Introduction to Statistics: Frequentist & Bayesian Approaches (for Non-Statisticians)
Refresher on Bayesian and Frequentist...
Transcript of Refresher on Bayesian and Frequentist...
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Refresher on Bayesian and Frequentist Concepts
Bayesians and FrequentistsModels, Assumptions, and Inference
George Casella
Department of Statistics
University of Florida
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ACCP 37th Annual Meeting, Philadelphia, PA [1]
Approaches to Statistics
◮ Frequentists: From Neymann/Pearson/Wald setup. An orthodox view that sampling is
infinite and decision rules can be sharp.
◮ Bayesians: From Bayes/Laplace/de Finetti tradition. Unknown quantities are treated
probabilistically and the state of the world can always be updated.
◮ Likelihoodists: From Fisher. Single sample inference based on maximizing the likelihood
function and relying on the Birnbaum (1962) Theorem. Bayesians - But they don’t
know it.
◮ So let’s look at some critical differences between Frequentists and Bayesians. . .
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Differences Between Bayesians and Non-BayesiansAccording to my friend Jeff Gill
Typical Bayesian Typical Non-Bayesian
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Differences Between Bayesians and Non-BayesiansWhat is Fixed?
Frequentist:
◮ Data are a repeatable random sample
- there is a frequency
◮ Underlying parameters remain con-
stant during this repeatable process
◮ Parameters are fixed
Bayesian:
◮ Data are observed from the realized
sample.
◮ Parameters are unknown and de-
scribed probabilistically
◮ Data are fixed
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Example: Application of Bayes Theorem toAminoglycoside-Associated Nephrotoxicity (AAN)
◮ Kim et al. (2004 Journal of Clinical Pharmacology )
◮ Examine the incidence of AAN related to
⊲ Extended-interval dosing (EID)
⊲ Individualized pharmacokinetic monitoring (IPM)
⊲ Multiple-daily dosing (MDD)
◮ Meta-analysis of published results
◮ Bayesian methods used
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Example: Application of Bayes Theorem to AAN-The Data-
EID IPM MDD
Incidence of AAN Incidence of AAN Incidence of AAN
Nephrotoxicity Related Nephrotoxicity Related Nephrotoxicity Related
34 8 80 13 66 11
179 25 62 0 1756 129
141 15 36 5 272 48
187 14 98 12 151 18
Studies ... ... ... ... ... ...
71 11 95 14 146 28
40 2 78 7 140 14
35 0 113 11
34 2 108 10
61 9
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Example: Application of Bayes Theorem to AAN-Histograms-
◮ Histograms of Relative Frequencies of AAN
◮ Protocols have similar means but different patterns
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Example: Application of Bayes Theorem to AAN-Sampling Models-
Frequentist:
◮ For Protocol i, = 1, 2, 3, X=AAN
frequency
◮ For Study j in Protocol i
⊲ Xj ∼ Binomial(nj, pi)
◮ pi is the same for each study
Bayesian:
◮ For Protocol i, = 1, 2, 3, X=AAN
frequency
◮ For Study j in Protocol i
⊲ Xj ∼ Binomial(nj, pi)
◮ pi can vary from study to study
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Example: Application of Bayes Theorem to AAN-Inference-
Frequentist:
◮ The true AAN rates p1, p2, p3 are
fixed
◮ The data are repeated
◮ Determine if p1, p2, p3 are different
Bayesian:
◮ The data from the studies are fixed
◮ The true AAN rates p1, p2, p3 can
vary
◮ Determine if p1, p2, p3 are different
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Differences Between Bayesians and Non-BayesiansWhat is Fixed?
Frequentist:
◮ Data are a repeatable random sample
- there is a frequency
-The studies are repeatable
◮ Underlying parameters remain con-
stant during this repeatable process
-The studies (in protocol) have same
AAN rate
◮ Parameters are fixed
Bayesian:
◮ Data are observed from the realized
sample
-The studies are fixed
◮ Parameters are unknown and de-
scribed probabilistically
-The studies (in protocol) have vary-
ing AAN rates
◮ Data are fixed
◮ We see why Kim et al. used Bayesian Inference
◮ Difficult to assume that this “experiment” is repeatable
◮ The collection of studies is a one-time phenomenon
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Differences Between Bayesians and Non-BayesiansGeneral Inference
Frequentist:
◮ Point estimates and standard errors
or 95% confidence intervals.
◮ Deduction from P (data|H0), by set-
ting α in advance.
◮ Accept H1 if P (data|H0) < α.
◮ Accept H0 if P (data|H0) ≥ α.
Bayesian:
◮ Induction from P (θ|data), starting
with P (θ).
