How to Test and Discover Cause-Effect Relationships in Biology using Path Analysis and Structural Equation Modeling
- Why is it important to test causal hypotheses? - What are the main methods and tools for doing so? H2: From cause to correlation and back - How to use correlation to infer causation - How to use causation to predict correlation - The limitations and assumptions of correlation analysis H2: Path analysis and structural equation modeling - What is path analysis and how does it work? - What is structural equation modeling and how does it extend path analysis? - How to use R software for path analysis and structural equation modeling H2: Measurement error and latent variables - What are measurement error and latent variables and how do they affect causal inference? - How to model measurement error and latent variables in path analysis and structural equation modeling - How to assess the reliability and validity of measurements H2: Multilevel models and multigroup models - What are multilevel models and multigroup models and why are they useful for causal inference? - How to fit multilevel models and multigroup models in R software - How to compare nested models and test hypotheses across groups or levels H2: Exploration, discovery and equivalence - How to explore data and generate causal hypotheses - How to discover causal relationships using graphical methods and d-separation - How to deal with equivalence classes and identify the most plausible causal model H1: Conclusion - A summary of the main points and takeaways from the article - A call to action for readers to download the book "Cause and Correlation in Biology" by Bill Shipley - A list of references for further reading H2: FAQs - Five frequently asked questions about cause and correlation in biology with brief answers # Article with HTML formatting Introduction
Biology is a science that studies the living world, from molecules to ecosystems. Biology is also a science that seeks to understand the causes of phenomena, such as why some species are more diverse than others, why some traits are inherited or not, or why some diseases are more prevalent or lethal than others.
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However, finding the causes of biological phenomena is not always easy. Often, we cannot perform randomized experiments to manipulate the potential causes and observe the effects. Instead, we have to rely on observational data, which may be affected by confounding factors, measurement errors, or hidden variables.
How can we test causal hypotheses using observational data? How can we distinguish between correlation and causation? How can we infer the direction, strength, and nature of causal relationships among variables?
In this article, we will introduce you to some of the main methods and tools for answering these questions. We will explain how you can use correlation analysis, path analysis, structural equation modeling, multilevel models, multigroup models, graphical methods, and d-separation to test and discover cause-effect relationships in biology. We will also show you how you can use R software, a free and powerful tool for statistical computing, to perform these analyses.
From cause to correlation and back
One of the most common ways to test causal hypotheses is to use correlation analysis. Correlation is a measure of how two variables are related to each other. For example, if we measure the height and weight of a sample of people, we may find that they are positively correlated: taller people tend to be heavier than shorter people.
But does this mean that height causes weight or vice versa? Not necessarily. Correlation does not imply causation. There may be other factors that influence both height and weight, such as genetics, nutrition, or exercise. These factors are called confounders or common causes. They can create a spurious correlation between two variables that are not causally related.
To infer causation from correlation, we need to rule out confounding factors. One way to do this is to use a randomized experiment, where we assign different values of one variable (the cause) to different groups of subjects and measure the other variable (the effect). For example, we can randomly assign different diets to different groups of people and measure their weight after a period of time. If we find that the diet groups differ in weight, we can conclude that diet causes weight.
However, randomized experiments are not always possible or ethical to perform in biology. For example, we cannot randomly assign different species to different habitats and measure their diversity. We have to use observational data, where we measure both the cause and the effect in a sample of subjects. In this case, we need to use statistical methods to control for confounding factors. For example, we can use regression analysis, where we model the effect as a function of the cause and other variables that may affect the effect. We can then estimate the causal effect of the cause by adjusting for the other variables.
Another way to test causal hypotheses is to use causation to predict correlation. If we have a causal model or theory that specifies how some variables are related to each other, we can derive predictions about how they should be correlated in data. For example, if we have a theory that says that smoking causes lung cancer, we can predict that smokers should have a higher risk of lung cancer than non-smokers.
To test these predictions, we need to collect data on the variables of interest and calculate their correlations. We can then compare the observed correlations with the predicted correlations and see if they match. If they do, we can say that the data are consistent with the causal model. If they don't, we can say that the data contradict the causal model.
However, correlation analysis has some limitations and assumptions that we need to be aware of. First, correlation is a symmetric measure: it does not tell us which variable is the cause and which is the effect. For example, if we find that height and weight are correlated, we cannot tell if height causes weight or weight causes height from correlation alone. We need to use other sources of information, such as prior knowledge, temporal order, or experimental manipulation, to infer causation from correlation.
Second, correlation is a linear measure: it only captures the degree of linear relationship between two variables. For example, if we find that height and weight are correlated, we cannot tell if the relationship is linear or nonlinear from correlation alone. We need to use other methods, such as scatterplots or nonlinear regression, to explore the shape of the relationship between two variables.
Third, correlation is a bivariate measure: it only captures the relationship between two variables at a time. For example, if we find that height and weight are correlated, we cannot tell if there are other variables that affect both height and weight from correlation alone. We need to use multivariate methods, such as multiple regression or path analysis, to account for the effects of other variables on the relationship between two variables.
