If the probability model is correctly specified, then we can estimate the covariance matrix of the asymptotic maximum likelihood estimate distribution using either the first or second derivatives of the likelihood function. Therefore, if the determinants of these two different covariance matrix estimation formulas differ this indicates model misspecification. This misspecification detection strategy is the basis of the Determinant Information Matrix Test ($GIMT_{Det}$). To investigate the performance of the $GIMT_{Det}$, a Deterministic Input Noisy And gate (DINA) Cognitive Diagnostic Model (CDM) was fit to the Fraction-Subtraction dataset. Next, various misspecified versions of the original DINA CDM were fit to bootstrap data sets generated by sampling from the original fitted DINA CDM. The $GIMT_{Det}$ showed good discrimination performance for larger levels of misspecification. In addition, the $GIMT_{Det}$ did not detect model misspecification when model misspecification was not present and additionally did not detect model misspecification when the level of misspecification was very low. However, the $GIMT_{Det}$ discrimation performance was highly variable across different misspecification strategies when the misspecification level was moderately sized. The proposed new misspecification detection methodology is promising but additional empirical studies are required to further characterize its strengths and limitations.

翻译：若概率模型设定正确，则我们既可使用似然函数的一阶导数，也可使用其二阶导数来估计渐近最大似然估计分布的协方差矩阵。因此，若这两种不同协方差矩阵估计公式的行列式存在差异，则表明模型存在误设。这种误设检测策略构成了行列式信息矩阵检验（$GIMT_{Det}$）的基础。为探究$GIMT_{Det}$的性能，本研究将确定性输入噪声与门（DINA）认知诊断模型（CDM）拟合至分数减法数据集。随后，将原始DINA CDM的多种误设版本拟合至通过从原始拟合DINA CDM中抽样生成的Bootstrap数据集。$GIMT_{Det}$在较大误设程度下展现出良好的判别性能。此外，当模型不存在误设时$GIMT_{Det}$未检测到误设，且在误设程度极低时同样未检测到误设。然而，在误设程度中等时，$GIMT_{Det}$的判别性能在不同误设策略间存在显著波动。所提出的新型误设检测方法前景可观，但需通过更多实证研究进一步明确其优势与局限。