#### Transcript Statistical Concepts and Methodologies for Data Analyses

```Statistical Concepts and
Methodologies for Data Analyses
Benilton Carvalho
Computational Biology and Statistics Group
Department of Oncology
University of Cambridge
FROM RANDOM VARIABLES TO
HYPOTHESIS TESTING
Random Variables
• Function that associates probability to:
– Countable items (discrete random variable);
• Tumor vs. Normal; Yes vs. No; Head vs. Tail;
– Uncountable items (continuous random variable):
• Log-expression; weight; height;
• Characterized by a distribution function:
– Bernoulli; Binomial; Geometric; NegativeBinomial; Poisson;
– Normal; Student’s t; Gamma;
Examples – Discrete Distributions
Examples – Continuous Distributions
Common Uses of
Different Distributions
• Bernoulli: probability of 1 success;
• Binomial: probability of K successes;
• Geometric: probability of K failures before 1st
success;
• Negative-Binomial: probability of K failures
before R successes;
• Poisson: probability of K rare events;
The Questions
• Investigation of populations or groups within a
– How does BRCAI behave across groups?
– Can genotype predict drug response?
– Does transcript abundance change as a function
of time?
The Experiment
• A procedure used to answer the questions;
• Comprised of multiple items:
– Population;
– Sample;
– Hypotheses;
– Test statistic;
– Rejection criteria;
Population
• Superset of subjects of interest;
• Ideally, every subject in the population is
surveyed;
• Issues with the “census approach”;
Sample
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Select some subjects from the population;
We refer to this subset as sample;
Subject in a sample can be called replicate;
Replicate: technical vs. biological;
Hypotheses
• Sets that define the “underlying truth”;
• Null Hypothesis (H0): default situation.
– Cannot be proven;
– Reject (in favor of H1) vs. fail to reject;
• Alternative Hypothesis (H1): alternative (duh!)
– Complements H0 on the parametric space;
– Assists on the definition of the rejection criteria.
Examples of Hypotheses – P1
• Comparing expression: Tumor vs. Normal:
– Expression on tumor is at most as high as on normal;
– Expression on tumor is higher than on normal;
Examples of Hypotheses – P2
• Comparing expression: Tumor vs. Normal:
– Expression on tumor is at least as low as on normal;
– Expression on tumor is lower than on normal;
Examples of Hypotheses – P3
• Comparing expression: Tumor vs. Normal:
– Expressions on tumor and normal are the same;
– Expressions on tumor and normal are different;
Test Statistic
•
•
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•
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Summary of the data;
Built “under H0”;
Independent of unknown parameters;
Known distributions;
Compatibility between data and H0;
Test Statistic
• What the statistician see…
Rejection Criteria
• Function of three factors:
– Test statistic;
– Hypotheses;
– Type I Error (False Positive), α;
• Determines thresholds used to reject H0:
– One threshold: one-sided tests;
– Two thresholds: two-sided tests;
• Defines what is “extreme” for the experiment;
Rejection Criteria
From Rejection Criteria to P-value!
p-value
Rejection Criteria
From Rejection Criteria to P-value!
p-value
Rejection Criteria
From Rejection Criteria to P-value!
p-value
Sampling and testing
Discrete
observations
Random sample of 10 balls
from the box
#red = 3
When do I think that I am not sampling from
this box anymore?
How many reds could I expect to get just by
chance alone!
10% red balls and 90%
blue balls
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Sample
Discrete
observations
Random sample of 10 balls
from the box
#red = 3
Test statistic
Rejection criteria
sample, do you have evidence
to reject the hypothesis that
you sampled from the null
population)
10% red balls and 90%
blue balls
Null hypothesis
being sampled)
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Sample
Continuous
observations
4, 2.3, 5.2, 4.7, 2.1, 3.5, ……..
mean = 3, sd = 0.6
Test statistic
Rejection criteria
sample, do you have evidence
to reject the hypothesis that
you sampled from the null
population)
Null hypothesis
being sampled)
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Summary of the Experiment
1) hypotheses
4) decision
2) sample
3) test statistic
Useful Facts
• The Law of the Large Numbers guarantees
that the larger the sample size is, the closer
the sample average is to the actual mean;
• Normality assumption isn’t that important
with large sample size;
• The Central Limit Theorem states that the
average is asymptotically normal;
Useful Facts
• The Z-score depends on the precise
knowledge of the variance term:
• Estimating the variance changes the
distribution of the test statistic:
Useful Facts
• The Student’s t distribution is similar to the
Normal distribution, but has heavier tails;
• Larger sample size, more d.f.;
• More d.f., closer to Normal;
Multiple Testing
• We are doing high-throughput experiments;
• Comparing thousands of units simultaneously;
• At this scale, we can observe several instances
of rare events just by chance:
– Event A: 1 in 1000 chance of happening;
– Event B: 999 in 1000 chance of happening;
– And the experiment is tried 20,000 times;
– We expect 20 occurrences of Event A to be
observed, although Event B is much more likely;
Multiple Testing
•
•
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•
•
•
Similar scenario, for example, with DE;
Most genes are not differentially expressed;
High-throughput experiments;
Differential expression is tested for 20K genes;
Need to protect against false positives;
Suggestion: use non-specific filtering;
DATA MODELING
What is a model?
