[{"data":1,"prerenderedAt":361},["ShallowReactive",2],{"content-query-fUkf9P9IUm":3},{"_path":4,"_dir":5,"_draft":6,"_partial":6,"_locale":7,"title":8,"description":9,"layout":10,"body":11,"_type":355,"_id":356,"_source":357,"_file":358,"_extension":359,"sitemap":360},"/documentation/algorithms-and-techniques/validity-and-model-fit/cta-pls","validity-and-model-fit",false,"","Confirmatory Tetrad Analysis in PLS-SEM (CTA-PLS)","CTA-PLS statistically tests whether a construct's measurement model is formative or reflective. Learn how the test works, its indicator requirements, and its settings in SmartPLS.","algorithm-description",{"type":12,"children":13,"toc":336},"root",[14,23,29,36,41,47,54,59,64,75,81,86,96,102,107,113,118,124,130,135,141,146,152,157,163,168,174,216,222],{"type":15,"tag":16,"props":17,"children":19},"element","h1",{"id":18},"confirmatory-tetrad-analysis-in-pls-cta-pls",[20],{"type":21,"value":22},"text","Confirmatory Tetrad Analysis in PLS (CTA-PLS)",{"type":15,"tag":24,"props":25,"children":26},"p",{},[27],{"type":21,"value":28},"Confirmatory tetrad analysis in PLS-SEM (CTA-PLS) is a statistical procedure that helps researchers decide whether a construct is more consistent with a formative or a reflective measurement model. Rather than relying only on conceptual reasoning, CTA-PLS tests the model-implied vanishing tetrads of a construct against the null hypothesis of a reflective measurement model, following the confirmatory approach of Bollen and Ting (1993, 2000). In the PLS-SEM context, a bootstrapping procedure is applied to test the significance of the model-implied tetrads (Cefis et al., 2025; Gudergan et al., 2008; Hair et al., 2024).",{"type":15,"tag":30,"props":31,"children":33},"h2",{"id":32},"how-cta-pls-works",[34],{"type":21,"value":35},"How CTA-PLS Works",{"type":15,"tag":24,"props":37,"children":38},{},[39],{"type":21,"value":40},"Gudergan et al. (2008) and Hair et al. (2024) describe the CTA-PLS procedure in detail. The implemented procedure requires at least 4 manifest variables per construct and can handle a maximum of 25 manifest variables per construct, because the number of tests needed to check whether a tetrad is redundant increases exponentially with the number of indicators.",{"type":15,"tag":30,"props":42,"children":44},{"id":43},"cta-pls-settings-in-smartpls",[45],{"type":21,"value":46},"CTA-PLS Settings in SmartPLS",{"type":15,"tag":48,"props":49,"children":51},"h3",{"id":50},"subsamples",[52],{"type":21,"value":53},"Subsamples",{"type":15,"tag":24,"props":55,"children":56},{},[57],{"type":21,"value":58},"In bootstrapping, subsamples are created with observations randomly drawn from the original set of data (with replacement). To ensure stability of results, the number of subsamples should be large.",{"type":15,"tag":24,"props":60,"children":61},{},[62],{"type":21,"value":63},"For an initial assessment, you may want to choose a smaller number of bootstrap subsamples (e.g., 500) to be randomly drawn and estimated with the PLS-SEM algorithm, since that requires less computation time. For the final results, however, use a large number of bootstrap subsamples (e.g., 10,000).",{"type":15,"tag":24,"props":65,"children":66},{},[67,73],{"type":15,"tag":68,"props":69,"children":70},"strong",{},[71],{"type":21,"value":72},"Note:",{"type":21,"value":74}," Larger numbers of bootstrap subsamples increase the computation time.",{"type":15,"tag":48,"props":76,"children":78},{"id":77},"parallel-processing",[79],{"type":21,"value":80},"Parallel Processing",{"type":15,"tag":24,"props":82,"children":83},{},[84],{"type":21,"value":85},"This option runs the bootstrapping routine on multiple processors, if your computer offers more than one core. Using parallel computing reduces computation time.",{"type":15,"tag":24,"props":87,"children":88},{},[89,94],{"type":15,"tag":68,"props":90,"children":91},{},[92],{"type":21,"value":93},"Important:",{"type":21,"value":95}," The number of processes should not be higher than the number of processors in your computer.",{"type":15,"tag":48,"props":97,"children":99},{"id":98},"test-type",[100],{"type":21,"value":101},"Test Type",{"type":15,"tag":24,"props":103,"children":104},{},[105],{"type":21,"value":106},"Specifies whether a one-sided or two-sided significance test is conducted.",{"type":15,"tag":48,"props":108,"children":110},{"id":109},"significance-level",[111],{"type":21,"value":112},"Significance Level",{"type":15,"tag":24,"props":114,"children":115},{},[116],{"type":21,"value":117},"Specifies the significance level of the test statistic.",{"type":15,"tag":30,"props":119,"children":121},{"id":120},"frequently-asked-questions",[122],{"type":21,"value":123},"Frequently