statistics for genomics

Statistics for genomicsMayo Clinic and Illinois Alliance Computational Genomics Course

Dave Zhao, [email protected] of StatisticsUniversity of Illinois Urbana-Champaign

What is statistics for

“Imagine a flashy spaceship lands in your backyard.

The door opens and you are invited to investigate

everything to see what you can learn. The

technology is clearly millions of years beyond what

we can make.

This is biology. That’s why it fascinates me so much.

Life is billions of years old technology, and we can

explore it to see how it works!”

We want to understand biology

https://berthub.eu/articles/posts/immune-system/

Our questions (usually) are:

1. Explanatory

2. Qualitative

3. General

https://berthub.eu/articles/posts/immune-system/

We will collect genomic data

https://www.illumina.com/techniques/sequencing/dna-sequencing/targeted-resequencing/amplicon-sequencing.html

Our data are:

1. Descriptive

2. Quantitative

3. SpecificOur questions (usually) are:

1. Explanatory

2. Qualitative

3. General

https://www.illumina.com/techniques/sequencing/dna-sequencing/targeted-resequencing/amplicon-sequencing.html

We need to bridge E ↔ D, ☺↔ #, and G ↔ S

☺ = qualitative, # = quantitative, D = descriptive, E = explanatory, G = general, S = specific

ExperimentQuestion

DataTheory

G☺E

G☺E

S#D

S#D

Deduction

Induction

?

?

We usually take several steps

ExperimentQuestion

Design

DataTheory

CollectPropose

Analyze

G☺E S#D

G☺E S#D

G☺D G#DFormulate

Model

Explain

Question1 Question2

ResultFinding

G☺D G#D

Interpret

Deduction

Induction


These steps are difficult

General qualitative explanatory question (G☺E) General qualitative descriptive question (G☺D)

1. How do cells in the habenula of zebrafish larvae give

rise to brain function?

1. What reductionist descriptive information would be

useful for systems-level theory?

General qualitative descriptive question (G☺D) General quantitative descriptive question (G#D)

1. How highly expressed is gng8 in zebrafish larval

habenular cells?

2. Are the expression levels of epcam and calb2a

associated in zebrafish larval habenula cells?

3. What are the distinct molecularly defined cells types in

zebrafish larval habenula?

1. How to define “expression” for all habenular cells in all

zebrafish larvae?

2. How to numerically define “association”?

3. How to numerically define “distinct” and “cell type”?

Specific quantitative descriptive answer (S#D) General quantitative descriptive answer (G#D)

1. The expression levels of gng8 in cells from the scRNA-

seq data are 102, 20, 21, 15, 2, etc.

1. How to make general conclusion about all habenular

cells in all zebrafish larvae from these observed data?

Pandey et al. (2018)

Statistics is a mathematical framework for bridging ☺↔ #, G ↔ S

ExperimentQuestion

Design

DataTheory

CollectPropose

Analyze

G☺E S#D

G☺E S#D

G☺D G#DFormulate

Model

Explain

Question1 Question2

ResultFinding

G☺D G#D

Interpret

Deduction

Induction


Concepts and language

Theory and

methods

Concepts and language

Theory and

methods

Principles of statistics

We will focus on principles instead of specific methods or code

“Why Structure and Interpretation of Computer Programs

matters” (https://people.eecs.berkeley.edu/~bh/sicp.html):

The language in which you'll spend most of your working life

hasn't been invented yet, so we can't teach it to you. Instead

we have to give you the skills you need to learn new languages

as they appear.

https://mitpress.mit.edu/sites/default/files/sicp/index.html

https://people.eecs.berkeley.edu/~bh/sicp.html

https://mitpress.mit.edu/sites/default/files/sicp/index.html

1. Identify population you want to generalize to

ExperimentQuestion

DataTheory

G☺E S#D

G☺E S#D

G☺D G#D

Question1 Question2

ResultFinding

G☺D G#D

Deduction

Induction


Exercise: zebrafish scRNA-seq

How do cells in the

habenula of zebrafish

larvae give rise to

brain function?

What is the population of interest?

All cells in the habenulae of all

zebrafish larvae


2. Population distribution function encodes all G#D information

ExperimentQuestion

DataTheory

G☺E S#D

G☺E S#D

G☺D G#D

Question1 Question2

ResultFinding

G☺D G#D

Deduction

Induction


𝑝(𝑥)

𝑝(𝑥) = proportion of the population whose variables take value 𝑥

The proportion of elements in the

population with

𝑋1 = 3.6599 and

𝑋2 = 93.4082 is

𝑝(3.6599, 93.4082) = 0.0081.

