Transcript Hypothesis Testing and Statistical Inference Using - Sortie-ND

Lecture 3
Hypothesis Testing and Statistical Inference using
Likelihood:
The Central Role of Models
Outline…
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Statistical inference:
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it’s what we use statistics for, but there are some surprisingly
tricky philosophical difficulties that have plagued statisticians
for over a century…
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The “frequentist” vs. “likelihoodist” solutions
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Hypothesis testing as a process of comparing alternate
models
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Examples – ANOVA and ANCOVA
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The issue of parsimony
Inference defined...
“a : the act of passing from one proposition,
statement, or judgment considered as true to
another whose truth is believed to follow from
that of the former
b : the act of passing from statistical sample
data to generalizations (as of the value of
population parameters) usually with calculated
degrees of certainty”
Source: Merriam-Webster Online Dictionary
Statistical Inference...
... Typically concerns inferring properties of an
unknown distribution from data generated by
that distribution ...
Components:
-- Point estimation
-- Hypothesis testing
-- Model comparison
Probability and Inference
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How do you choose the “correct inference” from your data,
given inevitable uncertainty and error?
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Can you assign a probability to your certainty in the
correctness of a given inference?
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(hint: if this is really important to you, then you should
consider becoming a Bayesian, as long as you can accept what I
consider to be some fairly objectionable baggage…)
How do you choose between alternate hypotheses?
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Can you assess the strength of your evidence for alternate
hypotheses?
The crux of the problem...
“Thus, our general problem is to assess the relative
merits of rival hypotheses in the light of
observational or experimental data that bear upon
them....” (Edwards, pg 1).
Edwards, A.W.F. 1992. Likelihood. Expanded Edition.
Johns Hopkins University Press.
Assigning Probabilities to Hypotheses
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Unfortunately, hypotheses (or even different parameter
estimates) can not generally be treated as “data” (outcomes
of trials)
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Statisticians have debated alternate solutions to this
problem for centuries
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(with no generally agreed upon solution)
One Way Out: Classical “Frequentist”
Statistics and Tests of Null Hypotheses
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Probability is defined in terms of the outcome of a series of
repeated trials..
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Hypothesis testing via “significance” of pre-defined “statistics”
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What is the probability of observing a particular value of a
predefined test statistic, given an assumed hypothesis about the
underlying scientific model, and assumptions about the
probability model of the test statistic...
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Hypotheses are never “accepted”, but are “rejected” (categorically)
if the probability of obtaining the observed value of the test
statistic is very small (“p-value”)
Limitations of Frequentist Statistics
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Do not provide a means of measuring relative strength of
observational support for alternate hypotheses (merely
helps decide when to “reject” individual hypotheses in
comparison to a single “null” hypothesis...)
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So you conclude the slope of the line is not = 0. How strong is
your evidence that the slope is really 0.45 vs. 0.50?
Extremely non-intuitive: just what is a “confidence interval”
anyway...
The “null hypothesis” approach
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When and where is “strong inference” really useful?
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When is it just an impediment to progress?
Platt, J. R. 1964. Strong inference. Science 146:347-353
Stephens et al. 2005. Information theory and hypothesis testing: a call
for pluralism. Journal of Applied Ecology 42:4-12.
Chamberlain’s alternative:
multiple working hypotheses
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Science rarely progresses through a series of dichotomously
branched decisions…
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Instead, we are constantly trying to choose among a large
set of alternate hypotheses
-
Concept is very old, but the computational power needed to
adopt this approach has only recently become available…
Chamberlain, T. C. 1890. The method of multiple working hypotheses.
Science 15:92.
Hypothesis testing and “significance”
Nester’s (1996) Creed:
•TREATMENTS: all treatments differ
•FACTORS: all factors interact
•CORRELATIONS: all variables are correlated
•POPULATIONS: no two populations are identical in any respect
•NORMALITY: no data are normally distributed
•VARIANCES: variances are never equal
•MODELS: all models are wrong
•EQUALITY: no two numbers are the same
•SIZE: many numbers are very small
Nester, M. R. 1996. An applied statistician’s creed. Applied Statistician
45:401-410
Hypothesis testing vs. estimation
“The problem of estimation is of more central
importance, (than hypothesis testing).. for in almost all
situations we know that the effect whose significance
we are measuring is perfectly real, however small;
what is at issue is its magnitude.” (Edwards, 1992, pg.
2)
“An insignificant result, far from telling us that the
effect is non-existent, merely warns us that the
sample was not large enough to reveal it.” (Edwards,
1992, pg. 2)
The most important point of
the course…
Any hypothesis test can be framed as
a comparison of alternate models…
(and being free of the constraints imposed by the alternate
models embedded in classical statistical tests is perhaps
the most important benefit of the likelihood approach…)
A simple example:
The likelihood alternative to 1-way ANOVA
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Basic model: a set of observations (j=1..n) that can be classified
into i = 1..a distinct groups (i.e. levels of treatment A)
yij    Ai   ij , for i  1..a groups
 ij  I .I .D .  N ( 0 , 2 )
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A likelihood alternative
yij  Ai   ij
 ij  I .I .D .  N ( 0 , 2 )
So, what would make sense as alternate
models?
Our first model
yij  Ai   ij
for i  1..a groups
 ij  I .I .D .  N ( 0 , 2 )
A “null” model:
yij  A   ij
 ij  N ( 0, 2 )
Could and should you test additional models that lump some
groups together (particularly if that lumping is based on looking
at the estimated group means)?
Remember that the error term is part
of the model…
And you don’t just have to accept that a simple, normally
distributed, homogeneous error is appropriate…
Estimate a separate error
term for each group
yij  Ai   ij
 ij  N ( 0, i )
Or an error term that
varies as a function of the
predicted value
yij  Ai   ij
 ij  N ( 0, 2 )
Or where the error isn’t
normally distributed
2
   ŷ
yij  Ai   ij
yij  Gamma( shape , scale )
A more general notation for the model…
The “scientific model”
ŷi  f ( xi )
and g(yi |  )  N ( yˆ i , 2 )
The “likelihood function”
The likelihood function [ g(yi|θ) ] specifies the probability of
observing yi, given the predicted value for that observation ( ŷi)
i.e. calculated as a function of the parameters in the scientific
model and the independent variables, and any parameters in
the PDF (i.e. σ)
Another Example: Analysis of Covariance
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A traditional ANCOVA model (homogeneous slopes):
yij  a j  bxij   ij
for j  1..A groups
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What is restrictive about this model?
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How would you generalize this in a likelihood framework?
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What alternate models are you testing with the standard
frequentist statistics?
What more general alternate models might you like to test?