Asked Questions",{"type":15,"tag":48,"props":125,"children":127},{"id":126},"what-does-cta-pls-test",[128],{"type":21,"value":129},"What does CTA-PLS test?",{"type":15,"tag":24,"props":131,"children":132},{},[133],{"type":21,"value":134},"CTA-PLS tests whether a construct's measurement model is more consistent with a formative or a reflective specification. It tests the model-implied vanishing tetrads against the null hypothesis of a reflective measurement model, using a bootstrapping procedure to assess statistical significance.",{"type":15,"tag":48,"props":136,"children":138},{"id":137},"how-many-indicators-does-a-construct-need-for-cta-pls",[139],{"type":21,"value":140},"How many indicators does a construct need for CTA-PLS?",{"type":15,"tag":24,"props":142,"children":143},{},[144],{"type":21,"value":145},"The implemented procedure requires at least 4 manifest variables per construct. It can handle a maximum of 25 manifest variables per construct, since the number of required tetrad tests grows exponentially with the number of indicators.",{"type":15,"tag":48,"props":147,"children":149},{"id":148},"how-many-bootstrap-subsamples-should-i-use-for-cta-pls",[150],{"type":21,"value":151},"How many bootstrap subsamples should I use for CTA-PLS?",{"type":15,"tag":24,"props":153,"children":154},{},[155],{"type":21,"value":156},"For an initial assessment, a smaller number of subsamples (e.g., 500) is sufficient and faster to compute. For final results intended for reporting, use a larger number of subsamples (e.g., 10,000) to ensure stable results.",{"type":15,"tag":48,"props":158,"children":160},{"id":159},"should-i-use-a-one-sided-or-two-sided-test-in-cta-pls",[161],{"type":21,"value":162},"Should I use a one-sided or two-sided test in CTA-PLS?",{"type":15,"tag":24,"props":164,"children":165},{},[166],{"type":21,"value":167},"SmartPLS lets you specify the test type (one-sided or two-sided) and the significance level for the CTA-PLS significance test. The appropriate choice depends on your research hypotheses and should be justified accordingly.",{"type":15,"tag":30,"props":169,"children":171},{"id":170},"related-smartpls-methods",[172],{"type":21,"value":173},"Related SmartPLS Methods",{"type":15,"tag":175,"props":176,"children":177},"ul",{},[178,189,198,207],{"type":15,"tag":179,"props":180,"children":181},"li",{},[182],{"type":15,"tag":183,"props":184,"children":186},"a",{"href":185},"/documentation/algorithms-and-techniques/validity-and-model-fit/confirmatory-composite-analysis/",[187],{"type":21,"value":188},"Confirmatory composite analysis (CCA)",{"type":15,"tag":179,"props":190,"children":191},{},[192],{"type":15,"tag":183,"props":193,"children":195},{"href":194},"/documentation/algorithms-and-techniques/validity-and-model-fit/discriminant-validity-assessment/",[196],{"type":21,"value":197},"Discriminant validity assessment (HTMT)",{"type":15,"tag":179,"props":199,"children":200},{},[201],{"type":15,"tag":183,"props":202,"children":204},{"href":203},"/documentation/algorithms-and-techniques/resampling-and-inference/bootstrapping/",[205],{"type":21,"value":206},"Bootstrapping",{"type":15,"tag":179,"props":208,"children":209},{},[210],{"type":15,"tag":183,"props":211,"children":213},{"href":212},"/documentation/algorithms-and-techniques/core-algorithm/pls/",[214],{"type":21,"value":215},"PLS-SEM algorithm",{"type":15,"tag":30,"props":217,"children":219},{"id":218},"references",[220],{"type":21,"value":221},"References",{"type":15,"tag":175,"props":223,"children":224},{},[225,238,257,285,311,327],{"type":15,"tag":179,"props":226,"children":227},{},[228,230,236],{"type":21,"value":229},"Bollen, K. A., & Ting, K.-f. (1993). Confirmatory tetrad analysis. In P. V. Marsden (Ed.), ",{"type":15,"tag":231,"props":232,"children":233},"em",{},[234],{"type":21,"value":235},"Sociological methodology",{"type":21,"value":237}," (pp. 147–175). American Sociological Association.",{"type":15,"tag":179,"props":239,"children":240},{},[241,243,248,250,255],{"type":21,"value":242},"Bollen, K. A., & Ting, K.-f. (2000). A tetrad test for causal indicators. ",{"type":15,"tag":231,"props":244,"children":245},{},[246],{"type":21,"value":247},"Psychological Methods",{"type":21,"value":249},", ",{"type":15,"tag":231,"props":251,"children":252},{},[253],{"type":21,"value":254},"5",{"type":21,"value":256},"(1), 3–22.",{"type":15,"tag":179,"props":258,"children":259},{},[260,262,270,272,277,278,283],{"type":21,"value":261},"Cefis, M., Angelelli, M., Carpita, M., & Ciavolino, E. (2025). 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