Key issues on partial least squares (PLS) in operations management research: A guide to submissions

Journal of Industrial Engineering and Management JIEM, 2019 – 12(2): 219-240 – Online ISSN: 2013-0953 – Print ISSN: 2013-8423 https://doi.org/10.3926/jiem.2944 Key Issues on Partial Least Squares (PLS) in Operations Management Research: A Guide to Submissions Juan A. Marin-Garcia1, Rafaela Alfalla-Luque2 1ROGLE. Dpto. de Organización de Empresas. Universitat Politècnica de València (Spain) 2GIDEAO Research Group, Departamento de Economía Financiera y Dirección de Operaciones Universidad de Sevilla (Spain) [email protected], [email protected] Received: February 2019 Accepted: May 2019 Abstract: Purpose: This work aims to systematise the use of PLS as an analysis tool via a usage guide or recommendation for researchers to help them eliminate errors when using this tool. Design/methodology/approach: A recent literature review about PLS and discussion with experts in the methodology. Findings: This article considers the current situation of PLS after intense academic debate in recent years, and summarises recommendations to properly conduct and report a research work that uses this methodology in its analyses. We particularly focus on how to: choose the construct type; choose the estimation technique (PLS or CB-SEM); evaluate and report the measurement model; evaluate and report the structural model; analyse statistical power. Research limitations: It was impossible to cover some relevant aspects in considerable detail herein: presenting a guided example that respects all the report recommendations presented herein to act as a practical guide for authors; does the specification or evaluation of the measurement model differ when it deals with first-order or second-order constructs?; how are the outcomes of the constructs interpreted with the indicators being measured with nominal measurement levels?; is the Confirmatory Composite Analysis approach compatible with recent proposals about the Confirmatory Tetrad Analysis (CTA)? These themes will the object of later publications. Originality/value: We provide a check list of the information elements that must contain any article using PLS. Our intention is for the article to act as a guide for the researchers and possible authors who send works to the JIEM (Journal of Industrial and Engineering Management). This guide could be used by both editors and reviewers of JIEM, or other journals in this area, to evaluate and reduce the risk of bias (Losilla, Oliveras, Marin-Garcia & Vives, 2018) in works using PLS as an analysis procedure. Keywords: PLS-SEM, Partial Least Squares, operations management, reporting standards To cite this article: Marin-Garcia, J.A., & Alfalla-Luque, R. (2019). Key issues on Partial Least Squares (PLS) in operations management research: A guide to submissions. Journal of Industrial Engineering and Management, 12(2), 219-240. https://doi.org/10.3926/jiem.2944 -219- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 1. Introduction This leading article is based on an article previously published by the authors in the journal WPOM (Marín-García & Alfalla-Luque, 2019). The original article was published in Spanish with further information to that mentioned herein. We republished this work with adaptations for JIEM and translated it into English for more widely disseminate it to a different group of researchers whose access to the original article would not be easy (Callahan, 2018), (similar examples are (González De Dios, 2011; Moher, Liberati, Tetzlaff, Altman & The, 2009; Urrútia & Bonfill, 2010, 2013), or (Welch, Petticrew, Petkovic, Moher, Waters, White et al., 2015, 2016a., 2016b; Welch, Petticrew, Tugwell, Moher, O’Neill, Waters et al., 2012, 2013), or (Moher, Shamseer, Clarke, Ghersi, Liberati, Petticrew et al., 2015, 2016). As some sections in both articles overlap and complement one another, we recommend citing both articles to refer to their contributions. Extending structural equations models based on composites (PLS-SEM) has been widely diffused of late in areas like marketing (Hair, Sarstedt, Ringle & Mena, 2012b; Henseler, Ringle & Sinkovics, 2009; Hult, Hair, Proksch, Sarstedt, Pinkwart & Ringle, 2018), information systems (Petter, 2018; Ringle, Sarstedt & Straub, 2012; Roldán & Sánchez-Franco, 2012; Sharma, Sarstedt, Shmueli, Kim & Thiele, 2019; Urbach & Ahleman, 2010), tourism (do Valle & Assaker, 2016; Duarte & Amaro, 2018; Kumar & Purani, 2018; Latan, 2018; Usakli & Kucukergin, 2018), health sciences (Avkiran, 2018) or human resources (Ringle, Sarstedt, Mitchell, & Gudergan, 2018), among