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Video transcript - [Voiceover] Let's now introduce ourselves to the idea of a differential equation.
And as we'll see, differential equations are super useful for modeling and simulating phenomena and understanding how they operate. But we'll get into that later. For now let's just think about or at least look at what a differential equation actually is. So if I were to write, so let's see here is an example of differential equation, if I were to write that the second derivative of y plus two times the first derivative of y is equal to three times y, this right over here is a differential equation.
Another way we could write it if we said that y is a function of x, we could write this in function notation. We could write the second derivative of our function with respect to x plus two times the first derivative of our function is equal to three times our function. Or if we wanted to use the Leibniz notation, we could also write, the second derivative of y with respect to x plus two times the first derivative of y with respect to x is equal to three times y. All three of these equations are really representing the same thing, they're saying OK, can I find functions where the second derivative of the function plus two times the first derivative of the function is equal to three times the function itself.
So just to be clear, these are all essentially saying the same thing. And you might have just caught from how I described it that the solution to a differential equation is a function, or a class of functions. It's not just a value or a set of values.
So the solution here, so the solution to a differential equation is a function, or a set of functions, or a class of functions. It's important to contrast this relative to a traditional equation.
So let me write that down. So a traditional equation, maybe I shouldn't say traditional equation, differential equations have been around for a while. So let me write this as maybe an algebraic equation that you're familiar with. An algebraic equation might look something like, and I'll just write up a simple quadratic.
Say x squared plus three x plus two is equal to zero. The solutions to this algebraic equation are going to be numbers, or a set of numbers. We can solve this, it's going to be x plus two times x plus one is equal to zero. So x could be equal to negative two or x could be equal to negative one. The solutions here are numbers, or a set of values that satisfy the equation. Here it's a relationship between a function and its derivatives. And so the solutions, or the solution, is going to be a function or a set of functions. Now let's make that a little more tangible. What would a solution to something like any of these three, which really represent the same thing, what would a solution actually look like?
Actually let me move this over a little bit. 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. Variables such as type and frequency of food eaten created the latent factor food intake; appetite and weight change loaded on the factor adaptation; swallowing and chewing ability loaded on the factor physiologic; and cooking and shopping ability formed the variable functional.
These factors were then loaded onto a higher level factor nutritional risk. Factor loadings varied between. It was, thus, concluded that these factors provide a comprehensive and valid indicator of nutritional risk for seniors. This framework was developed from previous research and presents confirmatory evidence for the nutrition behaviors used in the model of nutrition risk. SEM is a set of statistical methods that allows researchers to test hypotheses based on multiple constructs that may be indirectly or directly related for both linear and nonlinear models [ 36 ].
It is distinguished from other types of analyses in its ability to examine many relationships while simultaneously partialing out measurement error. It can also examine correlated measurement error to determine to what degree unknown factors influence shared error among variables - which may affect the estimated parameters of the model [ 37 ]. It also handles missing data well by fitting raw data instead of summary statistics.
SEM, in addition, can be used to analyze dependent observations e. It can, furthermore, manage longitudinal designs such as time series and growth models. For example, Dahly, Adair, and Bollen [ 22 ] developed a longitudinal latent variable medical model showing that maternal characteristics during pregnancy predicted children's blood pressure and weight approximately 20 years later while controlling for child's birth weight.
Therefore, SEM can be used for a number of research designs. A distinct advantage of SEM over conventional multiple regression analyses is that the former has greater statistical power probability of rejecting a false null hypothesis than does the latter. They were able to show that SEM statistics were more sensitive to changes in toxin exposure than were regression statistics, which resulted in estimates of lower, or safer, exposure levels than did the regression analyses.
SEM has sometimes been referred to as causal modeling; however, caution must be taken when interpreting SEM results as such. Several conditions are deemed necessary, but not sufficient for causation to be determined.
There must be an empirical association between the variables - they are significantly correlated. A common cause of the two variables has been ruled out, and the two variables have a theoretical connection. Also, one variable precedes the other, and if the preceding variable changes, the outcome variable also changes and not vice versa.
These requirements are unlikely to be satisfied; thus, causation cannot be definitively demonstrated. Rather, causal inferences are typically made from SEM results. Indeed, researchers argue that even when some of the conditions of causation are not fully met causal inference may still be justifiable [ 39 ]. As with any method, SEM has its limitations. Although a latent variable is a closer approximation of a construct than is a measured variable; it may not be a pure representation of the construct. Its variance may consist of, in addition to true variance of the measured variables, shared error between the measured variables.
Also, the advantage of simultaneous examination of multiple variables may be offset by the requirement for larger sample sizes for additional variables to derive a solution to the calculations. SEM cannot correct for weaknesses inherent in any type of study. Exploration of relationships among variables without a priori specification may result in statistical significance but have little theoretical significance. In addition, poor research planning, unreliable and invalid data, lack of theoretical guidance, and over interpretation of causal relationships can result in misleading conclusions.
With the development of SEM, medical researchers now have powerful analytic tools to examine complex causal models. It is superior over other correlational methods such as regression as multiple variables are analyzed simultaneously, and latent factors reduce measurement error. When used as an exploratory or confirmatory approach within good research design it yields information about the complex nature of disease and health behaviors.
It does so by examining both direct and indirect, and unidirectional and bidirectional relationships between measured and latent variables.