◮ Broad descriptions of the posterior
distribution such as means and quan-
tiles.
◮ Highest posterior density intervals in-
dicating region of highest posterior
probability, regardless of contiguity.
◮ Frequentist: P (data|H0) is the sampling distribution of the data given the parameter
◮ Bayesian: P (θ) is the prior distribution of the parameter (before the data are seen)
⊲ P (θ|data) is the posterior distribution of the parameter
⊲ Update of the prior with the data (more later)
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Differences Between Bayesians and Non-Bayesians90% Intervals
Frequentist:
◮ In repeated sampling
90% of realized intervals
cover the true parameter
Bayesian:
◮ For these data, with
probability 90% the parameter
is in the interval
◮ These are different probabilities
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Example: Application of Bayes Theorem to AAN-Construction of Confidence Intervals-
For Protocol i, = 1, 2, 3, X=AAN frequency
Frequentist:
◮ For Study j in Protocol i
⊲ Xj ∼ Binomial(nj, pi)
◮ pi is the same for each study
◮ Describe variability in Xj for fixed pi
Bayesian:
◮ For Study j in Protocol i
⊲ Xj ∼ Binomial(nj, pi)
◮ pi has a prior distribution
◮ Describe variability in pi for fixed Xj
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-Construction of Confidence Intervals-
Frequentist:
◮ Describe variability in Xj for fixed pi
Bayesian:
◮ Describe variability in pi for fixed Xj
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Three General Steps for Bayesian Modeling
I. Specify a probability model for unknown parameter values that includes some prior
knowledge about the parameters if available.
II. Update knowledge about the unknown parameters by conditioning this probability
model on observed data.
III. Evaluate the fit of the model to the data and the sensitivity of the conclusions to the
assumptions. (Another time)
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Where Do Priors Come From?
◮ Previous studies, published work.
◮ Researcher intuition.
◮ Substantive Experts
◮ Convenience (conjugacy, vagueness).
◮ Nonparametrics and other data sources.
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Kim et al. 2004 Priors
◮ First Prior
◮ From Review of Literature
◮ And Expert Judgement
◮ Second Prior
◮ Eliminate Extremes
◮ AAN > 35% Unlikely
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Priors and Posteriors -Posterior Interval-
◮ Prior Distribution
◮ Before Data are Seen
◮ Posterior Distribution
◮ Prior Updated with Data
◮ 90% “Credible” Interval
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Priors and Posteriors -Effect of the Prior-
◮ Prior has high variability
◮ Data information is very strong
◮ Intervals Similar
◮ But Inference Remains Different
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ACCP 37th Annual Meeting, Philadelphia, PA [19]
Interpretations of Confidence
◮ Frequentist: A collection of intervals
with 90% of them containing the true
parameter
Coverage
Confidence Inte
rvals
θ
◮ Bayesian: An interval that has a 90%
chance of containing the
true parameter.
• Which interpretation preferred?.
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ACCP 37th Annual Meeting, Philadelphia, PA [20]
So Why Did Frequency Win?
◮ 1950 − 1990 Nobody did Bayesian Analysis
⊲ Well Some, but on the fringe
◮ We want very automated, “cookbook” type procedures - or that is what we sold.
◮ Computers were slow and relatively unavailable.
◮ Bayesian Statistics need Lots of computation
And the everything changed....
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ACCP 37th Annual Meeting, Philadelphia, PA [21]
The History of Bayesian Statistics–Milestones
◮ Reverend Thomas Bayes (1702-
1761).
◮ Pierre Simon Laplace.
◮ Pearson (Karl), Fisher, Neyman
and Pearson (Egon), Wald.
◮ Jeffreys, de Finetti, Good, Savage,
Lindley, Zellner.
◮ A world divided (mainly over prac-
ticality).
◮ The revolution: Gelfand and Smith
(1990).
◮ Today. . .
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ACCP 37th Annual Meeting, Philadelphia, PA [22]
Technologies that have changed my life:
◮ microwave ovens
◮ ATM machines
◮ pay-at-the-pump
◮ Gibbs sampling
1.01.52.02.53.0
20
40
60
80
10
0
Lambda
k
0.40.60.81.01.2
20
40
60
80
10
0
Phi
k
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Markov Chain Monte CarloGibbs Sampling and Variations
◮ Computational algorithms
◮ Cracked open countless problems
◮ Research explosion 1990 − 2005
◮ Allowed solutions of
⊲ Practical Problems
⊲ Complex Models
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PKPD Medical Models - An Example from the Book
◮ Pharmacokinetics is the modeling of the relationship between the dosage of a drug and
the resulting concentration in the blood.
