Factor Analysis of the University Environment and Support System: A Bayesian Approach

University students represent a reservoir of entrepreneurial talent and an inherent source of creativity and innovation. One way to help unleash their talents as an engine of economic growth is by increasing our understanding of elements—internal or external, real or perceived—that lead to and influence the emergence of new ventures led by students. There is evidence in the literature that the university environment and support system (ESS) can affect the entrepreneurial intention of students attending the institution. The university ESS comprises the support mechanisms necessary for entrepreneurial activity and could motivate students to consider entrepreneurship a possible career choice. Kraaijenbrink et al. developed and validated a university ESS scale that helped them identify three motivational factors of the university ESS influencing entrepreneurial intention. Despite its increasing popularity, a detailed study and analysis of the Kraaijenbrink et al. university ESS scale are still lacking in the extant literature. This study fills that gap by conducting an extensive factor analysis of the university ESS scale using a Bayesian approach. To the best of our knowledge, this is the first time this scale has been the subject of such an extensive investigation. We capitalized on the added flexibility of the Bayesian approach to address novel substantive questions through the mathematical model. We used it to enrich our understanding of the university ESS and generate new ideas for possible scale and model modifications. Furthermore, this evaluation provided novel insight into the relationships among the many dimensions of the university ESS. In the future, similar studies conducted by aspiring entrepreneurial universities could help them interpret the efficacy of their efforts to promote entrepreneurial activities among students. Plain Language Summary Testing how well the measured variables represent the University Environment and Support System constructs University students are a great source of creativity and innovation that can lead to new businesses. To students apply their talents to generate new economic activity, it is essential to understand what helps or hinders their intentions of starting new businesses. Research shows that the university environment and support system (ESS) can influence whether students consider becoming entrepreneurs. Kraaijenbrink and colleagues developed and validated a survey that helped them identify three motivational factors of the university ESS influencing the intentions to start new businesses, although a detailed study and analysis of the survey by Kraaijenbrink and colleagues are still missing in the literature. Our study aims to fill this gap. We used a Bayesian approach to analyze the university ESS survey in depth. This approach allowed us to ask new questions and improve our understanding of how universities can encourage entrepreneurship. We also examined how different parts of the university support system are connected. Our findings can help universities that want to encourage their students to start new businesses. By understanding what works and what does not, these universities can be more effective in their efforts to support student entrepreneurs.


Plain Language Summary
Testing how well the measured variables represent the University Environment and Support System constructs University students are a great source of creativity and innovation that can lead to new businesses.To students apply their talents to generate new economic activity, it is essential to understand what helps or hinders their intentions of starting new businesses.Research shows that the university environment and support system (ESS) can influence whether students consider becoming entrepreneurs.Kraaijenbrink and colleagues developed and validated a survey that helped them identify three motivational factors of the university ESS influencing the intentions to start new businesses, although a detailed study and analysis of the survey by Kraaijenbrink and colleagues are still missing in the literature.Our study aims to fill this gap.We used a Bayesian approach to analyze the university ESS survey in depth.This approach allowed us to ask new questions and improve our understanding of how universities can encourage entrepreneurship.We also examined how different parts of the university support system are connected.Our findings can help universities that want to encourage their students to start new businesses.By understanding what works and what does not, these universities can be more effective in their efforts to support student entrepreneurs.

Introduction
Today, a broad consensus exists that a region's socioeconomic development depends upon increasing its entrepreneurial talent (Amentie & Negash, 2015).Thus, more entrepreneurship is necessary to bring about creativity and innovation that can translate into technological advances, economic growth, and employment for the region (Datta et al., 2022;Moremong-Nganunu et al., 2018).University students (hereafter students) represent a reservoir of entrepreneurial talent and an inherent source of creativity and innovation (Westhead & Solesvik, 2016).One way to help unleash their talents as an engine of economic growth is by increasing our understanding of elements-internal or external, real or perceived-that lead to and influence the emergence of new ventures led by students.In this regard, the literature argues that this entrepreneurial behavior, that is, starting a new business, is intentional; thus, the intention toward the behavior could help predict entrepreneurial activity better than attitudes, beliefs, personality, or demographics (Ajzen, 1991(Ajzen, , 2001;;Fayolle et al., 2006;Krueger et al., 2000).Numerous authors have studied the precursors (antecedents) of intention and their influencers to try and predict entrepreneurial intention (Autio et al., 2001;Devonish et al., 2010;Fayolle et al., 2006;Iakovleva et al., 2011;Kolvereid, 1996;Krueger et al., 2000;Tkachev & Kolvereid, 1999).
Furthermore, growing evidence in the literature suggests that contextual and situational factors affect entrepreneurial intention by influencing its precursors and the general motivation to behave (Boyd & Vozikis, 1994;Krueger et al., 2000;Lee & Wong, 2004).In addition, there is evidence that the university environment and support system (ESS) can influence the entrepreneurial intention of students (Bazan, Shaikh, et al., 2019;Lin˜a´n et al., 2011;Sesen, 2013;Shirokova et al., 2016;R. Trivedi, 2016;Zhang et al., 2014).The university ESS comprises the support mechanisms necessary for entrepreneurial activity, for example, entrepreneurship training, business incubation services, startup coaching, intellectual property protection, and technology transfer and commercialization.The university ESS has traditionally helped students develop entrepreneurial competencies and motivated them to consider an entrepreneurial career (Franke & L€ uthje, 2004;Henderson & Robertson, 1999;Kraaijenbrink et al., 2010;Peterman & Kennedy, 2003).Kraaijenbrink et al. (2010) have identified three motivational factors of the university ESS that could influence entrepreneurial intention.They suggested that educational, concept, and business development support correlate positively with students' entrepreneurial intention and behavior.Kraaijenbrink et al. (2010) developed and validated a scale for arriving at these conclusions.Several authors (Bazan, 2022;Bazan, Shaikh, et al., 2019;R. Trivedi, 2016;R. H. Trivedi, 2017) have adopted and adapted the scale to assess the effect of the university ESS on the precursors of the entrepreneurial intention of students.
Notwithstanding the increasing interest in how universities can influence the entrepreneurial intention of their students, the extant literature still lacks a detailed study and analysis of the university ESS scale.Most studies in the literature have used the scale to assess the students' perception of university support for entrepreneurial activities.The primary goal of those studies was to measure the relationship between the university ESS constructs, for example, educational, concept, and business development support, and the precursors of entrepreneurial intention.Before doing that, those studies conducted confirmatory factor analysis (CFA) to validate the factor structure of their set of observed variables (indicators) to verify if the data fit the hypothesized measurement model, where each construct is represented by its respective indicators.None of those studies assessed the relationship among the university ESS constructs themselves, which can provide a better understanding of the multiple support mechanisms provided by the university.
This study fills the abovementioned gap in the literature by conducting an extensive factor analysis of the university ESS scale using the Bayesian approach proposed by B. Muthe´n and Asparouhov (2012).Two questions guided the conception, design, and conduct of this study.( 1) What new insights concerning the phenomenon can this approach generate?(2) What general and ad hoc modifications to the university ESS scale can this approach inform?Kraaijenbrink et al. (2010) and subsequent adopters of the university ESS scale have provided sufficient support for its reliability, validity, and internal consistency.Thus, the goal of this study was not to confirm or disconfirm the scale.Instead, we capitalized on the added flexibility of the Bayesian approach to address novel substantive questions through the mathematical model.Consequently, after assessing the confirmatory factor analysis model fit, we used it to enrich our understanding of the university ESS and generate new ideas for possible scale and model modifications.Moreover, this study thoroughly evaluated the differences between the hypothesized CFA model and the data.This evaluation provided novel insight into the relationships among the many dimensions of the university ESS.As far as we are aware, this is the first time that a study has provided compelling evidence of the robustness of the Kraaijenbrink et al. (2010) scale and possible modifications that could also benefit researchers using a frequentist approach to studying the entrepreneurial intention of students.
In the future, similar studies conducted by aspiring entrepreneurial universities could help them interpret the efficacy of their efforts to promote entrepreneurial activities among students.Doing so could help transform technology innovations generated at the university into much-needed economic activity.That is, universities could assist in translating early-stage innovations into market-ready products and services and help create the startups that will spin-off from the university and commercialize the research outcomes.
We divided the remainder of this article into seven sections as follows.First, the Literature Review section provides an overview of studies that adopted and adapted the original university ESS by Kraaijenbrink et al. (2010).Second, the Bayesian Fundamentals section introduces the Bayesian approach to factor analysis and offers a comparison with the frequentist approach.Third, the Research Methodology section describes the CFA model of the university ESS informed by data from a previous study by others.Fourth, the Analysis of the Results section describes data sources and analyzes the results using the new approach.Fifth, the Limitations of the Study section describes possible shortcomings of the different methodologies used in the study.Sixth, the Discussion section interprets the results in the context of the phenomenon under study.Finally, the Conclusion section examines the implications of the data analysis for the research questions under investigation.

