[{"data":1,"prerenderedAt":551},["ShallowReactive",2],{"content-query-XteTUQSulJ":3},{"_path":4,"_dir":5,"_draft":6,"_partial":6,"_locale":7,"title":8,"description":9,"layout":10,"body":11,"_type":545,"_id":546,"_source":547,"_file":548,"_extension":549,"sitemap":550},"/documentation/algorithms-and-techniques/core-algorithm/pls","core-algorithm",false,"","The PLS-SEM Algorithm: How SmartPLS Estimates Models","The PLS-SEM algorithm iteratively estimates latent variable scores from indicator data through a sequence of regressions. Learn its stages and settings in SmartPLS.","algorithm-description",{"type":12,"children":13,"toc":524},"root",[14,23,29,36,41,46,48,115,120,127,132,138,145,150,151,212,217,223,228,234,260,266,271,277,296,302,308,313,319,324,330,335,341,346,352,357,363,394,400],{"type":15,"tag":16,"props":17,"children":19},"element","h1",{"id":18},"pls-sem-algorithm",[20],{"type":21,"value":22},"text","PLS-SEM Algorithm",{"type":15,"tag":24,"props":25,"children":26},"p",{},[27],{"type":21,"value":28},"The partial least squares (PLS) path modeling method, also known as PLS structural equation modeling (PLS-SEM), was developed by Wold (1982) and further refined by Lohmöller (1989). At its core, the PLS-SEM algorithm is a sequence of regressions expressed in terms of weight vectors; the resulting outer weights satisfy fixed point equations once the algorithm has converged. Researchers use this algorithm to estimate latent variable (construct) scores from the indicator data and the specified path model, which in turn provide the basis for estimating every other relationship in the model. It is the core estimation procedure underlying all PLS-SEM analyses in SmartPLS.",{"type":15,"tag":30,"props":31,"children":33},"h2",{"id":32},"how-the-pls-sem-algorithm-works",[34],{"type":21,"value":35},"How the PLS-SEM Algorithm Works",{"type":15,"tag":24,"props":37,"children":38},{},[39],{"type":21,"value":40},"The PLS algorithm is essentially a sequence of regressions in terms of weight vectors (Henseler et al., 2009). The weight vectors obtained at convergence satisfy fixed point equations (see Dijkstra, 2010, for a general analysis of such equations and the ensuing convergence issues). The objective of the PLS-SEM algorithm is to find a stable set of outer/indicator weights that iteratively maximizes both the explained variance between connected composites and the explained variance in the indicators by their respective composites (when using Mode A, correlation weights); see also Hair et al. (2024).",{"type":15,"tag":24,"props":42,"children":43},{},[44],{"type":21,"value":45},"The basic PLS algorithm, as suggested by Lohmöller (1989) and implemented in SmartPLS, includes the following three stages:",{"type":21,"value":47},"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n",{"type":15,"tag":49,"props":50,"children":51},"table",{},[52,71],{"type":15,"tag":53,"props":54,"children":55},"thead",{},[56],{"type":15,"tag":57,"props":58,"children":59},"tr",{},[60,66],{"type":15,"tag":61,"props":62,"children":63},"th",{},[64],{"type":21,"value":65},"Stage",{"type":15,"tag":61,"props":67,"children":68},{},[69],{"type":21,"value":70},"What happens",{"type":15,"tag":72,"props":73,"children":74},"tbody",{},[75,89,102],{"type":15,"tag":57,"props":76,"children":77},{},[78,84],{"type":15,"tag":79,"props":80,"children":81},"td",{},[82],{"type":21,"value":83},"Stage 1",{"type":15,"tag":79,"props":85,"children":86},{},[87],{"type":21,"value":88},"Iterative estimation of latent variable scores through a four-step procedure, repeated until convergence (or until the maximum number of iterations is reached): (1) outer approximation of the latent variable scores, (2) estimation of the inner weights, (3) inner