Cochran's Q statistic is routinely used for testing heterogeneity in meta-analysis. Its expected value is also used in several popular estimators of the between-study variance, ${\tau }^2$. Those applications generally have not considered the implications of its use of estimated variances in the inverse-variance weights. Importantly, those weights make approximating the distribution of Q (more explicitly, $Q_{\text{IV}}$) rather complicated. As an alternative, we investigate a new Q statistic, $Q_{\mathrm{F}}$, whose constant weights use only the studies' effective sample sizes. For the standardized mean difference as the measure of effect, we study, by simulation, approximations to distributions of $Q_{\text{IV}}$ and $Q_{\mathrm{F}}$, as the basis for tests of heterogeneity and for new point and interval estimators of ${\tau }^2$. These include new DerSimonian–Kacker-type moment estimators based on the first moment of $Q_{\mathrm{F}}$, and novel median-unbiased estimators. The results show that: an approximation based on an algorithm of Farebrother follows both the null and the alternative distributions of $Q_{\mathrm{F}}$ reasonably well, whereas the usual chi-squared approximation for the null distribution of $Q_{\text{IV}}$ and the Biggerstaff–Jackson approximation to its alternative distribution are poor; in estimating ${\tau }^2$, our moment estimator based on $Q_{\mathrm{F}}$ is almost unbiased, the Mandel – Paule estimator has some negative bias in some situations, and the DerSimonian–Laird and restricted maximum likelihood estimators have considerable negative bias; and all 95% interval estimators have coverage that is too high when ${\tau }^2=0$, but otherwise the Q-profile interval performs very well.

中文翻译：

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关于标准化均值差的恒定权重的 Q 统计量

Cochran 的Q统计量通常用于测试荟萃分析中的异质性。它的期望值也用于研究间方差的几个流行的估计中，${\tau }^2$. 这些应用程序通常没有考虑其在逆方差权重中使用估计方差的影响。重要的是，这些权重近似于Q的分布（更明确地说，${问}_{\text{四}}$) 相当复杂。作为替代方案，我们研究了一个新的Q统计量，${问}_{\mathrm{F}}$，其恒定权重仅使用研究的有效样本量。对于作为衡量效果的标准化平均差，我们通过模拟研究了${问}_{\text{四}}$和${问}_{\mathrm{F}}$, 作为异质性检验和新的点和区间估计量的基础${\tau }^2$. 其中包括新的 DerSimonian-Kacker 型矩估计器，它基于${问}_{\mathrm{F}}$，以及新颖的中值无偏估计量。结果表明：基于 Farebrother 算法的近似值同时遵循${问}_{\mathrm{F}}$相当好，而通常的卡方近似的零分布${问}_{\text{四}}$并且 Biggerstaff-Jackson 对其替代分布的近似很差；在估计${\tau }^2$, 我们的矩估计器基于${问}_{\mathrm{F}}$几乎是无偏的，Mandel-Paule 估计量在某些情况下有一些负偏，而 DerSimonian-Laird 和受限最大似然估计量有相当大的负偏；并且所有 95% 的区间估计器的覆盖率都过高${\tau }^2=0$，但除此之外Q -profile 区间表现得非常好。