Literature Review
Encouraging entrepreneurship is a topic of intense academic research, given the potential new economic activity that creating new ventures can generate (Sau´de et al., 2020).Traditionally, universities supported student entrepreneurship by offering entrepreneurship education programs.However, more recently, entrepreneurial universities have invested in promoting student entrepreneurship in many additional ways.For instance, entrepreneurial universities provide a wide range of extracurricular support mechanisms, that is, the university ESS, for students to consider entrepreneurship a possible career choice after graduation.In essence, entrepreneurial universities have been trying to help students develop their entrepreneurial intention.Consequently, researchers have been trying to gage students' perception of the university ESS to understand its influence on a student's entrepreneurial intention (Saeed et al., 2015).These efforts are consistent with the belief that with dynamic support and incentive for entrepreneurship, students will have more confidence in their ability to become entrepreneurs, increasing their entrepreneurial intention and possible behavior (Schwarz et al., 2009).
Most researchers have used structural equation modeling to combine the framework proposed by Kraaijenbrink et al. (2010) and the theory of planned behavior (Ajzen, 1991) to understand the university ESS's effect on a student's entrepreneurial intention.They have taken the university ESS as a potential contextual and situational element affecting entrepreneurial intention.Kraaijenbrink et al. (2010) proposed and validated a scale to measure students' perceived and desired university ESS.They argued that universities could support student entrepreneurship in many ways, although the effectiveness of the university ESS will vary based on how effectively they reach students.Kraaijenbrink et al. (2010) distinguish between three types of university ESS.Note that different authors use different terminology to describe the three types of university ESS.For example, as mentioned before, Kraaijenbrink et al. (2010) classify the university ESS as (1) business development support, (2) educational support, and (3) concept development support.R. Trivedi (2016) categorizes them as (1) noncognitive support, (2) educational support, and (3) cognitive support.While Bazan, Datta, et al. (2019) arrange the university ESS into (1) startup support, (2) entrepreneurship training, and (3) entrepreneurial milieu.This study uses the terminology by Bazan, Datta, et al. (2019), where the university ESS comprises three different, interrelated dimensions: startup support (SS), for example, seed funding, mentorship, business incubation; entrepreneurship training (ET), for example, courses, workshops, coaching; and entrepreneurial milieu (EM), for example, university policies, leadership, culture, awareness.
The study by Kraaijenbrink et al. (2010) used their scale to investigate students' perceptions of the university ESS and the extent to which students desire these support mechanisms.Thus, the primary focus of their study was on the two dependent variables: perceived and desired university ESS.They conducted the study with data from 2,417 students at five universities members of the European Consortium of Innovative Universities (ECIU): the University of Warwick (UK), Linko¨ping University (Sweden), Aalborg University (Denmark), the University of Twente (the Netherlands), and the University of Swinburne (Australia).Their scale comprised 13 indicators measuring the different dimensions of the university ESS.They conducted an exploratory factor analysis (EFA) with principal components extraction and varimax rotation to confirm the existence of the three factors representing the three types of support, SS, ET, and EM.After the factor and reliability analyses of the university ESS, they found composite reliability of 0.852, 0.897, and 0.862 for SS, ET, and EM, respectively.These results and the high factor loadings confirmed that the scale was sufficiently reliable and valid to measure perceived and desired university ESS.In addition, Kraaijenbrink et al. (2010) showed that students in those universities perceive and desire more ET than EM and more EM than SS.Davey et al. (2011) adapted the Kraaijenbrink et al. (2010) scale to identify the differences between African and European students concerning their entrepreneurial intention, attitudes toward entrepreneurship, role models, and entrepreneurial experience.They designed a 51indicator questionnaire containing nine from the scale by Kraaijenbrink et al. (2010) measuring the university ESS.They collected 1,055 valid responses from students at universities in three African countries that are either developing (Kenya and Uganda) or emerging (South Africa) and four developed European countries (Germany, Finland, Ireland, and Portugal).Their results indicate that students from developing or emerging countries are more likely to consider future entrepreneurial careers and are more positive toward entrepreneurship than students from industrialized European countries.Nonetheless, the motivators for employment or self-employment were similar across the two samples.Four years later, Saeed et al. (2015) also adapted the Kraaijenbrink et al. (2010) scale to form a 38-indicator questionnaire.They used it to test an integrative, multiperspective framework hypothesizing that the university ESS and institutional backing shape entrepreneurial selfefficacy among 805 students at five universities in Pakistan.In addition, they posited that entrepreneurial self-efficacy and individual motivations constitute the fundamental components of the intention to start a business.Their findings showed that ET exerts the most influence on self-efficacy, followed by EM, SS, and institutional support.
Afterward, Mustafa et al. (2016) used the Kraaijenbrink et al. (2010) scale, as modified by Saeed et al. (2015), to develop an empirical model that examines whether a student's proactive personality or the university ESS affects their entrepreneurial intention among 141 students at a Malaysian university.Additionally, they wanted to compare the relative strengths of the influences of a student's proactive personality and the university ESS.Their results indicated that a proactive personality and EM significantly impact a student's entrepreneurial intention.Also, they showed that a student's proactive personality had a greater effect on their entrepreneurial intention than the university ESS.In a somewhat different context, Yusoff et al. (2016) adapted Kraaijenbrink et al. (2010) scale to design a 38-indicator questionnaire to understand the role that the university ESS played in explaining ''agropreneurial'' intentions among 318 agriculture students at a Malaysian institution.Their results showed that the ''agropreneurship'' curriculum, agropreneurship experiential learning, and perceived desirability and feasibility significantly explained agropreneurial intention.However, the university ESS did not impact the agropreneurial intention of students.
In subsequent communications, R. Trivedi (2016) and R. H. Trivedi (2017) adapted and expanded the Kraaijenbrink et al. (2010) scale (increased to 18 indicators) to develop a 62-indicator questionnaire to examine the university ESS's role in fostering entrepreneurial intention among 1,097 students at universities in India, Malaysia and Singapore.In his analysis, R. Trivedi (2016) and R. H. Trivedi (2017) found that the university ESS indicators clustered into two factors he called targeted cognitive and non-cognitive support (SS + EM) and general educational support (ET).He then attempted to find relationships between these two factors and entrepreneurial intention.His studies found that the university ESS has a positive relationship with one of the precursors of intention (perceived behavioral control) and that the university ESS influences students from Malaysia, Singapore, and India differently (in that order of importance).Afterward, Bazan, Shaikh, et al. (2019) adapted the Kraaijenbrink et al. (2010) scale, as modified by R. Trivedi (2016) and R. H. Trivedi (2017) (reduced to 15 indicators), to design a 36-indicator questionnaire to assess the influence of the university ESS on the entrepreneurial intention of 405 students at a university in Canada.The study by Bazan, Shaikh, et al. (2019) was interested in the influence of the university ESS as a whole.Thus, it did not separate the indicators into factors (they used a single 15-indicator factor).Similarly to R. Trivedi's (2016) and R. H. Trivedi's (2017) study, Bazan, Shaikh, et al. (2019) found that the university ESS positively correlates with the perceived behavioral control precursor of intention.Soon after, Bazan, Datta, et al. (2019) expanded the study by Bazan, Shaikh, et al. (2019) by using the same data to assess the influence of the university ESS on female and male students.Their study found that the university ESS has a greater effect on the perceived behavioral control of female students than male students.However, the most novel contribution of the study by Bazan, Datta, et al. (2019) was using a second-order model to represent the assumption that the common underlying, higher-order university ESS factor can account for the seemingly distinct but related constructs, SS, ET, and EM.
More recently, Shi et al. (2019) adapted the Kraaijenbrink et al. (2010) scale and developed a 29-indicator questionnaire to clarify the relationship between the university ESS and the entrepreneurial intention of 374 students at a university in China.They also wanted to test the moderating role of the Chinese sense of face in the process.Their findings confirm the positive relationship between the university ESS and students' entrepreneurial intention.Additionally, they found differences in the moderating role of the Chinese sense of face in the relationships between self-efficacy and growth-oriented versus independence-oriented intention.Finally, Bazan (2022) also adapted the Kraaijenbrink et al. (2010) scale, as modified by Bazan, Datta, et al. (2019), to study the effect of the university ESS on the subjective social norms precursor of the entrepreneurial intention of 350 students at a university in Canada.Bazan (2022) argued that the university ESS could not only provide students with the support mechanisms to explore entrepreneurship as a possible career choice, but it can help students gain the support of their families and friends who influence their subjective social norms.His results support the hypothesis that the university ESS may influence students' perceptions of the opinions of important reference people regarding their prospects of becoming entrepreneurs.
As this brief literature review reveals, the Kraaijenbrink et al. (2010) scale has been adopted and adapted by numerous research studies to measure the influence of the university ESS on students' entrepreneurial intention.Thus, a more extensive analysis of the Kraaijenbrink et al. (2010) scale can provide new research questions and further insights into the ever-more vital phenomenon of student entrepreneurship.
Based on the literature and previous work by the authors, this study formulated the following hypotheses: H1: The model indicators will have large loadings on their hypothesized factors and several small crossloadings due to minor influence from other factors.H2: The model will be insensitive to the cross-loading prior parameters.H3: The model residuals will have several small correlations due to omitting several minor factors.H4: The model will be insensitive to the residual covariance prior parameters.