approximation of the latent variable scores, and (4) estimation of the outer weights.",{"type":15,"tag":57,"props":90,"children":91},{},[92,97],{"type":15,"tag":79,"props":93,"children":94},{},[95],{"type":21,"value":96},"Stage 2",{"type":15,"tag":79,"props":98,"children":99},{},[100],{"type":21,"value":101},"Estimation of outer weights/loadings and path coefficients.",{"type":15,"tag":57,"props":103,"children":104},{},[105,110],{"type":15,"tag":79,"props":106,"children":107},{},[108],{"type":21,"value":109},"Stage 3",{"type":15,"tag":79,"props":111,"children":112},{},[113],{"type":21,"value":114},"Estimation of location parameters.",{"type":15,"tag":24,"props":116,"children":117},{},[118],{"type":21,"value":119},"The following figure formally shows Wold's basic PLS-SEM algorithm as presented by Lohmöller (1989, p. 29):",{"type":15,"tag":121,"props":122,"children":126},"page-image",{":page":123,"image":124,"title":22,"width":125,"height":125},"path","/documentation/algorithms-and-techniques/core-algorithm/pls/pls-algorithm.jpg",650,[],{"type":15,"tag":24,"props":128,"children":129},{},[130],{"type":21,"value":131},"Hair et al. (2027), Henseler et al. (2012), and Lohmöller (1989) provide detailed explanations of how the basic PLS-SEM algorithm operates as implemented in SmartPLS.",{"type":15,"tag":30,"props":133,"children":135},{"id":134},"pls-algorithm-settings-in-smartpls",[136],{"type":21,"value":137},"PLS Algorithm Settings in SmartPLS",{"type":15,"tag":139,"props":140,"children":142},"h3",{"id":141},"weighting-scheme",[143],{"type":21,"value":144},"Weighting Scheme",{"type":15,"tag":24,"props":146,"children":147},{},[148],{"type":21,"value":149},"PLS-SEM allows you to apply three structural model weighting schemes:",{"type":21,"value":47},{"type":15,"tag":49,"props":152,"children":153},{},[154,170],{"type":15,"tag":53,"props":155,"children":156},{},[157],{"type":15,"tag":57,"props":158,"children":159},{},[160,165],{"type":15,"tag":61,"props":161,"children":162},{},[163],{"type":21,"value":164},"Weighting scheme",{"type":15,"tag":61,"props":166,"children":167},{},[168],{"type":21,"value":169},"Notes",{"type":15,"tag":72,"props":171,"children":172},{},[173,186,199],{"type":15,"tag":57,"props":174,"children":175},{},[176,181],{"type":15,"tag":79,"props":177,"children":178},{},[179],{"type":21,"value":180},"Factor weighting scheme",{"type":15,"tag":79,"props":182,"children":183},{},[184],{"type":21,"value":185},"Results differ little from the path weighting scheme.",{"type":15,"tag":57,"props":187,"children":188},{},[189,194],{"type":15,"tag":79,"props":190,"children":191},{},[192],{"type":21,"value":193},"Path weighting scheme (default)",{"type":15,"tag":79,"props":195,"children":196},{},[197],{"type":21,"value":198},"The recommended approach. It provides the highest R² value for endogenous latent variables and is generally applicable to all kinds of PLS path model specifications and estimations.",{"type":15,"tag":57,"props":200,"children":201},{},[202,207],{"type":15,"tag":79,"props":203,"children":204},{},[205],{"type":21,"value":206},"Principal component analysis (PCA)",{"type":15,"tag":79,"props":208,"children":209},{},[210],{"type":21,"value":211},"Uses the results of a PCA per construct to estimate the model.",{"type":15,"tag":24,"props":213,"children":214},{},[215],{"type":21,"value":216},"Note: SmartPLS no longer offers the centroid weighting scheme, which was available in SmartPLS 2 and 3.",{"type":15,"tag":139,"props":218,"children":220},{"id":219},"type-of-results",[221],{"type":21,"value":222},"Type of Results",{"type":15,"tag":24,"props":224,"children":225},{},[226],{"type":21,"value":227},"This option lets you choose between standardized (default), unstandardized, and