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Introduction to Bayesian Statistics Introduction to Bayesian Statistics and an Application and an Application Timothy M. Bahr Unconfounding the Confounded: Separating Introduction Treatment and Batch Effects in Confounded Definitions

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Introduction to Bayesian Statistics and an Application

Unconfounding the Confounded: Separating Treatment and Batch Effects in Confounded Microarray Experiments Timothy M. Bahr

Department of Statistics Brigham Young University

March 16, 2009

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Introduction Who am I?

Tim Bahr, Undergrad...

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Introduction Who am I?

Tim Bahr, Undergrad...

◮ 22, B.S. in Statistics,

emphasis: Biostat

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Introduction Who am I?

Tim Bahr, Undergrad...

◮ 22, B.S. in Statistics,

emphasis: Biostat

◮ My first intro to Statistics

in High School

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Introduction Who am I?

Tim Bahr, Undergrad...

◮ 22, B.S. in Statistics,

emphasis: Biostat

◮ My first intro to Statistics

in High School

◮ Fascination with the

Numerical Patterns in Science

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Introduction Who am I?

Tim Bahr, Undergrad...

◮ 22, B.S. in Statistics,

emphasis: Biostat

◮ My first intro to Statistics

in High School

◮ Fascination with the

Numerical Patterns in Science

◮ Future Goals

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Introduction Who are you?

Bioinformatics

◮ Majors? ◮ Math/Stat Background? ◮ Microarrays? ◮ Research? ◮ Why Bioinformatics? ◮ Can I tell you what I think

about Bioinformatics?

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Definitions

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Definitions

◮ Bayesian Statistics >>> statistical inferences on

experimental data + prior knowledge.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Definitions

◮ Bayesian Statistics >>> statistical inferences on

experimental data + prior knowledge.

◮ Classical (Frequentist) Statistics >>> data from

bservations or experiments only.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Definitions

◮ Bayesian Statistics >>> statistical inferences on

experimental data + prior knowledge.

◮ Classical (Frequentist) Statistics >>> data from

bservations or experiments only.

◮ Prior Distribution: The distribution we assume our

parameters come from.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Definitions

◮ Bayesian Statistics >>> statistical inferences on

experimental data + prior knowledge.

◮ Classical (Frequentist) Statistics >>> data from

bservations or experiments only.

◮ Prior Distribution: The distribution we assume our

parameters come from.

◮ Gibbs Sampling (simplification): An algorithm that

allows us to give interatively infer point estimates for “random” parameters.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Definitions

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Definitions

◮ Biostatistics: The application of statistics to a wide

range of topics in biology.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Definitions

◮ Biostatistics: The application of statistics to a wide

range of topics in biology.

◮ Gene Expression Microarray: A high-throughput

technology in molecular biology used to detect gene expression levels in a cellular sample.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Definitions

◮ Biostatistics: The application of statistics to a wide

range of topics in biology.

◮ Gene Expression Microarray: A high-throughput

technology in molecular biology used to detect gene expression levels in a cellular sample.

◮ Confounded Experiment: when two or more variables

vary together so that it is impossible to separate their unique effects.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

Probabilistic inference that computes the distribution of the model parameters and gives prediction for previously unseen input values probabilistically. Freqentist

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

Probabilistic inference that computes the distribution of the model parameters and gives prediction for previously unseen input values probabilistically. Freqentist

◮ θ, parameters, are fixed

and unknown

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

Probabilistic inference that computes the distribution of the model parameters and gives prediction for previously unseen input values probabilistically. Freqentist

◮ θ, parameters, are fixed

and unknown

◮ X, random variables

(data), are random Bayesian

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

Probabilistic inference that computes the distribution of the model parameters and gives prediction for previously unseen input values probabilistically. Freqentist

◮ θ, parameters, are fixed

and unknown

◮ X, random variables

(data), are random Bayesian

◮ θ, parameters, are random

and unknown

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

Probabilistic inference that computes the distribution of the model parameters and gives prediction for previously unseen input values probabilistically. Freqentist

