## Abstract

This is one of a series of monographs on research design and analysis. The purpose of this article is to describe a set of statistical procedures or techniques used to develop and test structural models that characterize the relationships and interrelationships between a group of concepts and variables. These procedures include multiple regression, exploratory and confirmatory factor analysis, path analysis, and structural equation modeling. The article describes the purpose of each of these procedures and how they relate to and build on one another. It also covers the different types of variables examined, including the distinction between endogenous, exogenous, and mediating variables, along with the distinction between measured and unmeasured (or latent) variables. Each procedure results in a set of statistical estimates, and the article presents the interpretation of these estimates, including regression coefficients (standardized and unstandardized), path coefficients, factor loadings, and coefficients of determination (or

*R*^{2}values). The article presents examples of how each procedure has been used in practice, along with additional resources for readers who wish to learn more.## Keywords

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Research Snapshot

**Research Question:**What are the key statistical procedures used to develop and test structural models and how are these procedures used in practice?

**Key Findings:**Researchers use multiple regression, exploratory and confirmatory factor analysis, path analysis, and structural equation modeling to develop and test structural models characterizing the relationships and interrelationships between concepts and variables. The structural models may include endogenous, exogenous, and mediating variables, and the variables may be measured or unmeasured (latent). Researchers use these procedures to examine which exogenous or mediating variables are most strongly related to the endogenous, or outcome, variables, although these procedures alone cannot assess cause-and-effect relationships.

*Journal of the Academy of Nutrition and Dietetics*intended to describe path analysis (PA), structural equation modeling (SEM), and related methods. Theories about nutritional concepts are usually more complicated than the relationship between 2 variables. Think of all the variables and concepts associated with the theory of how coronary heart disease develops in individuals. What can be done statistically to assess the relationships and interrelationships between a larger number of concepts and variables? The purpose of this article was to describe and explain statistical procedures that can be used to do this kind of assessment, including multiple regression, exploratory factor analysis (EFA), confirmatory factor analysis (CFA), PA, and SEM. It will describe the purposes of these procedures and address a variety of concepts related to the procedures. Although we will not describe in detail the actual mathematics and statistical analysis supporting the procedures, we will provide citations that cover those methods and list the most common software packages used to conduct these analyses. Figure 1 defines key concepts examined in this article. Additional resources found in the Summary section of this article are used as the theoretical basis for this article.

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Figure 1Definitions of commonly used terms in path analysis and structural equation modeling.

Term | Definition |
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Coefficient of determination | Statistical value indicating the proportion of the variation in the endogenous (dependent) variable in a model that is explained by the model’s exogenous (independent) variables. The coefficient of determination can be used as a goodness-of-fit statistic. Also known as the R^{2}. |

Confirmatory factor analysis (CFA) | Statistical procedure used to measure the relationships between a set of measured variables and a smaller number of unmeasured, latent variables, referred to as “factors.” Researchers typically use CFA to test an existing theory that identifies a specific set of factors along with the measured variables that contribute to each factor. |

Endogenous variable | Variable included in a multiple regression or related model that is influenced or affected by other (exogenous) variables included in the model. Also referred to as a dependent variable or outcome. Endogenous variables are those that are predicted by a model’s exogenous variables. |

Exogenous variable | Variable included in a multiple regression or related model that influences or affects another (endogenous) variable included in the model. Also referred to as an independent variable. Exogenous variables are those that attempt to predict outcome variables. |

Exploratory factor analysis (EFA) | Statistical procedure used to explore the relationships between a set of measured variables and a smaller number of unmeasured, latent variables, referred to as factors. Researchers may use EFA as a data reduction technique and typically do not have a strong theory about the number of underlying factors represented in a set of measured variables or about which measured variables contribute to each factor. |

Factor loading | Weight given to a particular measured variable in constructing a factor, representing the relationship between the measured variable and factor. Factor loadings, also known as structural coefficients, are determined by CFA or EFA and have values ranging from –1 to 1. Measured variables with positive factor loadings are positively correlated with the factor; those with negative values are negatively correlated with the factor. |

Goodness-of-fit parameter | A statistical value that measures how well a particular data set fits a model (based on an underlying theory) that is attempting to explain the data. It is used to determine the quality of the model, and hence, the quality of the underlying theory. The coefficient of determination, or R^{2} value, is an example of a goodness-of-fit parameter. |