Bayesian Fundamentals
A fundamental difference between frequentist and Bayesian approaches is how they view the model parameters.Frequentist analysis considers parameters as constants, while Bayesian analysis regards them as variables.Frequentist analysis, for example, using maximum likelihood (ML), finds parameter estimates by maximizing a likelihood computed for the data.Bayesian analysis uses apriori information by specifying evidence-based parameter distributions called priors.Depending on the researcher's knowledge of the parameters, these priors are diffuse (noninformative) or informative.Bayesian analysis uses the data distribution (likelihood) to modify the prior distributions into posterior distributions that provide the Bayesian parameter estimates.Accordingly, the prior distribution is a critical element of Bayesian analysis.Priors reflect the certainty (or lack thereof) in the parameter values.Prior distributions with large variances (diffuse priors) suggest little knowledge of the parameter distributions.In these cases, the likelihood contributes relatively more information to the formation of the posterior, and the estimate would be closer to a frequentist estimate, especially with large samples (Browne & Draper, 2006).Alternatively, small variances in informative priors reflect more certainty in (knowledge of) the parameter values.Thus, informative priors contribute more to forming the posterior distributions of the parameter estimates.Bayesian analysis routinely obtains the posterior distributions using Markov Chain Monte Carlo (MCMC) algorithms.
Frequentist approaches to factor analysis, for example, using ML and likelihood-ratio chi-square testing, may apply unnecessary restrictions to the models.For example, a specification such as exact zero cross-loadings or residual covariances often leads to model rejections because of over-simplified hypotheses derived from substantive theory or knowledge (Marsh et al., 2009;B. Muthe´n & Asparouhov, 2012).Researchers faced with unclear model rejections regularly resort to model modifications that might capitalize on chance or data idiosyncrasy (MacCallum et al., 1992).The Bayesian approach proposed by B. Muthe´n and Asparouhov (2012) relaxes some routinely fixed parameters (e.g., exact zeros or equality constraints) to reflect (less-simplified) substantive theory or knowledge more realistically.This relaxation involves, for example, replacing the parameter specification of exactly zero with approximate zeros by specifying informative priors for such parameters.Relaxing (freeing) these parameters during factor analysis employing a frequentist approach would usually result in unidentified models.The Bayesian approach resolves this lack of model identification by applying small-variance priors consistent with substantive theory or knowledge.
Furthermore, Bayesian analysis performs model assessment using posterior predictive checking (PPC) (Gelman et al., 1996).This technique is less sensitive to ignorable degrees of model misspecification than the likelihood-ratio chi-square testing (B.Muthe´n & Asparouhov, 2012).The PPC technique provides a posterior predictive (PP) p-value of model fit using a fit statistic f based on the likelihood-ratio chi-square test of an H 0 model against an unrestricted H 1 model.There is still no formal theory regarding what constitutes a good model fit using this approach.However, the higher the PP p-value, the better the model fit.As a rule of thumb, a PP p-value of around .50 and a zero f statistic difference close to the middle of the confidence interval indicate an excellent-fitting model.Nevertheless, simulation studies suggest that PP p-values ø 0:05 represent an adequate model fit (B.Muthe´n & Asparouhov, 2012).In addition, a valuable byproduct of Bayesian analysis is information for simultaneous model modifications, thus avoiding modifying a single parameter at a time, as in frequentist analysis.