others. The two most recent reviews in the operations management area (Kaufmann & Gaeckler, 2015; Peng & Lai, 2012) came prior to intense debate about the methodology that shook the Partial Least Squares (PLS) community between 2014 and 2018. New tools continue to be developed and modifications to the report standards of articles made using PLS as an analysis tool are established. This work aims to systematise the use of PLS as an analysis tool via a usage guide or recommendation for researchers to help them eliminate errors when using this tool. This guide could be used by editors and reviewers of JIEM, or other journals in the area, to evaluate and reduce the risk of bias (Losilla et al., 2018) of works using PLS as an analysis procedure. 2. PLS as an Analysis Tool The PLS method consists in a sequence of multiple regressions that allows the weights of construct components (when reaching the predefined level of convergence) and paths to be estimated between exogenous and endogenous constructs (Esposito-Vinzi, Chin, Henseler & Wang, 2010; Felipe, Roldán & Leal-Rodríguez, 2017; Henseler et al., 2009). The algorithm is developed in several stages. The first stage iteratively estimates the Latent Variable Scores (LVS). The second stage solves the measurement model by estimating outer weights and loadings (beginning with the LVS estimated in the first stage). The third stage estimates the parameters of the structural model (Hair, Hult, Ringle & Sarstedt, 2017; Ringle, Wende & Becker, 2015). PLS is one of the methods in the variance-based structural equations family (SEM). One of its most outstanding aspects is that it is a method based on composites instead of common (reflective) factors or causal formative constructs. However, this does not represent a limitation when attempting to test models whose constructs match a composites typology (Henseler, 2017a; Rigdon, 2012, 2014; Sarstedt, Hair, Ringle, Thiele & Gudergan, 2016). Along these lines, PLS allows weights based on correlations (Mode A) or regressions (Mode B) to be estimated, or to correct with PLSc (consistent PLS) the correlations of those constructs specified as common factors to make the results consistent with that measurement model (Dijkstra & Henseler, 2015a; Dijkstra & Henseler, 2015b). This provides versatility when analysing mixed models where the constructs that are present are composites (when indicators are poorly correlated and if indicators present a moderate-high correlation of the sample being worked with) and also common factors (Rigdon, Sarstedt & Ringle, 2017). The basic aspects of PLS can be consulted in various references (Hair et al., 2017; Hair, Hult, Ringle, Sarstedt, Castillo-Apraiz, Cepeda-Carrion et al., 2019b; Hair, Sarstedt, Ringle & Gudergan, 2018; Kaufmann & Gaeckler, 2015; Ringle et al., 2018). Just like any other type of structural equations procedure, it allows the simultaneous estimation of (direct or indirect) mediation effects or moderation effects with many constructs (Ringle et al., 2018). Yet to date, no PLS version allows non-recursive models to be dealt with (e.g., modelling the correlations between -220- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 predictor constructs), although alternatives based on Two-Stage Least Squares has become popular (Henseler & Roldan, 2017). We hope that debate will commence in the future about these and other alternatives to mathematically solve a usual conceptual problem in operations management, where we cannot always contemplate how exogenous constructs are orthogonal (no correlation among them). Another characteristic of a composite-based analysis is that it allows LVS to be calculated as an exact linear combination of the observed variables so that they are never indeterminate and can be used in subsequent analyses (either in aggregation stages in second-or third-order constructs or with other statistical procedures that differ from structural equations) (Chin, 1998; Hair, Risher, Sarstedt & Ringle, 2019b; Richter, Cepeda, Roldán & Ringle, 2016). One useful aspect to stress is that PLS allows exogenous (predictor) constructs to be used whose indicators are measured at any measurement level (nominal, ordinal, interval or ratio) (Ringle et al., 2018), although it is recommended that endogenous constructs have items with an interval or ratio measurement level, or variants of PLS would be used that fit the measurement levels of constructs (Bodoff & Ho, 2016; Schuberth, Henseler & Dijkstra, 2018b). Another interesting aspect to contemplate is that PLS allows models with first-order constructs (Lower-Order Constructs; LOC) or