Statistical Models
•
•
•
•
There is no “correct model”;
Models are approximations of the truth;
There is a “useful model”;
Understand the mechanisms of the system for
better choices of model alternatives;
Revisiting Microarrays
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•
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Scanned images;
Fluorescence intensities;
Proportional to target abundances;
Restricted dynamic range;
Asymmetrical distribution;
Log-Intensities behave better;
Revisiting Microarrays
Intensities
Log-Intensities
Back to Data Modeling
Linear Regression / ANOVA
• Nature of the data: continuous;
• Linear regression often used;
• For subject i, known factors/covariates are
candidates to predict log-intensities of a gene:
• Residuals expected to be Normal;
Interpreting Coefficients
• Statisticians indicate that a parameter is
estimated by using a “hat” on top of it:
• Assuming that X = 0 for normal tissue:
• Assuming that X = 1 for tumor tissue:
Interpreting Coefficients
Average log-intensity for normal tissue
Change in average log-intensity
associated to the tumor tissue
Average log-intensity for tumor tissue
GLM
•
•
•
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Generalized Linear Models;
Generic framework;
Accommodates different types of data;
Special cases: Linear regressions and ANOVAs;
Example – GLM Binomial Family
• Predictors: Gene expression / genotype / age;
• Example:
– Response: Cytogenetic abnormalities (Yes/No);
– Predictors: Log-expression of probeset 1059_at;
Log-Expression vs. Abnormalities
Modeling a Binary Response
• Response in the previous example:
– Observed cytogenetic abnormalities;
– Did not observe cytogenetic abnormalities;
• Linear regression does not work:
1059_at
1
0.8
0.6
0.4
0.2
0
3
3.2
3.4
3.6
3.8
4
Modeling a Binary Response
• Instead of modeling the actual response, we
model the probability of that response;
• Linear regression still fails;
Prob(Observe Cytogenetic Abnormalities)
200%
Invalid
Results
150%
100%
Valid
Results
50%
0%
-50%
-100%
2
2.5
3
3.5
4
4.5
5
Logistic Regression - Rationale
• Probability is restricted to the [0, 1] interval;
• Linear regression isn’t;
• Need to transform probability;
Logistic Regression - Rationale
• Instead of probability, model the odds:
• Odds range from 0 to Infinity;
• A linear regression approach would still fail;
Logistic Regression - Rationale
• Instead of odds, model the log-odds:
• Log-odds range from -Infinity to Infinity;
• An approach like linear regression, using the
log-odds scale, would work fine;
Back to GLM
• In the previous example:
Linear Predictor
Interpreting Coefficients
on a Logistic Model
• b0: average log-odds for normal tissue;
• b1: average change in log-odds on tumor;
• Suppose b0 = 10.87 and b1 = -3.46:
– How do we interpret?
Model Selection
• Likelihood measures the probability of
observing the data under a certain model;
• Given two models, M1 and M2 (M2⊃M1):
– Get L1: likelihood of the data under M1;
– Get L2: likelihood of the data under M2;
• LRT = -2 log(L1/L2) is known;
– Small LRT: choose M1;
– Large LRT: choose M2;
MODELING STRATEGIES FOR
SEQUENCING DATA
Sequencing – Rationale
Technical Replicate
• Sample j, transcript i is generated at rate λij;
• A fragment attaches to the flow cell with a
(low) probability pij;
• Number of observed tags, yij, is Poisson
distributed with rate proportional to λijpij;
Adapted from notes by Tom Hardcastle
Poisson
Probability function:
Analysis method: GLM
Expected count of
region i in sample j
Noise Part
Design matrix
Deterministic
Part
Library size
effect
(Differential) effect
for region i
Need to account for extra variability
technical rep – consistent with Poison
biol. rep – not consistent with Poison
Based on the data of Nagalakshmi et al.
Science 2008; slide adapted from Huber;
Sequencing – Rationale
Biological Replicates
• For subject j, on transcript i:
• Different subjects have different rates, which
we can model through:
• This hierarchy changes the distribution of Y:
Negative Binomial
Probability function:
source of variation
smooth dispersion-mean relation α
CONSIDERATIONS ON
EXPERIMENT DESIGN
Consideration
• Sample size is crucial. The larger, the better;
• With differential expression, one can observe
this more easily;
• Is RNA-Seq really worth it when we consider:
– Cost,
– Strategies for analysis, and
– Technical requirements?
Differential Expression Across Groups
Flow Cell Confounded With Group
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Flow Cell 1
Flow Cell 2
Flow Cell 3
Flow Cell 4
Group A
Group B
Group C
Group D
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Differential Expression Across Groups
Randomize Samples wrt Flow Cell
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Flow Cell 1
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Flow Cell 2
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Flow Cell 3
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Flow Cell 4
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Differential Expression Across Groups
Barcoding vs. Lane Effect
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Flow Cell 1
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Flow Cell 2
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Flow Cell 3
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Flow Cell 4
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CONSIDERATIONS ON
DATA PROCESSING
Normalization
• Samples are sequenced in different depths:
Gene
Gene 1
…
Gene N
Sample 1
Sample 2
500,000
…
500,000
…
0
500,000
15,000,00
0
30,000,00
0
• Genes with higher expression on Sample 2;