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					ASCA: analysis of multivariate data
  from an experimental design,



   Biosystems Data Analysis group
     Universiteit van Amsterdam
                  Contents
•   ANOVA
•   SCA
•   ASCA
•   Conclusions
                     ANOVA
  • different design factors contribute to the
    variation


For two treatments A and B the total sum of
squares can be split into several contributions


       SStotal  SS A  SS B  SS A B  SS within
x cdq  μ q  α cq  β dq αβ cdq
                                Example

Experiment:
Rats       are given Bromobenzene    that affects the liver


Measurements: NMR spectroscopy of urine
                                                         Rats

Experimental Design:                      6 hours

                                          24 hours
 Time: 6, 24 and 48 hours                 48 hours

Groups: 3 doses of BB                                                                    3.0275




 Vehicle group, Control group                                                                  2.055
                                                                              5.38       3.285
                                                                                         3.0475
 Animals: 3 rats per dose per time                                                    3.675
                                                                                     3.7525
                                                                                          2.7175
                                                                                             2.075
                                                                                          2.93
 point
                                                              10     8    6          4       2         0
                                                                   chemical shift (ppm)
               NMR Spectroscopy
0.7
                                             3.0275

                                                                     -   Each type of H-atom
0.6                                                                      has a specific Chemical
                                                                         shift
0.5
                                                                     -   The peak height is
                                                                         number of H-atoms at
0.4
                                                                         this chemical shift =
                                                         2.055           metabolite
0.3                   5.38              3.285
                                                                         concentration
                                             3.0475

                                                                     -   NMR measures
0.2
                                     3.675
                                   3.7525
                                                                         ‘concentrations’ of
                                                2.7175
                                                         2.075           different types of H-
0.1                                            2.93
                                                                         atoms

  0
   10   8       6             4                          2       0
            chemical shift (ppm)
                                          Different contributions
                                  Experimental Design
                                                                                                              Time
                            4

                           3.5                                         0   0.2   0.4          0.6   0.8   1
                                                                                       time
Metabolite concentration




                            3

                           2.5

                            2                                                                                 Dose
                           1.5

                            1
                                                                       0   0.2   0.4          0.6   0.8   1
                           0.5                                                         time

                            0

                           -0.5
                              0     0.2   0.4          0.6   0.8   1
                                                time
                                                                                                               Animal

                                   Trajectories                        0   0.2   0.4 time 0.6       0.8   1
                                                                                      Symbol   Meaning

                                                                                      k        Time

    The Method I: ANOVA                                                               h        Dose group

                                                                                      ih       Individual


     xhki     k   hk   hki
                  hk                                                         hk
                                                                                      xhih k   Data




    Estimates of these factors:

        xhkihk  x...  x..k  x...   xh.k  x..k   xhkihk  xh.k                                    
                                                                Constraints:
           0
                                                          
                                                          k
                                                                   k   0
0         0.2          0.4   time   0.6     0.8       1

                  0
                                                            
                                                           h
                                                                        hk   0
0         0.2          0.4   time   0.6     0.8       1

             0

    0           0.2     0.4 time 0.6      0.8     1
                                                            
                                                          ihk
                                                                         hkihk    0
                                  The Method II
ANOVA is a Univariate technique
         xhkihk     k   hk   hkihk
                                                    3.0275




                                                      2.055
                                              5.38 3.285
                                                   3.0475
                                                              For all values in
                                                 3.675
                                                3.7525
                                                   2.7175
                                                      2.075
                                                    2.93
                                                              the ANOVA
xhih k                                                        equation
                                                              e.g.:
 x                                            X
                                                              αk  X α
            MATRICES:
                                                                    Structured !
             X  1m  X α  X αβ  X αβγ
                            T


                 2              T 2                     2     2          2
             X  1m                     X α  X αβ  X αβγ
                        Multivariate Data
                                          NMR Spectroscopy

 0.7                                                            0.04
                            3.0275