Variable 𝑋1of interest

Variable 𝑋2of interest

Many useful descriptive quantities are properties of 𝑝(𝑥)

𝑝 𝑥2

𝑝 𝑥1𝑝(𝑥1 ∣ 𝑥2 = 93.4082)

𝑝(𝑥2 ∣ 𝑥1 = 3.6599)

mean1, median1

mean2, median2

• Variances

• Conditional mean of

𝑋2 among subgroup

of population with

𝑋1 = 𝑥1• Correlations

• Etc.

Conditional distributions Marginal distributions Other

*3*. Model G☺D question as a G#D question about 𝑝 𝑥

ExperimentQuestion

DataTheory

G☺E S#D

G☺E S#D

G☺D G#D

Question1 Question2

ResultFinding

G☺D G#D

Deduction

Induction


𝑝(𝑥)

Statistical methods can answer four types of G#D questions

Target

G#D question

“Is?”

Testing

“How much?”

Estimation

Underlying variables

Dimension reduction

Underlying groups

Clustering

Supervised learning

Unsupervised learning

Property of 𝑝(𝑥) of interest

Examples of testing questions

Li (2015)

Is Catalase differentially expressed

between ALL and AML patients?

Golub et al. (1999)

Which bacteria differ in abundance between normal and IBD patients?

Examples of estimation questions

Moffitt et al. (2015)

How much of a gene is expressed in different cell clusters?

Avalos et al. (2020)

How is minor allele frequency within a colony

related to colony defense response?

Examples of dimension reduction questions

Price et al. (2006)

What underlying patterns are captured in an

individual’s genetic code?

Lee and Seung (1999)

What canonical features can be combined to

form any face?

Examples of clustering questions

https://satijalab.org/seurat/articles/pbmc3k_tutorial.html

What types of molecularly defined cell types are present

in peripheral mononuclear blood cells?

Aure et al. (2017)

Are there clinically significant subtypes of the luminal A

subtype of breast cancer?

https://satijalab.org/seurat/articles/pbmc3k_tutorial.html

Generalized linear models make G☺D ⇒ G#D precise

𝑔−1 𝐸 𝑌 𝑋1 = 𝑥1, … , 𝑋𝑝 = 𝑥𝑝 = 𝛽0 +

𝑗=1

𝑝

𝑥𝑗𝛽𝑗

General qualitative descriptive question (G☺D) General quantitative descriptive ques. about 𝑝(𝑥) (G#D)

1. How highly expressed is gng8 in zebrafish larval

habenular cells?

2. Are the expression levels of epcam and calb2a

associated in zebrafish larval habenular cells?

3. What are the distinct molecularly defined cells types in

zebrafish larval habenula?

1. Let 𝑋1 = gng8 expression in a larval habenular cell. In

the model log 𝐸 𝑌 𝑋1 = 𝛽0, what is 𝛽0?

2. Let 𝑌 = epcam expression in a larval cell and 𝑋1 =

calb2a expression. In the model log 𝐸 𝑌 𝑋1 = 𝛽0 +𝛽1𝑋1, is 𝛽1 = 0?

3. Let 𝑋𝑗 be the expression of gene 𝑗 and let 𝐶𝑘 be

unknown group indicators that equal 1 if the cell is of

type 𝑘 and 0 otherwise, for 𝑘 = 1,… , 𝐾. In the models

log𝐸 𝑋𝑗 𝐶1, … , 𝐶𝐾 = 𝛽𝑗1𝐶1 +⋯+ 𝛽𝑗𝐾𝐶𝐾, what

definitions of 𝐶𝑘 make the 𝛽𝑗1, … , 𝛽𝑗𝐾 most distinct

from each other, for all genes 𝑗?

Generalized linear modeling

G☺D question ⇒ a G#D question about one or more of the 𝛽’s in:

4. Transform data into variables measured on samples

ExperimentQuestion

DataTheory

G☺E S#D

G☺E S#D

G☺D G#D

Question1 Question2

ResultFinding

G☺D G#D

Deduction

Induction


𝑿𝟏 𝑿𝟐 …

variables

sampling

units

𝑝(𝑥)

Exercise: zebrafish scRNA-seq

GSM2818521_larva_counts_matrix.txt:

10 days post fertilization larval heads from 25 gng8-GFP fish were dissected, and cells were dissociated as described

previously…The resulting single cell suspension was promptly loaded on the 10X Chromium system.

