Quantitative Genomics and Geneticsmezeylab.cb.bscb.cornell.edu/labmembers/documents... · Quantitative Genomics and Genetics BTRY 4830/6830; PBSB.5201.03 Lecture 2: Introduction to

Quantitative Genomics and Genetics

BTRY 4830/6830; PBSB.5201.03

Lecture 2: Introduction to probability basics

Jason [email protected]

Jan. 23, 2020 (Th) 8:40-9:55

mailto:[email protected]

Registering I• ANYONE at Cornell (Ithaca), Weill Cornell, Cornell Tech,

MSKCC, or Rockefeller is allowed to take this course (= you have my permission!) whether you officially register or not

• Please register if you can (even if you are Auditing!) but if you cannot register, you are welcome to take the class and - if you do the work - we will grade your work as though you are registered

• If you are a student (undergraduate or gradate) at Cornell (Ithaca), Weill Cornell (NYC) or Cornell Tech (NYC) you should be able to register through your class system

• If you are a graduate student at Rockefeller you can register for this course (contact Kristen Cullen [email protected] )

mailto:[email protected]

Registering II• If you are a postdoc, research associate, tech, employee, etc. at

Cornell, Ithaca you may be able to register - please check with your employer, the registrar, and / or your appropriate office (e.g. Human Resources) - also please note we are looking into whether you need to pay a registration fee (stay tuned)

• If you are a tech, employee, fellow, resident, etc. at Weill or MSKCC you may be able to register - please check with your PI / employer to get permission - please note we are waiting for confirmation from the Registrar on whether you can register (no guarantees but we are working on it!) - we will get you information about what forms (if any), if there are fees, how to register, as soon as we can

• As soon as information becomes available, we will email you through the class Piazza (!!) instance so please get on PIAZZA ASAP (see following slides)

• There is a REQUIRED computer lab (if you take the course for credit)

• Note that in Ithaca (= Labs taught by Rachel!):

• Lab 1 will meet 5-6PM on Thurs.

• Lab 2 will meet 8-9AM on Fri.

• The location of these labs will be in Weill 226 (!!)

• Note that in NYC (= Labs taught by Scott!):

• Labs will be Thurs. 4-5PM

• Thurs. labs will be in R [D236], S [D234] in 1300 York Ave Building

• If you are a Weill MS-CB student or Cornell Tech student please see me after class today (!!)

• Bring your laptop to every computer lab!

• NO COMPUTER LAB THIS WEEK (!!)

Times and Locations I

• Jason will hold office hours:

• On both campuses by video-conference Mon. 2-4PM or by appointment (!!) - first office hours this Mon. (!!)

• Office hours will be conducted using Zoom:

https://cornell.zoom.us/j/724550601

• Rachel and Scott will hold informal office hours after their computer labs

• Jason will NOT hold office hours this Mon. (Jan. 27) but office hours will start the following week (Feb. 3)

• You are always welcome to schedule individual help sessions with Jason, Rachel, and Scott (piazza us to set up session)

Times and Locations II

https://cornell.zoom.us/j/724550601

Class Resource 1: Website• The class website will be a under the “Classes” link on my

site: http://mezeylab.cb.bscb.cornell.edu/

http://mezeylab.cb.bscb.cornell.edu

Class Resources II: Piazza• MAKE SURE YOU SIGN UP ON PIAZZA (!!) ASAP

whether you officially register or not = all communication for the course!

• Class: https://piazza.com/class/k5lo4uj1ot279f

• Step 1: Sign up on Piazza (if you don’t have an account already)!

• Step 2: Enroll in BTRY 6830 (regardless if you are grad or undergrad!) - make sure you are on Spring 2020

• If you received a PIAZZA email from me last night you are all set!

• If you did receive an email, please get on PIAZZA, make sure you have activated your account, etc … and if you are still having trouble please email me directly

Class Resource III: CMS

• Assignments and computer labs (!!) will be posted on CUCS CMS (as BTRY 4830)

• If you have a NetID you should be able to access CMS

• If you do not have a NetID we will need to register you - PLEASE PIAZZA EMAIL ME THAT YOU NEED ME TO REGISTER YOU ON CMS - AND INCLUDE YOUR EMAIL WITH THE MESSAGE (!!) - and make sure I can see your name!

