Table_9_Investigating the Applicability of Alignment—A Monte Carlo Simulation Study.docx

Traditional multiple-group confirmatory factor analysis (multiple-group CFA) is usually criticized for having too restrictive model assumption, namely the scalar measurement invariance. The new multiple-group analysis methodology, alignment, has become an effective alternative. The alignment evaluates measurement invariance and more importantly, permits factor mean comparisons without requiring scalar invariance which is usually required in traditional multiple-group CFA. Some simulation studies and empirical studies have investigated the applicability of alignment under different conditions, but some areas remain unexplored. Based on the simulation studies of Asparouhov and Muthén and of Flake and McCoach, this current simulation study is broken into two sections. The first study investigates the minimal group sizes required for alignment in three-factor models. The second study compares the performance of multiple-group CFA, multiple-group exploratory structural equation model (multiple-group ESEM), and alignment by including different proportions and magnitudes of cross-loadings in the models. Study 1 shows that when the model has no noninvariant parameters, the alignment requires relatively lower group sizes. Explicitly, the minimal group size required for alignment was 250 when the amount of groups was three, the minimal group size was 150 when the amount of groups was nine, and 200 when the amount of groups was 15. When there are noninvariant parameters in the model and the amount of groups is low, a group size of 350 is a safe rule of thumb. When there are noninvariant parameters in the model and the amount of groups is high, a group size of 250 is required for trustworthy results. The magnitude of noninvariance and the noninvariance rate do not affect the minimal group size required for alignment. Study 2 shows that multiple-group CFA provides accurate factor mean estimates when each factor had 20% factor loading (1 factor loading) with small-sized cross-loading. Multiple-group ESEM provides accurate factor mean estimates when the magnitude of cross-loading is small or when each factor had 20% factor loading (1 factor loading) with medium-sized cross-loading. Alignment provides accurate factor mean estimates when there are only small-sized cross-loadings in the model. The parameter estimates, coverage rates and ratios of average standard error to standard deviation for each methodology are not influenced by the amount of groups. Recommendations are concluded for using multiple-group CFA, multiple-group ESEM, traditional alignment and aligned ESEM (AESEM) based on the results. Multiple-group CFA is more suitable for use when scalar invariance is established. Multiple-group ESEM works best when there are small-sized or only a few medium-sized cross-loadings in the model. Traditional alignment allows for small-sized cross-loadings and a few noninvariant parameters in the model. AESEM integrates the advantages of alignment and ESEM, can provide accurate estimates when noninvariant parameters and cross-loadings both exist in the model. Compared to multiple-group CFA, multiple-group ESEM, the alignment methodology performs well in more situations.

传统多组验证性因素分析（多组CFA）常因模型假设过于严格而受到批评，即标量测量不变性。新型的多组分析方法——对齐，已成为一种有效的替代方案。对齐方法评估测量不变性，更重要的是，允许在无需标量不变性的条件下进行因子均值比较，这在传统多组CFA中通常是必需的。一些模拟研究和实证研究已经探讨了在不同条件下对齐的应用性，但仍有一些领域尚未被深入探讨。基于Asparouhov和Muthén以及Flake和McCoach的模拟研究，本研究分为两个部分。第一部分研究探讨了三因子模型中对齐所需的最小组大小。第二部分通过在模型中包含不同比例和程度的交叉负荷，比较了多组CFA、多组探索性结构方程模型（多组ESEM）和对齐的性能。研究1表明，当模型没有非不变参数时，对齐所需的最小组大小相对较低。具体而言，当组数为三个时，对齐所需的最小组大小为250；当组数为九时，最小组大小为150；当组数为十五时，最小组大小为200。当模型中存在非不变参数且组数较少时，350个成员的组大小是安全的经验法则。当模型中存在非不变参数且组数较多时，需要250个成员的组大小以确保结果的可信度。非不变性的程度和非不变率不影响对齐所需的最小组大小。研究2表明，在交叉负荷较小，并且每个因子具有20%的因子负荷（1个因子负荷）时，多组CFA能够提供准确的因子均值估计。当交叉负荷的程度较小或每个因子具有20%的因子负荷（1个因子负荷）且交叉负荷为中等大小时，多组ESEM能够提供准确的因子均值估计。当模型中只有小规模交叉负荷时，对齐能够提供准确的因子均值估计。每种方法的参数估计、覆盖率和平均标准误与标准差之比均不受组数的影响。基于研究结果，提出了关于使用多组CFA、多组ESEM、传统对齐和对齐ESEM（AESEM）的建议。当标量不变性得到确立时，多组CFA更为适用。当模型中存在小规模或仅有少量中等规模交叉负荷时，多组ESEM表现最佳。传统对齐允许模型中有小规模交叉负荷和少量非不变参数。AESEM整合了对齐和ESEM的优势，在模型中同时存在非不变参数和交叉负荷时，能够提供准确的估计。与多组CFA和多组ESEM相比，对齐方法在更多情况下表现出良好的性能。