Exercises:

1. What are the variables?

Genes

2. What are the sampling units?

Cells

3. What is the population you can generalize to?

All cells from the habenula of only the 25 gng8-

GFP larva used in this experiment


5. Infer properties of 𝑝(𝑥) from data using statistical methods

ExperimentQuestion

DataTheory

G☺E S#D

G☺E S#D

G☺D G#D

Question1 Question2

ResultFinding

G☺D G#D

Deduction

Induction


𝑿𝟏 𝑿𝟐 …

variables

sampling

units

𝑝(𝑥)

Ƹ𝑝(𝑥)

Taxonomy of methods is more important than any specific method

Toolbox = taxonomy

Tools = methods

Methods can be categorized based on a few important factors

Target

G#D question

“Is?”

Testing

“How much?”

Estimation


Dimension reduction

Underlying groups

Clustering

Other factors:

1. Data structure

• Variable types

• Missingness

• Censoring

• Etc.

2. Model assumptions

• Parametric

• Semiparametric

• Nonparametric

3. Problem details

4. Cultural norms


Hardware

Operating system

Programs

The R ecosystem is useful for implementing analyses

https://www.r-project.org/

https://rstudio.com/products/rstudio/download/

https://git-scm.com/

User-contributed R packages

https://www.r-project.org/

https://rstudio.com/products/rstudio/download/

https://git-scm.com/

6. Use inferred ො𝑝(𝑥) to make qualitative conclusions

ExperimentQuestion

DataTheory

G☺E S#D

G☺E S#D

G☺D G#D

Question1 Question2

ResultFinding

G☺D G#D

Deduction

Induction


𝑿𝟏 𝑿𝟐 …

variables

sampling

units

𝑝(𝑥)

Ƹ𝑝(𝑥)

https://www.cusabio.com/pathway.html

https://www.cusabio.com/pathway.html

Generalized linear models make G#D ⇒ G☺D precise

𝑔−1 𝐸 𝑌 𝑋1 = 𝑥1, … , 𝑋𝑝 = 𝑥𝑝 = 𝛽0 +

𝑗=1

𝑝

𝑥𝑗𝛽𝑗

General quantitative descriptive result about 𝑝(𝑥) (G#D) General qualitative descriptive finding (G☺D)

1. Let 𝑋1 = gng8 expression in a larval habenular cell. In

the model log 𝐸 𝑌 𝑋1 = 𝛽0, 𝛽0 is estimated to equal

4.19 with a standard deviation of 0.90.

2. Let 𝑌 = epcam expression in a larval cell and 𝑋1 =

calb2a expression. In the model log 𝐸 𝑌 𝑋1 = 𝛽0 +𝛽1𝑋1, the p-value for testing 𝛽1 = 0 is less than 10−16.

3. Let 𝑋𝑗 be the expression of gene 𝑗. We can estimate

underlying group indicators 𝐶1, … 𝐶𝐾 such that in the

model log𝐸 𝑋𝑗 𝐶1, … , 𝐶𝐾 = 𝛽𝑗1𝐶1 +⋯+ 𝛽𝑗𝐾𝐶𝐾, the

estimated 𝛽𝑗1, … , 𝛽𝑗𝐾 are most distinct from each

other, for all genes 𝑗.

1. We’re fairly confident that the population average

expression level of gng8 in zebrafish larval habenular

cells is around 𝑒4.19.

2. Since the p-value is very low, we have enough

evidence to say that epcam and calb2a are associated

in zebrafish larval habenular cells.

3. The cells that have 𝐶𝑘 = 1 are part of the same cell

cluster and have a common structural or functional

role in the habenula.

Generalized linear modeling

G☺D question ⇒ a G#D question about one or more of the 𝛽’s in:

Case study

We will study zebrafish scRNA-seq data

G☺E

Deduction

Induction


How do cells in the


larvae give rise to

brain function?


*1*. Formulate a useful G☺D question

G☺E

G☺D

Deduction

Induction


How do cells in the


larvae give rise to

brain function?

What are the molecularly

distinct cell types in the

habenula?


*2*. Model G☺D question as a G#D question about 𝑝 𝑥

G☺D G#D

What underlying

groups can 𝑝(𝑥) be

divided into?Deduction

Induction


How do cells in the


larvae give rise to

brain function? 𝑝(𝑥)

G☺E



habenula?


3. Conduct experiment

G☺E S#D

G☺D G#D

What underlying



Induction


How do cells in the


larvae give rise to




habenula?


4. Transform raw data into data table

G☺E S#D

S#D

G☺D G#D

What underlying



Induction


How do cells in the


larvae give rise to


Bioinformatics

GSM2818521_larva_counts_matrix.txt



habenula?


5. Answer G#D question about 𝑝(𝑥)

G☺E S#D

S#D

G☺D G#D

What underlying


divided into?