Summary of lecture 2: Introduction to probability basics

• Last lecture, we provided a broad introduction to the field of quantitative genomics, which is concerned with modeling and the discovery of relationships between genomes (genotypes) and phenotypes

• In this class, we will be concerned with the most basic problem of quantitative genomics: how to identify genotypes where differences among individual genomes produce differences in individual phenotypes (i.e. genetic association studies)

• Today, we will discuss critical foundational concepts in biology and the modeling framework for quantitative genomics, which is developed from the fields of probability and statistics

Review: genomics-genetics connection

• Traditionally, studying the impact / relationship of the genome to phenotypes was the province of fields of “Genetics”

• Given this dependence on genomes, it is no surprise that modern genetic fields now incorporate genomics: the study of an organism’s entire genome (wikipedia definition)

• However, one can study genetics without genomics (i.e. without direct information concerning DNA) and the merging of genetics-genomics is quite recent

Review: advances in next-generation sequencing driving the field

The impact of Genomic Data on genetic analysis

• Before the “Genomic Era” genetic analysis was part of three different fields that used different analysis techniques: Medical Genetics, Agricultural Genetics, and Evolutionary Genetics

• The reason was they were analyzing different systems / interested in different questions AND they did not have the data available to do what they really wanted to do: identify which differences in a genome (genotypes) were responsible for differences in phenotypes of interest (!!)

• Once genomic data (i.e., data on the entire genome) became available the starting analysis of all of these fields became the same (i.e., analyzing which differences impacted phenotypes) and they started using the same set of methods (!!) = effectively unifying these fields into modern “Quantitative Genetics / Genomics”

• This is the reason the Quantitative Genetics literature before the Genomic Era is so difficult to follow / seems so diffuse… but after this class you will understand how to go back and figure out this literature (!!)

Why this is a good time to be learning about this subject

• Mapping (identifying) genotypes (genetic loci) with effects on important phenotypes is fast becoming the major use of genomic data and a major focus of genomics

• However, the data collection, experimental, and statistical analysis techniques for doing this are still being developed

• The current statistical approaches are the focus of this course (i.e., you will have a solid foundation by the end)

• The importance is just now starting to permeate broadly (i.e., we are now in the “internet generation” for genomics and the impact of genomics on biology)

• The basic statistical approaches are (=should be) applied in ANY analysis of ANY genomic data for ANY purpose

Motivating intro to prob & statistics: foundational biology concepts

• In this class, we will use statistical modeling to say something about biology, specifically the relationships between genotype (DNA) and phenotype

• Let’s start with the biology by asking the following question: why DNA?

• The structure of DNA has properties that make it worthwhile to focus on...

It’s the same in all cells

with a few exceptions (e.g. cancer, immune system...)

• In multicellular organisms, the structure of the genome is (almost) perfectly copied duringthe replication of cells.

• The genome is the same in every non-cancer cell of a multicellular organism, with just a fewexceptions; So, we may refer to the genome of an individual ; In cancers, the genome di↵ersfrom cell to cell, such that it is more problematic to refer to the genome of a cancer.

• The genome provides instructions for how biological processes proceed (e.g., development,metabolism, environmental response); So, the genome is an important determinant of themeasurable characteristics of an organism or cancer.

• In the production of a new organism or o↵spring, either the entire genome (e.g., bacteria)or a subset of the genome (e.g., half from each parent in humans) is copied almost perfectlyfrom parent to o↵spring; The copying of genomes from parents to o↵spring is the primaryreason why o↵spring tend to resemble their parents.

Figure 1: A simplified schematic showing genome organization in human cells. The DNA of agenome is located within the nucleus of a cell. The genome is organized in long strings that aretightly coiled around protein structures to form chromosomes. Each string is a double helix wherethe building blocks are A-T and G-C nucleotide pairs c� kintalk.org.

