# The Path Model or Path analysis or PA is a very useful multivariate technique which combines aspects of factor analysisThe Path Model or Path analysis or PA is a very useful multivariate technique which combines aspects of factor analysis

The Path Model or Path analysis or PA is a very useful multivariate technique which combines aspects of factor analysis (Hoyle, 1995) and multiple regression analysis (Cunningham and Wang, 2005) to yield appropriate estimations for a series of separate multiple regression equations simultaneously (Hair et al., 2010).

The spread of structural equation modelling
SEM has been widely used for PA to study causal relationships (Mitchell, 1992; Jöreskog and Sörbom, 2001; Loehlin, 2004).The use of a graphical method for modelling multivariate relations in PA is just one of SEM’s major advantages (Kline, 2005; Yuan et al., 2010; Oke et al., 2012). There are at least five other advantages as well of using SEM for modelling multivariate relations: (1) it has the capability to assist with the theory verification by effectively dealing with a large number of variables and complex phenomena (Ullman, 2007); (2) it is useful in survey research and hypothesis testing studies (Oke et al., 2012); (3) the calculations of path effects (Anderson and Gerbing, 1988) and measurement errors in SEM are computed simultaneously (Kline, 2005) which differ from the calculations in the traditional statistical programmes (e.g., SPSS or SAS), where the latter calculations are estimated separately (Schumacker and Lomax, 2004); (4) it has the capability to provide high validity and reliability model estimates by assessing or even correcting the measurement errors (Byrne, 2010); and (5) The programmes dedicated to SEM are readily available and have become increasingly user-friendly (Byrne, 2010). These advantages have put SEM much above other basic statistical methods for modelling multivariate relations. As Byrne (2010, p4) opines, “there are no widely and easily applied alternative methods for modelling multivariate relations, or for estimating point and/or interval indirect effects; these important features are available using SEM methodology”.

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Hence, SEM has been recognised as one of “the preeminent multivariate technique(s)” (Hershberger, 2003, p35) and been “applied to a diverse array of topics” (Loehlin, 2004, p116), “across all disciplines” (Schumacker and Lomax, 2004, p6), which includes the research in pedagogy, marketing (Hair et al., 2011), economics, criminology, demography (Li, 2004), public health, business management, sociology, psychology (Byrne, 2010), medicine, political science and biological science (Schumacker and Lomax, 2004). SEM has also acquired the status of first choice data analysis tool for conducting empirical research in business and management (e.g., Table 5.11) (Hult et al., 2006), including strategic management (e.g., Calantone and Zhao, 2001; Shook et al., 2004), logistics (e.g., Garver and Mentzer, 1999), supplier relationships (e.g., Cousins and Lawson, 2007), quality management (e.g., Lin et al., 2005), organisational research (e.g., Medsker et al., 1994) and production and operations management (e.g., Shah and Goldstein, 2006), SEM has been ranked as the top analysis methods in the Journal of Production and Operations Management during the period 1992 to 2005 (Gupta et al., 2006).

Literature Focus
(Chong et al., 2001; Anderson-Connolly et al., 2002; Fullerton and Wempe, 2005; Politis, 2005; Hoang et al., 2006; Cousins and Lawson, 2007; Dal Pont et al., 2008; Verworn et al., 2008; Wang and Cao, 2008; Fotopoulos and Psomas, 2009; Fullerton and Wempe, 2009; So and Sun, 2011; Agus and Hajinoor, 2012; AL-Tahat and Alkhalil, 2012; Bahri et al., 2012;
Chettiar et al., 2012; Fullerton et al., 2012; Habidin et al., 2012; Hong and Rawski, 2012; Zubir and Habidin, 2012)
The relationships between Lean related practices, shop floor management, and organisational/financial performance
(Kanji, 1998; Ahire and Dreyfus, 2000; Prajogo and Sohal, 2006; Maranto- Vargas and Gómez-Tagle Rangel, 2007; Chen et al., 2008; Dahlgaard-Park, 2009; Ni and Sun, 2009; Tian et al., 2010; Aloini et al., 2011; López and
Morales, 2011; Peng et al., 2011; Yang et al., 2011; AL-Tahat and Bataineh, 2012; Hashim et al., 2012; Oke and Kach, 2012; Zeng et al., 2013)
The relations between continuous improvement practices and organisation performance/business performance
(Hallgren and Olhager, 2009; Daniel et al., 2010; Nagati and Rebolledo, 2012; Vinodh and Joy, 2012; Zhou et al., 2012)
The critical success factors and Lean production implementation
(Corsten and Felde, 2005; Fynes et al., 2005; Li et al., 2006; Agbejule and Burrowes, 2007; Cousins and Lawson, 2007; Koh et al., 2007; Lawson et
al., 2009; Agus, 2011a; Agus, 2011b; Shamah, 2013a; Shamah, 2013b; Sukwadi et al., 2013; Zeng et al., 2013)
The relationships between supply chain management, quality
management, and improvement performance
(Rao, 2004; Rao and Holt, 2005; Shazali et al., 2013)
Lean production in other sector and
performance
Table 5.11 Selected business and management literature on structural equation modelling

