Copyright (c) 2008 All rights reserved. http://works.bepress.com/chris_lloyd Recent documents in Chris J. Lloyd en-us Wed, 02 Apr 2008 22:23:01 PDT 3600 http://works.bepress.com/chris_lloyd/10 http://works.bepress.com/chris_lloyd/10 Mon, 21 Jan 2008 19:24:39 PST Recent advances in likelihood asymptotics (Reid, 2003) lead to pivotal quantities that are closer to standard normal than standard pivotals and also respect some kind of conditionality. It is less clear the extent to which these methods work for discrete models. On the other hand, in the context of binomial trials conditional pivotals can lead to more efficient unconditional inferences, see Boschloo (1970) and Lloyd and Moldovan (2007). This suggests that second order pivotals that respect local conditionality might provide more powerful exact tests. For testing the rate ratio from independent binomial samples, we investigate 5 first order pivotals and the second order pivotal. Each of these is used to generate an exact test my maximising with respect to the nuisance parameter. We also consider the effect of pre-estimating the nuisance parameter. Chris Lloyd Exact testing Higher order asymptotics http://works.bepress.com/chris_lloyd/9 http://works.bepress.com/chris_lloyd/9 Thu, 06 Dec 2007 21:35:28 PST For testing canonical parameters in a continuous exponential family, P-values based on higher order asymptotic formulas such as p* approximate the exact conditional P-value with great accuracy. For discrete models, the conditional distribution can be extremely discrete or even degenerate which raises the questions (a) should one try to approximate the conditional P-value, (b) what does p* approximate? Pierce and Peters (1999) have argued that p* approximates an approximately conditional P-value and that this approximately conditional P-value is an inferentially sensible quantity worth approximating. Their arguments and numerical results are oriented towards problems where the conditioning variable has 3 or 4 dimensions. We investigate the performance and logic of approximately conditional P-values for the case of 2x2tables, as well as the extent to which p* functions as an approximation to these P-values. We conclude that approximately conditional P-values have rather erratic properties and suffer from a logical flaw. We also find that the mid-P value approximates them as well or better than p*, but that neither approximations work well when the observed data is near the boundary of the sample space. Chris Lloyd Higher order asymptotics http://works.bepress.com/chris_lloyd/8 http://works.bepress.com/chris_lloyd/8 Mon, 15 Oct 2007 21:13:41 PDT The Buehler upper confidence limit is as small as possible, subject to the constraints that (a) its coverage probability never falls below nimonal and (b) it is a non-decreasing function of a designated statistic. The designated statistic may have ties among its possible values.We prove that breaking such ties by a sufficiently smallmodification can never increase the Buehler limit. We also prove that, under commonly satisfied conditions, breaking ties by a sufficiently small modification will result in an improved i.e. smaller Buehler limit. We conclude that designated statistics should not contain ties, apart from ties justified by symmetry requirements. In particular, this principle excludes the most commonly suggested designated statistics in the literature. Chris Lloyd Exact confidence intervals http://works.bepress.com/chris_lloyd/7 http://works.bepress.com/chris_lloyd/7 Mon, 15 Oct 2007 21:06:55 PDT We construct exact and optimal one-sided upper and lower confidence bounds for the difference between two probabilities based on matched binary pairs using well-established optimality theory of Buehler (1957). Starting with five different approximate loer and upper limits, we adjust them to have coverage probability exactly equal to the desired nominal level and then compare the resulting exact limits by their mean size. Exact limits based on the signed root likelihood ratio statistic are preferred and recommended for practical use. Chris Lloyd Exact confidence intervals http://works.bepress.com/chris_lloyd/6 http://works.bepress.com/chris_lloyd/6 Sun, 23 Sep 2007 22:31:11 PDT For a correlated 2x2 table where the (01) cell is empty by design, the parameter of interest is typically the ratio of the probability of secondary response conditional on primary response to the probability of primary response, also known as a risk ratio. It is common to test whether or not the risk ratio equals one. One method of obtaining an exact P-value is to maximise the tail probability of the test statistic over the nuisance parameter. It is argued that better results are obtained by first replacing the nuisance parameter by its profile estimate in the calculation of its exact significance followed by maximisation - termed an E+M P-value. We consider four standard approximate test statistics with and without the common correction of adding 1/2 to each count. From a complete enumeration of the distributions of these P-values (for sample sizes 50 and 100), we recommend E+M P-values based on the uncorrected Wald statistic for testing the greater than alternative and on the corrected Wald statistic on the log-scale for testing the less than