Example for Presentation

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A RELATIVE MEASURE OF EFFECT SIZE FOR PAIRED DATA
GENERALISING THE WILCOXON MATCHED-PAIRS
SIGNED-RANKS TEST STATISTIC Robert G.
Newcombe Wales College of Medicine Cardiff
University Heath Park, Cardiff CF14 4YS,
UK. Email Newcombe_at_cf.ac.uk
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Non-parametric measures of effect size Unpaired
? U/mn AUROC ? 0, 1 (Newcombe 2006a,
b SIM) Paired ? of differences ve ( ½
_at_ 0) Define a more refined new measure ? based
on Wilcoxon T ? ?T / max (T1 T2) /
½n(n1) ? 1, 1 Preferable not to drop ties
i.e. zero differences.
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Example daily temperature data for 10
years Which of Madrid and Cape Town is more
consistently warmer than London? Mean
temperatures 52.9, 58.6, 62.0? F for London,
Madrid and CT Unpaired, ? U/mn 0.613 Madrid
v. London, 0.765 CT v. London BUT Paired, ? ?T
/ max 0.719 Madrid v. London, 0.573 CT v.
London Paired and unpaired effect size measures
answer different questions.
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Wilcoxon T is based on ranks and signs of paired
differences. Suppose d1 ... dn N(?, ?2). Wlog
?2 1. Let a d. ?(a) ? PrD gt 0 D a
1 / (1 exp(-2?a)) Replacing the as by their
ranks, fit single parameter logistic model ?(xi)
1 / (1 exp(-?xi)) Here xi i / n, i 1
... n. Data are sorted in increasing order of a
d.
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Folded distributions and Prd gt 0 in relation to
skewness (a) Symmetrical t B(6,6), d t -
0.4. Here ? 0.754, ? 0.684.Logit model
clearly appropriate here.
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Folded distributions and Prd gt 0 in relation to
skewness (b) Positive skew t B(3,7), d t -
0.2. Here ? 0.738, ? 0.687.Here the fit of
the logit model is reasonable.
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Folded distributions and Prd gt 0 in relation to
skewness (c) Negative skew t B(7,3), d t -
0.6. Here ? 0.768, ? 0.685.This is the case
in which model fit is unsatisfactory.
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Estimation Direct correspondence between ? and ?,
but depends also on n. For i 1 ... n, put yi
1 if di gt 0, 0 if di lt 0. Empirical estimate
From logistic model, When ,
, so ML and empirical estimates coincide.


.
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Relationship between ?,
?n and ??
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Example log T4 / T8 ratio in 20 Hodgkins
patients (Shapiro, Altman).
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Evaluation of coverage etc. 84 parameter space
points (PSPs) n 5, 10, 20, 50, 100, 1000 B(6,6)
with fold point 0.5, 0.45, 0.4, 0.35, 0.3 B(3,7)
with fold point 0.25, 0.2, 0.15, 0.1 B(7,3) with
fold point 0.7, 0.65, 0.6, 0.55, 0.5 Fold point
determines value of ? - these range across (0,
1). 100000 runs for each PSP.
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Output for a typical PSP iopt 0 nruns 100000
n 100 alpha 7.0 beta 3.0 foldpt 0.50
psi 0.9433155 Summary statistics for randomly
generated psi values Mean 0.940877 SD 0.027787
Min 0.697030 Max 1.000000 Median 0.944146
Quartiles 0.923997 and 0.961159 Asympt? Method
Mean width CP LNCP RNCP 1 1
0.10296418 0.91145 0.04760 0.04095 1
2 0.12132708 0.92379 0.00826 0.06795 2
1 0.10380934 0.91225 0.04493
0.04282 2 2 0.12228121 0.92089
0.00826 0.07085
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Results Coverage tends to be anticonservative
usually slightly. For low n with high ?, coverage
satisfactory for likelihood but not score
interval. Otherwise, coverage better for score
interval. Interval width For 0 lt ? lt 0.5 score
interval slightly narrower than likelihood For
0.5 lt ? lt 1 score interval wider than
likelihood Location score interval is more often
too mesially located Likelihood interval has
better location. Ease of use Likelihood interval
readily obtained in SAS.
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References Newcombe RG. Confidence intervals for
an effect size measure based on the Mann-Whitney
statistic. Part 1 General issues and tail area
based methods. Statistics in Medicine 2006 25
543-557. Newcombe RG. Confidence intervals for
an effect size measure based on the Mann-Whitney
statistic. Part 2 Asymptotic methods and
evaluation. Statistics in Medicine 2006 25
559-573. Shapiro CM, Beckmann E, Christiansen N,
et al. Immunologic status of patients in
remission from Hodgkins disease and disseminated
malignancies. Am J Med Sci 1986 293
366-370. Altman DG. Practical Statistics for
Medical Research. Chapman Hall London, 1991
204.