.597785903    -0.51452261    5.947316659
249      40       59.9         78.03       0.767653467       -0.264416863      4.224884937




                                                         6
(There are 8832 lines in the data editor, one for each gene, times 4)
346       635679     2566.06         2595.57        0.988630628          -0.011434497       7.855844176
346       635726     503.88         1426.14        0.353317346          -1.040388628       6.742532459
;
proc anova data=data1;
class slide gene ;
model M = slide gene ;
means slide gene ;
run;
proc reg;
by slide ;
model M = A ;
run;




               /*The following is parts of the SAS Output:*/

                                    Analysis of Variance Procedure

Dependent Variable: M

Source                      DF          Sum of Squares                  Mean Square     F Value       Pr > F

Model                   2210             7426.21031233                   3.36027616       19.58       0.0001

Error                   6621             1136.07842842                   0.17158714

Corrected Total         8831             8562.28874075

                   R-Square                          C.V.                  Root MSE                   M Mean

                   0.867316                     -313.1447                 0.41423078              -0.13228093


Source                      DF                  Anova SS                Mean Square     F Value        Pr > F

SLIDE                    3            4696.80875487           1565.60291829   9124.24     0.0001
GENE                  2207            2729.40155747              1.23670211      7.21     0.0001
------------------------------------------ SLIDE=249 -------------------------------------------

Model: MODEL1
Dependent Variable: M

                                         Analysis of Variance

                                             Sum of             Mean
                  Source           DF       Squares           Square       F Value         Prob>F

                  Model             1      0.84394          0.84394         1.662         0.1974
                  Error          2206   1119.87844          0.50765
                  C Total        2207   1120.72238

                      Root MSE        0.71250        R-square          0.0008
                      Dep Mean       -0.25918        Adj R-sq          0.0003
                      C.V.         -274.90481

                                         Parameter Estimates

                                  Parameter        Standard        T for H0:
               Variable     DF     Estimate           Error       Parameter=0   Prob > |T|

               INTERCEP      1    -0.333268      0.05942840            -5.608          0.0001
               A             1     0.014724      0.01141940             1.289          0.1974




                                                              7
------------------------------------------ SLIDE=286 -------------------------------------------

Model: MODEL1
Dependent Variable: M

                                       Analysis of Variance

                                         Sum of            Mean
                Source          DF      Squares          Square         F Value      Prob>F

                Model             1    217.38230      217.38230         645.239      0.0001
                Error          2206    743.20587        0.33690
                C Total        2207    960.58817

                    Root MSE        0.58043        R-square         0.2263
                    Dep Mean       -0.32170        Adj R-sq         0.2260
                    C.V.         -180.42542

                                       Parameter Estimates

                                Parameter       Standard        T for H0:
              Variable    DF     Estimate          Error       Parameter=0   Prob > |T|

              INTERCEP     1    -1.619827     0.05257581           -30.809        0.0001
              A            1     0.220763     0.00869093            25.402        0.0001




------------------------------------------ SLIDE=287 -------------------------------------------

Model: MODEL1
Dependent Variable: M

                                       Analysis of Variance

                                         Sum of            Mean
                Source          DF      Squares          Square         F Value      Prob>F

                Model             1    233.84467      233.84467         562.351       0.0001
                Error          2206    917.32905        0.41583
                C Total        2207   1151.17372

                    Root MSE       0.64485         R-square         0.2031
                    Dep Mean       1.03215         Adj R-sq         0.2028
                    C.V.          62.47651

                                       Parameter Estimates

                                Parameter       Standard        T for H0:
              Variable    DF     Estimate          Error       Parameter=0    Prob > |T|

              INTERCEP     1     2.417441     0.06000700            40.286        0.0001
              A            1    -0.225042     0.00948987           -23.714        0.0001


------------------------------------------ SLIDE=346 -------------------------------------------

Model: MODEL1
Dependent Variable: M

                                       Analysis of Variance

                                         Sum of            Mean
                Source          DF      Squares          Square         F Value       Prob>F




