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Opinionated

Opinionated. Lessons. in Statistics. by Bill Press. #15 The Towne Family – Again. Now that we’re so adept at p-value stuff, let’s go back to the Towne family. N=1. D =0. N=3. bin(0,3x37,r). We used these data points to estimate of the parameter r.

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Opinionated

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  1. Opinionated Lessons in Statistics by Bill Press #15 The Towne Family – Again Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  2. Now that we’re so adept at p-value stuff, let’s go back to the Towne family. N=1 D=0 N=3 bin(0,3x37,r) We used these data points to estimate of the parameter r Are T2 and T11 descendents or were there “non-paternal events”? N=6 D=0 bin(0,3x37,r) And T13? N=9 D=0 bin(0,6x37,r) D=5 N=10 D=4 (of 12) D=23 D=3 N=11 bin(3,10x37,r) D=0 D=1 D=1 Professor William H. Press, Department of Computer Science, the University of Texas at Austin bin(1,5x37,r) bin(0,5x37,r) bin(1,11x37,r)

  3. 3 7 X ( ) ( j ) ( ) ( ) ( ) ( ) b k i d b b b b P 9 3 7 i i i i 0 3 3 7 0 3 3 7 1 5 3 7 0 5 3 7 t £ £ £ £ £ p n r r a a n r n r n r n r = l = i 1 1 t a ; ; ; ; ; ; ; ; ; ; ; k 5 ( ) ( ) ( ) = b b b i i i 0 6 3 7 1 1 1 3 7 3 1 0 3 7 = £ £ £ £ n r n r n r r ; ; ; ; ; ; If we really knew r, then a p-value (tail) test on T2, T11, and T13 would be straightforward, notice how the “neglect backmutation” assumption comes in here The problem is we have only Bayesian (uncertain) knowledge about r A common frequentist practice is to use the maximum likelihood estimate of r. This is just wrong (except asymptotically if the distribution of r were very narrow) because T11’s extreme tail probabilities will be dominated by the extreme (but possible) values of r. Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  4. 3 7 Á 1 1 Z Z X ( ) ( j ) ( j ) b k d d d d P P i 9 3 7 t t £ p n r r a a r r a a r = l i 1 1 t a ; ; ; 0 0 k 5 = One “modern” way to proceed is to integrate the p-value over the posterior probability of all estimated quantities. This is called the “posterior predictive p-value” and is an example of a set of methods loosely called “empirical Bayes”. So the three questionables are all unlikely to be descendents. D=5p=.01 D=23p=1.0e-13 D=3p=.12 D=4 (of 12)p=.0013 Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  5. ( j ) ( j ) ( ) ( ) d d b b P P i i 5 9 3 7 4 1 0 1 2 t t £ £ £ r a a / r a a n r n r i p r e v o u s ; ; ; ; This would be a satisfactory end to the Towne story, except that we tainted the data by tail trimming. While T2 is hopeless, what if we had included T11 and T13? t11tail = 0.0953 t2tail = 3.2348e-011 t13tail = 0.0122 the extra data drags up the distribution for r So suddenly there is hope for T11.T2 and T13 still strongly ruled out. This is an actual methodological problem with “posterior predictive p-value”. Data is being used twice: once to get the posterior, then again to test itself. Often you can get away with this (e.g., try posterior both with and without questionable data). Doing so in this example, we find that T11 is left ambiguous. This is when we need real (We’ll return to the Towne family one more time, later.) Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  6. (Let me explain where we’re going here…) • Building up prerequisites to do a fairly sophisticated treatment of model fitting • Bayes parameter estimation • p-value tail tests • really understand multivariate normal and covariance • really understand chi-square • Then, we get to appreciate the actual model fitting stuff • fitted parameters • their uncertainty expressed in several different ways • goodness-of-fit • And it will in turn be a nice “platform” for learning some other things • bootstrap resampling Professor William H. Press, Department of Computer Science, the University of Texas at Austin

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