second-order constructs (Higher-Order Constructs; HOC) (Ringle et al., 2018) to be analysed. Indeed when PLS is used with specific constructs as composites, we consider LOC (or HOC) to be a mediator (aggregator) among indicators (values of the items observed or captured during data collection) or subdimensions (the LVS of the LOC composing the HOC) (Bollen & Bauldry, 2011; Muller, Schuberth & Henseler, 2018; Schuberth, Henseler & Dijkstra, 2018a). So we can construct more parsimonious models (Ringle et al., 2018) by grouping lists of sets of variables with a certain joint theoretical sense (Grace & Bollen, 2008) that can be interpreted as a unit, but without losing the analysis of the effect of each one separately. This is particularly relevant when the number of variables increases, the correlation among them also increases and/or sample size reduces. In these situations, multiple regression models without SEM can be seriously affected by the net suppression conditions among the variables that are moderately/highly correlated to one another (Alfalla-Luque, Machuca & Marin-Garcia, 2018; Ato & Vallejo, 2011). To illustrate how a PLS construct is an aggregator among its indicators and the rest of the model, but not an exact mediator (as it has no “disturbance” term), we can, for example, take a set of data simulated with the code of Annex. With this, a set of 50000 data was simulated with four predictor variables (“V1” to “V4”) with normal distribution and preset means (4.0; 3.0; 2.5 and 3.2), and with a standard deviation that equalled 1.0 in all four, and a correlation of 0.35 among them all. With V1, V2, V3 and V4, an unstandardised variable “y” was created as a linear combination with paths (0.4; 0.3; 0.25; 0.25) and a random error term so that the four predictors explained R2 = 0.25. In Figure 1 we observe the descriptives of the generated dataset. As the standard deviation of all the predictors equals 1, and that of variable “y” is 1.715, the standardised paths, which are those reported in programmes like SmarPLS, would respectively be 0.23; 0.17; 0.15, 0.15 (Betai unstad =(Sd Y/Sd Vi).betai std) (Fassott, Henseler & Coelho, 2016). In Figure 2 we can see that the variables fit the design parameters. Figure 2 summarises the results of the analysis done with SPSS v22 and shows the standardised beta values and their confidence intervals. We can also see that the unstandardised coefficients coincide with those designed. Figure 3 shows that SmartPLS exactly reproduces the (standarised) regression results in both the estimated values and their confidence intervals because the contemplated model emulates multiple regression. Throughout this article, hexagons are used to represent based constructs (Bollen & Bauldry, 2011) (the origin of this notation apparently lies in the work by Grace and Bollen, 2008). However, the time when PLS models actually display part of their value is when we can transform the model and consider two constructs defined as Mode B composite (or Mode A when indicators are highly correlated). Each one groups two of the predictor variables (in a real model, it should be justified why they are taken as composites). We can verify in Figure 4 and Figure 5 how PLS-SEM enabled us to draw conclusions about the aggregate effect of the variables that compose a construct (paths in the model in Figure 4) and also the role played by each variable separately (by multiplying the value of the weight by that of the path in Figure 5). For -221- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 instance for V1, we can see that 0.666*0.337= 0.222, a value that comes very close to the original standardised path (0.231), but is not exactly the same. PLS does not report non-standardised values, but they can be calculated by starting with the descriptives of standard deviations or, more conveniently, the reports of the results using the IPMA of the SmartPLS procedure can be used for the total effects of the indicators of the model’s constructs (Figure 5) Figure 1. The descriptives of the simulated dataset Figure 2. The multiple regression results with SPSS -222- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 Figure 3. The results of the model when emulating multiple regression with SmartPLS Figure 4. PLS “standardised paths coefficients”, “standardised outer weights”, R2adj of “latent variable Y” and the “model fit” in SmartPLS -223- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 Figure 5. “Unstandardised path coefficients” and “rescaled outer weights” with the IPMA procedure in SmartPLS Between 2013 and 2016, several articles criticising the suitability of