Latent variable (factor) | A variable that represents an underlying concept but that is not directly observed or measured. Researchers indirectly infer the presence of latent variables and their values based on a set of measured variables examined through statistical analyses, such as factor analysis or structural equation modeling. Latent variables are sometimes referred to as factors or principal components. |

Mediating variable | Variable included in path analysis and structural equation modeling that serves as both an endogenous and exogenous variable in the model. In other words, a mediating variable is both affected by 1 or more exogenous variables and affects 1 or more endogenous variables. It is said to mediate 1 relationship between the exogenous variables that affect it and the endogenous variables it affects. |

Multiple regression analysis | A statistical procedures that aims to examine the relationship between an endogenous (or outcome) variable and a set of exogenous (or independent) variables. The procedure generates estimates of regression coefficients that indicate how changes in each exogenous variable would be predicted to be associated with changes in the endogenous variable. The coefficient of determination from multiple regression measures the extent to which the exogenous variables in the regression explain variation in the endogenous variable. |

Path analysis | A statistical procedure that examines the relationships and interrelationships between a set of measured variables that may include exogenous, mediating, and endogenous variables. Path analysis produces path coefficients that characterize the associations between variables and a set of coefficients of determination that indicate the extent to which variation in the endogenous variables is explained by the exogenous and mediating variables in the model. |

Path coefficient | Numerical value representing the association between 2 variables in a structural model as estimated by path analysis or structural equation modeling. Path coefficients are analogous to standardized regression coefficients in multiple regression, with possible values ranging between –1 and +1. Values further away from 0 represent stronger (positive or negative) relationships. |

Regression coefficient (standardized or unstandardized) | Numerical value representing the relationship between an exogenous variable and an endogenous variable in a multiple regression model. Unstandardized regression coefficients represent the expected change in the endogenous associated with each 1-unit change in the exogenous variable. Standardized regression coefficients, which are analogous to path coefficients, represent the strength of the relationship and range between –1 and +1. |

Statistical error | Numerical values that represent factors affect endogenous variables that are not represented by any of a model’s exogenous or mediating variables. Also known as a residual. A random statistical error or residual could reflect systematic factors that are difficult to observe or measure, or transitory factors whose influence on the endogenous variable are difficult to foresee. The influence of random statistical error on variation in the endogenous variable is the unexplained portion of this variation. |

Structural equation modeling | A statistical procedure that examines the relationships and interrelationships between a set of measured and latent variables that may include exogenous, mediating, and endogenous variables. Similar to path analysis except that it allows for the inclusion of latent variables in the structural model being estimated. Produces path coefficients that characterize associations between variables along with coefficients of determination that indicate the extent to which variation in the model’s endogenous variables is explained by exogenous and mediating variables. |

Structure coefficient | Value produced by structural equation modeling or factor analysis that represents the relationship between a measured variable and unmeasured latent variable, or factor. This value, also known as a factor loading, has a value ranging from –1 to 1. Measured variables with positive structure coefficients are positively correlated with the factor; those with negative structure coefficients are negatively correlated with the factor. |

## Multiple Regression

All of the statistical procedures discussed in this article derive from multiple regression. Multiple regression is a statistical procedure that allows one to analyze the relationships of multiple variables (referred to as independent or exogenous variables) to an outcome variable (known as a dependent or endogenous variable). Exogenous variables are those that attempt to predict the outcome variables. Endogenous variables are those that are predicted. These variable terms will also be used when discussing the other statistical procedures in this article.

As illustrated in Figure 2, suppose that a nutrition researcher wanted to examine the relationships between some measures of diet quality, aerobic fitness, muscular strength, age, and sex (exogenous variables) and percent body fat (endogenous variable). Multiple regression can be used to assess how well the exogenous variables—as a group—predict or explain the overall variation in percent body fat among individuals in a sample. The statistical parameter, called the “coefficient of determination” (abbreviated as