2

It’s passed on to the next generation

Credit: Watson et al., Molecular Biology of the Gene, CSHL Press, 2004

The Structure of DNA

Credit: Jones and Pevzner, An Introduction to Bioinformatics Algorithms, MIT Press, 2004

Credit: Watson et al., Molecular Biology of the Gene, CSHL Press, 2004 Credit: Watson et al., Molecular Biology of the Gene, CSHL Press, 2004

Credit: Watson et al., Molecular Biology of the Gene, CSHL Press, 2004 Credit: Watson et al., Molecular Biology of the Gene, CSHL Press, 2004

It has convenient structure for quantifying differences

It’s almost the same in each individual in a species

It’s responsible for the construction and maintenance of organisms

Note: other regions of genomes can impact phenotypes...

Statistics and probability I

• Quantitative genomics is a field concerned with the modeling of the relationship between genomes and phenotypes and using these models to discover and predict

• We will use frameworks from the fields of probability and statistics for this purpose

• Note that this is not the only useful framework (!!) - and even more generally - mathematical based frameworks are not the only useful (or even necessarily “the best”) frameworks for this purpose

Statistics and probability II

• A non-technical definition of probability: a mathematical framework for modeling under uncertainty

• Such a system is particularly useful for modeling systems where we don’t know and / or cannot measure critical information for explaining the patterns we observe

• This is exactly the case we have in quantitative genomes when connecting differences in a genome to differences in phenotypes

Statistics and probability III

• We will therefore use a probability framework to model, but we are also interested in using this framework to discover and predict

• More specifically, we are interested in using a probability model to identify relationships between genomes and phenotypes using DNA sequences and phenotype measurements (=Data)

• For this purpose, we will use the framework of statistics, which we can (non-technically) define as a system for interpreting data for the purposes of prediction and decision making given uncertainty

Definitions: Probability / Statistics

• Probability (non-technical def) - a mathematical framework for modeling under uncertainty

• Statistics (non-technical def) - a system for interpreting data for the purposes of prediction and decision making given uncertainty

These frameworks are particularly appropriate for modeling genetic systems, since we are missing information concerning the complete set of components and relationships among components that determine genome-phenotype relationships

Conceptual OverviewSystem

Questi

on

Experiment

Sample

Assumptions

InferencePr

ob. M

odels

Statistics

Starting point: a system

• System - a process, an object, etc. which we would like to know something about

• Example: Genetic contribution to height

Genome Height

SNP {A

T

Taller (on average)

Shorter (on average)?

Starting point: a system

• System - a process, an object, etc. which we would like to know something about

• Examples: (1) coin, (2) heights in a population

Coin - same amount of metal on both sides?

Heights - what is the average height in the US?

Experiments (general)

• To learn about a system, we generally pose a specific question that suggests an experiment, where we can extrapolate a property of the system from the results of the experiment

• Examples of “ideal” experiments (System / Experiment):

• SNP contribution to height / directly manipulate A -> T keeping all other genetic, environmental, etc. components the same and observe result on height

• Coin / cut coin in half, melt and measure the volume of each half

• Height / measure the height of every person in the US

Experiments (general)

• To learn about a system, we generally pose a specific question that suggests an experiment, where we can extrapolate a property of the system from the results of the experiment

• Examples of “non-ideal” experiments (System / Experiment):

• SNP contribution to height / measure heights of individuals that have an A and individuals that have a T

• Coin / flip the coin and observe “Heads” and “Tails”

• Height / measure some people in the US

Experiments and samples

• Experiment - a manipulation or measurement of a system that produces an outcome we can observe

• Experimental trial - one instance of an experiment

• Sample outcome - a possible outcome of the experiment

• Sample - the results of one or more experimental trials

• Example (Experiment / Sample outcomes):

• Coin flip / “Heads” or “Tails”

• Two coin flips / HH, HT, TH, TT

• Measure heights in this class / 1.5m, 1.71m, 1.85m, ...

