One should demonstrate the principle of model postulation in addition to research design. Indeed, there were very few papers discussing the technique and principle of their models.
A well-applied principle of parsimony for model users emphasizes the simplicity of a model. According to this principle, the users should justify if a model could present a phenomenon by a few variables.
Cover and Thomas had proposed other modeling principles. Model identification was often overlooked, with only Most publications provided the df values in their SEMs and we estimated the df of those that did not report. All publications had a df greater than zero. All the models with CFA met the requirement that each latent variable should have at least two indicators.
However, many studies skipped over scaling the latent variables before estimation, resulting in non-robust results. The unscaled latent variable can hardly provide useful information to the causal test. Otherwise, it is likely that the user had just fit the model by chance. Many estimation methods in SEM exist, such as maximum likelihood ML , generalized least squares, weighted least squares, and partial least squares.
Maximum likelihood estimation is the default estimation method in many SEM software Kline ; Hoyle All of the publications stated the estimation methods were based on ML, which assumes that 1 no skewness or kurtosis in the joint distribution of the variables exists e. However, very few publications provided this key information about their data. Instead, they simply ignored the data quality or chose not to discuss the raw data.
Some papers briefly discussed the multivariate normality of their data, but none discussed the data screening and transformation i. We assume that most of their ecological data was continuous, yet one needs to assure the continuity of the data to support their choice of estimation methods. The partial least square method requires neither continuous data nor multi-normality. We did not find a publication with sufficient explanation for its CFA in regard to the prior knowledge or preferred function i.
Factor analysis is a useful tool for dimension reduction. The prior knowledge of a measurement model includes two parts: 1 the prior knowledge of indicators for a latent variable and 2 the prior knowledge of the relationships between the latent variable and its indicators Bentler and Chou For example, the soil fertility of a forest as a latent variable was estimated based on two types of prior knowledge, including 1 the observation of tree density, water resources, and presence of microorganisms and 2 the positive correlations among the three observed variables.
If the estimation of a latent variable is performed without prior knowledge, CFA will become a method only for data dimension reduction. In addition, we did not find any CFAs in the ecological publications explaining the magnitude of the latent variable.
Therefore, these latent variables lack a meaningful explanation in regard to the hypothesis of an SEM Bollen ; Duncan et al The reasons are very flexible for measuring a latent variable, but they require the user to explain the application of CFA carefully. SEM requires measurement models to be based on prior knowledge so that latent variables can be interpreted correctly Bentler and Chou SEM is not a method to only reduce data dimensions.
Instead, one should explain the magnitude and importance of indicators and latent variables. Therefore, users should base their explanations on theory when discussing the associated changes between latent variables and indicators.
The explanation should include the analysis of the magnitude of the latent variable, indicators, and factor loadings. Reporting of fit indices in any SEM is strongly recommended and needed. Approximately However, none justified their usage of the chosen fit indices. Those that did not report model fit indices also did not provide the reason for doing so. Fit indices are important indicators of model performances.
Due to their different properties, they are sensitive to many factors, such as data distribution, missing data, model size, and sample size Hu and Bentler ; Fan and Sivo ; Barrett Most fit indices i. Selection of model fit indices in an SEM exercise is key to explaining the model e.
Users should at least discuss the usage of fit indices to ensure that they are consistent with their study objectives. An SEM report should include all the estimation and modeling process reports. However, most publications did not include a full description of the results for their hypothesis tests. Some publications provided their SEMs based on a covariance matrix Table 1 , while even fewer studies reported the exact input covariance or correlation matrix.
No study reported the multivariate normality, absence, or outliers of their data. A small percentage 8. The basic statistics i. The reporting guidelines are comprised of five components McDonald and Ho ; Jackson et al.
Model specification: Model specification process should be reported, including prior knowledge of the theoretically plausible models, prior knowledge of the positive or negative direct effects among variables, data sampling method, sample size, and model type. Data preparation: Data processing should be reported, including the assessment of multivariate normality, analysis of missing data, method to address missing data, and data transformations. Estimation of SEM: The estimation procedure should be reported, including the input matrix, estimation method, software brand and version, and method for fixing the scale of latent variables.
Model evaluation and modification: The model evaluation should be reported, including fit indices with cutoff values and model modification. Reports of findings: All of the findings from an SEM analysis should be reported, including latent variables, factor loadings, standard errors, p values, R 2 , standardized and unstandardized structure coefficients, and graphic representations of the model. The estimation of sample size is another issue for the SEM application.