*R*^{2}), indicates the explanatory level of the overall model and ranges from 0 to 1. The*R*^{2}value can be interpreted as the percentage of the overall variation in the endogenous variable that is explained by the set of exogenous variables in the regression model. Suppose the*R*^{2}= 0.40 for the model that examines the relationship between the various factors listed above and percent body fat. This means that 40% of the variation in percent body fat can be explained by the exogenous variables in total.Multiple regression also reveals information about how much each exogenous variable contributes to explaining variation in percent body fat. Software packages will calculate both unstandardized and standardized (β) regression coefficients for each exogenous variable, numerical values that look at how each variable relates to the outcome variable. An unstandardized coefficient can take on any value and indicates the amount of change in the outcome or endogenous variable associated with a 1-unit change in an exogenous variable, while holding the other exogenous variables constant. For example, in school-aged children, an unstandardized coefficient on the variable age (measured in years) of 0.15 would indicate that for each year increase in age there is a 0.15 percentage point increase in percent body fat, holding constant variables such as hours of screen time and hours of physical activity. If age were measured in months, an unstandardized coefficient of 0.15 would indicate that for each year in age there is a 1.8 percentage point (12 × 0.15) increase in percent body fat. However, because different exogenous variables have different units of measurement, these unstandardized coefficients do not allow us to easily compare how much each exogenous variable affects the outcome. Standardized (β) regression coefficients are calculated to do this by removing variation in the units of measurement (ie, by standardizing the way that variables’ values are interpreted) and allowing comparison of exogenous variable contributions. Higher coefficients can be compared with the lower ones to see which factors are most important in the model. These standardized coefficients range between –1 and 1 and are used in the examples throughout this article. For the example in Figure 2, suppose the standardized (β) regression coefficients were –.25, –.26, –.21, .45, and .47 for diet quality, aerobic fitness, muscular strength, age, and biological sex, respectively. This indicates that the variables that have the strongest relationships with percent body fat within the regression, which holds constant other factors, are age and sex. In diagrams such as Figure 2, these coefficients are presented on the arrows. The

*R*^{2}value is presented above the endogenous variable box. Note that all of the variables are presented in rectangles. This indicates that all of the variables are measured directly.Multiple regression is a useful way to explore simple relationships between multiple exogenous variables and a single outcome (endogenous). However, it is laborious to use multiple regression to attempt to look at more complex relationships between variables, such as when a theory indicates the presence of mediating variables. Mediating variables are affected by 1 or more exogenous variables, but they also affect an outcome variable. In a more complex model of body fat than shown in Figure 2, for example, the theory may indicate that physical exercise and diet are mediating variables that mediate the relationship between age and percent body fat—older individuals might exercise less and change their diets, which in turn affects their percent body fat. Other statistical techniques derived from multiple regression can be used to examine these more complex relationships, such as those involving mediating variables.

A good example of the use of multiple regression published in the exploring the relationship between adolescent sugar-sweetened beverage (SSB) intake and adolescent demographic, intrapersonal, interpersonal, and home availability variables. Standardized β coefficients and

*Journal*is the study conducted by Yuhas and colleagues^{7}

*R*^{2}values are presented, and the authors found that home and parental factors had the greatest association with SSB intake. Exogenous variables, such as male sex, non-Hispanic Black vs non-Hispanic White, parents’ permission to consume SSBs on bad days, and availability of SSBs at home were associated with higher SSB intake.## EFA

Sometimes researchers face situations in which they have data for several measured variables and they want to determine how these variables organize into certain unidentified or unmeasured concepts. In that situation, when there are several measured variables and there is an intention to group them based on underlying concepts, EFA can be used to estimate relationships between the measured variables and unmeasured underlying concepts. Suppose you have a dietary questionnaire to assess dietary behaviors. The questionnaire is comprehensive with various categories of questions. It would be valuable to see whether the data from various questions could be grouped into larger conceptual categories. These conceptual categories are called latent variables, or factors, hence the reference to “factor” in EFA (

*factor*will be used for the rest of the discussion). EFA is used to determine which questions (variables) group into which factors. For instance, 5 questions might group into a factor that might be called cooking practices, another 3 as shopping practices, and so on. EFA can also be used to shorten a long questionnaire and make it more efficient. Suppose 5 questions group as cooking practices. Three of the questions may have the strongest relationship to the underlying concept of cooking practices, which would allow the elimination of 2 questions in future iterations of the survey.Figure 3 represents an example of how EFA might be used in an analysis of body inflammation. Please note that the unmeasured factors are presented in ovals and the measured variables in rectangles. Arrows are used to show potential relationships. A conceptual model analyzed using EFA may also include statistical error terms (described in the next section), but they are not shown here for simplicity. Suppose a questionnaire is drafted to collect information on a wide range of possible factors that are related to body inflammation. Each of the individual questions may be related to a broader set of underlying lifestyle concepts that are hypothesized to influence body inflammation. However, these concepts—diet, smoking behavior, alcohol consumption, and exercise—are not directly measured themselves and are unspecified in the beginning of the analysis. In EFA, these variables will be measured and factors or grouping determined in the data analysis.