Modeling the results of (non-ideal) experiments

• Mathematics (while not the only approach!) provides a particularly valuable foundation for describing or modeling a system or the outcomes of an experiment

• The reason is that a considerable amount of mathematics is constructed (on purpose!) to provide a good representation of how we think about the world in a way that matches our intuition

• Once constructed, we can use this modeling approach to formalize our intuition in a manner that has currency for others and develop deeper understanding

• In general, mathematics useful for modeling (including probability) can be developed from foundations developed in set theory

• A lot of assumptions, called axioms, are at the foundation of set theory put in place so that set theory produces logical constructions that match our intuition

Sets / Set Operations / Definitions• Set - any collection, group, or conglomerate

• Element - a member of a set

• Set Operations:

• Important Definitions:

• A Special Set:

Union (⇧) � an operator on sets which produces a single set containing all elementsof the sets.

Intersection (⌃) � an operator on sets which produces a single set containing all ele-ments common to all of the sets.

Note that we can think of these as ‘or’ and ‘and’. A simple example of applying the unionoperator is {5�, 5�3��} ⇧ {5�3��, 5�5��} = {5�, 5�3��, 5�5��} and a simple example of intersectionis {5�, 5�3��} ⌃ {5�3��, 5�5��} = {5�3��}. Note that we can write the following generalizationsof these operators:

⇥�

i=1

Ai = A1 ⇧A2 ⇧ ... (1)

⇥⇥

i=1

Ai = A1 ⌃A2 ⌃ ... (2)

where each Ai is a set. Before we leave sets and sample spaces, let’s provide a few otherimportant definitions:

Subset (⇥) � a set that is contained within another set, e.g. {H} ⇥ {H,T}

Complement (Ac) � the set containing all other elements of a set other than A, e.g.{H}c = {T}.

Empty Set (⇤) � the set with no elements.

The empty set is unique and is sometimes represented as { }.

Disjoint Sets � sets with no elements in common.

Note that for disjoint sets Ai and Aj , the following holds: Ai ⌃Aj = ⇤.

5 Probability Functions

To use sample spaces in probability, we need a way to map these sets to the real numbers.To do this, we define a function. Before we consider the specifics of how we define a prob-ability function or measure, let’s consider the intuitive definition of a function:

Function (intuitive def.) � a mathematical operator that takes an input and produces anoutput.

5

Union ([) ⌘ an operator on sets which produces a single set containing all elementsof the sets.

Intersection (\) ⌘ an operator on sets which produces a single set containing all ele-ments common to all of the sets.

Note that we can think of these as ‘or’ and ‘and’. A simple example of applying the unionoperator is {50, 50300} [ {50300, 505000} = {50, 50300, 505000} and a simple example of intersectionis {50, 50300} \ {50300, 505000} = {50300}. Note that we can write the following generalizationsof these operators:

1[

i=1

Ai = A1

[A2

[ ... (1)

1\

i=1

Ai = A1

\A2

\ ... (2)


Subset (⇢) ⌘ a set that is contained within another set, e.g. {H} ⇢ {H,T}

Complement (Ac) ⌘ the set containing all other elements of a set other than A, e.g.{H}c = {T}.

Disjoint Sets ⌘ sets with no elements in common.

Empty Set (;) ⌘ the set with no elements (the empty set is unique and is sometimesand is sometimes represented as { }).


Note that for disjoint sets Ai and Aj , the following holds: Ai \Aj = ;.


To use sample spaces in probability, we need a way to map these sets to the real numbers.To do this, we define a function. Before we consider the specifics of how we define a prob-

ability function or measure, let’s consider the intuitive definition of a function:

Function (intuitive def.) ⌘ a mathematical operator that takes an input and produces anoutput.

