The aims of the first edition of Matrix Structural Analysis were to place proper emphasis on the methods of matrix structural analysis used in practice and to lay the groundwork for more advanced subject matter. This extensively revised Second Edition accounts for changes in practice that have taken place in the intervening twenty years. It incorporates advances in the science and art of analysis that are suitable for application now, and will be of increasing importance in the years ahead. It is written to meet the needs of both the present and the coming generation of structural engineers.
RONALD D. ZIEMIAN, PhD, Professor of Civil Engineering, Bucknell University. He is engaged in the development of structural analysis software and active as a researcher and consulting engineer. He is the editor of the reference book Guide to Stability Design Criteria for Metal Structures. He is the recipient of the American Society of Civil Engineers' premier award, the Norman Medal, for a paper based on the inelastic behaviour of steel structures, and ASCE's Shortridge Hardesty Award.
SEM is a combination of two statistical methods: confirmatory factor analysis and path analysis. Confirmatory factor analysis, which originated in psychometrics, has an objective to estimate the latent psychological traits, such as attitude and satisfaction (Galton 1888; Pearson and Lee 1903; Spearman 1904). Path analysis, on the other hand, had its beginning in biometrics and aimed to find the causal relationship among variables by creating a path diagram (Wright 1918, 1920, 1921). The path analysis in earlier econometrics was presented with simultaneous equations (Haavelmo 1943). In the early 1970s, SEM combined the two aforementioned methods (Joreskog 1969, 1970, 1978; Joreskog and Goldberger 1975) and became popular in many fields, such as social science, business, medical and health science, and natural science.
Path analysis was developed to quantify the relationships among multiple variables (Wright 1918, 1920, 1921). It was the early name for SEM before there were latent variables, and was very powerful in testing and developing the structural hypothesis with both indirect and direct causal effects. However, the two effects have recently been synonymized. Path analysis can explain the causal relationships among variables. A common function of path analysis is mediation, which assumes that a variable can influence an outcome directly and indirectly through another variable. For example, light intensity (PAR), air temperature (Ta), and aboveground temperature (Ts) can influence net ecosystem exchange (NEE) indirectly through respiration (Re); yet PAR and Ts can influence Re directly (Fig. 1, Shao et al. 2016). Santibáñez-Andrade et al. (2015) applied mediation to evaluate the direct and indirect causes of degradation in the forests of the Magdalena river basin adjacent to Mexico City. The study sought to integrate abiotic controls and disturbance pressure with ecosystem conservation indicators to develop strategies in preserving biodiversity. In another study with SEM, a 23-year field experiment on a plant community in an Alaskan floodplain, found that alder inhibited spruce growth in the drier site directly, while at the wetter site it inhibited growth indirectly through effects mediated by competition with other vegetation and herbivory (Chapin et al. 2016).
The basic usage of structural equation modeling (SEM) in path analysis with mediation. The causal relationships include both indirect and direct effects, where Re is a mediator that intervenes with the causal relationships (modified from Shao et al. 2016). The acronyms in the models are photosynthetically active radiation (PAR), air temperature (Ta), soil temperature (Ts), net ecosystem exchange (NEE), and respiration (Re)
Confirmatory factor analysis (CFA) is the method for measuring latent variables (Hoyle 1995; 2011; Kline 2010; Byrne 2013). It extracts the latent construct from other variables and shares the most variance with related variables. For example, abiotic stress as a latent variable is measured by the observation of soil changes (i.e., soil salinity, organic matter, flooding height; Fig. 2, Grace et al. 2010). Confirmatory factor analysis estimates latent variables based on the correlated variations of the dataset (e.g., association, causal relationship) and can reduce the data dimensions, standardize the scale of multiple indicators, and account for the correlations inherent in the dataset (Byrne 2013). Therefore, to postulate a latent variable, one should be concerned about the reason to use a latent variable. In the abiotic stress example given above, community stress and disturbance are latent variables that account for the correlation in the dataset. Shao et al. (2015) applied CFA to constrict the soil-nutrition features to one variable that accounted for soil organic carbon, litter total nitrogen, and carbon-to-nitrogen ratio. Also, Capmouteres and Anand (2016) defined the habitat function as an environmental indicator that explained both plant cover and native bird abundance for the forest ecosystems by using CFA.
SEM is composed of the measurement model and the structural model. A measurement model measures the latent variables or composite variables (Hoyle 1995, 2011; Kline 2010), while the structural model tests all the hypothetical dependencies based on path analysis (Hoyle 1995, 2011; Kline 2010).
In addition to LGC, SEM can be incorporated into a time series analysis (e.g., autoregressive integrated moving average model). For example, Almaraz (2005) applied a time series SEM to predict the population growth of the purple heron (Ardea purpurea). The moving average process was used as a matrix of time-based weights for analyzing the seasonal changes and autocorrelations.
SEM requires measurement models to be based on prior knowledge so that latent variables can be interpreted correctly (Bentler and Chou 1987). 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.
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.
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 Framework Method for the management and analysis of qualitative data has been used since the 1980s . The method originated in large-scale social policy research but is becoming an increasingly popular approach in medical and health research; however, there is some confusion about its potential application and limitations. In this article we discuss when it is appropriate to use the Framework Method and how it compares to other qualitative analysis methods. In particular, we explore how it can be used in multi-disciplinary health research teams. Multi-disciplinary and mixed methods studies are becoming increasingly commonplace in applied health research. As well as disciplines familiar with qualitative research, such as nursing, psychology and sociology, teams often include epidemiologists, health economists, management scientists and others. Furthermore, applied health research often has clinical representation and, increasingly, patient and public involvement . We argue that while leadership is undoubtedly required from an experienced qualitative methodologist, non-specialists from the wider team can and should be involved in the analysis process. We then present a step-by-step guide to the application of the Framework Method, illustrated using a worked example (See Additional File 1) from a published study  to illustrate the main stages of the process. Technical terms are included in the glossary (below). Finally, we discuss the strengths and limitations of the approach.
There are a number of approaches to qualitative data analysis, including those that pay close attention to language and how it is being used in social interaction such as discourse analysis  and ethnomethodology ; those that are concerned with experience, meaning and language such as phenomenology [17, 18] and narrative methods ; and those that seek to develop theory derived from data through a set of procedures and interconnected stages such as Grounded Theory [20, 21]. Many of these approaches are associated with specific disciplines and are underpinned by philosophical ideas which shape the process of analysis . The Framework Method, however, is not aligned with a particular epistemological, philosophical, or theoretical approach. Rather it is a flexible tool that can be adapted for use with many qualitative approaches that aim to generate themes. 2b1af7f3a8