A sample of individuals is chosen to whom the questionnaire is administered. The data from the questionnaire will be used to conduct EFA. The items in Figure 3 in the rectangles are associated with questions on the questionnaire, so they are measured variables. EFA is used to group questions into factors. These factors are not measured directly. They are created based on the analysis of gathered data examining how questions and measured variables can be grouped into conceptual ideas. Figure 3 proposes these groupings as a result of EFA statistical analysis.

In the data analysis for EFA, the first step is to look at a table called a correlation matrix, which shows the Pearson correlation coefficients between each of the variables included in the analysis. Table 1 illustrates a portion of the correlation matrix for the example. The Pearson correlation coefficients for the relationships among dietary variables, as well as those for the relationships among smoking variables are high, but for relationships between the dietary variables and smoking variables they are low. Variables that are highly correlated to one another are likely to contribute to the same factor. The next step in EFA is called data extraction. In this step, related variables are grouped into categories, which are called factors. For example, variables related to the concept of depression are grouped together and those related to anxiety are grouped together in a study of a proposed emotion analysis tool. The final step in EFA is called factor rotation. There are different types of factor rotation, but each adjusts the way different variables contribute to each factor in order to enhance the extent to which the factors are distinct. The general goal of factor rotation is to make the factors easier to interpret.

Table 1Correlation matrix used in conducting exploratory factor analysis involving dietary- and smoking-related measures

Variable | Dietary fiber | Anthocyanins | Frequency of smoking | Smoking intensity | Second-hand smoke |
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Saturated fat | –0.80 | –0.87 | 0.35 | 0.33 | 0.20 |

Dietary fiber | 0.93 | –0.32 | –0.20 | –0.25 | |

Anthocyanins | –0.25 | –0.21 | –0.23 | ||

Frequency of smoking | 0.85 | 0.75 | |||

Smoking intensity | 0.72 |

a Model being estimated is the same model as shown in Figure 3.

Once the software conducts EFA, the results will be presented in a factor matrix that includes structure coefficients (sometimes called factor loadings) showing how each variable is related to each potential factor. Table 2 shows an excerpt of the factor matrix for the dietary and smoking variables. The structure coefficients can range between –1 and 1. The structure coefficients for diet-related variables are high for factor 1, and those for the smoking-related variables are low, close to 0. This indicates which variables contribute to this first factor and which do not, and these results would lead a researcher to conclude that the first factor represents key aspects of the quality of the individual’s diet. For factor 2, the structure coefficients are high for the smoking-related variables and low for the diet-related variables, indicating that the second factor represents the smoking concept. The structure coefficients are correlations between individual measured variables and unmeasured factors. Often in diagrams such as those presented in Figure 3, the structure coefficients will be associated with the arrows from the factor to the measured variable. The Factor Analysis application of the standard

*Statistical Package for the Social Sciences*(SPSS) software can be used for EFA data analysis. Other statistical packages, such as*SAS*software, could be used for this purpose as well.Table 2Factor matrix from exploratory factor analysis involving dietary- and smoking-related measures

Variable | Factor 1 (diet) | Factor 2 (smoking) |
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Saturated fat | 0.97 | 0.03 |

Dietary fiber | 0.95 | 0.01 |

Anthocyanins | 0.91 | 0.04 |

Frequency of smoking | 0.04 | 0.96 |

Smoking intensity | 0.04 | 0.99 |

Second-hand smoke | 0.02 | 0.98 |

a Model being estimated is the same model as shown in Figure 3.

EFA is sometimes referred to as a data reduction technique because it can be used to eliminate some variables from inclusion in a given factor when their relationship with the factor is not strong. Suppose alcohol type has a low structure coefficient for alcohol consumption. In this case, alcohol type might be excluded from the alcohol factor in the model. EFA is often used for scale or test construction when multiple questions are asked to determine how the questions group together in order to eliminate questions that have lower structure coefficients. With EFA, the researcher creates a measured version of the factor using the values of the structure coefficients to create a scale or index variable representing the factor. Thus, an assessment is created with subscales for measurement of a variety of concepts. It is referred to as “exploratory” factor analysis because the researcher does not predetermine which measured variables will contribute to each underlying factor. It does not even predetermine the number of factors to be created, although the researcher may have a strong hypothesis about this. Instead, the researcher determines the number of factors to retain in the analysis based on the extent to which the factors that are created explain the total variation in the questions or variables that are included in the EFA. In making this determination, the researcher may create a scree plot that shows the amount of variation explained by each created factor, as explained by Gleason and colleagues.

