1. Exploring Microarray data
Javier Cabrera

2. Outline Exploratory Analysis Steps.
Microarray Data as Multivariate Data.
Dimension Reduction
Correlation Matrix
Principal components Geometrical Interpretation
Linear Algebra basics
How many principal componets
Biplots
Other graphical software for EDA : Ggobi

3. Process
Assume the data has gone the QC process, normalization, outlier detection. At this point we are using an have an exprSet:
Array of expressions (rows are genes, columns are samples)
Select Gene Subset by one of the methods. This will bring you down to some small subset of genes (hundred’s or less)
Use PCA to further reduce the dimension : from 1 to 10’s
Apply data analysis method: biplot, clustering, classification, mds

4. Microarray data as Multivariate Data
Microarray Data: Gene expression Matrix = Gxp matrix
Genes are the variables =>
G >p Many more Variables than Samples
This makes microarray data very different the data that is found in other applications.
Most Multivariate analysis methods rely on more observations than variables : G < p this means that the
Standard multivariate methods must be reexamined.
Dimension reduction becomes very important and requires:
Gene subset selection.
Principal Components for further dimension reduction

5. Dimension Reduction: gene subset selection
Use sample grouping:
Response: Calculate the F-statistic for each individual gene and select those genes with the highest F-value.
Group of genes related to some pathway.
Correlated subsets
(a) Maximum correlation statistic. For each gene calculate the maximum correlation between that gene and any of the the others. Select those genes that have the highest maximum correlation.
(b) Maximum eigenvalue. Select random subsets of genes of a prefixed size and calculate the largest or two largest eigenvalues of their covariance matrix. Chose the subset with largest eigenvalues.
4. Coefficient of Variation

6. Correlation Matrix Use covariance or correlation matrix?
- It depends on our way of thinking about microarray data.
- Two genes are highly correlated but in very different scales.
They belong in the same group?  Use Correlation
Dim(R) = GxG and G is between 1000 and 25000, this is too big
 Dimension reduction.
Rank (R) = p
Gene expression matrix X:
Rows = Genes = Variables
Columns = Microarrays = Subjects = Observations

7. Sample
Correlation
Matrix
Scatterplot
Matrix

8. Principal Components Geometrical Intuition
- The data cloud is approximated by an ellipsoid
- The axes of the ellipsoid represent the natural components of the data
- The length of the semi-axis represent the variability of the component.
Variable X2
Component1
Component2
Data
Variable X1

9. DIMENSION REDUCTION
When some of the components show a very small variability they can be omitted.
The graphs shows that Component 2 has low variability so it can be removed.
The dimension is reduced from dim=2 to dim=1
Variable X2
Component1
Component2
Data
Variable X1

10. Linear Algebra
Linear algebra is useful to write computations in a convenient way. Since the number of genes (G) is very large we need to write the computations so we do not generate any GxG matrices.
Notice that the rows of X are the genes = variables.
Singular Value Decomposition: X = U D V’
Gxp Gxp pxp pxp
In standard Multivariate Analysis X would be transposed so the variables correspond to columns of X. But if we do it that way D and V would both be GxG matrices and that is what we are trying to avoid.

11. Linear Algebra Singular Value Decomposition: X = U D V’
Gxp Gxp pxp pxp
The Covariance Matrix takes the form: S = U D2 U’
GxG Gxp pxp pxG
S is GxG but we do not need to write it down to do the dimension reduction.
Correlation Matrix: Subtract mean of rows of X and divide by
standard deviation and calculate the covariance
Principal Components(PC): p Columns of U.
Eigenvalues (Variance of PC’s): p Diagonal elements of D2
The first data reduction is to expressed S or R (GxG) as a function of U (Gxp) and D(pxp).

12. Principal Components Table
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
Standard deviation
Proportion of Variance
Cumulative Proportion
Comp.9 Comp10 Comp11 Comp12 Comp13 Comp14 Comp15 Comp16
Standard deviation
Proportion of Variance
Cumulative Proportion

13. Choosing the number of PC’s
Dimension reduction:
Choosing the number of PC’s
k components explain some percentage of the variance: 60%, 70%, 80%.
k eigenvalues are greater than the average (1)
Scree plot: Graph the eigenvalues and look for the last sharp decline and choose k as the number of points above the cut off.
Test the null hypothesis that the last m eigenvalues are equal (0)
The dfs= (m-1)(m+2)/2 and it is possible to start with a smaller p.

14. average The top 3 eigenvalues explain 70% of variability.
13 eigenvalues greater than the average 1
Scree Plot
Test statistic highly significant for 3.
average

15. Principal Components Graph: PC3 Vs PC2 Vs PC1
The four tumor
groups are
represented by
different colors.

16. Biplots Combination of two graphs into one:
1. Graph of the observations in the coordinates of the two principal components. (Scores)
Graph of the Variables projected into the plane of the two principal components. (Loadings)
The variables are represented as arrows, the observations as points or labels.

17. Biplots: Linear Algebra
From SVD: X = UDV’  X2 = U2D2V2’
A = U2D2a and B=V2D2b, a+b=1 so X=AB’
The biplot is a Graphical display of X in which two sets of markers are plotted.
One set of markers a1,…,aG represents the rows of X
The other set of markers, b1,…, bp, represents the columns of X.
The biplot is the graph of A and B together in the same graph.
If the number of genes is too big it is better to omit and plot them in a separate graph or to invert the graph.

18. Biplots of the first two principal components
- The data cloud is divided into 4 clear clusters
The arrows representing the genes fall in approximately three groups
Next step is to identify the gene groups and check their biological information.

19. Ggobi display finding four clusters of tumors using the PP index on the set of 63 cases. The main panel shows the two dimensional projection selected by the PP index with the four clusters in different colors and glyphs. The top left panel shows the main controls and the left bottom panel displays the controls and the graph of the PP index that is been optimized. The graph shows the index value for a sequence of projection ending at the current one.

20. Exploratory Analysis Steps
Dimension Reduction: Gene subset selection.
Principal Components for further dimension reduction.
Biplot and Graphs
For samples: Select natural clusters of samples. Identify sample grouping with natural clusters.
For genes: Identify gene clusters and their function.