1.
Predicting Outcomes When Your
Outcomes are Graphs (or functions)
Bill Shannon, PhD, MBA
Co-Founder and CEO, BioRankings
Professor Emeritus of Biostatistics in Medicine, WUSM
bill@biorankings.com, 314-704-8725

2.
With big data come new complex data
formats – data as graphs
Functional MRI Data
• Brains are inserted into MRI
scanner
• 30 gigabytes raw data
• Parcellation
• Networks
– Nodes are regions of the
brain
– Edges are the correlations
between pairs of nodes

3.
Connectome Graph

4.
With big data come new complex data
formats – data as graphs
Data Microbiome
• Sample from human,
animal, field (soil),
environment
• Next Generation
Sequencing (write once,
read never data)
• Genomic analysis
processing
– Annotation to taxonomic
label (i.e., genus, species)

5.
Microbiome Tree

6.
Statistics is interested in inferring
things about everything from a sample
Sample to Population Inference
• Collect a bunch of graphs – 1
per subject
• Plot graphs
• Estimate mean and variance
(or g* and tau)
• Does this plot teach us about
the graphs in terms of how
they are distributed and what
the central tendency is?

7.
Does this plot teach us anything?

8.
Graphs are too complex – let’s simplify
Network metrics
Average connectivity
Small world network
Species diversity
Taxa counts
Enterotype

9.
Many-to-one mapping is not necessarily a good
way to simplify data for analysis

10.
Simplifying in fMRI and Microbiome
fMRI
• Average Node Connectivity
• Consider two brain scans
– Patient 1
• Right half ANC = 10
• Left half ANC = 0
– Patient 2
• Right half ANC = 5
• Left half ANC = 5
• Both whole brain ANC = 5
Microbiome
• Species Diversity
• Consider two samples
– Patient 1
• Proportion Taxa A, B, C = 1/3
• Proportion Taxa D, E, F = 0
– Patient 2
• Proportion Taxa A, B, C = 0
• Proportion Taxa D, E, F = 1/3
• Both have Simpson diversity
= 0.33

11.
We analyze graphical data the same
way as we analyze columns of data
Gibbs distribution
• Let G be a finite set of graphs and denote the
elements of G by g. Let 𝑑 be an arbitrary
distance metric on G. The Gibbs distribution
on the graphs G is denoted by
ℙ 𝒈; 𝒈∗
, 𝝉 = 𝒄 𝒈∗
, 𝝉 𝒆𝒙𝒑 −𝝉𝒅 𝒈∗
, 𝒈 , ∀𝒈 𝝐 𝐆,
with parameters g∗
the central or average
graph, and 𝜏 a non-negative number that is a
measure of the dispersion of the observed
connectome data around g∗
. 𝑐 g∗
, 𝜏 is the
normalizing constant.
ℙ 𝑔𝑖; g∗
, 𝜏 is the probability of observing a
specific graph 𝑔𝑖 given the parameters
g∗
, 𝜏 .
Statistics on Graphs

12.
We analyze graphical data the same
way as we analyze columns of data
Recursive partitioning
• Regress the graphs on
covariates
• In this example of Parkinson's
disease
– Y = connectome
– X = group, sex, age
• RP splits the connectomes into
homogeneous groups based
on likelihood of Gibbs
Statistics on Graphs

13.
What else can be analyzed with
graphical OODA?
IoT
Blockchain
Cybersecurity

14.
What about data which are functional
objects?
Untargeted Metabolomics
• Liquid chromatography and
mass spec – LC/MS
• RT x m/z plots
• Which peaks correspond to
metabolites (known or
unknown), and which peaks
are different in patients
who live and die?

15.
RT x m/z plots are too complex – let’s
simply
Looking for things that look
different and then testing them
statistically is wrong – P values
don’t mean anything in these
cases.

16.
Why not analyze functions using
functional OODA?

17.
Why not analyze functions using
functional OODA?

18.
Field Enabling
Technology
Bioinformatics Exploratory Analysis Translational
Statistics
Microbiome Next generation
Sequencing
Assembly,
annotation, chimera
checking
Cluster analysis,
multidimensional
scaling, heatmaps
Dirichlet-
multinomial for taxa
counts
Gibbs distribution
for taxonomic
trees
Brain Imaging Functional MRI
(fMRI)
Image registration,
parcellation
Generalized linear
models with
multiple testing
adjustment, graph
metrics
Gibbs distribution
for connectome
Metabolomics LC/MS Peak detection,
centering
Mass univariate
testing with multiple
testing adjustment
Functional data
analysis, Gibbs
distribution, Co-
Inertia, and the
Exploratory-
Validation Model for
experimental design
Projects in object oriented data analysis