EFA is similar to principal components analysis, with a few key differences. Both approaches involve examining relationships between a set of measured variables and 1 or more unobserved latent variables (factors or components). However, EFA typically uses a reflective measurement structure, whereby the model assumes that a latent variable causes the measured variables to take on certain values and includes error terms or residuals. In Figure 3, for example, an individual’s decisions about their diet—what foods they eat—affects the values of the dietary intake measures, such as saturated fat. Principal components analysis, by contrast, uses a formative measurement structure, whereby the model assumes that the measured variables cause the latent variable to take on certain values and does not include error terms or residuals. For example, a researcher might use a formative measurement structure and principal components analysis to capture the effect that a set of variables that measure the foods individuals consume has on a latent variable reflecting the type of diet or dietary pattern they follow.

A useful example of EFA is an article by Vaughn and colleagues, who used it in the process of developing a survey to measure parents’ practices and children’s behaviors in families. After administering the survey with a sample of parents, the authors used EFA to determine which survey questions contributed in an important way to the factors that the authors determine to be essential to their analysis, and which survey questions did not contribute in an important way to these factors. It is useful to note that a survey item that did not contribute to one factor may have contributed in an important way to another factor, or the researchers may have determined that it was an important individual item that should be included on its own in the analysis. If not, the researchers discarded the item. In this example, the survey started with 124 items and was ultimately reduced to 86 items. The analysis led to the identification and construction of 24 latent variables, including 5 coercive control practices (16 items), 7 autonomy support practices (24 items), and 12 structure practices (46 items).

## CFA

Suppose a thorough review of the literature on the development of disordered eating is conducted and it warrants the development of a theory. A theory is then drafted with disordered eating among adolescents who identify as female as the underlying concept of interest. In this case, disordered eating is defined as a range of irregular eating behaviors that may or may not warrant a diagnosis of a specific eating disorder. Data are collected and measures are drafted that could relate to disordered eating, such as desire for thinness, body dissatisfaction, disruptive parental relationships, dieting behavior, emotional disturbance, and cognitive distortions. A structural model is then developed illustrating proposed interrelationships and relationships (Figure 4).

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Ideally, this model would be derived from a combination of a previously developed theory from a thorough review of the literature and EFA. The EFA can verify through data collection the relevance of the concepts and identify key variables contributing to those concepts. The result of this process of developing the theory or structural model based on the literature review and EFA would be a well-defined set of key unmeasured concepts (factors) and a specific set of variables (survey items) that contribute to each factor. CFA can then be used to test this structural model using data collected in a different context or from a different sample than was used for the EFA. For example, a questionnaire with the measured variables could be administered to a sample of female adolescents in a different location or at a different time than the original data. Alternatively, a researcher who wanted to conduct both an EFA and CFA with a single dataset could split the sample into 2 randomly determined groups, conduct EFA with one sample, and then conduct CFA with the second sample. In either case, CFA allows the testing of the fit of the proposed structural model to the collected data. It is used to validate a structural model.

Standard

*SPSS*software cannot be used for CFA. An add-on called*Amos*can be added to do the analysis. The initial data analysis looks at the overall fit of the structural model to the data. There are several goodness-of-fit statistics calculated by*SPSS Amos*. A researcher could also use the CALIS procedure in*SAS*or a software program designed for structural equation modeling, such as the lavaan package in*R*to conduct CFA.The structural model in Figure 4 is similar to Figure 3 in that ovals indicate unmeasured latent variables (factors) and the rectangles indicate measured variables. Figure 4 also adds the concept of random statistical errors (e), shown in circles, also known as the uniqueness or residual term, which reflect the fact that the value of a measured variable for a given sample member can have influences other than the latent variable it represents. For example, the response of an female adolescent to a survey item about body dissatisfaction could represent more than disordered eating, such as concerns about health related to health problems experienced by family members with similar body types or something as simple as the respondent’s mood on the day of the survey. It is noteworthy that the arrows are directed from the latent variable to the measured variables, indicating the model starts with the conceptual (latent) variable, which is assumed to influence the values of the measured variables. The numbers in boxes associated with the arrows are called “path coefficients.” The higher they are, the stronger the relationship between the latent variable (conceptual variable disordered eating) and the measured one. Values higher than .30 are considered significant relationships, but measured variables that best fit the latent variable have path coefficients higher than .60. As one can see the relationships between body dissatisfaction, emotional disturbance, and cognitive distortions and disordered eating are the strongest.