mean-centered PLS-SEM outcomes. These settings affect the construct scores provided by the PLS-SEM algorithm and the estimated coefficients (e.g., standardized coefficients in the structural model, or unstandardized coefficients with an intercept in the structural model).",{"type":15,"tag":139,"props":229,"children":231},{"id":230},"initial-outer-weights",[232],{"type":21,"value":233},"Initial Outer Weights",{"type":15,"tag":235,"props":236,"children":237},"ul",{},[238,250],{"type":15,"tag":239,"props":240,"children":241},"li",{},[242,248],{"type":15,"tag":243,"props":244,"children":245},"strong",{},[246],{"type":21,"value":247},"Standard",{"type":21,"value":249},": As the default (i.e., the SmartPLS setting), the initial outer weights are set to +1.",{"type":15,"tag":239,"props":251,"children":252},{},[253,258],{"type":15,"tag":243,"props":254,"children":255},{},[256],{"type":21,"value":257},"Individual",{"type":21,"value":259},": SmartPLS lets you define individual initial outer weights for every indicator in the PLS path model. For example, an indicator that is assumed a priori to have a particularly strong positive relationship with the latent variable can be given an initial weight of +1, while the other indicators of the same measurement model receive 0.",{"type":15,"tag":139,"props":261,"children":263},{"id":262},"maximum-iterations",[264],{"type":21,"value":265},"Maximum Iterations",{"type":15,"tag":24,"props":267,"children":268},{},[269],{"type":21,"value":270},"This option to change the settings for running the PLS-SEM algorithm is no longer available in SmartPLS 4. The permanent setting is 3,000 iterations. This parameter represents the maximum number of iterations used for calculating the PLS results. This number should be sufficiently large (e.g., 3,000 iterations). When checking a PLS-SEM result, make sure that the algorithm did not stop because the maximum number of iterations was reached, but because the stop criterion was met. Note: Selecting 0 for the maximum number of iterations lets you obtain results from the sum scores approach.",{"type":15,"tag":139,"props":272,"children":274},{"id":273},"stop-criterion",[275],{"type":21,"value":276},"Stop Criterion",{"type":15,"tag":24,"props":278,"children":279},{},[280,282,288,290,294],{"type":21,"value":281},"This option to change the settings for running the PLS-SEM algorithm is no longer available in SmartPLS 4. The permanent setting is 10",{"type":15,"tag":283,"props":284,"children":285},"sup",{},[286],{"type":21,"value":287},"-7",{"type":21,"value":289},". The PLS algorithm stops when the change in the outer weights between two consecutive iterations is smaller than this stop criterion value (or the maximum number of iterations is reached). This value should be sufficiently small (e.g., 10",{"type":15,"tag":283,"props":291,"children":292},{},[293],{"type":21,"value":287},{"type":21,"value":295},").",{"type":15,"tag":30,"props":297,"children":299},{"id":298},"frequently-asked-questions",[300],{"type":21,"value":301},"Frequently Asked Questions",{"type":15,"tag":139,"props":303,"children":305},{"id":304},"what-is-the-pls-sem-algorithm",[306],{"type":21,"value":307},"What is the PLS-SEM algorithm?",{"type":15,"tag":24,"props":309,"children":310},{},[311],{"type":21,"value":312},"The PLS-SEM algorithm is the estimation procedure behind partial least squares structural equation modeling. It is a sequence of regressions in terms of weight vectors that iteratively estimates latent variable scores from the indicator data and the specified path model, which are then used to estimate all relationships in the model.",{"type":15,"tag":139,"props":314,"children":316},{"id":315},"who-developed-the-pls-sem-algorithm",[317],{"type":21,"value":318},"Who