                                                           8
                  Model             1     70.00958          70.00958       274.325        0.0001
                  Error          2206    562.98614           0.25521
                  C Total        2207    632.99572

                      Root MSE        0.50518        R-square          0.1106
                      Dep Mean       -0.98039        Adj R-sq          0.1102
                      C.V.          -51.52833

                                         Parameter Estimates

                                  Parameter       Standard         T for H0:
              Variable      DF     Estimate          Error        Parameter=0    Prob > |T|

              INTERCEP       1    -2.401480     0.08647113             -27.772         0.0001
              A              1     0.195344     0.01179415              16.563         0.0001




These output statements for the 4 slides (249, 286, 287 and 346) are then used to normalize our
response variables, which is done by generating a new response variable, M’ by using (Model A’ )
                                 Slide 249:     M’ = M – 0.014724*A ;
                                 Slide 286:     M’ = M – 0.220763*A ;
                                 Slide 287:     M’ = M + 0.225042*A ;
                                 Slide 346:     M’ = M – 0.195344*A ;


From the new variable M’, the same procedure repeats (iterates) until steady state is reached.
This step is followed by a new cycle (with two way ANOVA etc.), from which it is possible to
examine the MSe to generate confidence limits.


Note: In the present case I have only run the procedure a single time, although it isn’t optimal
conditions!




              /*The following is parts of the 2nd SAS Output:*/

                                    Analysis of Variance Procedure

Dependent Variable: M’

Source                      DF          Sum of Squares                   Mean Square    F Value    Pr > F

Model                  2210             32292.58289147                   14.61202846     147.59    0.0001

Error                  6621               655.49326130                    0.09900215

Corrected Total        8831             32948.07615277

                   R-Square                          C.V.                   Root MSE               M’ Mean




                                                              9
                   0.980105                       -64.97132              0.31464608              -0.48428460


Source                  DF                        Anova SS             Mean Square    F Value        Pr > F

SLIDE                    3               29604.67664758               9868.22554919   99676.88       0.0001
GENE                  2207                2687.90624388                  1.21790043      12.30       0.0001




From this SAS output, we are able to create a confidence interval on both sides (absolute value) of
the mean value of every single gene.


We recall that: We want to test the hypothesis,                    H0 : loge[ZgHeart/ZgIBEC ]  0
From Sidák’s adjustment (P < 0.05) we find that Padjusted < 0.000023
which is equal to a t-value of 4.235; DFe= 6621


                                - CL1-/2 < Yg  t1-/2 (MSe) < CL1-/2
                                                   4.235 (0.099)
                                                   4.2350.3146
                                Loge(Zg(Heart)/Zg(IBEC))  1.33


    -    are considered significant, which indicate that the gene (g = 1, ….., 2208) is expressed
         different for Heart and IBEC !


I found 115 genes that are expressed significantly different between Heart- and IBEC mRNA. The
codes of those genes are listed below.

                              Level of           --------------M’-------------
                              GENE       N           Mean              SD
                              96         4        -1.77706561       1.73504983
                              118        4        -1.42050163       1.86470806
                              120        4        -2.26124846       1.71616866
                              334        4         2.45132825       2.31903167
                              359        4        -1.71602782       2.04587237
                              388        4        -1.45170947       1.93259376
                              444        4        -1.71665568       2.02282346
                              454        4        -1.40136929       2.38519972
                              476        4        -2.50204893       1.72009338
                              483        4         2.38481770       2.53358889
                              577        4        -1.42956460       1.79847840
                              578        4        -1.40729397       2.39270125
                              615        4        -2.04795280       1.68799200
                              633        4        -1.50352218       2.25108857
                              678        4        -2.37132493       2.14877464
                              679        4        -1.52520470       2.25786694
                              678        4        -2.37132493       2.14877464