CFA is also useful to determine whether a structural model applies to groups with other characteristics. Initially, the CFA was solely done using data from a sample of female adolescents. It might be valuable to have adolescents who identify as male complete the measures and see the overall fit of the model to their data. This may lead to a modification of the model so that it is more appropriate for male adolescents.

As noted above, CFA is used to test theoretical models rather than to develop models.

Unlike the case with EFA, in CFA the model prespecifies the latent variables being examined and the measured variables that contribute to those latent variables. The data are used to measure the relative strength of these relationships, as well as how well the overall model explains the data. If the fit of the model to the data is not good, the model could be reassessed and changed, and the new model tested.

A good example of combining EFA with CFA is the construction and validation of the Chinese Preschoolers’ Caregivers’ Feeding Behavior Scale. Caregiver feeding behavior has been found to be associated with child weight status. The authors used EFA to develop the structural model that would determine the variables that would contribute to this scale. They used CFA to test and finalize the structural model and the feeding behavior scale that came from this model. This article demonstrates the process of developing a reliable and valid scale.

## PA

PA assesses the relationships and interrelationships between a series of measured variables. With PA, more complex relationships can be examined in which some variables can serve as both exogenous and endogenous variables. For example, PA allows the examination of mediating variables (variables that are affected by one variable in the model and go on to affect another). A basic illustration of a model that can be examined using PA with exogenous, mediating, and endogenous variables is the connection between vitamin D supplementation, serum levels of 25-hydroxycholecalciferol, and measured depression as a depression score on a validated scale (Figure 5). In this case, the supplement will only be effective if enough vitamin D is absorbed to reach blood threshold levels of 25-hydroxycholecalciferol in order that brain levels of vitamin D can affect brain function. In another example, Freedman and colleagues uses PA to examine a model in which the frequency of food shopping was the mediating variable between perceived control of healthy eating and Healthy Eating Index 2010 score. Finally, Figure 6 provides a useful hypothetical model to illustrate the value of PA, as discussed below. For a validated general model examining factors related to behavior use, see DeNicola and colleagues.

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As illustrated in Figure 6, relationships between variables in a structural model can be evaluated using PA. As mentioned previously, exogenous variables are those that influence other variables (eg, the external variables in Figure 6); endogenous variables are influenced by other variables in the model (eg, vegetarian or not); and mediating variables serve as both exogenous and endogenous variables (eg, social values about health and food). Single-direction arrows indicate potential cause-and-effect relationships. The curved 2-headed arrows connecting variables indicate potential correlations between the variables (eg, personality and spiritual preference). To use PA to estimate the relationships in a structural model, all of the variables in the model must be measured (ie, measurement tools must be available and used for all variables in the model).

Once data are collected on all variables in the model from a sample, PA tests the overall fit of the model; that is, how well the model explains the data that have been collected to test the model. When the fit is good, the exogenous and mediating variables in the model will do a good job of explaining the variance in the final endogenous variable (vegetarian or not). In addition, PA evaluates each of the relationships in the model and the explanatory power of each variable. Relationships between variables are presented as numerical standardized path coefficients that allows one to compare the strongest relationships between variables. These path coefficients are analogous to standardized regression coefficients in multiple regression models. They range from –1 to 1, with positive values representing a positive relationship and negative values a negative relationship. The closer to 1 or –1 a path coefficient is, the stronger the relationship it represents; coefficients of 0 or close to 0 indicate there is no relationship between the variables being examined. If you square the path coefficient it tells you the percent of the variation in the affected (endogenous or mediating) variable in the relationship that is explained by the (exogenous or mediating) variable that affects it. In addition,

*R*^{2}values are calculated for each endogenous variable, indicating the amount of its variance that is explained by the full set of exogenous and/or mediating variables that affect it. Finally, fit statistics are calculated indicating overall model fit to the data collected.It would be possible for a researcher to estimate all of the relationships in a structural model, such as that represented by Figure 6 using multiple regression. However, the advantage of using PA and a program such as

