Exact distributions of intraclass correlation and Cronbach's alpha with Gaussian data and general covariance
Exact distributions of intraclass correlation and Cronbach's alpha with Gaussian data and general covariance
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DOI：
10.1007/bf02295646
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发表时间：
2004-09-01
期刊：
影响因子：
3
通讯作者：
Muller, KE
中科院分区：
文献类型：
--
作者：
Kistner, EO;Muller, KE
类内相关性和Cronbach's alpha被广泛用于描述测试和测量的可靠性。即使使用高斯数据，精确分布也只对复合对称协方差（等方差和等相关性）已知。最近，对分布函数导出了大样本高斯近似。新的精确结果允许计算精确的分布函数以及类内相关性和Cronbach's alpha的其他属性，适用于任何协方差模式的高斯数据，而不仅仅是复合对称性。概率是根据独立卡方随机变量加权和的分布函数计算的。类内相关分布函数和Cronbach's alpha的新F近似比精确形式更简单，计算速度更快。假设协方差矩阵是已知的，即使只有10个观测值，这种近似通常也能提供足够的精度。精确分布或近似分布都可用于在可靠性估计值周围创建置信区间。蒙特卡罗模拟得出了许多结论。正确地假设协方差矩阵是复合对称的，可以得到准确的置信区间，正如先前已知结果所期望的那样。然而，假设和估计一个一般的协方差矩阵产生一些乐观的窄置信区间与10个观测值。将样本量增加到100，基本上可以获得无偏的覆盖。不正确地假设复合对称导致悲观的大置信区间，悲观情绪随着样本量的增加而增加。相反，错误地假设一般协方差只会在小样本中引入适度的乐观偏差。因此，新方法似乎更适合于创建置信区间，除非复合对称绝对成立。
Intraclass correlation and Cronbach's alpha are widely used to describe reliability of tests and measurements. Even with Gaussian data, exact distributions are known only for compound symmetric covariance (equal variances and equal correlations). Recently, large sample Gaussian approximations were derived for the distribution functions.New exact results allow calculating the exact distribution function and other properties of intraclass correlation and Cronbach's alpha, for Gaussian data with any covariance pattern, not just compound symmetry. Probabilities are computed in terms of the distribution function of a weighted sum of independent chi-square random variables.New F approximations for the distribution functions of intraclass correlation and Cronbach's alpha are much simpler and faster to compute than the exact forms. Assuming the covariance matrix is known, the approximations typically provide sufficient accuracy, even with as few as ten observations.Either the exact or approximate distributions may be used to create confidence intervals around an estimate of reliability. Monte Carlo simulations led to a number of conclusions. Correctly assuming that the covariance matrix is compound symmetric leads to accurate confidence intervals, as was expected from previously known results. However, assuming and estimating a general covariance matrix produces somewhat optimistically narrow confidence intervals with 10 observations. Increasing sample size to 100 gives essentially unbiased coverage. Incorrectly assuming compound symmetry leads to pessimistically large confidence intervals, with pessimism increasing with sample size. In contrast, incorrectly assuming general covariance introduces only a modest optimistic bias in small samples. Hence the new methods seem preferable for creating confidence intervals, except when compound symmetry definitely holds.