working with confidence and credible intervals. standard normal distribution μ=0 and σ 2 =1
TRANSCRIPT
Working with CONFIDENCE and CREDIBLE intervals
Standard Normal Distribution• μ=0 and σ2=1
Confidence Intervals
• Scientists often use a sample’s standard deviation to construct a confidence interval around the mean.
• For a normally distributed random variable:
approximately 67% of the observations occur within + 1 standard deviation of the mean
approximately 96% of the observations occur within + 2 standard deviations of the mean
What does it mean?
nsswhere
sYsYP
Y
YY
95.0)96.196.1(
Because our sample mean and sample standard error of
the mean are derived from a single sample, this confidence interval WILL change if we sample again. Thus, this expression asserts that the “true” population mean μ will fall within a single calculated confidence in 95% of the iterations..
By extension:
•If we were to repeatedly sample the population (keeping sample size and all conditions equal), 5% of the time we would expect that the true population mean μ would fall outside of this confidence interval!!
Interpretation
“There is a 95% chance that the true population mean μ occurs within this interval.”
WRONG
“95% of the realizations, a confidence interval calculated in this way will contain the “true” value of μ.”
Right
Bad news
• This is not satisfying!!!!• It is not exactly what you like to assert
when you construct a confidence interval!!• You would like to say how confident you
are that the confidence interval contains the population mean.
• A frequentist statistician, however, can’t assert that !!!!
Good news
• A Bayesian approach turns this around.• Because the confidence interval is fixed
(by your sample data), a Bayesian statistician can calculate the probability that the population mean (itself a random variable) occurs within the confidence interval.
• Bayesians refer to this as: Credibility or credible intervals
More news
• Bayesian credibility intervals and frequentist confidence intervals are usually numerically similar if the Bayesian prior probability distribution is uninformative.
• Note that:– When the intervals are identical, the choice does not
matter. – When the intervals are different, only the Bayesian
approach provides logical results.
Generalized Confidence Intervalsare calculated with reference to the
T-distribution
kX
k
s
Xt
)1()( ]1[]1[ YnYn stYstYP
-5 -4 -3 -2 -1 0 1 2 3 4 50
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-5 -4 -3 -2 -1 0 1 2 3 4 50
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-5 -4 -3 -2 -1 0 1 2 3 4 50
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20 df
4 df 8 df
Some t-distributions:
What do degrees of freedom mean? What happens as we increase them?
t- distribution
http://www.statsoft.com/textbook/sttable.html#t
Calculating the tail probability: Student’s t table
df\p 0.4 0.25 0.1 0.05 0.025 0.01 0.005 0.0005
1 0.32492 1 3.077684 6.313752 12.7062 31.82052 63.65674 636.6192
2 0.288675 0.816497 1.885618 2.919986 4.30265 6.96456 9.92484 31.5991
3 0.276671 0.764892 1.637744 2.353363 3.18245 4.5407 5.84091 12.924
4 0.270722 0.740697 1.533206 2.131847 2.77645 3.74695 4.60409 8.6103
5 0.267181 0.726687 1.475884 2.015048 2.57058 3.36493 4.03214 6.8688
6 0.264835 0.717558 1.439756 1.94318 2.44691 3.14267 3.70743 5.9588
7 0.263167 0.711142 1.414924 1.894579 2.36462 2.99795 3.49948 5.4079
8 0.261921 0.706387 1.396815 1.859548 2.306 2.89646 3.35539 5.0413
http://www.statsoft.com/textbook/sttable.html#t
PS. Why use R?
• Search of Google Scholar from 2002-2008 with the search phrase “R Development” and Publication Name of “Ecology* or Evolution*”
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d2002 2003 2004 2005 2006 2007 2008
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Why learn programming?
• “One of the most important things you can do is to take the time to learn a real programming language…
• Unfortunately, learning to program is like learning to speak a foreign language—it takes time and practice, and there is no immediate payoff…But if you can overcome the steep learning curve, the scientific payoff is tremendous [emphasis added].”
• Excerpted from footnote on p. 116 of Gotelli & Ellison (2004)
Why learn programming? (cont.)
• “In offering advice to graduate students in almost any branch of ecology, one of the most important recommendations is to acquire at least some programming skills.”
Excerpted from p. 320 of Fortin, M.-J. & M. Dale. 2005. Spatial Analysis: A Guide for Ecologists. Cambridge University Press.
NOTE: Gotelli, Ellison, Fortin, and Dale are all field ecologists, not “just theoretical modelers”!