The application of AMOS in developing and analysing SEM
Many softwares, like LISREL (SSI, 2012), CALIS in SAS/STAT (Yung, 2010), EQS (Bentler and Wu, 2012), AMOS (Arbuckle, 2007), Mplus ( 2013) and many others, can be used to perform SEM (Hox, 1995; McDonald and Ho, 2002; Kline, 2005). To be fair, the choice of the programme for SEM is often a matter of personal preference (Worthke, 2013). In this current research, AMOS(Analysis of Moment Structures) was selected for the data analysis. It is a package within the IBM SPSS family (Arbuckle, 2007). It provides a very rich visual framework that permits users to easily compare, confirm and refine models (IBM, 2011). A five-step process for developing and analysing structural equation models (Figure 5.12) was developed by Schumacher and Lomax in2004.

Figure 5.12 A five-step process in developing and analysing SEM (Schumacker and Lomax, 2004)

Model specification
This step deals with the development of a theoretical model on the basis of available relevant theory, research, and information. SEM is an ‘a priori’ technique in which the theory drives the development of the model (Valluzzi et al., 2003). This is in contrast to mining the data to develop a model (Kline, 2005). In this way, a theoretical model involving the determining relationships (paths) and/or parameters (variables) will be specified and developed on the basis of the findings from the literature survey and theories before any data collection.

Model identification
This step is the second step. In this step, the parameters are identified and specified and also, the degrees of freedom (D.f) is calculated before the estimation of the particular model. The D.f. of a model is “the difference between the number of observations (or distinct sample moments in AMOS, calculated as m × (m + 1) ? 2), and the number of its parameters (free or estimated)” (Kline, 2005, p100; Byrne, 2010). The following is the formula to calculate the D.f. of a SEM path model (Rigdon, 1994, p276):

D.f. = m × (m + 1) ? 2 – 2 × m – ? × (? – 1) ? 2 – g – b
m: observed variables
?: latent variables (constructs/factors)
g: direct paths of exogenous constructs on endogenous constructs
b: direct paths of endogenous constructs on each other

A SEM model requires to be either just- or over-identified (D.f. ? 0) to generate accurate parameter estimates. Thus, an under-identified (D.f. 0) SEM model that has more number of observations than free parameters to be estimated. It has
positive degrees of freedom.
Just-identified (D.f. = 0) SEM model containing just enough degrees of freedom to estimate all free parameters.
Underidentified
(D.f. ; 0) SEM model with more parameters to be estimated than there are observations. It has negative
degrees of freedom.
Table 5.12 The three levels of model identifications by Hair et al (2010) and Schumacker and Lomax (2004)

After identifying the model, the statistical power (?) of a model requires to be examined to establish the probability of errors. In statistics, there are two types of probability errors: Type I error and Type II error (Table 5.13). Type I error refers to the failure to accept a true null hypothesis (or H0, a default hypothesis, i.e., there is no relationship between two measured phenomena), whereas a Type II error refers to the failure to reject a false null hypothesis (H0) (Neyman and Pearson, 1933).

Decision True population
H0 is true H0 is false
Reject H0 Wrong decision, Type I error Correct decision
Fail to reject H0 Correction decision Wrong decision, Type II error

Table 5.13 The two types of probability errors (Walker and Maddan, 2012, p330)

The statistical power (?) is required to calculate the probability of rejecting a false hypothesis correctly (Cohen, 1988). The following formula can be used for the calculation in a SEM model:

Statistical power (?) = 1 – ?
?: the probability of a Type II error

The sample size as well as the degrees of freedom affects the statistical power of the model estimations (McQuitty, 2004). In order to achieve a desired level of statistical power for a reliable model, it is necessary to determine the required sample size with associated degrees of freedoms of the model (Hoe, 2008).Thus, the minimum required sample size must be considered for forming the SEM to estimate measures or develop theory (Table 5.14).

D.f. ? = 0.60, N? ? = 0.70, N? ? = 0.80, N? ? = 0.90, N?
5 885 1132 1643 1994
20 280 346 435 572
50 145 175 214 274
100 92 110 132 165
150 72 85 101 125
200 61 71 84 104
400 41 48 56 68
Table 5.14 Minimum sample size required to achieve specified statistical power (McQuitty, 2004, p181)