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EXACT DISTRIBUTIONS OF INTRACLASS CORRELATION AND CRONBACH’S ALPHA WITH GAUSSIAN DATA AND GENERAL COVARIANCE

机译：高斯数据和一般协方差的类内相关性和Cronbach Alpha的精确分布

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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.
机译：类内相关和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，也仅对于复合对称协方差（均方差和均等相关）才知道确切的分布。最近，为分布函数推导了大样本高斯近似值。新的精确结果允许计算具有任何协方差模式的高斯数据的精确分布函数和类内相关性和Cronbach's alpha的其他属性，而不仅仅是复合对称性。概率是根据独立卡方随机变量的加权和的分布函数来计算的。类内相关和Cronbach's alpha的分布函数的新F近似比精确形式更容易计算。假设协方差矩阵是已知的，即使只有十个观测值，这些近似值通常也可以提供足够的准确性。精确分布或近似分布都可以用来创建围绕可靠性评估的置信区间。蒙特卡洛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，这将导致准确的置信区间，如先前已知结果所预期的那样。但是，假设并估计一般协方差矩阵会产生10个观测值，在某种程度上是乐观的窄置信区间。将样本大小增加到100可提供基本无偏的覆盖范围。错误地假设化合物对称性会导致悲观的置信区间过大，悲观情绪随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，新方法似乎可用于创建置信区间，除非复合对称性绝对成立。

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