developed the PLS-SEM algorithm?",{"type":15,"tag":24,"props":320,"children":321},{},[322],{"type":21,"value":323},"The method was developed by Wold (1982) and further refined by Lohmöller (1989), whose three-stage version of the algorithm is the one implemented in SmartPLS.",{"type":15,"tag":139,"props":325,"children":327},{"id":326},"what-are-the-three-stages-of-the-pls-sem-algorithm",[328],{"type":21,"value":329},"What are the three stages of the PLS-SEM algorithm?",{"type":15,"tag":24,"props":331,"children":332},{},[333],{"type":21,"value":334},"Stage 1 iteratively estimates latent variable scores through outer approximation, inner weight estimation, inner approximation, and outer weight estimation, repeated until convergence. Stage 2 estimates the outer weights/loadings and path coefficients. Stage 3 estimates the location parameters.",{"type":15,"tag":139,"props":336,"children":338},{"id":337},"which-weighting-scheme-should-i-use-in-smartpls",[339],{"type":21,"value":340},"Which weighting scheme should I use in SmartPLS?",{"type":15,"tag":24,"props":342,"children":343},{},[344],{"type":21,"value":345},"The path weighting scheme is the default and recommended option because it provides the highest R² value for endogenous latent variables and applies to all kinds of PLS path model specifications. The factor weighting scheme produces very similar results. The PCA option instead estimates the model using per-construct PCA results.",{"type":15,"tag":139,"props":347,"children":349},{"id":348},"can-i-change-the-maximum-number-of-iterations-or-the-stop-criterion",[350],{"type":21,"value":351},"Can I change the maximum number of iterations or the stop criterion?",{"type":15,"tag":24,"props":353,"children":354},{},[355],{"type":21,"value":356},"No, not in SmartPLS 4. Both settings are fixed (3,000 iterations and a stop criterion of 10^-7^) and are no longer user-configurable. Setting the maximum number of iterations to 0 returns results from the sum scores approach instead of the iterative PLS algorithm.",{"type":15,"tag":30,"props":358,"children":360},{"id":359},"related-smartpls-methods",[361],{"type":21,"value":362},"Related SmartPLS Methods",{"type":15,"tag":235,"props":364,"children":365},{},[366,376,385],{"type":15,"tag":239,"props":367,"children":368},{},[369],{"type":15,"tag":370,"props":371,"children":373},"a",{"href":372},"/documentation/algorithms-and-techniques/core-algorithm/consistent-pls/",[374],{"type":21,"value":375},"Consistent PLS-SEM (PLSc)",{"type":15,"tag":239,"props":377,"children":378},{},[379],{"type":15,"tag":370,"props":380,"children":382},{"href":381},"/documentation/algorithms-and-techniques/core-algorithm/weighted-pls/",[383],{"type":21,"value":384},"Weighted PLS algorithm (WPLS)",{"type":15,"tag":239,"props":386,"children":387},{},[388],{"type":15,"tag":370,"props":389,"children":391},{"href":390},"/documentation/algorithms-and-techniques/core-algorithm/pls-pca/",[392],{"type":21,"value":393},"PLS-PCA",{"type":15,"tag":30,"props":395,"children":397},{"id":396},"references",[398],{"type":21,"value":399},"References",{"type":15,"tag":235,"props":401,"children":402},{},[403,420,449,470,491,503,515],{"type":15,"tag":239,"props":404,"children":405},{},[406,408,418],{"type":21,"value":407},"Hair, J. F., Hult, G. T. M., Ringle, C. M., & Sarstedt, M. (2027). ",{"type":15,"tag":409,"props":410,"children":411},"em",{},[412],{"type":15,"tag":370,"props":413,"children":415},{"href":414},"/documentation/must-reads/pls-sem-book",[416],{"type":21,"value":417},"A primer on partial least squares structural equation modeling (PLS-SEM)",{"type":21,"value":419}," (4th