While some Technically, sample size for an SEM varies depending on many factors, including fit index, model size, distribution of the variables, amount of missing data, reliability of the variables, and strength of path parameters Fan et al.
Some researchers recommend a minimum sample size of — or five cases per free parameter in the model Tabachnick and Fidell ; Kline One should be cautious when applying these general rules, however. Increasingly, use of model-based methods for estimation of sample size is highly recommended, with sound methods based on fit indices or power analysis of the model. Kim developed equations to compute the sample size based on model fit indices for a given statistical power.
We did not find that SEM was validated in the reviewed ecological studies, even though it is a necessary process for quantitative analysis. This is probably because most SEM software is developed without model validation features. The purpose of model validation is to provide more evidence for the hypothetical model. The basic method of model validation is to test a model by two or more random datasets from the same sample.
Therefore, the validation requires a large sample size. The principle of the model validation is to assure that the parameters are similar when a model is based on different datasets from the same population.
This technique is a required step in many learning models. However, it is still unpopular in SEM applications. SEM is a powerful multivariate analysis tool that has great potential in ecological research, as data accessibility continues to increase. However, it remains a challenge even though it was introduced to the ecological community decades ago. Regardless of its rapidly increased application in ecological research, well-established models remain rare.
In fact, well-established models can serve as a prior model, as this has been extensively used in psychometrics, behavioral science, business, and marketing research. There is an overlooked yet valuable opportunity for ecologists to establish an SEM representing the complex network of any ecosystem.
Many ecological studies are characterized by large amounts of public data, which need multivariate data analysis. SEM users are provided with this opportunity to look for suitable public data and uncover patterns in research. However, big data will also inevitably bring new issues, such as the uncertainty of data sources. Therefore, improved data preparation protocols for SEM research are urgently needed.
Fortunately, the exponential growth of usage in data-driven models, such as machine learning, provides SEM users a promising opportunity to develop creative methods to combine hypothesis-based and data-driven models together. The growing availability of big data is transforming studies from hypothesis-driven and experiment-based research to more inductive, data-driven, and model-based research. Causal inference derived from data itself with learning algorithms and little prior knowledge has been widely accepted as accurate Hinton et al.
The original causal foundation of SEM was based on a hypothesis test Pearl , , ; Bareinboim and Pearl However, with the advancement of data mining tools, the data-driven and hypothesis-driven models may be mixed in the future. Here, we emphasize the importance of utilizing hypothesis-based models that are from a deductive-scientific stance, with prior knowledge or related theory.
Meanwhile, we also agree that new technologies such as machine learning under big data exploration will stimulate new perspectives on ecological systems. On the other hand, the increased data availability and new modeling approaches—as well as their possible marriage with SEM—may skew our attention towards phenomena that deliver easily accessible data, while consequently obscuring other important phenomena Brommelstroet et al.
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Wright S Correlation and causation. J Agric Res 20 7 — Download references. We thank Dr. It also includes path analysis regression analysis whereby equations representing the effect of one or more variables on others can be solved to estimate their relationships. Factor analysis is another special case of SEM whereby unobserved variables factors or latent variables are calculated from measured variables.
These analyses can usually be performed using data in the form of means or correlations and covariances i. These data, moreover, may be obtained from experimental, nonexperimental and observational studies. All of these techniques can be incorporated into the following example. Several symptoms of a disease are measured and used in a factor model that represents these symptoms.
The impact of different types of medication on the factor s is then compared across the measured behavioral and environmental conditions. To conduct the above analyses, both a structural i. The structural model refers to the relationships among latent variables, and allows the researcher to determine their degree of correlation calculated as path coefficients.
That is, path coefficients were defined by Wright , p. Each structural equation coefficient is computed while all other variances are taken into account. Thus, coefficients are calculated simultaneously for all endogenous variables rather than sequentially as in regular multiple regression models. To determine the magnitude of these coefficients, the researcher specifies the structure of the model.
This is depicted in Figure 1. As shown, the researcher may expect that there is a correlation between variables A and B, as shown by the double headed arrow. There may be no expected relationship between variables A and C, so no arrow is drawn. Finally, the researcher may hypothesize that there is a unidirectional relationship of variable C to B, as indicated by an arrow pointing from C to B.
The relationships among variables A, B, and C represent the structural model. Researchers detail these relationships by writing a series of equations, hence the term 'structural equation' referring to the relationships between the variables.
