Center for Biofilm Engineering
Some statistical
considerations in
molecular methods
Al Parker
Statistician and Research Engineer
Montana State University
BSTM– July 2009
Acknowledgments
Colleagues in the CBE:
 James Moberly, Seth D’Imperio, Brent Peyton
 Markus Dieser
 Marty Hamilton
The problem
How to extract useful information from hundreds
to thousands of response variables (eg. microarray analysis) measured from only a few
replicates (experiments or environmental
samples)
Statistical thinking
 Multivariate Statistics attempts to organize and
summarize data sets with large numbers of
response variables
 “organize and summarize” = dimension reduction
 In this talk, I will focus on abundance data,
estimated for example from micro-array or clone
analysis of PCR
Statistical thinking
 Hierarchical Clustering
 Principle Components
 Canonical Correlation
Hierarchical Clustering
(38 variables, 9 replicates)
Hierarchical Clustering
(38 variables, 9 replicates)
Similarity or Distance
Linkage:
How the
similarity
measure
determines
clusters
Two different ways to
generate clusters with
the same similarity
measure
A Distance or Similarity Measure
Correlation measures the strength and direction
of a linear relationship between paired variables x
and y
Corr(x,y) =
Σ(xi – mean(x))(yi – mean(y))
(n-1)SxSy
 Unitless
 Values between -1 and 1
An example
(2 variables, 9 replicates)
Corr(Actinobacteria, Acidobacteria) = .7833
Another (made up) example
Corr(species 1, species 2) = 0.000
A matrix of scatterplots for 6 variables
A correlation matrix of 6 variables
Acidobacteria
Acidobacteria
Actinobacteria Bacteroidetes Chloroflexi
Proteobacteria Verrucomicrobia
1
0.7833
0.7589
0.8556
0.8444
0.7975
Actinobacteria
Bacteroidetes
Chloroflexi
0.7833
0.7589
0.8556
1
0.8993
0.8257
0.8993
1
0.7901
0.8257
0.7901
1
0.9698
0.9393
0.8704
0.8230
0.8392
0.9699
Proteobacteria
0.8444
0.9698
0.9393
0.8704
1
0.8621
Verrucomicrobia
0.7975
0.8230
0.8392
0.9699
0.8621
1
Principle Components Analysis (PCA)
 PCA uses the correlation matrix formed by the
original variables to optimally construct a smaller
number of new variables which capture the
maximum amount of variability in the original
variables
 PCA applied to the correlation matrix is not
affected by disparate units between the different
variables
 The number of new variables is only as large as
the number of replicates
Original variable #1
PCA with 2 (standardized) responses
Original variable #2
PCA with 2 (standardized) responses
1st PC - 78%
Original variable #1
1st PC is
loaded by
Orig Var #1
2nd PC – 22%
2nd PC is
loaded by
Orig Var #2
Original variable #2
PCA terminology
 The new variables are called principle components
 The amount of variability of the original data
captured by each component is given
 The correlation between the original variables and
the principle components are
principle component loadings
Reducing 7 original variables to 2 PCs
Original variables:
1. Water depth
2. Core depth
3. Fe
4. Mn
New variables =
Principle Components
1st PC: Metals
2nd PC: Water depth
and Core depth
55%
5. Cu
6. Pb
7. Zn
18%
Reducing 7 original variables to 2 PCs
1st PC - 55%
2nd PC – 18%
Total:
73%
PCA is another way to cluster
Canonical Correlation Analysis (CCA)
 CCA uses the correlation matrix to determine the
(linear) relationship between input variables (eg.
environmental variables) and response variables
(eg. phylogenic data)
 CCA simultaneously finds new variables from the
input and response variables which have maximal
correlation
 The number of new variables (canonical
components) can be no larger than the number of
replicates
CCA Example
(7 inputs, 6 outputs, 9 replicates)
Original environmental
variables:
1. Water depth
2. Core depth
3. Fe
4. Mn
5. Cu
6. Pb
7. Zn
Original microbial
variables:
1. Acidobacteria
2. Actinobacteria
3. Bacteroidetes
4. Chloroflexi
5. Proteobacteria
6. Verrucomicrobia
CCA
(7 inputs, 6 outputs, 9 replicates)
Original environmental
variables:
1. Water depth
2. Core depth
3. Fe
4. Mn
5. Cu
6. Pb
Original microbial
variables:
1. Acidobacteria
2. Actinobacteria
3. Bacteroidetes
4. Chloroflexi
5. Proteobacteria
6. Verrucomicrobia
7. Zn
1st CC: Water depth
and Core depth
1st CC: Acidobacteria,…,
Verucomicrobia
2nd CC: Metals
2nd CC: Bacteroidetes
CCA
(7 inputs, 6 outputs, 9 replicates)
1st CC: Water depth
and Core depth
1st CC: Acidobacteria,…,
Verucomicrobia
2nd CC: Metals
2nd CC: Bacteroidetes
Summary
PROBLEM: Lots of variables measured from a few
samples
SOME APPROACHES:
 Cluster similar variables together
 Principle component analysis creates a few new
variables which optimally represent the data
 Canonical correlation analysis describes the
optimal (linear) relationship between input and
output variables
Fin
•
Principal Component Analysis: water depth , core depth (, Mn-Total, Fe-Total, C
•
Eigenanalysis of the Correlation Matrix
•
•
•
Eigenvalue 3.8467 1.2443 1.0043 0.6628 0.1567 0.0830 0.0023
Proportion 0.550 0.178 0.143 0.095 0.022 0.012 0.000
Cumulative 0.550 0.727 0.871 0.965 0.988 1.000 1.000
•
•
•
•
•
•
•
•
Variable
PC1 PC2 PC3 PC4 PC5 PC6 PC7
water depth (cm) 0.090 -0.529 -0.732 0.338 0.131 0.201 -0.062
core depth (cm) -0.193 0.702 -0.154 0.558 0.194 0.313 0.009
Mn-Total
0.488 0.163 -0.171 -0.016 -0.366 0.084 0.752
Fe-Total
0.477 0.228 -0.126 -0.057 -0.504 0.154 -0.651
Cu-Total
0.227 -0.358 0.608 0.633 -0.119 0.188 0.004
Zn-Total
0.463 0.019 0.147 -0.326 0.634 0.505 -0.026
Pb-Total
0.474 0.142 -0.055 0.253 0.376 -0.735 -0.080
CCA
(7 input variables, 9 replicates)
1st CC: Water depth
Core depth
2nd CC: Metals
CCA
(6 response variables, 9 replicates)
1st CC:
Acidobacteria,…,
Verucomicrobia
2nd CC:
Bacteroidetes
Hierarchical Clustering
 The large number of variables are organized into a
smaller number of similar clusters
 One can choose a representative variable from
each cluster (eg. a mean)