Posts Tagged ‘ Probability and Statistics ’

Interim analysis, futility monitoring, and predictive probability

October 19, 2016
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Interim analysis, futility monitoring, and predictive probability

An interim analysis of a clinical trial is an unusual analysis. At the end of the trial you want to estimate how well some treatment X works. For example, you want to how likely is it that treatment X works better than the control treatment Y. But in the middle of the trial you want to know something more subtle. It’s […]

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Uncertainty in a probability

September 20, 2016
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Uncertainty in a probability

Suppose you did a pilot study with 10 subjects and found a treatment was effective in 7 out of the 10 subjects. With no more information than this, what would you estimate the probability to be that the treatment is effective in the next subject? Easy: 0.7. Now what would you estimate the probability to be […]

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Insufficient statistics

September 12, 2016
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Experience with the normal distribution makes people think all distributions have (useful) sufficient statistics [1]. If you have data from a normal distribution, then the sufficient statistics are the sample mean and sample variance. These statistics are “sufficient” in that the entire data set isn’t any more informative than those two statistics. They effectively condense […]

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Mittag-Leffler function and probability distribution

July 17, 2016
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Mittag-Leffler function and probability distribution

The Mittag-Leffler function is a generalization of the exponential function. Since k!= Γ(k + 1), we can write the exponential function’s power series as and we can generalize this to the Mittag=Leffler function which reduces to the exponential function when α = β = 1. There are a few other values of α and β for […]

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Continuum between anecdote and data

March 4, 2016
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Continuum between anecdote and data

The difference between anecdotal evidence and data is overstated. People often have in mind this dividing line where observations on one side are worthless and observations on the other side are trustworthy. But there’s no such dividing line. Observations are data, but some observations are more valuable than others, and there’s a continuum of value. I believe […]

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The empty middle: why no one is average

February 20, 2016
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The empty middle: why no one is average

In 1945, a Cleveland newspaper held a contest to find the woman whose measurements were closest to average. This average was based on a study of 15,000 women by Dr. Robert Dickinson and embodied in a statue called Norma by Abram Belskie. Out of 3,864 contestants, no one was average on all nine factors, and fewer than 40 […]

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Improving on Chebyshev’s inequality

February 12, 2016
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Improving on Chebyshev’s inequality

Chebyshev’s inequality says that the probability of a random variable being more than k standard deviations away from its mean is less than 1/k2. In symbols, This inequality is very general, but also very weak. It assumes very little about the random variable X but it also gives a loose bound. If we assume slightly more, […]

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Connection between hypergeometric distribution and series

February 8, 2016
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Connection between hypergeometric distribution and series

What’s the connection between the hypergeometric distributions, hypergeometric functions, and hypergeometric series? The hypergeometric distribution is a probability distribution with parameters N, M, and n. Suppose you have an urn containing N balls, M red and the rest, N – M blue and you select n balls at a time. The hypergeometric distribution gives the probability of selecting k red balls. The probability generating function […]

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Reproducible randomized controlled trials

February 1, 2016
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Reproducible randomized controlled trials

“Reproducible” and “randomized” don’t seem to go together. If something was unpredictable the first time, shouldn’t it be unpredictable if you start over and run it again? As is often the case, we want incompatible things. But the combination of reproducible and random can be reconciled. Why would we want a randomized controlled trial (RCT) to […]

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Random number generator seed mistakes

January 29, 2016
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Random number generator seed mistakes

Long run or broken software? I got a call one time to take a look at randomization software that wasn’t randomizing. My first thought was that the software was working as designed, and that the users were just seeing a long run. Long sequences of the same assignment are more likely than you think. You […]

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