PLS as a scientific analysis method were published. These articles gave way to academic dispute, which led the Editor-in-Chief of some scientific journals in the operations management area to state that any article in which PLS was used as an analysis tool would quite likely be rejected by the Editor with no more ado. After reviewing the literature, all the objections made about PLS were answered in the responses that appear in other scientific journals (Cepeda-Carrion, Cegarra-Navarro & Cillo, 2019; Henseler, Dijkstra, Sarstedt, Ringle, Diamantopoulos, Straub et al., 2014; Sarstedt et al., 2016). In any case, an emphatic stance that denies any positive aspect of a technique or that blindly backs the exclusive appropriateness of a methodological approach apparently cannot be taken as good scientific practice, nor helps scientific disciplines to advance (Hair, Ringle & Sarstedt, 2012a; Kaufmann & Gaeckler, 2015). After intense debate, what clearly came over was that the weaknesses of composite-based PLS-SEM do not mean that methods based on covariance (CB-SEM) are the best analysis tool for all cases (Kaufmann & Gaeckler, 2015). Indeed, biases emerge when PLS is used to estimate models in which only constructs exist as common factors, but also when CB-SEM is followed to estimate models in which constructs are composites. Therefore, it is necessary to select and carefully justify the method (PLS-SEM or CB-SEM) that is more suitable for the dataset to be analysed (Kaufmann & Gaeckler, 2015). If in doubt about the nature of constructs (if we are not clear about them being common factors or composites), it would be preferable to use PLS because it provides the least biased solutions (Sarstedt et al., 2016). What all this debate has generated is a series of publications that attempt to systematise PLS being used as an analysis tool by means of usage guides or recommendations to make them more robust (Hair et al., 2019b; Henseler, Hubona & Ray, 2016a). Indeed, we believe it is interesting to summarise these guides. 3. Practical Guide. How Do We Avoid Frequent Errors In Articles Sent to Academic Journals? 3.1. Choice of Constructs The first step when constructing analysis models must always consist in explicitly defining the meaning of the constructs to be used from both a theoretical definition perspective (including the possible relevant subdimensions of the constructs) and the specification of the items or indicators to be used to measure these constructs or dimensions (Grace & Bollen, 2008). Then it is necessary to justify that the employed items conceptually match the theoretical definition chosen for the construct (Bollen, 2011). The next aspect involves specifying the measurement model; i.e., explaining how indicators are associated with or connected to constructs. On the one hand, this helps us to explain which information the model provides (common variance, in CB-SEM, or total variance, in PLS-SEM) -224- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 and, on the other hand, provides us with patterns to evaluate one of the criteria to be used to select the analysis procedure: CB-SEM or PLS-SEM. Items or indicators are variables with which we obtain a measure or observation during fieldwork, which helps us to estimate the parameters of latent/emergent variables (unobserved variables, but they represent a concept with an entity that is relevant for our analysis either because they naturally exist or are designed by people) (Grace & Bollen, 2008; Henseler, 2015, 2017a). To date, three types of indicators have been identified, which have given way to different measurement models in constructs (we will use constructs as a way to indistinctively refer to latent/emergent variables) (Figure 6): effect (or reflective) indicator (used in common factor constructs), causal indicator and composite indicator (it give rise to composite constructs) (Bollen & Bauldry, 2011; Grace & Bollen, 2008; Henseler, Ringle & Sarstedt, 2016b). Figure 6. The three types of constructs (Bollen & Bauldry, 2011) In common factor constructs, the researcher assumes that the cause of variation in the indicators is the common factor (latent variable) and a random mean error (not correlated with other variables or measurement errors in the model) (Henseler et al., 2016b). Indicators must have a conceptual unit (which refers to one of the concepts that the researcher has defined in the theoretical consideration). Indicators must also show a high correlation to one another (over 0.6). If indicators also display similar reliability and validity (very similar loadings in the model), they can be taken