*SPSS Amos*to do the calculations over multiple regression is that*SPSS Amos*can calculate all parameters simultaneously for all paths. This sort of software can also calculate indirect effects that account for 2 path coefficients, along with appropriate estimates of the SEs of these indirect effects that allow for statistical testing when samples are very large (with smaller samples, bootstrapping or Monte Carlo methods are recommended). The software also produces measures of the overall fit of the model to the data. With multiple regression, separate analyses must be done for each path in the model, SEs of indirect effects are not straightforward to calculate, and no fit statistics are produced.Figure 7 presents a simplified version of a theory called the “Theory of Planned Behavior” with the standardized path coefficients shown adjacent to each path and the The

*R*^{2}values provided at the top of the figure for both intention and behavior.^{18}

*R*^{2}of 0.51 for intention means that the model explains 51% of the variation in intention. The*R*^{2}of 0.25 for behavior means that the model, as a whole, explains 25% of the variation in behavior. The strongest path coefficients are for the relationships between perceived behavioral control and intention and intention and behavior, meaning these are the strongest relationships between variables.PA starts with a theory (represented by the structural model) and the analysis provides a test of the theory; therefore, its purpose is for theory testing rather than theory development. With PA, a researcher can examine both direct and indirect relationships between measured variables. PA produces path coefficients that tell us the relationship between the variables in the model, but cannot typically be interpreted as causal relationships.

A useful example is a study conducted by Murphy and colleagues that tested a model examining the relationship between aerobic fitness, eating behavior, and physical activity to body composition in college students. They found that physical activity in the form of aerobic exercise was most effective compared with other variables in maintaining body composition.

## SEM

Whereas PA assesses the relationships and interrelationships between measured variables, SEM allows the evaluation of directional relationships and interrelationships between measured variables and conceptual unmeasured variables (latent variables). In other words, SEM is similar to PA, but allows researchers to include latent variables in the structural model that is being tested with the data. Figure 8 presents a nutrition example of an SEM model.

As illustrated in this model, stress, socioeconomic status, genetic propensity, and physical activity are latent variables (in ovals). All of the others are measured variables (in rectangles). Because SEM includes latent variables, their associated errors are included in the model and factored into the analysis (usually shown as circles but not included here). In addition, there are 2-headed arrows that indicate correlations between variables (eg, low-density lipoprotein and high blood pressure). SEM differs from CFA in that the structural model typically proposes causal relationships and can include mediating variables. In CFA, correlations between variables are examined, but these relationships are not necessarily causal. Because of this, the structural models examined with SEM are often more complex than those examined with CFA—to credibly account for all possible ways variables might affect one another the structural models in SEM tend to have multiple possible paths between variables. These models tend to include many different exogenous variables and endogenous variables, as well as mediating variables that can serve as both exogenous and endogenous variables (eg, large waist circumference).

*SPSS Amos*is an example of statistical software that can be used to conduct the data analysis. As with the other procedures, the software will determine the overall fit of the model to the data and present path coefficients that indicate the strength of relationships in the model. In Figure 8, the relationships that stress, large waist circumference, high alcohol intake, and high sodium/low potassium have with high blood pressure are all relatively strong. The software also generates

*R*

^{2}values (not shown) for endogenous variables, indicating the amount of variance in the data explained by that variable, as well as path coefficients for mediated pathways (eg, socioeconomic status, high alcohol intake, and high blood pressure). Once the data analysis is complete, investigators will often modify the model to eliminate those variables with very low path coefficients and then they will retest the model.

SEM combines elements of multiple regression, CFA, and PA. It examines the fit of complex models that include both measured and latent variables and also allows the evaluation of relationships between latent variables. There are many similarities between SEM and PA. As with PA, SEM starts with a theory and uses data to test the theory (how well the model explains variation in the variables). It may include variables that serve as both endogenous and exogenous variables. It generates path coefficients interpreted in the same way as in PA. Direct and indirect relationships can be examined. There are also differences between SEM and PA. Unlike PA, the SEM model includes latent variables in addition to measured variables. In addition, the model includes error terms for each endogenous variable, representing factors not included in the model that affect those variables.