Research Methodology
This study collected secondary data from two previous, more extensive studies of the influence of the university ESS on students' entrepreneurial intention (Bazan, 2022;Bazan, Shaikh, et al., 2019).Those previous studies collected the data 2 years apart (2018 and 2020) by sampling the same population of students at a university in Canada using the same structured, non-disguised questionnaire.Both previous studies employed convenience sampling to collect the data.In general, convenience samples do not produce representative results.However, convenience samples may provide accurate correlations.In addition, those studies employed cross-sectional data, that is, no temporal link between the outcome and the exposure.The purpose of the cross-sectional studies was to examine the presence of an outcome and the presence of an exposure (prevalence) at a specific time.(We refer the reader to the original articles for more information on the research instrument and data collection and screening.)Table 1 shows the select demographics of the two samples.
We used the 2018 sample (Bazan, Shaikh, et al., 2019) for prior elicitation and the 2020 sample (Bazan, 2022) to conduct the study to understand the university ESS more exhaustively.For this study, we only retrieved the section of the two samples that collected responses related to the university ESS.In other words, we were interested in understanding the university ESS comprising the three interrelated dimensions: SS, ET, and EM.Table 2 shows the 15 indicators of the university ESS and the originally theorized factor breakdown.
This study conducted CFA of the university ESS by following recommendations by Depaoli (2021) and B. Muthe´n and Asparouhov (2012).We evaluated the CFA model in Figure 1 for an observed (15-dimensional) vector y of indicators, where n: intercept vector, L: loading matrix, h: factor vector, e: residual vector, a: factor mean vector, C: factor covariance matrix, Y: residual covariance matrix V e ð Þ.Additionally, we initially assumed e and h normally distributed and uncorrelated.
We conducted the first CFA of the university ESS using the 2018 sample (Bazan, Shaikh, et al., 2019) with noninformative priors to elicit the priors of interest for the second CFA using the 2020 sample (Bazan, 2022).These priors, 15 indicator loadings, 3 3 3 factor variancecovariance matrix, and 15 indicator error term variances endowed the second CFA of the university ESS with informative priors.In addition to the elicited priors, this study considered informative priors for two other measurement model specifications: cross-loadings and residual correlations.We conducted all analyses in this study using Bayesian estimation in the Mplus version 8.8 software program (L.Muthe´n & Muthe´n, 2022) with standardized variables.
Again, the first CFA used the 2018 sample with default diffuse prior settings.Note that the normal distribution N m, s 2 ð Þ= N 0, 10 10 À Á is the default prior for intercepts, regression slopes and loading parameters.The default prior for variance parameters is the inversegamma distribution IG a, b ð Þ= IG 0, À 1 ð Þ for variancecovariance blocks of size q = 1 and inverse-Wishart distribution IW c, d ð Þ= IW 0, À q À 1 ð Þ for variancecovariance blocks of size q.1.We used Gibbs sampling and four chains with an initial seed value of zero and starting values based on the ML estimates.We also used M = 100, 000 MCMC iterations per chain, of which we discarded the first half as the burn-in phase.We monitored convergence using the potential scale reduction factor (PSRF) criterion developed by Gelman and Rubin (1992).We used a much stricter cutoff for the PSRF (1.01) than the default software setting (1.05).We also examined all trace plots for evidence against convergence and verified that all parameters converged according to the PSRF by 50,000 iterations.In addition, we estimated the CFA model again with double the number of iterations M = 200, 000 and twice as many iterations for the corresponding burn-in phase.Once again, the first model satisfied the PSRF criterion, and the trace plots converged after doubling the MCMC iterations.Then, we calculated the percent of relative deviation to assess how similar the results were across the two analyses (with M and 2M MCMC iterations).We found similar results across both analyses where the relative deviation levels .arrangesconferences and workshops on entrepreneurship ?0 0 ET4 .organizes mentoring and advisory services for student entrepreneurs ?0 0 ET5 .offers to work on projects that focus on entrepreneurship ?0 0 Startup Support (SS) My university.SS1 .organizes business idea competitions 0 ?0 SS2 .has many resources to support a startup company 0 ?0 SS3 .provides students with ideas to start a new business 0 ?0 SS4 .arranges meetings with successful entrepreneurs to share their experiences 0 ?0 SS5 .provides students with the financial means needed to start a new business 0 ?0 Entrepreneurial Milieu (EM) My university.EM1 .provides a creative atmosphere to develop ideas for new business startups 0 0 ?EM2 .helps students build the required network for starting a business 0 0 ?EM3 .motivates students to start a new business 0 0 ?EM4 .creates awareness of entrepreneurship as a possible career choice 0 0 ?EM5 .brings entrepreneurial students in contact with each other 0 0 ?
Note.A question mark (?) specifies that the indicator loads on the factor, while a zero denotes that the indicator does not load on the factor.

