310 likes | 452 Views
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
E N D
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. micro-array 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) 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 Σ(xi – mean(x))(yi – mean(y)) Corr(x,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
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
PCA with 2 (standardized) responses Original variable #1 Original variable #2
PCA with 2 (standardized) responses 1st PC - 78% 1st PC is loaded by Orig Var #1 Original variable #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 5. Cu 6. Pb 7. Zn New variables = Principle Components 55% 1st PC: Metals 2nd PC: Water depth and Core depth 18%
Reducing 7 original variables to 2 PCs 1st PC - 55% 2nd PC – 18% Total: 73%
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 microbial variables: Original environmental variables: 1. Water depth 2. Core depth 3. Fe 4. Mn 5. Cu 6. Pb 7. Zn 1. Acidobacteria 2. Actinobacteria 3. Bacteroidetes 4. Chloroflexi 5. Proteobacteria 6. Verrucomicrobia
CCA (7 inputs, 6 outputs, 9 replicates) Original microbial variables: Original environmental variables: 1. Water depth 2. Core depth 3. Fe 4. Mn 5. Cu 6. Pb 7. Zn 1. Acidobacteria 2. Actinobacteria 3. Bacteroidetes 4. Chloroflexi 5. Proteobacteria 6. Verrucomicrobia 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
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)