example, the probability p

of a ‘‘yes’’ response to Ques-

tion B will be approximately p

¼ 0:5. Note that although

birth rates vary slightly across months [24], it can be shown

that UQM estimates are rather robust when the true prob-

ability p

deviates from the assumed probability [25].

In addition to being psychologically more accept-

able than Warner’s original method, the UQM also has

more favorable statistica l properties (see [25, 26]). With

the UQM, the maximum likelihood estimator of p

computed from the observed proportion k of total ‘‘yes’’

responses (see [23], p. 533) as

k  1  pð Þ  p

: ð1Þ

With a sample of n respo ndents, the standard error of the

estimate

is computed as

SE ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

k  1  kð Þ

n  p

ð2Þ

With adequate sample size, this technique yields high

statistical power [25].

Randomized response techniques such as the UQ M have

been used in several prior studies of athletes in Germany

[

27–33], which all revealed a much higher prevalence of

doping among athletes than that obtained with conven-

tional survey methods that lacked a guarantee of anon-

ymity. The ﬁrst such study RRT study, to our knowledge,

was performed by Musch and Plessner in 2002 [27]. In an

online survey, these authors employed a forced-answer

version of RRT and surveyed 467 German athletes; this

study reported a 42% prevalence of doping for the athletes

during their careers. In another online survey in 2007,

Pitsch, Emrich, and Klein [28] also employed the forced-

answer version of RRT to assess doping among a group of

448 German top-level athletes. These researchers reported

that between 26% and 48% of this group admitted to

having used banned substances or methods during their

athletic careers. In 2012, using a much larger sample, the

same investigators replicated their previous ﬁnding and

concluded that between 10% and 35% of German athletes

use doping in a given season. In a recent RRT online study,

Frenger, Pitsch, and Emrich estimated the doping

Fig. 1 Probability tree for the unrelated question model (UQM).

a The respondent is asked an initial personal question for which only

the respondent knows the answer, but for which the probability p of a

‘‘no’’ response and the complementary probability q = 1 - p of a

‘‘yes’’ response in the overall population are known (e.g., a birthdate;

see Fig. 2). Respondents answering ‘‘yes’’ to the initial question are

directed to a subsequent non-sensitive question A, for which the

probability p

of a ‘‘yes’’ answer is again known (e.g., another

birthdate). Respondents answering ‘‘no’’ on the initial question are

directed to a sensitive question B (e.g., past-year doping), where the

probability p

of a ‘‘yes’’ answer is unknown and represents the target

of the investigation. b The expected proportion k of ‘‘yes’’ answers

from the total group of respondents is therefore a function of p, p

and p

. In the present study, the probability p of being directed to the

sensitive question was approximately 2/3. In addition, the probability

of answering the non-sensitive question with ‘‘yes’’ was approx-

imately 1/2. Using these values and solving for p

, the prevalence of

past-year doping would be expected to be approximately (k - 0.167)/

0.667. See the main text for exact numbers

Doping in Elite Sports Assessed by Randomized-Response Surveys 213

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