to be interchangeable because, basically, they are merely manifestations of the latent variable (Bollen, 2011; Jarvis, MacKenzie & Podsakoff, 2003). In this case, the important point lies in several manifestations being present in the measurement model so that the estimation is more reliable, but which one is selected is not actually the important point. In principle, any combination of indicators will lead to a very similar estimation of parameters (its loadings will be stable and will configure an equivalent measurement model because adding or removing an indicator does not change the essential nature of the construct to be measured) (Bollen, 2011; Roberts & Thatcher, 2009). In such indicators, loadings are stable regardless of the measurement model being analysed separately or it being connected to a structural model with other constructs (a nomological net), and even regardless of connected constructs being changed. A wide variation of loadings can be taken as evidence to indicate that the common factor model does not fit the real construct structure. Thus common factors represent the common variance of -225- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 their indicators (Henseler et al., 2016b). This is why researchers are only interested in analysing the relation of this common variance with other parts of the model in the models they are present in. Indicators themselves are no object of interest, except for allowing parameters to be estimated because they are not considered levers on which to act and be used to modify constructs. Quite the opposite in fact as they are merely manifestations of the concept of interest (Bollen & Bauldry, 2011), and the variations of the existing construct, although we cannot measure it, are those that cause the values of indicators to vary. Attitudes, beliefs, intelligence test dimensions and personality traits are some paradigmatic examples of the social sciences constructs that tend to be modelled with effect indicators (Bollen & Bauldry, 2011). Generally speaking, the manifestations of the latent variables subject to the natural laws of any field (physics, mathematics, psychology, social sciences, etc.) tend to act as a suitable model with effect indicators (Henseler, 2015). For instance, anxiety of digitalisation (“computer anxiety”), understood as fears stemming from using computers, often takes an operational definition based on the effect indicators of the type “I avoid using computers for fear of making mistakes I can’t correct” or “I don’t feel happy about using computers” (Roberts & Thatcher, 2009). Causal indicators, conversely, affect the latent variable (they cause it, and not vice versa). Hence causal indicators must have a conceptual unit or a shared meaning because they must all closely correspond to the definition selected for the unobserved construct (Bollen, 2011). We do not expect these indicators to be over-correlated, but no restriction is imposed in this case (Bollen & Bauldry, 2011; Roberts & Thatcher, 2009). At any rate, the construct exists in itself (even though it cannot be observed) (Grace & Bollen, 2008) and causal indicators compose a census, which is as complete as possible, of all the causes leading to it. This is why they are not interchangeable because, if one is suppressed, a considerable part of the causes is lost and the measurement error increases (Henseler, 2017a; Roberts & Thatcher, 2009). If, precisely, not all the causes are present, the construct will be measured with some error, represented by the term disturbance associated with the construct. In other words, the R2 of the construct will surely not be explained 100% by the indicators measured in research (Bollen, 2011), while the error term (disturbance) represents all the other causes not represented by the causal indicators included in the analysis (Roberts & Thatcher, 2009). If all these possible causes are contemplated with indicators, disturbance can be considered to equal zero (Diamantopoulos & Siguaw, 2006), in which case we could analyse the construct as if it were a composite (see later on). From a practical point of view, if the R2 of the latent construct explained by the indicators exceeds 0.26, the census of the indicators can be considered complete enough to contemplate the model with no disturbance (Diamantopoulos, 2006; Roberts & Thatcher, 2009). Moreover, as the construct itself exists, the weights of the indicators used to build the latent construct are relatively stable, despite the outcome variables associated with this construct changing (Bollen & Bauldry, 2011). For instance, the latent variable “exposure