The combination of these equations specifies the pattern of relationships [ 12 ]. The second component to be specified is the measurement model. As represented in Figure 1 , it consists of the measured variables e. Latent variables are factors like those derived from factor analysis, which consist of at least two inter-related measured variables.
They are called latent because they are not directly measured, but rather are represented by the overlapping variance of measured variables. They are said to better represent the research constructs than are measured variables because they contain less measurement error. As indicated in Figure 1 , for example, measurement model A depicts a latent variable A, which is the construct underlying measured variables 1 and 2.
To further explicate the process of developing and analyzing a model, the following steps are outlined next. Why should we use SEM?
Pros and cons of structural equation modeling. Meth Psychol Res Online , 8 The researcher develops hypotheses about the relationships among variables that are based on theory, previous empirical findings or both [ 15 ].
These relationships may be direct or indirect whereby intervening variables may mediate the effect of one variable on another. The researcher must also determine if the relationships are unidirectional or bidirectional, by using previous research and theoretical predictions as a guide.
The researcher outlines the model by determining the number and relationships of measured and latent variables. Care must be taken in using variables that provide a valid and reliable indicator of the constructs under study. The use of latent variables is not a substitute for poorly measured variables. A path diagram depicting the structural and measurement models will guide the researcher when identifying the model, as described next.
Identifying the model is a crucial step in model development as decisions at this stage will determine whether the model can be feasibly evaluated. For each parameter in the model to be estimated, there must be at least as many values i. This problem also occurs when variables are highly intercorrelated multicollinearity c , the scales of the variables are not fixed the path from a latent variable to one of the measured variables must be set as a constant , or there is no unique solution to the equations because the underidentification results in more parameters to be estimated than information provided by the measured variables.
In underidentified models there are an infinite number of solutions and therefore no unique one. These problems may be remedied with the addition of independent variables, which requires that the model be conceptualized before data are collected. There are many further issues to consider when managing parameters that cannot be addressed in this primer. For further details on model identification, readers are encouraged to see Kline [ 16 ].
There are many estimation procedures available to test models, with three primary ones discussed here. ML is set as the default estimator in most SEM software.
It is an iterative process that estimates the extent to which the model predicts the values of the sample covariance matrix, with values closer to zero indicating better fit.
The name maximum likelihood is based on its calculation. The estimate maximizes the likelihood that the data were drawn from its population. The estimates require large sample sizes, but do not usually depend on the measurement units of the measured variables. It is also robust to non-normal data distributions [ 17 ].
Another widely used estimate is least squares LS , which minimizes the sum of the squares of the residuals in the model. LS is similar to ML as it also examines patterns of relationships, but does so by determining the optimum solution by minimizing the sum of the squared deviation scores between the hypothesized and observed model.
It often performs better with smaller sample sizes and provides more accurate estimates of the model when assumptions of distribution, independence, and asymptotic sample sizes are violated [ 18 ]. The third, asymptotically distribution free ADF estimation procedures also known as Weighted Least Squares are less often used but may be appropriate if the data are skewed or peaked. ML, however, tends to be more reliable than ADF. This method also requires sample sizes of to to obtain reliable estimates for simple models and may under-estimate model parameters [ 16 , 19 ].
For further details see Hu et al. These estimation procedures determine how well the model fits the data. Fitting the latent variable path model involves minimizing the difference between the sample covariances and the covariances predicted by the model. The population model is formally represented as:. This simple equation allows the implementation of a general mathematical and statistical approach to the analysis of linear structural equation system through the estimation of parameters and the fitting of models.
Estimation can be classified by type of distribution multinormal, elliptical, arbitrary assumed of the data and weight matrix used during the computations. The function to be minimized is given by:. W is the weight matrix that can be specified in several ways to yield a number of different estimators that depend on the distribution assumed.
Essentially the researcher attempts to represent the population covariance matrix in the sample variables. Then, an estimation procedure is selected, which runs through an iterative process until the best solution is found.
Another source of information in the output is the fit indices. There are many indices available, with most ranging from 0 to 1 with a high value indicating a great degree of variance in the data accounted for by the model [ 21 ]. A good fit is also represented by low residual values e.
This statistic, however, varies as a function of sample size, cannot be directly interpreted because there is no upper bound , and is almost always significant. It is useful, however, when directly comparing models on the same sample. Dahly, Adair, and Bollen [ 22 ], for example, tested various fit indices for different models depicting the relationship between maternal height and arm fat area with fetal growth.