ed.). Sage.",{"type":15,"tag":239,"props":421,"children":422},{},[423,425,433,435,440,442,447],{"type":21,"value":424},"Hair, J. F., Sarstedt, M., Ringle, C. M., Sharma, P. N., & Liengaard, B. D. (2024). ",{"type":15,"tag":370,"props":426,"children":430},{"href":427,"rel":428},"https://www.emerald.com/ejm/article/58/13/81/1222222/Going-beyond-the-untold-facts-in-PLS-SEM-and",[429],"nofollow",[431],{"type":21,"value":432},"Going beyond the untold facts in PLS-SEM and moving forward.",{"type":21,"value":434}," ",{"type":15,"tag":409,"props":436,"children":437},{},[438],{"type":21,"value":439},"European Journal of Marketing",{"type":21,"value":441},", ",{"type":15,"tag":409,"props":443,"children":444},{},[445],{"type":21,"value":446},"58",{"type":21,"value":448},"(13), 81–106.",{"type":15,"tag":239,"props":450,"children":451},{},[452,454,461,463,468],{"type":21,"value":453},"Henseler, J., Ringle, C. M., & Sarstedt, M. (2012). ",{"type":15,"tag":370,"props":455,"children":458},{"href":456,"rel":457},"http://www.elgaronline.com/view/9781848448582.00023.xml",[429],[459],{"type":21,"value":460},"Using partial least squares path modeling in international advertising research: Basic concepts and recent issues.",{"type":21,"value":462}," In S. Okazaki (Ed.), ",{"type":15,"tag":409,"props":464,"children":465},{},[466],{"type":21,"value":467},"Handbook of research in international advertising",{"type":21,"value":469}," (pp. 252–276). Edward Elgar Publishing.",{"type":15,"tag":239,"props":471,"children":472},{},[473,475,482,484,489],{"type":21,"value":474},"Henseler, J., Ringle, C. M., & Sinkovics, R. R. (2009). ",{"type":15,"tag":370,"props":476,"children":479},{"href":477,"rel":478},"http://www.emeraldinsight.com/books.htm?chapterid=1775963",[429],[480],{"type":21,"value":481},"The use of partial least squares path modeling in international marketing.",{"type":21,"value":483}," In R. R. Sinkovics & P. N. Ghauri (Eds.), ",{"type":15,"tag":409,"props":485,"children":486},{},[487],{"type":21,"value":488},"Advances in international marketing",{"type":21,"value":490}," (pp. 277–320). Emerald.",{"type":15,"tag":239,"props":492,"children":493},{},[494,496,501],{"type":21,"value":495},"Lohmöller, J.-B. (1989). ",{"type":15,"tag":409,"props":497,"children":498},{},[499],{"type":21,"value":500},"Latent variable path modeling with partial least squares",{"type":21,"value":502},". Physica.",{"type":15,"tag":239,"props":504,"children":505},{},[506,508,513],{"type":21,"value":507},"Wold, H. (1982). Soft modeling: The basic design and some extensions. In K. G. Jöreskog & H. Wold (Eds.), ",{"type":15,"tag":409,"props":509,"children":510},{},[511],{"type":21,"value":512},"Systems under indirect observations: Part II",{"type":21,"value":514}," (pp. 1–54). North-Holland.",{"type":15,"tag":239,"props":516,"children":517},{},[518],{"type":15,"tag":370,"props":519,"children":521},{"href":520},"/documentation",[522],{"type":21,"value":523},"More literature ...",{"title":7,"searchDepth":525,"depth":525,"links":526},2,[527,528,536,543,544],{"id":32,"depth":525,"text":35},{"id":134,"depth":525,"text":137,"children":529},[530,532,533,534,535],{"id":141,"depth":531,"text":144},3,{"id":219,"depth":531,"text":222},{"id":230,"depth":531,"text":233},{"id":262,"depth":531,"text":265},{"id":273,"depth":531,"text":276},{"id":298,"depth":525,"text":301,"children":537},[538,539,540,541,542],{"id":304,"depth":531,"text":307},{"id":315,"depth":531,"text":318},{"id":326,"depth":531,"text":329},{"id":337,"depth":531,"text":340},{"id":348,"depth":531,"text":351},{"id":359,"depth":525,"text":362},{"id":396,"depth":525,"text":399},"markdown","content:documentation:algorithms-and-techniques:core-algorithm:pls:index.md","content","documentation/algorithms-and-techniques/core-algorithm/pls/index.md","md",{"loc":4},1784805324531]