6




1[

i=1

Ai = A1

[A2

[ ... (1)

1\

i=1

Ai = A1

\A2

\ ... (2)


Element of (2) ⌘ an object within a set, e.g. H 2 {H,T}







N = {1, 2, 3, ...} (3)

Z = {...� 3,�2,�1, 0, 1, 2, 3, ...} (4)

R = { 0!} (5)

�1 > x >1 (6)

6

Some Special Sets

• The following sets have properties that align with our intuitive conception about how we represent and use groups

• The Natural Numbers and the Integers:

• The Reals:

• Note that these sets are infinite (although they represent two different “sizes” of infinite: countable and uncountable), where we often make use of the following symbols in both cases:




1[

i=1

Ai = A1

[A2

[ ... (1)

1\

i=1

Ai = A1

\A2

\ ... (2)








N = {1, 2, 3, ...} (3)

Z = {�3,�2,�1, 0, 1, 2, 3, ...} (4)

R = { 0!} (5)

�1 > x >1 (6)

6




1[

i=1

Ai = A1

[A2

[ ... (1)

1\

i=1

Ai = A1

\A2

\ ... (2)








N = {1, 2, 3, ...} (3)

Z = {...� 3,�2,�1, 0, 1, 2, 3, ...} (4)

R = { 0!} (5)

�1 > x >1 (6)

6

0-1-3 1-2 2 3

l(✓̂1

|y) = l(�̂µ, �̂a, �̂d|y) (187)

l(✓̂0

|y) = l(�̂µ, 0, 0|y) (188)

x =

2

6664

1 x1,a x

1,d

1 x2,a x

2,d...

.... . .

1 xn,a xn,d

3

7775

�[t] =

2

64�[t]µ

�[t]a

�[t]d

3

75

F[2,n�3]

(y,xa,xd) = f

✓ SSE(

ˆ✓0

)�SSE(

ˆ✓1

)

2

SSE(

ˆ✓1

)

n�3

◆(189)

Pr(µ|y) / N

✓( �2

+Pn

i yi�2

)

( 1

�2

+ n�2

), (

1

�2

+n

�2

)�1

◆(190)

Pr(�a,�d|y) =Z 1

0

Z 1

�1Pr(�µ,�a,�d,�

2

✏ |y)d�µd�2

✏ (191)

�↵ = �a

✓a+

�d2(p

1

� p2

)

◆(192)

�̂µ,0 (193)

H0

: Cov(Y,X) (194)

;R =To see how this is accomplished in a permutation analysis, let’s first describe a permutation.If we write our data in a matrix as follows:

Data =

2

64z11

... z1k y

11

... y1m x

11

... x1N

......

......

......

......

...zn1 ... znk yn1 ... ynm x

11

... xnN

3

75

where the latter columns are the genotypes, a permutation is produced by randomizing thephenotype samples y keeping the genotypes in the same order, e.g.:

Y = �µ +Xa�a +Xd�d +Xz,1�z,1 +Xz,2�z,2 + ✏ (195)

21

Sample Spaces

• Sample Space ( ) - set comprising all possible outcomes associated with an experiment

• Examples (Experiment / Sample Space):

• “Single coin flip” / {H, T}

• “Two coin flips” / {HH, HT, TH, TT}

• “Measure Heights” / any actual measurement OR we could use

• Events - a subset of the sample space

• Examples (Sample Space / Examples of Events):

• “Single coin flip” / , {H}, {H, T}

• “Two coin flips” / {TH}, {HH, TH}, {HT, TH, TT}

• “Measure Heights” / {1.7m}, {1.5m, ..., 2.2m} OR [1.7m], (1.5m,1.8m)

⌦ (7)

F (8)

; 2 F (9)

This A 2 F then Ac 2 F

A1

,A2

, ... 2 F thenS1

i=1

Ai 2 F





This concept is often introduced to us as Y = f(X) where f() is the function that mapsthe values taken by X to Y . For example, we can have the function Y = X2 (see figurefrom class).

We are going to define a probability function which map sample spaces to the real line(to numbers):

Pr(S) : S ! R (10)

where Pr(S) is a function, which we could have written f(S).