alternative. A good compromise statistic for both kinds of alternatives is the likelihood ratio statistic. Chris Lloyd Exact testing http://works.bepress.com/chris_lloyd/5 http://works.bepress.com/chris_lloyd/5 Sun, 23 Sep 2007 22:24:17 PDT We compare various one-sided confidence limits for the odds ratio in a 2x2 table. The first group of limits relies on first order asymptotic approximations and includes limits based on the (signed) likelihood ratio, score and Wald statistics. The second group of limits is based on the conditional tilted hypergeometric distribution, with and without mid-P correction. All these limits have poor unconditional coverage properties and so we apply the general transformation of Buehler (1957) to obtain limits which are unconditionally exact. The performance of these competing exact limits is assessed across a range of sample sizes and parameter values by looking at their mean size. The results indicate that Buehler limits generated from the conditional likelihood have the best performance, with a slight preference for the mid-P version. This confidence limit has not been proposed before and is recommended for general use, especially when the underlying probabilities are not extreme. Chris Lloyd Exact confidence intervals http://works.bepress.com/chris_lloyd/4 http://works.bepress.com/chris_lloyd/4 Thu, 20 Sep 2007 18:26:39 PDT In constructing exact tests, one must deal with the possible dependence of the P-value on the nuisance parameter/s psi as well as discreteness of the sample space. A classical but heavy handed approach is to maximise over psi. We prove what has previously been understood informally, namely that maximisation produces the uinique and smallest possible P-value subject to the ordering induced by the underlying test statistic and test validity. On the other hand, allowing for the worst case will be more attractive when the P-value is less dependent on psi. We investigate the extent to which estimating psi under the null reduces this dependence. An approach somewhere between full maximisation and estimation is partial maximisation, with appropriate penalty, as introduced by Berger and Boos (1994). It is argued that estimation followed by maximisation is an attractive, but computationally more demanding, alternative to partial maximisation. We illustrate the ideas on a range of low dimension but important examples where the alternative methods can be investigated completely numerically. Chris Lloyd Exact testing http://works.bepress.com/chris_lloyd/3 http://works.bepress.com/chris_lloyd/3 Thu, 20 Sep 2007 18:15:02 PDT We consider the problem of testing for a difference in the probability of success from matched binary pairs. Starting with three standard inexact tests, the nuisance parameter is first estimated and then the residual dependence is eliminated by maximisation, producing what I call an E+M P-value. The E+M P-value based on McNemar's statistic is shown numerically to dominate previous suggestions, including partially maximised P-values as described in Berger and Sidik (2003). The latter method however may have computational advantages for large samples. Chris Lloyd Exact testing http://works.bepress.com/chris_lloyd/2 http://works.bepress.com/chris_lloyd/2 Thu, 20 Sep 2007 16:37:15 PDT This paper examines exact one-sided confidence limits for the risk ratio in a 2x2 table with structural zero. Starting with four approximate lower and upper limits, we adjust each using the algorithm of Buehler (1957) to arrive at lower (upper) limits that have exact coverage properties and are as large (small) as possible subject to coverage, as well as an ordering, constraint. Different Buehler limits are compared by their mean size, since all are exact in their coverage. Buehler limits based on the signed root likelihood ratio statistic are found to have the best performance and recommended for practical use. Chris J. Lloyd Exact confidence intervals http://works.bepress.com/chris_lloyd/1 http://works.bepress.com/chris_lloyd/1 Wed, 19 Sep 2007 23:34:14 PDT This paper describes a method of estimating the performance of a multiple screening test where those who test negative do not have their true disease status determined. The methodology is motivated by a dataset on 49,927 subjects who were given K=6 binary tests for bowel cancer. A complicating factor is that individuals may have polyps present in the bowel, a condition that the screening test is not designed to detect but which may be worth diagnosing. The methodology is based on a multinomial logit model for Pr(S|R_6), the probability distribution of patient status S (healthy, polyps or diseased) conditional on the results R_6 from six binary tests. An advantage of the described methodology is that the modeling is data driven. In particular, we require no assumptions about (i) correlation within subjects, (ii) the relative sensitivity of the K tests, (iii) the conditional independence of the tests. The model leads to simple estimates of the trade-off between different errors as the number of tests is varied, presented graphically using ROC curves. Finally, the model allows us to estimate better protocols for assigning subjects to the disease group, as well as the gains in accuracy from these protocols. Chris J. Lloyd Diagnostic testing