Bazan
where \ 1% j j.The largest absolute difference corresponded to the SS factor variance (0:49%).Table 3 shows the model parameters' unstandardized posterior mean estimate and posterior standard deviation using the 2018 sample and default diffuse prior settings.Note that we set the loading of the first indicator of each factor to one during the CFA using the 2018 sample.
The unstandardized results directly map onto the prior settings for the CFA using the 2020 sample.For instance, we used the mean parameter estimate for the factor loading SS2 on SS in Table 3 (1.083) to define the prior for the loading SS2 on SS, N 1:083, 0:1 ð Þ , in the CFA using the 2020 sample.Similarly, we used the mean parameter estimate for the covariance SS with ET in Table 3 (0.652) to define the prior for the covariance SS with ET, IW 1:304, 6 ð Þ , in the CFA using the 2020 sample.For this, we used the mean of the inverse-Wishart distribution m ¼ c= d À q À 1 ð Þ , where d is the number of degrees of freedom taken as d.q + 1.Finally, we used the mean parameter estimate for the residual variances SS1 in Table 3 (0.410) to define the prior for the residual variance SS1, IG 3:681, 1:099 ð Þ , in the CFA using the 2020 sample.For this, we used the relationships a = 2 + m 2 =s 2 and b = m + m 3 =s 2 , where m and s 2 are the mean and variance of the parameter estimate.Note that we set the variance to s 2 = 0:1.We chose these variances for the set of priors for them to inform the second CFA without being too prescriptive.Table 4 shows the informed priors derived from the CFA using the 2018 sample.
This study also implemented informative priors for two additional measurement model specifications: crossloadings and residual covariances.As a comparison, a frequentist approach would specify zero cross-loadings between the indicators and the factors that, by hypothesis, do not reflect on those indicators.However, this specification is an enormously simplified representation of the actual measurement structure and is not essential or necessary to the notion that, by design, each indicator should measure a specific factor (Depaoli, 2021;B. Muthe´n & Asparouhov, 2012).More realistically, researchers can expect each indicator to have a relatively large loading on its hypothesized factor and possibly small cross-loadings due to a minor influence from some other factors (B.Muthe´n, 2010).Thus, following the Bayesian approach by B. Muthe´n and Asparouhov (2012), we specified priors prescribing a mean of zero and a normal distribution with a small variance for the cross-loadings in L to reflect substantive knowledge more realistically (see Equation 1).This prior choice is consistent with the abovementioned statement that the SS, ET, and EM factors might reflect on indicators other than their own hypothesized indicators to some extent.In other words, we recognize that an indicator, for example, SS2, loading on its hypothesized factor, that is, SS, might also load slightly on a different factor, for example, ET.More concretely, we specified cross-loadings drawing from N 0, 0:01 ð Þ so that 95% of the loading variation will be between 20.2 and 0.2 (i.e., small loadings when using standardized variables).This study also assessed the CFA model sensitivity to the Similarly, a frequentist approach would specify a diagonal residual covariance matrix Y (see Equation 1).It would assume the absence of residual covariances among indicators within and outside their reflecting factors.The university ESS is a complex, multifaceted effort by the institution to promote and encourage entrepreneurial behavior among students.Most attempts to represent the university ESS's effect on a student's entrepreneurial intention have modeled it using either three factors (Kraaijenbrink et al., 2010;Saeed et al., 2015), two factors (R. Trivedi, 2016;R. H. Trivedi, 2017), or an overarching factor comprised of many indicators (Bazan, Shaikh, et al., 2019).Because of the university ESS's complexity, some residuals might correlate slightly because simplified models need to omit several minor factors.In practice, it is not easy to foresee which residuals will covary.Thus, instead of assuming a diagonal residual covariance matrix Y, this study prescribed a more realistic covariance structure model where Y = O + Y Ã , and where O is a non-diagonal covariance matrix of the possible minor factors and Y Ã is a diagonal covariance matrix.As mentioned above, it is routine in Bayesian analysis (Depaoli & van de Schoot, 2017) to provide O informative priors following an inverse-Wishart distribution, thus allowing Y to contain residual covariates that deviate slightly from zero means.More specifically, we chose prior means of zero and small variances for the potential magnitude of residual covariances.We specified these restrictions by selecting the degrees of freedom value (d) of an inverse-Wishart distribution.As seen above, the parameter d affects the marginal prior variance through d À q, where the block size is q = 15 in this case.Note that using an inverse-Wishart prior IW c, d ð Þwith d = q + 6 or d = 15 + 6 = 21 gives a prior variance of 0.01 so that 95% of the residual covariance variation will be between 20.2 and 0.2 (again, small residual covariances when using standardized variables).As before, this study also assessed the model sensitivity to the residual covariance prior by subjecting it to distributions with several prior variances.
This study conducted the second CFA using the same Bayesian settings as the first CFA except for a few different strategies.First, we assessed the stability of the parameter values across the iterations and avoided local convergence by comparing estimates after M, 2M, and 0:5M MCMC iterations.Second, although the second CFA model looks straightforward, it might not be easy to estimate, that is, the MCMC iterations might not cover all parts of the posterior distribution, leading to poor mixing and high autocorrelation (Muthe´n et al., 2016).Also, CFA with residual correlations often leads to heavy computation because of slow MCMC convergence.Thus, we implemented thining to prevent complications in the MCMC process.

Analysis of the Results
This study followed several recommendations by Depaoli and van de Schoot (2017) to report the Bayesian results.Table 5 presents ML and Bayesian model fit statistics for comparison and completeness.In this context, a measurement model fits the data well if it largely accounts for any patterns in the means, variances and covariances among the indicators and generates reasonable