Path analysis and structural equation modeling
One of the most powerful and versatile methods for testing and discovering cause-effect relationships in biology is path analysis. Path analysis is a method that allows us to represent a causal model as a set of equations and a diagram. The equations specify how each variable in the model is related to its causes and effects. The diagram shows how the variables are connected by arrows that indicate the direction and magnitude of causal effects.
For example, suppose we have a causal model that says that plant growth depends on light intensity, soil moisture, and nutrient availability. We can write this model as a set of equations:
y = b0 + b1x1 + b2x2 + b3x3 + e x1 = c0 + c1z + u1 x2 = d0 + d1z + u2 x3 = f0 + f1z + u3
where y is plant growth, x1 is light intensity, x2 is soil moisture, x3 is nutrient availability, z is elevation (a common cause of x1, x2, and x3), e is measurement error in y, u1 is measurement error in x1, u2 is measurement error in x2, u3 is measurement error in x3, b0 is the intercept of y, b1 is the effect of x1 on y, b2 is the effect of x2 on y, b3 is the effect of x3 on y, c0 is the intercept of x1, c1 is the effect of z on x1, d0 is the intercept of x2, d1 is the effect of z on x2, f0 is the intercept of x3, f1 is the effect of z on x3.
We can also draw this model as a diagram:
z --> x1 --> y
z --> x2 --> y ^ v z --> x3 --> y
The arrows in the diagram represent the causal effects of one variable on another. The coefficients in the equations represent the magnitude of these effects. The variables without arrows pointing to them are called exogenous variables. They are assumed to be independent of the other variables in the model. The variables with arrows pointing to them are called endogenous variables. They are assumed to be dependent on the other variables in the model.
Path analysis allows us to estimate the causal effects of each variable in the model using data. We can use a statistical technique called maximum likelihood estimation to find the values of the coefficients that best fit the data. We can also test hypotheses about the causal effects using statistical tests such as t-tests or chi-square tests.
For example, we can test if light intensity has a positive effect on plant growth, if soil moisture has a negative effect on plant growth, or if elevation has an indirect effect on plant growth through light intensity, soil moisture, and nutrient availability.
Path analysis also allows us to compare different causal models using data. We can use a statistical technique called likelihood ratio test to compare the fit of two nested models: one that is a special case of the other. For example, we can compare a model that includes all the causal effects with a model that excludes some of them. We can also use a statistical technique called information criteria to compare the fit of two non-nested models: one that is not a special case of the other. For example, we can compare a model that assumes a linear relationship between the variables with a model that assumes a nonlinear relationship.
Path analysis is a powerful method for testing and discovering cause-effect relationships in biology, but it also has some limitations and assumptions that we need to be aware of. First, path analysis requires that we have a causal model or theory to start with. We cannot use path analysis to discover causal relationships from data alone. We need to use other methods, such as exploration, discovery, and equivalence, to generate and evaluate causal hypotheses.
Second, path analysis requires that we have measured all the relevant variables in the model. If we have omitted some important variables, such as confounders or mediators, we may get biased or misleading estimates of the causal effects. We need to use other methods, such as measurement error and latent variables, to account for unobserved or imperfectly observed variables.
Third, path analysis requires that we have a large and representative sample of data. If we have a small or biased sample of data, we may get inaccurate or unreliable estimates of the causal effects. We need to use other methods, such as multilevel models and multigroup models, to account for the variability and heterogeneity of data across groups or levels.
Measurement error and latent variables
One of the challenges of testing and discovering cause-effect relationships in biology is that we often have to deal with unobserved or imperfectly observed variables. These variables can affect our causal inference in different ways.
One type of unobserved or imperfectly observed variable is measurement error. Measurement error is the difference between the true value and the observed value of a variable. For example, if we measure plant growth using a ruler, we may introduce some error due to human or instrument error.
Measurement error can bias our estimates of the causal effects if it affects both the cause and the effect variables. For example, if we measure light intensity and plant growth using noisy instruments, we may underestimate their correlation and their causal effect.
To account for measurement error, we need to model it explicitly in our path analysis or structural equation modeling. We can do this by adding an error term to each equation and an error variable to each diagram. For example, we can write this model as a set of equations:
y = b0 + b1x1 + b2x2 + b3x3 + e x1 = c0 + c1z + u1 x2 = d0 + d1z + u2 x3 = f0 + f1z + u3
where e is measurement error in y, u1 is measurement error in x1, u2 is measurement error in x2, u3 is measurement error in x3.
We can also draw this model as a diagram:
z --> x1 --> y ^ ^ v z --> x2 --> y ^ v z --> x3 --> y v v u1 e u2 e u3 e
The error variables are represented by circles in the diagram. They are assumed to be independent of each other and of the other variables in the model. They are also assumed to have a mean of zero and a variance that can be estimated from the data.