To be useful, we need some rules for how probability functions are defined (that is, not allfunctions on sample spaces are probability functions). These rules are are called the axioms

of probability (note that an axiom is a rule that we assume). There is some variation inhow these are presented, but we will present them as three axioms:

Axioms of Probability

1. For A ⇢ S, Pr(A) > 0.

2. Pr(S) = 1.

3. For A1

,A2

, ... 2 S, if Ai\Aj = ; (disjoint) for each i 6= j: Pr(S1

i Ai) =P1

i Pr(A).

These axioms are necessary for many of the logically consistent results built upon proba-bility. Intuitively, we can think of these axioms as matching how we tend to think about

7

l(✓̂1

|y) = l(�̂µ, �̂a, �̂d|y) (187)

l(✓̂0

|y) = l(�̂µ, 0, 0|y) (188)

x =

2

6664

1 x1,a x

1,d

1 x2,a x

2,d...

.... . .

1 xn,a xn,d

3

7775

�[t] =

2

64�[t]µ

�[t]a

�[t]d

3

75

F[2,n�3]

(y,xa,xd) = f

✓ SSE(

ˆ✓0

)�SSE(

ˆ✓1

)

2

SSE(

ˆ✓1

)

n�3

◆(189)

Pr(µ|y) / N

✓( �2

+Pn

i yi�2

)

( 1

�2

+ n�2

), (

1

�2

+n

�2

)�1

◆(190)

Pr(�a,�d|y) =Z 1

0

Z 1

�1Pr(�µ,�a,�d,�

2

✏ |y)d�µd�2

✏ (191)

�↵ = �a

✓a+

�d2(p

1

� p2

)

◆(192)

�̂µ,0 (193)

H0

: Cov(Y,X) (194)

;To see how this is accomplished in a permutation analysis, let’s first describe a permutation.If we write our data in a matrix as follows:

Data =

2

64z11

... z1k y

11

... y1m x

11

... x1N

......

......

......

......

...zn1 ... znk yn1 ... ynm x

11

... xnN

3

75


Y = �µ +Xa�a +Xd�d +Xz,1�z,1 +Xz,2�z,2 + ✏ (195)

21

l(✓̂1

|y) = l(�̂µ, �̂a, �̂d|y) (187)

l(✓̂0

|y) = l(�̂µ, 0, 0|y) (188)

x =

2

6664

1 x1,a x

1,d

1 x2,a x

2,d...

.... . .

1 xn,a xn,d

3

7775

�[t] =

2

64�[t]µ

�[t]a

�[t]d

3

75

F[2,n�3]

(y,xa,xd) = f

✓ SSE(

ˆ✓0

)�SSE(

ˆ✓1

)

2

SSE(

ˆ✓1

)

n�3

◆(189)

Pr(µ|y) / N

✓( �2

+Pn

i yi�2

)

( 1

�2

+ n�2

), (

1

�2

+n

�2

)�1

◆(190)

Pr(�a,�d|y) =Z 1

0

Z 1

�1Pr(�µ,�a,�d,�

2

✏ |y)d�µd�2

✏ (191)

�↵ = �a

✓a+

�d2(p

1

� p2

)

◆(192)

�̂µ,0 (193)

H0

: Cov(Y,X) (194)

;R To see how this is accomplished in a permutation analysis, let’s first describe a permu-tation. If we write our data in a matrix as follows:

Data =

2

64z11

... z1k y

11

... y1m x

11

... x1N

......

......

......

......

...zn1 ... znk yn1 ... ynm x

11

... xnN

3

75


Y = �µ +Xa�a +Xd�d +Xz,1�z,1 +Xz,2�z,2 + ✏ (195)

21

Functions

• Now that we have formalized the concept of a sample space, we need to define what “probability”means

• To do this, we need the concept of a mathematical function

• Function (formally) - a binary relation between every member of a domain to exactly one member of the codomain

• Function (informally) - ?