When adding and removing variables, as well as specifying varying relationships between variables, each corresponding fit index was calculated. This allowed the researchers to determine factors in the fetal environment that are most significantly related to systolic blood pressure of young adults.
A comparison of indices was conducted by Hu and Bentler [ 18 ] on data that violated assumptions of normal distribution, independence of observations, and symmetry. Many of these are provided by standard SEM software packages e. To determine the model's goodness-of-fit, sample size is an important consideration.
It must be large enough to obtain stable estimates of the parameters. Many recommendations have been published, suggesting that there is no precise decision rule. Monte Carlo studies provide guidance that sample sizes of 10 for a one-factor, five-observed variable model, and 30 for a two-factor, five-observed variable model provide robust results [ 24 ].
More general guidelines are used in current research with the suggestion that at least but preferably cases are needed to obtain stable results [ 16 ]. Using a large sample reduces the likelihood of random variation that can occur in small samples [ 25 ], but may be difficult to obtain in practice. To obtain improved fit results, the above sequence of steps is repeated until the most succinct model is derived i. A recommended procedure to improve the model estimations is through examination of the size of the standardized residual values between variables.
Large residuals may suggest inadequate model fit. This can be addressed by the addition of a path link, or inclusion of mediating or moderating variables if theoretically supported. Once the model is re-calculated, its fit may show improvement and residual may be reduced.
These results then need to be confirmed on an alternate sample, and through further studies. This replication strengthens confidence in the inferences, and provides implications for theoretical development and practical application. Before executing SEM procedures, there are many additional topics to consider.
As for any research study, careful planning of design, sampling, and measures is needed to develop valid models. SEM can be used in either cross-sectional or longitudinal studies, whereby the former are identified by links among variables measured at the same point in time, and the latter are specified by the links among variables measured at different points in time.
These models often include autoregressive effects where a variable measured at two time points is correlated with itself. This corrects for an over-estimation of the relationship among exogenous independent and endogenous dependent variables [ 26 ].
While this is a distinct advantage of SEM, it is often disregarded. Types of measures must also be considered [ 27 ]. Calculations of variances, covariances and product-moment correlations all assume that values are measured on an interval scale. Measures that include, for example, rating scales without equal distances between data points, are not necessarily considered appropriate [ 28 ].
Researchers must be prudent in selecting the appropriate procedures for particular levels of measurement including, for example, dichotomous and polytomous data [ 29 ]. Indeed, another advantage of SEM is the ability to manage continuous and binary data simultaneously. SEM can be employed for both exploratory and confirmatory models.
An exploratory approach is more traditional in that a detailed model specifying the relationships among variables is not made a priori. All latent variables are assumed, therefore, to influence all observed variables so that the number of latent variables are not pre-determined, and measurement errors are not allowed to correlate [ 29 ].
Although both exploratory and confirmatory factor analyses are a subset of SEM involving the measurement model only, the latter is more frequently used to test hypothetical constructs. The following section presents three examples of application of SEM in medical and health sciences research. SEM has been applied in psychiatry to understanding patients' experiences of schizophrenia. Loberg and colleagues examined the role of positive symptoms and duration of schizophrenia on dichotic listening of patients [ 30 ].
Dichotic listening tasks are used as a means of assessing functioning within the left temporal lobe language areas. Previous research suggested increased impairment in left temporal lobe language processing among patients with a high number of positive symptoms e.
Loberg, Jorgensen, Green, Rund et al [ 30 ] attempted to replicate these results as well as determine whether duration of illness further decreases language functioning. A total of patients from clinics in Norway and California diagnosed with schizophrenia were included. The Extended Brief Psychiatric Rating Scale and Positive and Negative Syndrome scale were completed by blind observers to measure symptoms of schizophrenia, and the duration of the disease was calculated based on initial onset of symptoms.
Dichotic listening was measured by patients' responses to consonant and vowel blends spoken through headphones. In one condition patients were told which ear to listen with attention and in another they were not laterality. The theoretical model tested is shown in Figure 2. Analysis of this model using SEM indicated it fit the data well. The CFI was 0. Close inspection of the model Figure 2 shows that all the path coefficients to the predicted latent variables are moderate to high range from.