SS ET EM
Informative priors for the factor loadings SS2;N 1:083, 0: Informative priors for elements of the factor variance-covariance matrix parameter estimates (Roos & Bauldry, 2022).Table 5 (upper panel) shows that the ML analysis rejects the CFA and EFA models based on the likelihood-ratio chisquare test, although the adjusted x 2 =d\3:842 test supports both models.Additionally, the ML analysis supports the CFA and EFA models based on the popular fit criteria root-mean-square error of approximation (RMSEA), comparative fit index (CFI), and standardized root mean square residual (SRMR).We expected this discrepancy given the hypersensitivity of the chisquare to large samples (n = 350 in this case).In addition, the CFA solution shows that all the loadings are statistically significant at the p\0:001 level ranging between 0.647 (SS5) and 0.810 (EM2).However, the CFA solution also shows three modification indices larger than 10.Those are EM1 with SS3 (17.339),EM2 with SS3 (12.025), and EM2 with EM1 (12.428).Large modification indices (which they are not in this case) may indicate some potentially significant cross-loadings (Brown, 2015).Likewise, the EFA solution with Geomin rotation confirms that the major loadings of the EFA correspond to the hypothesized three-factor loading pattern.However, the test also shows several small statistically significant cross-loadings on other factors.Both phenomena above might partly explain the discrepancy in the fit criteria of the CFA and EFA models after the ML analysis.Table 5 (middle panel) shows the Bayesian model fit statistics of the CFA model A and CFA model B for M = 100, 000.The CFA model A specified the informative priors listed in Table 4 only.In contrast, the CFA model B specified the informative priors listed in Table 4 (except for the inverse-Gamma priors) and the informative priors for cross-loadings and residual correlations.Note that the RMSEA and CFI fit indices in Table 5 (middle panel) correspond to the Bayesian adaptation of these popular frequentist metrics provided for rough comparisons (Asparouhov & Muthe´n, 2021;Garnier-Villarreal & Jorgensen, 2020).The CFA of model A yields results similar to the CFA rendered by the ML analysis, and all the parameter results are statistically significant.In turn, the CFA of model B generates an excellent model fit.We checked the convergence of the chains by inspecting the trace plots, histograms, autocorrelation, and kernel density plots for every parameter.Figure 2 shows sample plots of the posterior distribution of the  Þ to assess the model sensitivity to the residual covariance priors.The CFA model seems marginally more sensitive to changes in the variance of the priors for the residual covariances.That is, the informativeness of the priors affects the parameters they influenced slightly more.[Support for H4] We generally expect this sensitivity when the residual variances are small (Schuurman et al., 2016), as seems to be the case here where all residual covariances are \0:5.These findings provide additional evidence concerning the ML model fit statistics discrepancy.
Tables 6 and 7 show the Bayesian standardized model results.In this context, we use the term statistically significant to mean that the 95% credibility interval does not contain zero.Table 6reveals that all major loadings are statistically significant and higher than 0.7, except for two corresponding to SS1, ''My university organizes business idea competitions'' (0.665) and SS5, ''My university provides students with the financial means needed to start a new business'' (0.688), which are marginally lower.The average absolute major loading (range) is 0.764 (0.665-0.864).The average absolute major loadings (ranges) on SS, ET, and EM are 0.731, 0.767, and 0.794 (0.665-0.804, 0.709-0.858,and 0.718-0.864),respectively.Thus, the indicators explain (on average) more than 50% of the variance of the corresponding factors.On the other hand, all crossloadings are not statistically significant and are very small in absolute value (they are all \0:050).The average absolute cross-loading (range) is 0.008 (20.010 to 0.034).
[Support for H1] Thus, as anticipated above and in hindsight, cross-loadings do not seem important enough to incorporate into the model.In addition, all factor correlations are statistically significant and moderately high.The average absolute factor correlation (range) is .866(.846-.884).Such correlations make sense given the sometimes unperceivable differences in the factors in the eyes of the respondents.
The residual covariances are relatively small, and none is .0:5 in absolute value.The average absolute residual correlation (range) is .092(2.290 to .296).Of the 105 residual covariances, only eight residual correlations are statistically significant.[Support for H3] This finding suggests that incorporating a small number of residual correlations could render the CFA model more appropriate for representing the data.A closer look at the eight statistically significant residual correlations gives additional insight into the relationships among the indicators.The residual correlation between: SS2, ''My university has many resources to support a startup company,'' and SS3, ''My university provides students with ideas to start a new business,'' is 20.280.SS3, ''My university provides students with ideas to start a new business,'' and SS4, ''My university arranges meetings with successful entrepreneurs to share their experiences,'' is 20.246.SS3, ''My university provides students with ideas to start a new business,'' and EM1, ''My university provides a creative atmosphere to develop ideas for new business startups,'' is 0.296.SS3, ''My university provides students with ideas to start a new business,'' and EM2, ''My university helps students build the required network for starting a business,'' is 0.262.SS4, ''My university arranges meetings with successful entrepreneurs to share their experiences,'' and SS5, ''My university provides students with the financial means needed to start a new business,'' is 20.221.ET1, ''My university provides students with the knowledge necessary to create a new business,'' and ET4, ''My university organizes mentoring and advisory services for student entrepreneurs,'' is 20.257.ET2, ''My university offers training in entrepreneurship,'' and EM4, ''My university creates awareness of entrepreneurship as a possible career choice,'' is 0.233.EM3, ''My university motivates students to start a new business,'' and EM5, ''My university brings entrepreneurial students in contact with each other,'' is 20.290.
Interestingly, two of the eight statistically significant residual covariances correspond to two of the three indicator correlations whose modification indices were .10(SS3 with EM1 and SS3 with EM2).
When using Bayesian CFA for hypothesis testing, we relied on the evidence from a combination of the statistical outputs, model diagnostics, and theoretical justifications.The analysis of the results supports three and partially supports one of the four hypotheses formulated above.We started the analysis by confirming that the specified factor structure aligns with the observed data.The Bayesian model fit statistics of the CFA model B (PP Limit, PP-p, RMSEA, CFI, and DIC) indicate an excellent model fit.Furthermore, the convergence of the chains, characteristics of the posterior distributions, and prior sensitivity analysis all suggest that the proposed CFA model represents the data very well.The individual assessments of the hypotheses are as follows.
Hypothesis H1: The model indicators will have large loadings on their hypothesized factors and several small cross-loadings due to minor influence from other factors is partially supported.All major loadings are large (average 0.764) and statistically significant, while all minor (cross-) loadings are small (average 0.008) but not statistically significant.These large indicators explain more than 50% of the variance of their hypothesized factor but do not contribute to explaining the variance of other factors, that is, the small cross-loadings are not statistically significant, thus partly confirming the assumptions formulated in H1.Hypothesis H2: The model will be insensitive to the cross-loading prior parameters is supported.The model is not sensitive to changes in the variance (N 0, 0:001 ð Þ , N 0, 0:005 ð Þ , N 0, 0:02 ð Þ, and N 0, 0:04 ð Þ) of the priors for the cross-loadings.After reducing or increasing the cross-loading prior parameters, the Bayesian model fit statistics, convergence of the chains, and characteristics of the posterior distributions remain unchanged.In other words, we confirmed the assumptions formulated in H2 by replication.Replication is a valid scientific method and ensures that the results are consistent and not just a one-time occurrence or due to random chance.By replicating an experiment and obtaining the same outcomes, we strengthened the validity and reliability of the findings.
Hypothesis H3: The model residuals will have several small correlations due to omitting several minor factors is supported.The residual covariances are small (average of absolute values 0.092), and eight out of 105 residual correlations are statistically significant.The university ESS is a complex and multifaceted effort by universities to promote and encourage entrepreneurial behavior, which may comprise several minor factors not accounted for by the traditional CFA model.As hypothesized, these few small residual correlations compensate for these omissions.Hypothesis H4: The model will be insensitive to the residual covariance prior parameters is supported.The model is not sensitive to changes in the variance Þ) of the priors for the residual covariances.After reducing or increasing the residual covariance prior parameters, the Bayesian model fit statistics, convergence of the chains, and characteristics of the posterior distributions remain unchanged.Again, we confirmed the assumptions formulated in H4 by replication.
To our knowledge, this is the first time that a study has provided such compelling evidence of the robustness of the Kraaijenbrink et al. (2010) scale and possible modifications for use in models studying the entrepreneurial intention of students.We discuss these possibilities further below.