Zeidi and colleagues used SEM to examine psychosocial factors affecting the adoption of a healthy nutrition style in women with a history of gestational diabetes. Psychosocial factors were 8 measured variables with healthy nutrition style as a measured endogenous variable. A model was drafted proposing a connection between these 8 variables and healthy nutrition style. The model predicted 23% and 51% of measured intention to eat a healthy diet variance and nutrition style variance, respectively. Action self-efficacy and risk perception were the most important predictors of intention. In addition, planning and recovery self-efficacy significantly predicted a healthy nutrition style.

^{20}

## Limitations of These Statistical Procedures

It is important to understand that these procedures alone cannot assess cause-and-effect relationships. Although a structural model may depict cause-and-effect relationships in theory, these statistical procedures are designed primarily to assess relationships among measured and latent variables, which may or may not be causal. It is about model building and evaluating the strengths of models and theories. It is about refining models and looking at the variations in applications of the models with different populations and settings. To truly assess causal relationships, a researcher needs to design a study specifically for that purpose, such as a randomized controlled trial. However, as a result of procedures like PA and SEM with observational data, a researcher might identify particular relationships worthy of designing randomized controlled trials to test cause-and-effect relationships.

## Statistical Packages

There are various software packages that allow these statistical procedures to be completed. and

*SPSS Amos*,*LISREL*,^{21}

*Mplus*are the most common ones. Diagrams can be drawn using these packages and the sophisticated statistical analyses can be conducted.## Summary

In this article, a series of statistical techniques with the purpose of model building and testing have been outlined. Figure 9 summarizes key features of these techniques. The actual statistical analysis that underlies these techniques is beyond the scope of this article. Multiple regression and path analysis can be used to test models that include exclusively measured variables and assess mediation pathways. EFA is used to build models from measured variables derived from literature review leading to latent variables representing concepts derived from the measured variables. CFA is used to test proposed models for their fit with the data. It includes both measured and latent variables. Finally, SEM tests models and directional relationships between a mix of latent and measured variables and may include mediation pathways. The procedures are used to build assessments with subscales and test the facility of proposed models. These assessments and model may be modified to have more reliable and valid assessments and models that better explain data variance. For resources that include further information about these statistical techniques, please refer to Figure 10.

Figure 9Key features of statistical techniques used in examining structural models.

Feature of statistical technique | Exploratory factor analysis | Confirmatory factor analysis | Path analysis | Structural equation modeling |
---|---|---|---|---|

Main purpose is model building/development | ✓ | |||

Main purpose is model testing | ✓ | ✓ | ✓ | |

Includes endogenous (outcome) and exogenous variables | ✓ | ✓ | ✓ | ✓ |

May include mediating variables | ✓ | ✓ | ||

May include unmeasured (latent) variables | ✓ | ✓ | ✓ | |

Includes estimation of random statistical error terms | ✓ | ✓ | ||

Estimates do not necessarily represent causal relationships | ✓ | ✓ | ✓ | ✓ |

Figure 10Resources for more information about path analysis and structural equations modeling

Resources |

Applied Multivariate Research Design and Interpretation by Meyers and colleagues |

“Publishing nutrition research: A review of multivariate techniques-part 3: data reduction methods,” by Gleason and colleagues |

“Path analysis: An introduction and analysis of a decade of research” by Stage and colleagues |

“Finding our way: An introduction to path analysis” by Streiner |

“Building a better model: An introduction to structural equation modelling” by Streiner |

“Understanding power and rules of thumb for determining sample sizes” by VanVoorhis and Morgan |

## Acknowledgements

The authors thank Mary Naglak, PhD, MMSc, RD, LDN and Linda Snetselaar, PhD, RD, LD, FAND for insightful comments on a draft of this article. The authors also thank all members of the

*Journal of the Academy of Nutrition and Dietetics*statistics team for useful discussions.### Author Contributions

J. E. Harris and P. M. Gleason contributed equally to writing the initial draft of the article and on subsequent revisions after receiving comments.

## Supplementary Data

- Supplementary Figures

## References

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## Biography

J. E. Harris is a professor and chair of the department of nutrition, West Chester University, West Chester, PA.

P. M. Gleason is a senior fellow, Mathematica Policy Research, Geneva, NY.

## Article Info

### Publication History

Published online: July 18, 2022

Accepted:
July 14,
2022

Received:
December 9,
2021

### Footnotes

**STATEMENT OF POTENTIAL CONFLICT OF INTEREST** No potential conflict of interest was reported by the authors.

**FUNDING/SUPPORT** There is no funding to disclose.

### Identification

### Copyright

© 2022 by the Academy of Nutrition and Dietetics.