 0.6                                                            0.03

 0.5
                                                                0.02




                                                     6.01 ppm
 0.4
                                     2.055                      0.01
 0.3             5.38      3.285
                            3.0475
                                                                  0
 0.2
                         3.675
                        3.7525
                                 2.7175
                                 2.93
                                      2.075                     -0.01
 0.1

  0                                                             -0.02
  10   8     6       4        2               0                      -0.2 -0.1   0   0.1 0.2 0.3   0.4   0.5
                                                                                      2.05 ppm
           chemical shift (ppm)

Or:
Relationship                                                      Covariance between the
between                   X                                       variables
the columns of
X
                  The Method III: Principal Component
                               Analysis
                                                                                               Loading PC 1
                                                                                               Loading PC 2
      3

     2.5    Loading PC 2                   Loading PC 1
      2

     1.5                                                                             X
x3




      1

     0.5
                                                                Scores
      0
      1
                                                       1               0.6
                  0.5
                                         0.5                           0.4
             x2             0   0
                                               x1                      0.2


       X  1m  TP  E  T            T              residuals   PC 2
                                                                   -0.2
                                                                        0

                        scores      loadings
                                                                   -0.4

           3D  2D … Imagine!                                      -0.6

           350D  2D !!!                                           -0.8-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
                                                                                          PC 1
The Method IV: ANOVA and PCA  ASCA

  X  1m  Xα  Xαβ  Xαβγ
           T                                     Column spaces
                                                 are
                                                 Orthogonal


                       Pα         Pαβ          Pαβγ

       X                                              E
                  Tα        Tαβ         Tαβγ
                                                       Parts of the
                                                       data not
                                                       explained by
                                                       the
                                                       component
      X  1mT  TαPα  TαβPαβ  TαβγPαβγ  E
                   T       T         T
                                                       models
              In Words:
• ASCA models the different contributions
  to the variation in the data
• ASCA takes the covariance between
  the variables into account
• ASCA gives a solution for the problem
  at hand.
                Results I


                             0.5                          control
                                                          vehicle
                                                          low
      Xαβγ                   0.4
                                                          medium
Xα                                                        high
                             0.3
                                        αβ -scores
     Xαβ            Scores   0.2

                             0.1


             40 %              0

                             -0.1

                             -0.2


                                    6       24                      48
                                           Time (Hours)
                                   Results II



         0.5                         control
                                     vehicle
                                                    • Quantitative
         0.4                         low
                                     medium
                                     high
                                                      effect!
         0.3


         0.2
                                                    • No effect of
Scores




         0.1                                          vehicle
           0
                                                    • Scores are in
         -0.1
                                                      agreement with
         -0.2
                                                      visual inspection
                6    24                        48
                    Time (Hours)
               Results III  biomarkers
                                          3.0475
                        5.38


                                 3.7525
                                  3.675
                                                                   Unique to the α submodel

α
                                                                   Differences
                               3.9675        2.735
                                                       2.055       between submodels
                                                 2.5425


                                              2.5825
                                             2.6975
                                                     2.055
                                                                   Interesting for Biology

                                                     2.075
                                                                   Interesting for
                                            2.91                   Diagnostics
αβ
                                            2.93
                                          3.0275



                               3.9675        2.735
                                             2.6975
                                              2.5825


                                        3.285
                                        3.2625


                                                     2.075

αβγ
                                            2.93
                                          3.0475     2.055
                                   3.73
                                3.8875



                                             2.735
                                          3.0275




                                        3.285

      10   8       6            4                    2         0
                chemical shift (ppm)
            Conclusions
• Metabolomics (and other –omics)
  techniques give multivariate datasets
  with an underlying experimental design
• For this type of data, ASCA can be used
• The results observed for this experiment
  are in accordance with clinical
  observations
• The metabolites that are responsible for
  this variation can be found using ASCA
   BIOMARKERS
             Discussion
1. How can I perform statistics on the
   ASCA model? (e.g. Significance
   testing)
2. Are there other constraints possible for
   this model? (e.g. stochastic
   independence)
3. Are there alternative methods for
   solving this problem?

				
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posted:10/24/2012
language:Latin
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