                                                              10
679    4   -1.52520470        2.25786694
686    4   -1.44179038        2.42870911
709    4   -1.58019562        2.28518023
711    4   -1.51502202        2.12457047
723    4   -1.34480726        2.41626528
739    4   -1.35949234        1.92563179
759    4   -1.57488816        2.01243322
770    4   -2.32499648        1.96242649
794    4   -1.43985042        2.45410611
799    4   -1.48797712        2.42778473
809    4   -1.33434804        1.70056162
855    4   -2.10995666        1.73030837
871    4   -1.61398086        2.28944950
885    4    2.41969034        2.32293005
887    4   -2.50112000        1.84981440
933    4    2.26010378        2.24350151
958    4    1.92778691        2.39551563
959    4   -1.62086664        2.26731640
967    4   -1.74608039        2.07221323
987    4   -1.47042080        2.46540126
1031   4   -2.39563869        1.82980338
1034   4   -1.46248592        2.40007732
1067   4    2.08096289        2.67522106
1099   4    2.44351166        2.28541313
1104   4   -1.65684651        2.09215447
1108   4    1.70954915        2.53606826
1125   4   -2.04283789        1.65232604
1128   4   -1.40847924        2.01101305
1147   4    1.72347634        2.44664088
1151   4   -1.34697217        1.90574357
1179   4   -1.91065343        2.17633976
1183   4    2.17431987        2.58022198
1187   4   -1.45402919        2.03550297
1188   4   -1.55378624        2.48806810
1189   4   -1.37586213        2.12068623
1307   4   -1.68733341        2.19122401
1339   4   -1.42647236        2.22604891
1352   4   -2.38261710        2.20000304
1413   4   -1.88232954        1.86929083
1433   4   -2.19352001        1.73566700
1463   4   -2.18860179        1.99317671
1515   4   -1.70990896        2.33093347
1517   4   -1.41773078        2.52510968
1524   4   -1.47733504        2.52347329
1529   4   -1.49298790        2.17291139
1553   4    2.50489576        2.32930453
1710   4    2.14336801        2.60169161
1728   4    2.37921286        2.58530001
1760   4    2.11736703        2.44116018
1779   4   -1.62847674        2.44949063
1807   4    2.30083460        2.28785772
1811   4   -1.39042424        1.90765922
1833   4   -1.69608750        1.68554939
1841   4   -1.57271217        2.43594157
1851   4   -2.59788936        1.94429717
1861   4   -1.98104773        2.45878339
1869   4    2.08103173        2.57000343
1882   4   -1.34556324        1.91221477
1902   4    2.30358993        2.33281576
1909   4    2.33876502        2.57121868
1911   4    2.34263121        2.45076489
1940   4    2.38394178        2.45381613
2055   4   -1.51223441        2.33356717
2063   4    1.53859244        2.13995288
2066   4   -1.39350661        2.42007881
2072   4    2.18627622        2.60723914
2087   4   -1.34288358        1.69906235
2094   4    1.98553889        2.77823313
2103   4   -1.69560531        1.96589747
2105   4   -1.56933925        1.98501220
2198   4    2.07518916        2.75892279




                         11
                        2200       4     -1.51264190        2.39964722
                        2202       4     -1.50012546        2.08198123
                        2205       4     -1.53663403        2.40720351
                        2348       4      2.31418773        2.45148774
                        2361       4     -1.46131749        2.37670218
                        2366       4      2.33003389        2.72712673
                        2370       4     -1.36629538        2.27619187
                        2371       4      1.85604930        2.58134105
                        2373       4     -1.59193834        1.87400389
                        2398       4     -2.39354930        2.07290769
                        313981     4      2.54179171        2.43782196
                        316082     4      2.19755988        2.46187709
                        333633     4     -1.40968211        2.50381096
                        334438     4     -1.52660832        2.20936398
                        367062     4     -1.69800726        1.89878641
                        481408     4      2.49329173        2.41310003
                        482270     4      2.11229665        2.48069470
                        483232     4      1.80929860        2.40772366
                        517625     4      1.38981251        2.13102859
                        533961     4     -2.24291503        2.13020547
                        534143     4      1.50963798        2.31931527
                        535688     4     -2.09953625        2.30368923
                        537504     4     -1.54040353        1.73944632
                        618431     4      1.85063707        2.20984390
                        634838     4     -1.88573912        2.12259336
                        635173     4     -1.67097214        1.78623670
                        635508     4     -2.04690098        1.91969363