Example of a function

X

Y

Y = X2

Probability functions (intuition)

• Probability Function (intuition) - we would like to construct a function that assigns a number to each event such that it matches our intuition about the “chance” the event will happen (as a result of an experiment)

• To be useful, we need to assign a number not just to each individual element of the set but to EVERY event

• To accomplish this, we will need the concept of a Sigma Algebra (or Sigma Field)

• What’s more, we need to make sure the function that we use to assign these numbers adheres to a specific set of “rules” (axioms)

Sample Spaces / Sigma Algebra

• Sigma Algebra ( ) - a collection of events (subsets) of of interest with the following three properties: 1. , 2. , 3.

Note that we are interested in a particular Sigma Algebra for each sample space...

• Examples (Sample Space / Sigma Algebra):

• {H, T} /

• {HH, HT, TH, TT} /

• / more complicated to define the sigma algebra of interest (see next slide…)

• Note that the pair is referred to as a measurable space

⌦ (7)

F (8)

; 2 F (9)


A1

,A2

, ... 2 F thenS1

i=1

Ai 2 F







Pr(S) : S ! R (10)





1. For A ⇢ S, Pr(A) > 0.

2. Pr(S) = 1.

3. For A1

,A2


i Ai) =P1

i Pr(A).


7

⌦ (7)

F (8)

; 2 F (9)


A1

,A2

, ... 2 F thenS1

i=1

Ai 2 F







Pr(S) : S ! R (10)





1. For A ⇢ S, Pr(A) > 0.

2. Pr(S) = 1.

3. For A1

,A2


i Ai) =P1

i Pr(A).


7

⌦ (7)

F (8)

; 2 F (9)


A1

,A2

, ... 2 F thenS1

i=1

Ai 2 F







Pr(S) : S ! R (10)





1. For A ⇢ S, Pr(A) > 0.

2. Pr(S) = 1.

3. For A1

,A2


i Ai) =P1

i Pr(A).


7

⌦ (7)

F (8)

; 2 F (9)


A1

,A2

, ... 2 F thenS1

i=1

Ai 2 F







Pr(S) : S ! R (10)





1. For A ⇢ S, Pr(A) > 0.

2. Pr(S) = 1.

3. For A1

,A2


i Ai) =P1

i Pr(A).


7

⌦ (7)

F (8)

; 2 F (9)


A1

,A2

, ... 2 F thenS1

i=1

Ai 2 F

;, {H}, {T}, {H,T} (10)







Pr(S) : S ! R (11)





1. For A ⇢ S, Pr(A) > 0.

2. Pr(S) = 1.

3. For A1

,A2


i Ai) =P1

i Pr(A).

7

⌦ (7)

F (8)

; 2 F (9)


A1

,A2

, ... 2 F thenS1

i=1

Ai 2 F







Pr(S) : S ! R (10)





1. For A ⇢ S, Pr(A) > 0.

2. Pr(S) = 1.

3. For A1

,A2


i Ai) =P1

i Pr(A).


7

l(✓̂1

|y) = l(�̂µ, �̂a, �̂d|y) (187)

l(✓̂0

|y) = l(�̂µ, 0, 0|y) (188)

x =

2

6664

1 x1,a x

1,d

1 x2,a x

2,d...

.... . .

1 xn,a xn,d

3

7775

�[t] =

2

64�[t]µ

�[t]a

�[t]d

3

75

F[2,n�3]

(y,xa,xd) = f

✓ SSE(

ˆ✓0

)�SSE(

ˆ✓1

)

2

SSE(

ˆ✓1

)

n�3

◆(189)

Pr(µ|y) / N

✓( �2

+Pn

i yi�2

)

( 1

�2

+ n�2

), (

1

�2

+n

�2

)�1

◆(190)

Pr(�a,�d|y) =Z 1

0

Z 1

�1Pr(�µ,�a,�d,�

2

✏ |y)d�µd�2

✏ (191)

�↵ = �a

✓a+

�d2(p

1

� p2

)

◆(192)

�̂µ,0 (193)

H0

: Cov(Y,X) (194)

;R To see how this is accomplished in a permutation analysis, let’s first describe a permu-tation. If we write our data in a matrix as follows:

Data =

2

64z11

... z1k y

11

... y1m x

11

... x1N

......