◮ θ, parameters, are fixed

and unknown

◮ X, random variables

(data), are random Bayesian

◮ θ, parameters, are random

and unknown

◮ X, random variables

(data), are random “If you want to work on really interesting problems [Bayesian Inference] is where those problems lie”

Don Rubin, Ph.D., Dept. Chair, Harvard Statistics

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference The idea of a prior

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference The idea of a prior

◮ Frequentists assume a parameter is fixed:

◮ For example X ∼ N(µ, σ2) ◮ µ is a fixed unknown value

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference The idea of a prior

◮ Frequentists assume a parameter is fixed:

◮ For example X ∼ N(µ, σ2) ◮ µ is a fixed unknown value

◮ What if µ is not fixed? What if it too can assume a

distribution with variation

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference The idea of a prior

◮ Frequentists assume a parameter is fixed:

◮ For example X ∼ N(µ, σ2) ◮ µ is a fixed unknown value

◮ What if µ is not fixed? What if it too can assume a

distribution with variation

◮ We assume a prior on µ. i.e. µ ∼ N(mµ, s2 µ)

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

Bayes’ Theorem, based on basic theories of probability: π(θ|x) = f(x|θ)π(θ)

f(x|θ)π(θ)dθ

(1)

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

Bayes’ Theorem, based on basic theories of probability: π(θ|x) = f(x|θ)π(θ)

f(x|θ)π(θ)dθ

(1)

◮ π(θ|x) is the posterior distribution of our parameters, θ.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

Bayes’ Theorem, based on basic theories of probability: π(θ|x) = f(x|θ)π(θ)

f(x|θ)π(θ)dθ

(1)

◮ π(θ|x) is the posterior distribution of our parameters, θ. ◮ f(x|θ) is the likelihood of the data

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

Bayes’ Theorem, based on basic theories of probability: π(θ|x) = f(x|θ)π(θ)

f(x|θ)π(θ)dθ

(1)

◮ π(θ|x) is the posterior distribution of our parameters, θ. ◮ f(x|θ) is the likelihood of the data ◮ π(θ) is the prior distribution assumed on our parameters,

θ.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Bayesian Inference

In the End: Estimate Parameters

◮ We solve for the posterior of the parameters ◮ Use different methods to estimate an “optimum” value of

ur parameters.

◮ Take the Expected Value of a Parameter ◮ Gibbs Sampling ◮ Metropolis-Hastings

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Microarrays What is a Microarray?

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Microarrays What is a Microarray?

◮ We use microarrays to detect gene expression levels for a

given cellular sample.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Microarrays

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Microarrays

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Microarrays

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments

◮ Consider a fertilizer experiment with corn:

First, An “unconfounded” experiment.

◮ 1 plot of corn; left half- control (no fertilizer), right half-

treatment (Fertilizer)

◮ Differences in corn quality can be attributed to the

treatment effect.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments What is a Confounded Experiment?

◮ Consider a fertilizer experiment with corn:

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments What is a Confounded Experiment?

◮ Consider a fertilizer experiment with corn: ◮ Plot 1 (Batch 1)

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments What is a Confounded Experiment?

◮ Consider a fertilizer experiment with corn: ◮ Plot 1 (Batch 1) ◮ Control (no fertilizer)

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments What is a Confounded Experiment?

◮ Consider a fertilizer experiment with corn: ◮ Plot 1 (Batch 1) ◮ Control (no fertilizer) ◮ Plot 2 (Batch 2) - 1 mi. away

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments What is a Confounded Experiment?

◮ Consider a fertilizer experiment with corn: ◮ Plot 1 (Batch 1) ◮ Control (no fertilizer) ◮ Plot 2 (Batch 2) - 1 mi. away ◮ Treatment (New Fertilizer)

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments What is a Confounded Experiment?

◮ Consider a fertilizer experiment with corn:

If we observe a significant difference between the corn quality

f the two plots (batches), can we attribute this difference to

the fertilizer?

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments What is a Confounded Experiment?

◮ Consider a fertilizer experiment with corn:

No. The difference may be due to the treatment effect, the

plot (batch effect), or a combination of the two.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments What is a Confounded Experiment?