Limitations of the Study
This study is subject to some limitations.(1) The original authors collected their primary data using convenience samples.Unfortunately, convenience samples are prone to selection bias because they do not ensure that the sample represents the broader population accurately.A possible approach to overcome this limitation would be using random or stratified sampling.(2) The collected data represent the perceptions of students.Therefore, it is possible that a difference between ''perception'' and ''reality'' exists.However, analyzing how students perceive the university ESS is equally important since this might shape their entrepreneurial intention (Turker & Sonmez Selcuk, 2009).A possible way to minimize this limitation would be to ensure carefully written neutral wording, not just in the indicators but also in the recruitment letter and informed consent form.(3) CFA is widely popular for testing and validating the hypothesized factor structure of a set of observed variables.However, CFA has several limitations when certain assumptions are unmet (especially when using a frequentist approach)-for example, assuming a linear relationship between observed and latent variables.Ways to minimize the limitations of CFA are relying on a solid theoretical foundation and prior knowledge when specifying the model, making sure that the measurement model captures all measurement errors, and using large sample sizes that rely on random sampling methods.(4) The Bayesian approach to CFA offers several advantages over traditional frequentist methods but has some limitations.For example, choosing appropriate priors can be challenging, and subjectivity in the prior specifications could introduce potential researcher bias into the analysis.In addition, interpreting the estimates for model parameters can be more complicated than interpreting frequentist estimates, as the uncertainty in the parameter estimates is captured by the entire posterior distribution rather than a single-point estimate.A possible solution to the subjectivity in choosing priors would be to average parameters from multiple and diverse samples from previous studies.This approach can also add to the generalizability of the model.

Implications of the Study
The Kraaijenbrink et al. (2010) scale used in this study captures students' perceptions of the three interrelated dimensions of the university ESS, that is, SS, ET, and EM.Our results show that all the indicators have large statistically significant loadings on their hypothesized factors and very small nonstatistically significant loadings on other factors.In other words, the three factors of the university ESS have statistically significant associations with all their respective indicators and no statistically significant associations with other indicators.These findings imply that the CFA model of the university ESS would not benefit from modeling potential crosscorrelations among indicators and factors they do not measure by design.Therefore, researchers using a frequentist approach do not need to relax (free) these parameters during factor analysis, thus avoiding unidentified models and yielding other benefits such as the following.(1) A more parsimonious model with fewer parameters to estimate.Parsimonious models are often more generalizable and less prone to overfitting.(2) A model with more unambiguous relationships between indicators and factors that is more straightforward to interpret.(3) A model with enhanced discriminant validity of the constructs provided by the distinctness of the latent constructs.(4) A model that accurately represents the underlying theoretical structure of the data.
Furthermore, the findings of this study also show that all error covariances are small.These small-sized error covariances suggest that the factor pattern is well specified.In addition, only eight out of 105 are statistically significant.These error covariances capture shared sources of variation between pairs of indicators that remain after accounting for any variation due to their factors (Roos & Bauldry, 2022).These results suggest that the CFA model of the university ESS would benefit from modeling potential error covariances among a few indicators.We can interpret these additional model specifications as nuisance parameters hypothesized to be zero, although perhaps not precisely zero (Asparouhov et al., 2015).In essence, these error covariances would account for potential methods effects.First, the university ESS scale might include similarly worded indicators, for example, SS3, ''My university provides students with ideas to start a new business,'' and EM1, ''My university provides a creative atmosphere to develop ideas for new business startups.''Second, another source of methods effects that could be present in the university ESS scale is indicators with similar reading difficulty levels.This source could be of great importance in universities with many international students whose mother languages are not the language of the survey.Lastly, in some contexts, some indicators of the university ESS scale might be subject to social desirability bias, for example, SS4, ''My university arranges meetings with successful entrepreneurs to share their experiences,'' and SS5, ''My university provides students with the financial means needed to start a new business.''The Bayesian approach used in this study allows the inclusion of error covariances to account for shared variances that are always present in any scale.Researchers using a frequentist approach may only need to judiciously include a few error covariances to balance the empirical data with theory and model parsimony.In an ideal model, the associated constructs should explain all variance in their indicators.In reality, there are often other sources of shared variance between indicators.Allowing for error covariances after conducting a Bayesian CFA can be empirically warranted and theoretically justified, yielding several additional benefits like the following.(1) Incorporating error covariances can provide a better-fitting model.(2) Error covariances can account for shared method variance due to, for example, similarly worded indicators or closely related concepts.(3) Error covariances can help capture method effects, for example, two negatively worded indicators.(4) Error covariances can help minimize model misspecification, resulting in more accurate and unbiased estimates.(5) Recognizing and modeling nuance parameters can make the model more reflective of complex realworld phenomena.
In practice, small error covariances and small crossloadings provide additional ways of representing correlations among the university ESS indicators.Consequently, in hindsight, this study could have given similar results should it had implemented only the error covariances specifications with informative priors.Furthermore, it could have specified informative priors Bazan only on the eight statistically significant residual correlations.However, despite using prior elicitation from an earlier study, there was no way of knowing that that would be the case.In addition, this phenomenon might not be identical for all datasets collected using the university ESS instrument.Further researchers could investigate these eight indicators (especially SS3, EM1, and EM2) and their implications in methods effects in more detail for possible improvements to this widely used university ESS scale.In turn, entrepreneurial universities could use the new insight provided by studies on their campuses to improve their university ESS and provide the resources that students could use to further their entrepreneurial aspirations.

