The tetrachoric correlation is a popular measure of association for binary data and estimates the correlation of an underlying normal latent vector. However, when the underlying vector is not normal, the tetrachoric correlation will be different from the underlying correlation. Since assuming underlying normality is often done on pragmatic and not substantial grounds, the estimated tetrachoric correlation may therefore be quite different from the true underlying correlation that is modeled in structural equation modeling. This motivates studying the range of latent correlations that are compatible with given binary data, when the distribution of the latent vector is partly or completely unknown. We show that nothing can be said about the latent correlations unless we know more than what can be derived from the data. We identify an interval constituting all latent correlations compatible with observed data when the marginals of the latent variables are known. Also, we quantify how partial knowledge of the dependence structure of the latent variables affect the range of compatible latent correlations. Implications for tests of underlying normality are briefly discussed.

中文翻译：

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二进制数据潜在相关性的部分识别

四项相关性是二元数据关联的常用度量，可估计潜在正态潜在向量的相关性。但是，当底层向量不正常时，四项相关性将与底层相关性不同。由于假设潜在的正态性通常是基于实用而非实质性的理由，因此估计的四项相关性可能与在结构方程建模中建模的真实潜在相关性完全不同。当潜在向量的分布部分或完全未知时，这激发了研究与给定二进制数据兼容的潜在相关性范围。我们表明，除非我们知道的不仅仅是可以从数据中得出的内容，否则就无法谈论潜在的相关性。当潜在变量的边缘已知时，我们确定一个区间，该区间构成与观察到的数据兼容的所有潜在相关性。此外，我们量化了潜在变量依赖结构的部分知识如何影响兼容潜在相关性的范围。简要讨论了对基本正态性检验的影响。