and
Stata Technical Bulletin
11
Q0 = n - 1/2 - √2n2 - 4Tn + 1/4
Duval and Tweedie provide the mean and variance of each estimator as follows (the reader should refer to the original paper
for the derivation):
E[¾] = k0, var[7⅞] = 2k0 + 2
E[Z0] = fc0 — fcθ∕(2n — 1),
var[Z0] = 16 var(Tn)∕(2n — I)2
where
var(Tn) = (n(n + l)(2n⅛ 1) ⅛ 10fcθ + 27fcθ + 17Z⅞ — 18nfcθ — 18nk0 + 6n2⅛o)∕24
and
FJOJ →n +________2 vad7"i________
k° + ((n - 1/2)2 - fc0(2n - k0 - I))3/2 ’
4 var(Tn)
(n — 1/2)2 — k0(2n — k0 — 1)
The authors also report that for n large and k0 of a smaller order than n, then asymptotically:
E[7⅛] = k0,
E[Z0] ~ k0,
E[Qo] ~ k0 + 1/6,
var[7⅛] = o(n);
var[Z0] ~ n/3;
var[Qo] ~ n/3.
These results suggest that L0 and Q0 should have similar behavior, but the authors report that in practice Q0 is often larger,
sometimes excessively so. They also note that L0 generally has smaller mean square error than Q0 when k0 ≥ n/4 — 2.
Duval and Tweedie remark that the Rо run estimator is rather conservative and nonrobust to the presence of a relatively
isolated negative term at the end of the sequence of ranks. They suggest that the estimators based on Tn seem more robust to
such a departure from the suppressed Bernoulli hypothesis. They also note that the Qо quadratic estimator is defined only when
Trt < n2∕2 + 1/16, and that simulations show this to be violated quite frequently when the number of studies, n, is small and
when the number of suppressed studies, k0, is large relative to n. These concerns leave the L0 linear estimator as the best all
around choice.
Because only whole studies can be trimmed, the estimators are rounded in practice to the nearest nonnegative integer, as
follows:
Rq = max{0, 7¾∕}
Lq = max
{°,i0+∣y
Q0 =
max {θ, Q0 ÷ 2}
where [.r] is the integer part of x.
The Iterative trim and fill algorithm
Because the global “effect size” Δ is unknown, the number and position of any missing studies is correlated with the
true value of Δ. Therefore, Duval and Tweedie developed an iterative algorithm to estimate these values simultaneously. The
algorithm can be used with any of the three estimators of k0 defined in the previous section (the metatrim program allows
the user to specify which one is to be used through the estimât () option). Likewise, either a fixed-effects or random-effects
meta-analysis model can be used to estimate within each iteration (Z) of the algorithm (the default model in metatrim is
fixed effects, but random effects is used when option reflect is specified). Note that the meta program of Sharp and Sterne
(1997, 1998) is called by metatrim to carry out the meta-analysis calculations.
The algorithm proceeds as follows:
1. Starting with values Yt, estimate ∆W using the chosen meta-analysis model. Construct an initial set of centered values
yiω = yi-∆W, i = ι,...,n
and estimate k^ using the chosen estimator for k0 applied to the set of values