to stress” could take an operational definition based on causal indicators of these types, “changing jobs”, “getting married or divorced”, “suffering a serious disease” or “moving house”, which would act as a census of the possible causes for exposure to stress (Bollen & Bauldry, 2011). Each cause may have a distinct weight depending on its relative importance for producing stress. In this example, we may logically think that when someone suffers some of these causes (as it is not normal for everyone to suffer them all at once, the correlation among them would not be high), exposure to stress grows rather than someone being more predisposed to moving house, getting married, becoming ill and changing job (all at once) when stress grows. Indicators are the cause, and not the consequence, of the latent variable. So, it is foreseeable that the relative weight of the indicators is more or less stable because the intensity of the cause depends on its relation with the latent variable, and not with the constructs that the latent variable intends to explain. Composite indicators share certain similarities with the causal indicators that we have just looked at, but differ as far as several aspects are concerned (Bollen, 2011). Conceptually speaking, composite indicators are not coefficients of causality (Bollen & Bauldry, 2011), but of composition (Henseler et al., 2016b). They are the operational definition of the emergent construct that mediates all its effects in the model to be considered (Henseler, 2015). Therefore, these indicators have to share the same consequences (or the effects deriving from them) (Henseler, 2017a), although they may not be unidimensional and might not share a conceptual unit, which was the case of effect or causal indicators. Thus, composite indicators may represent different aspects relating to the construct. In any case, what they share with causal indicators is that they are not interchangeable with one another and they all contribute a unique aspect to the construct definition. In this case, however, suppressing an -226- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 indicator could modify the meaning of the construct. The emergent construct did not previously exist, but someone investigating creates/designs this construct and defines it as a linear combination of indicators (Whitt, 1986), and it is assumed by the restriction that R2 is 1; i.e., without the error term (Bollen, 2011; Grace & Bollen, 2008). Assigning the weight to the indicators of an emergent construct can be done in several ways, ranging from fixed weights (the unit or another value) to weights being estimated by some statistics procedure (correlations –Mode A- or regression coefficients –Mode B– are the commonest). If the weights of the indicators about the emergent construct are calculated with Mode B, they depend on the outcome variables (endogenous) included in the model, which we wish to explain. If the outcome variables are changed, it is quite likely that the weights of the composite indicators change (Bollen & Bauldry, 2011). In other words, if we take the composite indicators to be levers that we can move to achieve certain effects (effects that are assumed to be completely mediated by the emergent construct), the importance of these levers will depend on the effects that we wish to analyse. For instance, the relative weights of the components of a triple-A supply chain (adaptability, agility and alignment) may be different when they try to explain the competitive advantage than when they connect to the customer support and service level (Marin-Garcia, 2018). In the same way, if a profile of competences (emergent construct) is composed of indicators of creative behaviour, the manifestation of critical thought, technical knowledge about the job post and teamwork competency (Marin-Garcia, 2018), then it is foreseeable that the weights of these indicators differ when the competence profile is connected to explain the variance of a variable like “innovation results” from where it is connected to “organisational commitment” or to “piecework production capacity”. Generally, regardless of human designs being business practices, management models, performance indices, or even personal skills, as they are objects composed by elements, they may have suitable modelling with composite indicators, independently of other modellings being admitted. Thus, their data fit must be confirmed by the relevant analysis (Henseler, 2015; Henseler, 2017a). 