......

......

......

...zn1 ... znk yn1 ... ynm x

11

... xnN

3

75


Y = �µ +Xa�a +Xd�d +Xz,1�z,1 +Xz,2�z,2 + ✏ (195)

21

l(✓̂1

|y) = l(�̂µ, �̂a, �̂d|y) (187)

l(✓̂0

|y) = l(�̂µ, 0, 0|y) (188)

x =

2

6664

1 x1,a x

1,d

1 x2,a x

2,d...

.... . .

1 xn,a xn,d

3

7775

�[t] =

2

64�[t]µ

�[t]a

�[t]d

3

75

F[2,n�3]

(y,xa,xd) = f

✓ SSE(

ˆ✓0

)�SSE(

ˆ✓1

)

2

SSE(

ˆ✓1

)

n�3

◆(189)

Pr(µ|y) / N

✓( �2

+Pn

i yi�2

)

( 1

�2

+ n�2

), (

1

�2

+n

�2

)�1

◆(190)

Pr(�a,�d|y) =Z 1

0

Z 1

�1Pr(�µ,�a,�d,�

2

✏ |y)d�µd�2

✏ (191)

�↵ = �a

✓a+

�d2(p

1

� p2

)

◆(192)

�̂µ,0 (193)

H0

: Cov(Y,X) (194)

;R =(⌦,F) To see how this is accomplished in a permutation analysis, let’s first describe apermutation. If we write our data in a matrix as follows:

Data =

2

64z11

... z1k y

11

... y1m x

11

... x1N

......

......

......

......

...zn1 ... znk yn1 ... ynm x

11

... xnN

3

75


Y = �µ +Xa�a +Xd�d +Xz,1�z,1 +Xz,2�z,2 + ✏ (195)

21

Problem 2 (Medium)

Assume that the system we are interested in is a coin. The experiment we will consider is two flipsof the coin. Note: for parts (d)-(j) use the probability model in part (c).

a. What is the sample space of this experiment?

⌦ = {HH,HT, TH, TT}

b. What is the Sigma-algebra (containing all events) for this sample space? Which of theseevents is the event ‘the first flip is heads’? Which of these events is the event ‘the second flipis heads’?

F =;, {HH}, {HT}, {TH}, {TT}, {HH,HT}, {HH,TH}, {HH,TT}, {HT, TH}, {HT, TT},{TH, TT}{HH,HT, TH}, {HH,HT, TT}, {HH,TH, TT}, {TH,HT, TT}{HH,TH,HT, TT}

{H1st} = {HH,HT}

{H2nd} = {HH,TH}

c. Define a probability model such that the probability of a ‘heads’ on the first flip and thesecond flip is Pr(H1st) = Pr(H2nd) = 0.5, where the probability of heads on both the firstand second flip is Pr(H1st\H2nd) = 0.3. Write out the probabilities for all possible outcomesof an experimental trial. Write down the formulas or relationships you used to calculate theseprobabilities as part of your answer.

H2nd T2nd

H1st Pr(H1st \H2nd) Pr(H1st \ T2nd) Pr(H1st)

T1st Pr(T1st \H2nd) Pr(T1st \ T2nd) Pr(t1st)

Pr(H2nd) Pr(T2nd)

H2nd T2nd

H1st 0.3 Pr(H1st) - Pr(H1st \H2nd) 0.5

T1st Pr(H2nd) - Pr(H1st \H2nd) Eq 0.5

0.5 0.5

Eq = 1 - Pr(H1st \H2nd) + Pr(H1st \ T2nd) + Pr(T1st \H2nd)

H2nd T2nd

H1st 0.3 0.2 0.5

T1st 0.2 0.3 0.5

0.5 0.5

2

That’s it for today

• Next lecture, we will introduce random variables, and random vectors

Quantitative Genomics and Geneticsmezeylab.cb.bscb.cornell.edu/labmembers/documents... · Quantitative Genomics and Genetics BTRY 4830/6830; PBSB.5201.03 Lecture 2: Introduction to

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