The score test statistic computed using the observed information is easy to compute numerically. Its large sample distribution under the null hypothesis is well known and is equivalent to that of the score test based on the expected information, the likelihood-ratio test and the Wald test. However, several authors have noted that under the alternative hypothesis this no longer holds and in particular the score statistic computed using the observed information can take negative values. We extend the body of work on the score test to a problem of interest in ecology when studying the occurrence of species. This is the comparison of two zero-inflated binomial random variables from two independent samples under imperfect detection. An analysis of eigenvalues associated with the score test in this setting assists in understanding why using the observed information matrix in the score test can be problematic. We demonstrate through a combination of simulations and theoretical analysis that the power of the score test calculated under the observed information decreases as the populations being compared become more dissimilar. In particular, the score test based on the observed information is inconsistent. Finally, we propose a modified rule that rejects the null hypothesis when the score statistic is computed using the observed information is negative or is larger than the usual chi-square cut-off. In simulations in our setting this has power that is comparable to the Wald and likelihood ratio tests and consistency is largely restored. Our new test is easy to use and inference is possible. Supplementary material for this article is available online as per journal instructions.

中文翻译：

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二样本占用模型的得分检验

使用观察到的信息计算的分数测试统计量很容易用数字计算。其在原假设下的大样本分布是众所周知的，相当于基于预期信息的评分检验、似然比检验和Wald检验。然而，几位作者指出，在替代假设下，这不再成立，特别是使用观察到的信息计算的分数统计可能会取负值。在研究物种发生时，我们将分数测试的工作范围扩展到生态学中感兴趣的问题。这是在不完美检测下来自两个独立样本的两个零膨胀二项式随机变量的比较。在此设置中对与评分测试相关联的特征值进行分析有助于理解为什么在评分测试中使用观察到的信息矩阵可能会出现问题。我们通过模拟和理论分析的结合证明了在观察到的信息下计算的分数测试的功效随着被比较的人群变得更加不同而降低。尤其是基于观察到的信息的分数测试是不一致的。最后，我们提出了一个修改后的规则，当使用观察到的信息计算得分统计量为负或大于通常的卡方截止值时，该规则拒绝零假设。在我们设置的模拟中，这具有与 Wald 和似然比测试相当的能力，并且在很大程度上恢复了一致性。我们的新测试易于使用并且可以进行推理。根据期刊说明，可在线获取本文的补充材料。