3.2. Choosing the Estimation Technique After defining which type of constructs appears in the model to be analysed, the next step consists in choosing the estimation technique that best fits the model (PLS-SEM, CB-SEM or others) (Sarstedt et al., 2016). Table 2 summarises the situations in which each analysis method would be recommendable (Dijkstra & Henseler, 2015a; Hair et al., 2017; Richter et al., 2016; Rigdon et al., 2016, 2017). Use PLS-SEM if… Use CB-SEM if… The model only has composites (PLSc is recommended for a mixture of composites and common factors) The model is composed only of reflective constructs (common factors) or constructs with causal indicators The objective is to estimate the relative importance of the indicators (and not just the construct) to predict the outcome construct The objective is to know the relations among constructs, without having to analyse the specific effects/contributions of their indicators The intention is to use the LVS in subsequent analyses The LVS are not required for other analyses and the research questions can be answered only with the parameters estimated by CB-SEM Use of secondary data or file data The primary data where the variables to be collected and the way to measure them are decided The model is very complex (many indicators and lots of constructs) Use of data analytics or big data (predict effects with the models emerging from data rather from former theories) Table 2. Recommendations for choosing between CB-SEM and PLS-SEM as an analysis method -227- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 Declaration of Conflicting Interests The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The authors received no financial support for the research, authorship, and/or publication of this article. References Alfalla-Luque, R., Machuca, J.A.D., & Marin-Garcia, J.A. (2018). Triple-A and competitive advantage in supply chains: Empirical research in developed countries. International Journal of Production Economics, 203(September), 48- 61. https://doi.org/10.1016/j.ijpe.2018.05.020 Ato, M., & Vallejo, G. (2011). Los efectos de terceras variables en la investigación psicológica. Anales de Psicología, 27(2), 550-561. Avkiran, N.K. (2018). 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Industrial Management & Data Systems, 116(9), 1887-1900. https://doi.org/10.1108/IMDS-06-2016- 0248 # Specify population means and standard deviation of four predictor variables that is sampled from std<-c(1,1,1,1) #Standard deviations values desired by design for V1 to V4 var<-std^2 mu<-c(4,3,2.5,3.2) #mean values desired by design for V1 to V4 corxx<-0.35 #correlations desired by design between V1 to V4 (all equal) sigma.1 <- matrix(c(1,corxx,corxx,corxx, corxx,1,corxx,corxx, corxx,corxx,1,corxx, corxx,corxx,corxx,1),nrow=4,ncol=4) mu.1 <- rep(0,4) # Specify sample size, true regression coefficients, and explained variance n.obs <- 500 # 500000 recommended to avoid sampling error problems intercept <- 0 #value desired to add to the weighted means of Vs for mean of y (unstandardized) -239- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2944 beta <- c(0.4, 0.3, 0.25, 0.25) #unstandardized Beta coefficients for lineal combination of V1 to V4 to create y (unstandardized) r2 <- 0.25 # # Create sample with four predictor variables library(MASS) sample1 <- as.data.frame(mvrnorm(n = n.obs, mu.1, sigma.1, empirical=FALSE)) 'standardized Vi variables' apply(sample1, 2, mean) apply(sample1, 2, sd) # dataframe with independent values without standardization (y variable it is not standardized. Mean= intercept and variance depends on equation above sample1$y) sample2<-as.data.frame(sample1) sample2$V1<-sample2$V1*std[1]+mu[1] sample2$V2<-sample2$V2*std[2]+mu[2] sample2$V3<-sample2$V3*std[3]+mu[3] sample2$V4<-sample2$V4*std[4]+mu[4] # Add error variable based on desired r2 var.epsilon <- (beta[1]^2+beta[2]^2+beta[3]^2+beta[4]^2+corxx)*((1 - r2)/r2) #original si todas las corr son iguales sample1$epsilon <- rnorm(n.obs, sd=sqrt(var.epsilon)) # Add y variable based on true coefficients and desired r2 sample2$y <- intercept + beta[1]*sample2$V1 + beta[2]*sample2$V2 + beta[3]*sample2$V3 + beta[4]*sample2$V4 + sample1$epsilon 'variables unstandardized' apply(sample2, 2, mean) apply(sample2, 2, sd) # Inspect model summary(lm(scale(y)~scale(V1)+scale(V2)+scale(V3)+scale(V4), data=sample2)) summary(lm(y~V1+V2+V3+V4, data=sample2)) cor(sample1) cor(sample2) # Write CSV in R write.csv(sample2, file = "MyData500.csv",row.names=FALSE, na="-999" Journal of Industrial Engineering and Management, 2019 (www.jiem.org) Article’s contents are provided on an Attribution-Non Commercial 4.0 Creative commons International License. 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