Conclusion
The literature shows that the university ESS is a potential contextual and situational element affecting students' entrepreneurial intention.Several researchers have studied this phenomenon by combining the framework proposed by Kraaijenbrink et al. (2010) and the theory of planned behavior (Ajzen, 1991).All those studies in the literature have used the university ESS scale to assess the students' perception of university support for entrepreneurial activities.Thus, their primary goal was to measure the relationship between the university ESS constructs, for example, educational, concept, and business development support, and the precursors of entrepreneurial intention.Those studies validated the factor structure of their indicators by verifying that the data fit the hypothesized measurement model.However, none of those studies assessed the relationship among the university ESS constructs themselves, which can provide a better understanding of the multiple support mechanisms provided by the university.
This study conducted a detailed factor analysis of the university ESS scale proposed by Kraaijenbrink et al. (2010) as modified by a few other authors (Bazan, Shaikh, et al., 2019;R. Trivedi, 2016;R. H. Trivedi, 2017).In addition, we used the added flexibility of a Bayesian approach to incorporate previous knowledge about the CFA model consisting of informative priors elicited from an earlier study.One of the hallmarks of Bayesian analysis is integrating previous information and beliefs about a phenomenon into the mathematical model before seeing the new data.To the best of our knowledge, this is the first time this popular university ESS scale has been the subject of such an extensive investigation.This investigation allowed us to answer the two research questions guiding the study.( 1) What new insights concerning the phenomenon can this approach generate?(2) What general and ad hoc modifications to the university ESS scale can this approach inform?
This study produced several novel insights concerning the university ESS scale routinely used to study entrepreneurial intention.First, the three distinct but interrelated constructs of the university ESS (SS, ET, and EM) reflect only on their hypothesized indicators.(All the cross-loadings the Bayesian approach allows are very small and not statistically significant.)This finding has important implications for universities and researchers studying student entrepreneurship.Universities have traditionally helped students develop an entrepreneurial mindset, skills, and competencies by offering entrepreneurship education programs.Researchers have been studying the impact of entrepreneurship education programs on the precursors of entrepreneurial intention for many years with inconsistent results (Fayolle et al., 2006;Rae & Woodier-Harris, 2013;Rauch & Hulsink, 2015;Sa´nchez, 2011;Souitaris et al., 2007).The inconsistent results likely stem from students' perception of the presence or absence of additional support mechanisms necessary for entrepreneurial activity, for example, business incubation services, startup coaching, intellectual property protection, technology transfer, and commercialization.Using the university ESS scale within the entrepreneurial intention and behavior assessments could help institutions interpret the efficacy of their three different efforts (SS, ET, and EM) to promote entrepreneurial activities among students.Second, researchers can use the added flexibility of the Bayesian approach to address novel substantive questions through the mathematical model.For example, why do the statistically significant residuals correlate negatively with residuals within their factors and positively with residuals outside their factors?Although the correlations are weak, researchers could investigate whether the factor model may not accurately capture the underlying structure of the data because the relationships between the observed variables may be more complex than what the model assumed.
Universities and researchers can consider general and ad hoc modifications to the university ESS scale to more accurately capture the phenomenon under study.For example, two of the eight statistically significant residual covariances correspond to two indicator correlations whose modification indices were .10(SS3 with EM1 and SS3 with EM2).These indicators belong to two different factors and show positive residual correlations.This positive residual correlation often suggests potential cross-loading.For example, indicator SS3 (''My university provides students with ideas to start a new business'') has weak residual correlations with indicators EM1 (''My university provides a creative atmosphere to develop ideas for new business startups'') and EM2 (''My university helps students build the required network for starting a business'').However, indicators EM1 and EM2 do not have statistically significant residual correlations despite belonging to the same factor.One alternative to test in this instance would be to reword the SS3 indicator to prevent possible reading confusion, for example, between SS3 and EM1, where both refer to ideas to start a new business.Furthermore, researchers using a frequentist approach could specify correlated residuals between the indicators in question, that is, SS3 with EM1 and SS3 with EM2, to help them develop a model that accurately reflects the relationships among factors and indicators.
In the future, similar studies conducted by aspiring entrepreneurial universities could help them assess the efficacy of their efforts to promote entrepreneurial activities among students.Doing so could help transform technology innovations generated at the university into much-needed economic activity.That is, universities could assist students in translating early-stage innovations into market-ready products and services and help create the startups that will spin off from the university and commercialize the research outcomes.

Figure 1 .
Figure 1.Confirmatory factor analysis model of the university environment and support system.
loadings for the second indicator (SS2, ET2, and EM2) of each factor (SS, ET, and EM).In addition, the posterior distributions of all the parameters are smooth, make substantive sense, do not have posterior standard deviations greater than the scale of the original parameters, do not have a range of credible intervals greater than the underlying scale of the original parameters, and do not show large fluctuations in the variances(Depaoli & van de Schoot, 2017).[Support for H1-H4] Table5 (lower panel) shows the local convergence assessment and prior sensitivity analysis for model B and M = 100, 000 as the base model specifications.Note that all the model specifications remained the same unless mentioned otherwise.(1) We changed the MCMC iterations to 2M and 0:5M to assess local convergence.The stability of the fit parameters shown in Table 5 (lower panel) and a visual evaluation of the MCMC sequences confirm the convergence of the parameter values.[Support for H1-H4] (2) We changed the priors N 0, 0:001 ð Þ , N 0, 0:005 ð Þ , N 0, 0:02 ð Þ, and N 0, 0:04 ð Þ to assess the model sensitivity to the cross-loading priors.Table 5 (lower panel) shows that the CFA model is not sensitive to changes in the variance of the priors for the cross-loadings.In other words, the model parameters affected by these priors do not change substantially with less or more informative priors.[Support for H2] This result indicates that cross-loadings may not be the leading reason for the discrepancy in ML model fit results.(3) We changed the priors to IW c, 11 ð Þ, IW c, 16 ð Þ, IW c, 26 ð Þ, and IW c, 31 ð

Table 7 .
Standardized Model Results: Residual Variances (Diagonals) and Covariances (Off-Diagonals).this context, we use the term statistically significant to mean that the 95% credibility interval does not contain zero.

Table 1 .
Select Demographics of the Student Samples.

Table 2 .
Traditionally Proposed Three-Factor Solution for the University Environment and Support System.

Table 3 .
Unstandardized Confirmatory Factor Analysis Parameter Estimates for the Three-Factor Solution Using the 2018 Sample and Noninformative Priors, n = 405.
cross-loading prior by subjecting it to distributions with several prior variances.

Table 4 .
Informative Priors Elicited From the Confirmatory Factor Analysis Using the 2018 Sample.

Table 5 .
Maximum Likelihood and Bayesian Model Fit Results, n = 350.
Brown (2015)2015)recommends a parsimony correction index RMSEA close to 0.06 or less, a comparative fit index CFI close to 0.95 or greater, and an absolute fit index SRMR close to 0.08 or less.The deviance information criterion (DIC) is the most popular criterion for Bayesian model selection and comparison.Lower DIC values suggest a better model fit.

Table 6 .
Standardized Model Results: Factor Loadings and Correlations.
*In this context, we use the term statistically significant to mean that the 95% credibility interval does not contain zero.