Positive symptoms were measured by hallucinations, disorganized thoughts, and unusual thought content. Dichotic listening was measured by accuracy of sounds identified in each ear according to the condition in which the patients heard the sounds. In terms of the relationship between dichotic listening and schizophrenia, duration of schizophrenia and number of positive symptoms were related to accuracy of sound detection. That is, patients who have had schizophrenia for a longer duration and experience more positive symptoms, the poorer their identification of vowel-consonant blends.
These results support findings from previous research suggesting impaired language processing and structural abnormalities in the left superior temporal gyrus for patients with schizophrenia. The advantage of this research over other studies is that it examines three types of positive symptoms and duration of schizophrenia simultaneously, rather than separately, in relation to dichotic listening.
In other words, the model also suggests that patients with many positive symptoms are likely to have difficulty identifying sounds accurately, especially if the duration of the illness is long. Greater confidence can be placed in these results than other regression models because more than one indicator of the constructs of interest was used in the model. Identifying basic underlying latent variables positive symptoms and dichotic listening is another advantage over interpreting simple correlations among measured variables.
Because this is a cross-sectional model, it is unknown whether the language processing deficit existed before, at the same time, or after the onset of schizophrenia. Direction of cause in the model is, thus, unknown. Given that time was an important variable in this model, we can explore the advantages of longitudinal modeling, or measuring variables at more than one point in time.
That is, using the same measures of positive symptoms of schizophrenia and language processing taken at Time 1 and Time 2, path coefficients between the two latent variables at both points in time can be simultaneously examined to determine those that are significant. Previously a cross-lagged design would have been used whereby positive symptoms at Time 1 are correlated with language processing at Time 2.
This correlation is then compared to the correlation between language processing at Time 1 and positive symptoms at Time 2. This comparison does not account for autoregression, does not include latent variables, and cannot be easily applied to multiple time points or multiple variables.
An alternate method is multiple regression analysis whereby the positive symptoms of schizophrenia and language processing measured at Time 1 are used to predict language processing at Time 2.
The magnitude of the regression weights would indicate the strength of the relationship between schizophrenia and language processing while controlling for initial language processing. Although this takes autoregression into account and includes multiple measured variables, latent variables cannot be used, and reciprocal patterns impact of language processing on positive symptoms cannot be examined. A second example of SEM is of a model in population health that depicts the relationship between childhood victimization and school achievement.
Beran and Lupart postulated that children who are targeted by acts of aggression from their peers may be at risk for poor achievement [ 31 ]. This argument is supported by Eccles' Expectancy-Value theory [ 32 ]. Accordingly, achievement involves the culture, socialization, and the environmental "fit" of schools for students.
When children are exposed to positive experiences within this environment they are likely to gain academic and social competence [ 33 ]. Exposure to aggressive initiations from peers, however, may reduce a child's sense of competence for interpersonal interactions.
Given that learning at school takes place in a social environment these harmful interactions may reduce learning behaviors such as volunteering answers and asking questions. Rather, children who are targeted may become discouraged and disengaged from peers and classroom learning [ 33 ].
All of these factors were simultaneously examined to determine the likelihood of targeted adolescents experiencing poor achievement. The theoretical SEM model is depicted in Figure 3. Achievement was measured by four report sources including the language arts and math teachers, who reported on performance in those subjects, and the parent and child's report of overall achievement. Victimization was measured by adolescents' reports of frequency of attack and threats received from peers as well as degree of discomfort they feel among their peers.
Victimization and achievement were used as latent variables in the model and were found to be mediated by disruptive behaviors and friendship experiences. This is shown by the arrows and coefficients whereby there is no arrow directly linking victimization with achievement.
Rather, harassment was related to friendships and conduct problems, indicating that adolescents who were harassed reported having few or no friends as shown by the negative sign and exhibited conduct problems. These factors were also related to achievement. These combined results suggest that adolescents who are targeted by their peers are at risk of experiencing poor school achievement if they exhibit disruptive behavior problems and poor peer interactions.
A third example applies SEM within the field of clinical epidemiology by examining how health nutrition behaviors can serve to reduce risk of illness within a senior population.
Specifically, Keller [ 35 ] examined behaviors that constitute risk of poor nutrition among seniors as part of a screening intervention. A measurement model of risk factors that constitute poor nutrition was developed a priori based on exploratory results from a previous study that identified four factors from 15 measured variables.
A total of 1, Canadian seniors were interviewed or self-administered 15 questions about eating behaviors that matched those used previously.
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