◮ Estimate pharmacokinetic parameters using mixed-effects model and
◮ For a given dose di administered at time 0 to patient i, the measured log concentration
in the blood at time tij, Xij, is assumed to follow a normal distribution
Xij ∼ N(log gij, σ2),
gij(λi) =di
Vi
exp
(
−Ci tij
Vi
)
.
◦ Ci represents clearance
◦ Vi represents volume for patient i.
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ACCP 37th Annual Meeting, Philadelphia, PA [25]
PKPD Medical Models - Cadralazine Concentration
◮ Wakefield et al. 1994 applied Statistics
◮ Data on 10 Cardiac Failure Patients
◮ Plasma Concentration after 30mg Dose
Hours After Administration
Patient 2 4 6 8 10 24
1 1.09 0.7 0.53 0.34 0.23 0.02
2 2.03 1.28 1.2 1.02 0.83 0.28
3 1.44 1.3 0.95 0.68 0.52 0.06
4 1.55 0.96 0.8 0.62 0.46 0.08
5 1.35 0.78 0.5 0.33 0.18 0.02
6 1.08 0.59 0.37 0.23 0.17 0
7 1.32 0.74 0.46 0.28 0.27 0.03
8 1.63 1.01 0.73 0.55 0.41 0.01
9 1.26 0.73 0.4 0.3 0.21 0
10 1.3 0.7 0.4 0.257 0.14 0
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PKPD Medical ModelsEstimates of Log Clearance and Log Volume
Patient Log Clearance Log Volume
1 1.269 2.854
2 0.2877 2.624
3 0.6723 2.721
4 0.8287 2.71
5 1.219 2.642
6 1.472 2.763
7 1.257 2.666
8 0.8884 2.599
9 1.309 2.682
10 1.328 2.624
◮ Std Error ≈ .07
◮ Variability in Clearance
◮ Lesser variability in Volume?
◮ With MCMC we can see more
◮ Have the entire posterior distribution for C and V
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PKPD Medical ModelsPosterior Distribution of Log Clearance For All Patients
0.8 1.2 1.6
02
46
0.0 0.4 0.8
02
46
0.6 1.0
02
46
8
0.7 0.9 1.1
02
46
8
0.8 1.0 1.2 1.4
02
46
810
1.0 1.4
02
46
0.8 1.2
02
46
0.7 1.0 1.3
02
46
8
0.9 1.2 1.5
02
46
8
0.9 1.2 1.5
02
46
8
◮ Large Mean Differences ◮ Evidence of Bimodality
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PKPD Medical ModelsPosterior Distribution of Log Volume For All Patients
1.0 2.0 3.0
01
23
4
1.0 2.0
01
23
1.0 2.0 3.0
01
23
45
1.0 2.0 3.0
01
23
4
1.0 2.0
01
23
45
1.0 2.0 3.0
01
23
4
1.0 2.0 3.0
01
23
45
1.0 2.0
01
23
45
1.0 2.0 3.0
01
23
41.0 2.0
01
23
4
◮ Small Mean Differences ◮ Evidence of Bimodality
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PKPD Medical Models - Shape of the Curves
◮ Multimodal parameter distribution ⇒ distinct curves
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Differences Between Bayesians and Frequentists
Frequentist:
◮ The parameters of interest are fixed
and unchanging under all realistic cir-
cumstances.
◮ No information prior to the model
specification.
Bayesian:
◮ View the world probabilistically,
rather than as a set of fixed phe-
nomena that are either known or
unknown.
◮ Prior information abounds and it is
important and helpful to use it.
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ACCP 37th Annual Meeting, Philadelphia, PA [31]
Differences Between Bayesians and Frequentists
Frequentist:
◮ Statistical results assume that data
were from a controlled experiment.
◮ Nothing is more important than
repeatability, no matter what we pay
for it.
Bayesian:
◮ Very careful about stipulating assump-
tions and are willing to defend them.
◮ Every statistical model ever created in
the history of the human race is sub-
jective; we are willing to admit it.
◮ Berger and Berry “Statistical Analysis and the Illusion of Objectivity”
American Scientist 1988
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ACCP 37th Annual Meeting, Philadelphia, PA [32]
But in the End
• We are Statisticians
• We should use all of our tools
Frequentist:
◮ Evaluative Paradigm
◮ Repeatability can be Important
Bayesian:
◮ Modeling Paradigm
◮ Inference can be appropriate
• Bring what is needed to Solve the Problem
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ACCP 37th Annual Meeting, Philadelphia, PA [33]
Thanks for your Attention
Thank You and Go Gators