G#D

Deduction

Induction


How do cells in the


larvae give rise to


Bioinformatics


1. Preprocessing

a. Quality control

b. Normalization

2. Analysis and visualization

Ƹ𝑝(𝑥)



habenula?


We will be using the Seurat R package

Quality control and normalization are technology-specific

## ##############################################################

## load data

## ##############################################################

file = "/home/user/data/stat530_2021/GSM2818521_larva_counts_matrix.txt"

pandey <- read.table(file, header = TRUE)

dim(pandey)

library(Seurat)

## set random seed for reproducibility

set.seed(1)

s_obj <- CreateSeuratObject(counts = pandey, min.cells = 30, min.features = 500)

## ##############################################################

## quality control

## ##############################################################

s_obj[["percent.mt"]] = PercentageFeatureSet(s_obj, pattern = "^MT-")

VlnPlot(s_obj, features = c("nCount_RNA", "nFeature_RNA", "percent.mt"))

s_obj <- subset(s_obj, percent.mt <= 6)

## ##############################################################

## normalization

## ##############################################################

s_obj <- NormalizeData(s_obj)

s_obj <- FindVariableFeatures(s_obj)

s_obj <- ScaleData(s_obj, vars.to.regress = "percent.mt")Estimate variances

Estimate associations


We cluster using the Louvain algorithm on an SNN graph

Target

G#D question

“Is?”

Testing

“How much?”

Estimation


Dimension reduction

Underlying groups

Clustering

Ertoz, Steinbach, Kumar (2003), Waltman and van Eck (2013)

𝑝(𝑥)

Algorithm:

## ##############################################################

## analysis

## ##############################################################

## dimension reduction

s_obj <- RunPCA(s_obj)

## clustering

s_obj <- FindNeighbors(s_obj)

s_obj <- FindClusters(s_obj, resolution = 0.5)

We first reduce variables for speed and accuracy using PCA


Target

G#D question

“Is?”

Testing

“How much?”

Estimation


Dimension reduction

Underlying groups

Clustering

𝑝(𝑥)

𝑋 𝑌 new 𝑍 = position on orange line

https://online.stat.psu.edu/stat200/lesson/3/3.4/3.4.1

Algorithm:

## ##############################################################

## analysis

## ##############################################################


s_obj <- RunPCA(s_obj)

## clustering

s_obj <- FindNeighbors(s_obj)

s_obj <- FindClusters(s_obj, resolution = 0.5)

https://online.stat.psu.edu/stat200/lesson/3/3.4/3.4.1

We visualize using UMAP to reduce variables


Target

G#D question

“Is?”

Testing

“How much?”

Estimation


Dimension reduction

Underlying groups

Clustering

𝑝(𝑥)

Rhys (2020)

Algorithm:

## ##############################################################

## visualization

## ##############################################################


s_obj <- RunUMAP(s_obj, dims = 1:20)

## visualization

DimPlot(s_obj)

6. Answer G☺D question about population

G☺E S#D

S#D

G☺D G#D

What underlying


divided into?

G☺D G#D

Deduction

Induction


How do cells in the


larvae give rise to


Bioinformatics


1. Preprocessing

a. Quality control

b. Normalization


Ƹ𝑝(𝑥)



habenula?

There are at least 12

distinct cell types in

the habenula.


Combine biological knowledge with results to arrive at finding

## ##############################################################

## interpretation

## ##############################################################

## visualize marker of olfactory cells

FeaturePlot(s_obj, features = "EPCAM")

Not a cluster of

habenular cells

7. Reconcile finding with existing theory

G☺E S#D

S#D

G☺D G#D



habenula?

What underlying


divided into?



the habenula.

G☺D G#D

Deduction

Induction


How do cells in the


larvae give rise to


Bioinformatics


1. Preprocessing

a. Quality control

b. Normalization


Ƹ𝑝(𝑥)

Theory G☺E


8. Continue

Theory G☺E

Deduction

Induction


G☺E

G☺D

How do cells in the


larvae give rise to

brain function?

What are the biological

functions of the different

habenular cell types?




the habenula.

G☺D

Find differentially expressed genes and active pathways

## ##############################################################

## additional analysis

## ##############################################################

markers = FindMarkers(s_obj, ident.1 = 5)

gene_names = rownames(markers)[markers$p_val_adj <= 0.05]

length(gene_names)

cat(gene_names, sep = "\n")

## https://david.ncifcrf.gov/

Questions

Dave Zhao, [email protected]

Department of Statistics

University of Illinois Urbana-Champaign

http://publish.illinois.edu/sdzhao/

statistics for genomics

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