# Common Errors in Statistics and How to Avoid Them - Good P.I

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For a given size sample, bootstrap estimates of percentiles in the tails will always be less accurate than estimates of more centrally located percentiles. Similarly, bootstrap interval estimates for the variance of a distribution will always be less accurate than estimates of central location such as the mean or median because the variance depends strongly upon extreme values in the population.

One proposed remedy is the tilted bootstrap3 in which instead of sampling each element of the original sample with equal probability, we weight the probabilities of selection so as to favor or discourage the selection of extreme values.

If we know something about the population distribution in advancefor example, if we know that the distribution is symmetric, or that it is chi-square with six degrees of freedomthen we may be able to take advantage of a parametric or semiparametric bootstrap as described in Chapter 4. Recognize that in doing so, you run the risk of introducing error through an inappropriate choice of parametric framework.

Problems due to the discreteness of the bootstrap statistic are usually evident from plots of bootstrap output. They can be addressed using a smooth bootstrap as described in Davison and Hinkley [1997, Section 3.4].

BAYESIAN METHODOLOGY

Since being communicated to the Royal Society in 1763,4 Bayes Theorem has exerted a near fatal attraction on those exposed to it.5 Much as a bell placed on the cat would magically resolve so many of the problems of the average house mouse, Bayes straightforward, easily grasped mathematical formula would appear to provide the long-awaited basis for a robotic judge free of human prejudice.

On the plus side, Bayes Theorem offers three main advantages:

3 See, for example, Hinkley and Shi [1989] and Phipps [1997].

4 Philos. Tran. 1763; 53:376-398. Reproduced in Biometrika 1958; 45: 293-315.

5 The interested reader is directed to Keynes [1921] and Redmayne [1998] for some accounts.

CHAPTER 6 LIMITATIONS OF SOME MISCELLANEOUS STATISTICAL PROCEDURES 79

1. It simplifies the combination of a variety of different kinds of evidence, lab tests, animal experiments, and clinical trials, and it serves as an effective aid to decision making.

2. It permits evaluating evidence in favor of a null hypothesis. And with very large samples, a null hypothesis is not automatically rejected.

3. It provides flexibility during the conduct of an experiment; sample sizes can be modified, measuring devices altered, subject populations changed, and endpoints redefined.

Suppose we have in hand a set of evidence E = {E^ E2,..., En} and thus have determined the conditional probability Pr{A | E} that some event A is true. A might be the event that O.J. killed his ex-wife, that the Captain of the Valdez behaved recklessly, or some other incident whose truth or falsehood we wish to establish. An additional piece of evidence En+1 now comes to light. Bayes Theorem tell us that

Pr{A|Ei,...,E,E+ij =

___________________Pr{E+i | A} Pr{A| Ei,..., E}___________________

Pr{E+i | A} Pr{A| Ei,..., En} + Pr{E+i | ~A} Pr{~A| Ei,..., E}

where ~A, read not A, is the event that A did not occur. Recall that Pr{A} + Pr{~A} = i. Pr{A | Ei,..., En} is the prior probability of A, and Pr{A | Ei,..., En, En+i} the posterior probability of A once the item of evidence En+i is in hand. Gather sufficient evidence and we shall have an automatic verdict.

The problem with the application of Bayes Theorem in practice comes at the beginning when we have no evidence in hand, and n = 0. What is the prior probability of A then?

Applications in the Courtroom6

Bayes Theorem has seen little use in criminal trials as ultimately the theorem relies on unproven estimates rather than known facts.7 Tribe [i97i] states several objections including the argument that a jury might actually use the evidence twice, once in its initial assessment of guiltthat is, to determine a prior probabilityand a second time when the jury applies Bayes Theorem. A further objection to the theorems application is that if a man is innocent until proven guilty, the prior probability of his guilt must be zero; by Bayes Theorem the posterior probability of his

6 The majority of this section is reprinted with permission from Applying Statistics in the Courtroom, by Phillip Good, Copyright 200i by CRC Press, Inc.

7 See, for example, People v. Collins, 68 Cal .2d 3i9, 36 ALR3d ii76 (i968).

80 PART II HYPOTHESIS TESTING AND ESTIMATION

guilt would be zero also, rendering a trial unnecessary. The courts of several states have remained unmoved by this argument.8

In State v. Spann,9 showing the defendant had fathered the victims child was key to establishing a charge of sexual assault. The States expert testified that only 1% of the presumed relevant population of possible fathers had the type of blood and tissue that the father had and, further, that the defendant was included within that 1%. In other words, 99% of the male population at large was excluded. Next, she used Bayes Theorem to show that the defendant had a posterior probability of fathering the victims child of 96.5%.

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