                                              4. Discussion


 The normalization process for microarray analysis is a complex affair because of the large
diversity   between      the     genes    and     the       lack   of    distribution.   Yang   et   al.
(www.berkeley.EDU/users/terry) discussed several methods to describe and remove systematic
variation in microarray experiments. Variation which otherwise can affect the measured gene
expression levels. Like Yang et al., Dudoit et al. (2000) and Scheidl et al. (2001) I found that
presenting data in a scatter plot, provides a good information if the graph is presented as Log-Ratio
versus the “Log-Mean”, because by using these plots a stepwise reduction of the systematic
variation seems less abstract.
 Unlike the most other authors contributing to this kind of research, I haven’t used the adjusted p-
values from permutation or any other non-parametric methods (e.g. p-values generated from ranks)
instead I have used an iterative method in which the systematic variation is attempted removed,
while improving the variance homogeneity and thereby hopefully the power of the design.
 Due to the multiple comparisons of genes, an “ordinary p-value” (e.g. at 5%) is obviously not
valid, because of the probability of making type I Errors. Instead an adjustment has to be done to
the level of significance. In this study I have used the Sidák adjustment due to the simplicity,
although “The Westfall and Young step-down adjusted p-values” (Dudoit et al., 2000) probably are



                                                       12
more correct. I found the level of significance to be P < 0.000023 and the degrees of freedom, DF e=
6621.
  Whether I have found the same genes to be differentially expressed as Scheidl et al. (2001) – I
don’t know, because the data I got was given in coded names and I couldn’t find any “nomenclature
key”.
  One thing is for certain; the discipline of using statistics and other calculating tools in microarray
analysis, needs further work e.g. via integration to some advanced biochemistry courses, so that the
complexity of the analysis becomes more “hidden” before clinicians (e.g. oncologists) and other
non-statisticians “dare” using these methods on their own.
Maybe a total MANAVA (Microarray Analysis of Variance) package can be generated to software
like SAS, SPSS, Sigma Stat etc. in the future !!!


                                                   References
From the World Wide Web:
                               www-stat.Stanford.edu/~hastie/Papers/
                               www.stat.Berkeley.EDU/users/terry/zarray/HTML/index.html


Dudoit, S. Yang, YH. Callow, MJ. and Speed, TP. “Statistical methods for identifying differentially expressed genes
in replicated cDNA microarray experiments”, Technical report #578, 2000.


B. Dutilh. “Analysis of data from Microarray experiments, the state of the art in gene network reconstruction”. From
http://www-binf.bio.uu.nl/~dutilh/gene-networks, 1999.


M. Rudemo. “Excerpts from Image Analysis”. Unpublished; Department of Mathematical Statistics, Chalmers
University of Technology, 1-3, 2001.


Scheidel, SJ. Nilsson, S. Kalén, M. Hellström, M. Takemoto, M. Håkansson, J. and Lindahl, P.
“mRNA Expression Profiling of Laser Microbeam Microdissected Cells from embryonic structures”. Submitted, 2001.


M. Schena, D. Shalon, R.W. Davis, and P.O. Brown. “Quantitative monitoring of gene expression patterns with a
complementary DNA Microarray”. Science, 270:467-470, 1995.


E. Southern. “Detection of specific sequences among DNA fragments separated by gel electrophoresis.” J. Mol. Biol.,
98:503-517, 1975.


Yang, YH. Dudoit, S. Luu, P. and Speed, TP. “Normalization for cDNA Microarray Data”.




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Submitted, 2001.




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