◮ Consider a fertilizer experiment with corn:

The Treatment Effect is confounded with the Plot or Batch Effect.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments What is a Confounded Experiment?

◮ Consider a fertilizer experiment with corn:

The same principle applies to microarray experiments.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

◮ Microarrays prepared at different times, in different places,

by different people etc. ... are often confounded by batch effects.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

◮ Microarrays prepared at different times, in different places,

by different people etc. ... are often confounded by batch effects.

◮ We are not interested in the the batch effect. We want to

subtract it out.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

◮ Microarrays prepared at different times, in different places,

by different people etc. ... are often confounded by batch effects.

◮ We are not interested in the the batch effect. We want to

subtract it out.

◮ Our algorithm uses statistical methods to adjust for the

Batch effect in confounded microarray experiments.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

Why?

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

Why?

◮ Often times biologists can save money by using data that

was obtained in previous experiments.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

Why?

◮ Often times biologists can save money by using data that

was obtained in previous experiments.

◮ Inter-lab collaboration becomes much more reliable when

batch effects are accounted for.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

Our Solution

Our method allows precise estimation of the batch effect and the treatment effect.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

Our Solution

Our method allows precise estimation of the batch effect and the treatment effect.

◮ A dynamic linear model

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

Our Solution

Our method allows precise estimation of the batch effect and the treatment effect.

◮ A dynamic linear model ◮ Novel yet Appropriate Assumptions

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Confounded Experiments: Microarrays

Our Solution

Our method allows precise estimation of the batch effect and the treatment effect.

◮ A dynamic linear model ◮ Novel yet Appropriate Assumptions ◮ Bayesian Statistical Methods

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Our Model The Model: yig = µg + Xiαg + Ziτg + εig (2)

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Our Model The Model: yig = µg + Xiαg + Ziτg + εig (2)

◮ yig - the “expression level” for a sample i from gene g

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Our Model The Model: yig = µg + Xiαg + Ziτg + εig (2)

◮ yig - the “expression level” for a sample i from gene g ◮ µg - an overall average for gene g

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Our Model The Model: yig = µg + Xiαg + Ziτg + εig (2)

◮ yig - the “expression level” for a sample i from gene g ◮ µg - an overall average for gene g ◮ αg - the Treatment Effect for gene g

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Our Model The Model: yig = µg + Xiαg + Ziτg + εig (2)

◮ yig - the “expression level” for a sample i from gene g ◮ µg - an overall average for gene g ◮ αg - the Treatment Effect for gene g ◮ τg - the Batch Effect for gene g

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Our Model The Model: yig = µg + Xiαg + Ziτg + εig (2)

◮ yig - the “expression level” for a sample i from gene g ◮ µg - an overall average for gene g ◮ αg - the Treatment Effect for gene g ◮ τg - the Batch Effect for gene g ◮ εig - error for sample i from gene g

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Solution Formulation and Assumptions

First, an “unconfounded” formulation.

Difference between treatment and control can be attributed to “treatment effect.”

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Solution Formulation and Assumptions

We can’t differentiate the values of αg and τg.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Solution Formulation and Assumptions

We assume treatment, αg, has no effect on group 2 genes

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Solution Formulation and Assumptions

Determine which genes in each group >>> estimate αg and τg.

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

How do we estimate αg and τg?

Gibbs Sampling

◮ A Bayesian Method ◮ Gives us the power to estimate which genes are in each

group

◮ Iteratively estimates values until sequence converges

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Estimating αg

5 10 15 20 25 30 35 2 4 6 8 10 12

Estimating Alpha

iterations alpha actual

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Estimating τg

5 10 15 20 25 30 35 2 3 4 5

Estimating Tau

iterations tau actual

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Application

Possible Applications

◮ Microarrays in Cancer Research ◮ Clinical use of microarrays for diagnosis ◮ Possible applications in non-array experiments

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Introduction to Bayesian Statistics and an Application Timothy M. Bahr Introduction Definitions Bayesian Statistics Microarrays Confounded Experiments Model Gibbs Sampling Application

Acknowledgments

◮ W. Evan Johnson, mentor ◮ Nathaniel Gustafson, programmer ◮ BYU Dept. of Statistics ◮ Johnson Lab