By modeling measurement error, we can obtain more accurate and reliable estimates of the causal effects. We can also assess the reliability and validity of our measurements. Reliability is the degree to which a measurement is consistent and free from error. Validity is the degree to which a measurement measures what it is supposed to measure.
Another type of unobserved or imperfectly observed variable is latent variable. Latent variable is a variable that is not directly measured but inferred from other variables. For example, if we want to measure plant health, we may not have a direct measure of it, but we may have some indicators of it, such as leaf color, leaf area, or chlorophyll content.
Latent variable can improve our causal inference if it represents a common cause or a common effect of the observed variables. For example, if plant health is a common cause of leaf color, leaf area, and chlorophyll content, we can use it to control for confounding factors. If plant health is a common effect of light intensity, soil moisture, and nutrient availability, we can use it to measure the overall effect of these factors.
To account for latent variable, we need to model it explicitly in our path analysis or structural equation modeling. We can do this by adding a latent variable to each equation and each diagram. For example, we can write this model as a set of equations:
y = b0 + b1x1 + b2x2 + b3x3 + e x1 = c0 + c1z + u1 x2 = d0 + d1z + u2 x3 = f0 + f1z + u3 y1 = g0 + g1y + v1 y2 = h0 + h1y + v2 y3 = i0 + i1y + v3
where y is plant health (a latent variable), y1 is leaf color (an indicator of y), y2 is leaf area (an indicator of y), y3 is chlorophyll content (an indicator of y), v1 is measurement error in y1, v2 is measurement error in y2, v3 is measurement error in y3, g0 is the intercept of y1, g1 is the effect of y on y1, h0 is the intercept of y2, h1 is the effect of y on y2, i0 is the intercept of y3, i1 is the effect of y on y3.
We can also draw this model as a diagram:
z --> x1 --> y --> y1 ^ ^ ^ v z --> x2 --> y --> y2 ^ ^ v z --> x3 --> y --> y3 v v v u1 e v1 u2 e v2 u3 e v3
The latent variable is represented by an ellipse in the diagram. It is assumed to be dependent on its causes and effects and independent of its indicators' errors. It is also assumed to have a mean and a variance that can be estimated from the data.
By modeling latent variable, we can obtain more comprehensive and meaningful estimates of the causal effects. We can also assess the validity and reliability of our latent variable. Validity is the degree to which the latent variable reflects the underlying construct that it is supposed to measure. Reliability is the degree to which the indicators measure the latent variable consistently and accurately.
Multilevel models and multigroup models
Another challenge of testing and discovering cause-effect relationships in biology is that we often have to deal with data that are not homogeneous or independent. These data can affect our causal inference in different ways.
: the plant level and the plot level. The data at the plant level are nested within the data at the plot level.
Multilevel data can improve our causal inference if we account for the variability and heterogeneity of data across levels or groups. For example, if we want to estimate the effect of light intensity on plant growth, we need to account for the fact that light intensity may vary across plots and that plants within the same plot may be more similar to each other than plants across plots.
To account for multilevel data, we need to use multilevel models or hierarchical models in our path analysis or structural equation modeling. We can do this by adding random effects or variance components to each equation and each diagram. For example, we can write this model as a set of equations:
yij = b0 + b1x1ij + b2x2ij + b3x3ij + u0j + eij x1ij = c0 + c1zj + u1j x2ij = d0 + d1zj + u2j x3ij = f0 + f1zj + u3j
where yij is plant growth of plant i in plot j, x1ij is light intensity of plant i in plot j, x2ij is soil moisture of plant i in plot j, x3ij is nutrient availability of plant i in plot j, zj is elevation of plot j, eij is measurement error in yij, u0j is random effect of plot j on yij, u1j is measurement error in x1ij, u2j is measurement error in x2ij, u3j is measurement error in x3ij.
We can also draw this model as a diagram:
z --> x1 --> y ^ ^ v z --> x2 --> y ^ v z --> x3 --> y v v u1 u0 u2 e u3 e
The random effects are represented by circles with a subscript j in the diagram. They are assumed to be independent of each other and of the other variables in the model. They are also assumed to have a mean of zero and a variance that can be estimated from the data.
By using multilevel models, we can obtain more precise and generalizable estimates of the causal effects. We can also test hypotheses about the variability and heterogeneity of the causal effects across levels or groups. For example, we can test if the effect of light intensity on plant growth varies across plots or if it is constant across plots.
Another type of data that are not homogeneous or independent are multigroup data. Multigroup data are data that are divided into subgroups based on some criteria. For example, if we measure plant growth in different seasons, we may have data for four groups: spring, summer, autumn, and winter. The data for each group are independent of the data for other groups.
Multigroup data can improve our causal inference if we account for the differences and similarities of data across groups. For example, if we want to estimate the effect of light intensity on plant growth, we need to account for the fact that light intensity may differ across seasons and that plant growth may