Chapter
2 From cause to correlation and back
2.1 Translating from causal to statistical models
2.5 Probability distributions
2.6 Probabilistic independence
2.8 The translation from causal models to observational models
2.9 Counterintuitive consequences and limitations of d-separation: conditioning on a causal child
2.10 Counterintuitive consequences and limitations of d-separation: conditioning due to selection bias
2.11 Counterintuitive consequences and limitations of d-separation: feedback loops and cyclic causal graphs
2.12 Counterintuitive consequences and limitations of d-separation: imposed conservation relationships
2.13 Counterintuitive consequences and limitations of d-separation: unfaithfulness
2.14 Counterintuitive consequences and limitations of d-separation: context-sensitive independence
2.15 The logic of causal inference
2.16 Statistical control is not always the same as physical control
2.17 A taste of things to come
3 Sewall Wright, path analysis and d-separation
3.2 Why Wright’s method of path analysis was ignored
3.4 Independence of d-separation statements
3.5 Testing for probabilistic independence
Case 1: X and Y are both normally distributed and any relationship between them is linear
Case 2: X and Y are continuous but not normally distributed and any relationship between them is only monotonic
Case 3: X and Y are continuous and any relationship between them is not even monotonic
3.6 Permutation tests of independence
3.8 Conditional independence
3.9 Spearman partial correlations
3.10 Seed production in St Lucie’s Cherry
3.11 Specific leaf area and leaf gas exchange
4 Path analysis and maximum likelihood
4.1 Testing path models using maximum likelihood
Step 1: Translate the hypothetical causal system into a path diagram
Step 2: Translate the causal model into an observational model in the form of a set of structural equations
Step 3: Derive the predicted variance and the covariance between each pair of variables in the model using covariance algebra
Step 4: Estimate the free parameters by minimising the difference between the observed and predicted variances and covariance
Step 5: Calculate the probability of having observed the measured minimum difference, assuming that the observed and…
Step 6: If the calculated probability is sufficiently small (say below 0.05) then one concludes that the model was wrong. If…
4.2 Decomposing effects in path diagrams
Ancestor–descendant relationships
4.3 Multiple regression expressed as a path model
4.4 Maximum likelihood estimation of the gas-exchange model
5 Measurement error and latent variables
5.1 Measurement error and the inferential tests
5.2 Measurement error and the estimation of path coefficients
5.4 The nature of latent variables
5.5 Horn dimensions in Bighorn Sheep
5.6 Body size in Bighorn Sheep
6 The structural equations model
6.1 Parameter identification
6.2 Structural underidentification with measurement models
6.3 Structural underidentification with structural models
6.4 Behaviour of the maximum likelihood chi-squared statistic with small sample sizes
6.5 Behaviour of the maximum likelihood chi-squared statistic with data that do not follow a multivariate normal distribution
Multivariate measures of skew and kurtosis
6.6 Solutions for modelling non-normally distributed variables
6.7 Alternative measures of ‘approximate’ fit
6.8 Bentler’s comparative fit index
6.9 Approximate fit measured by the root mean square error of approximation
6.10 An SEM analysis of the Bumpus House Sparrow data
7 Nested models and multilevel models
7.3 The dangers of hierarchically structured data
8 Exploration, discovery and equivalence
8.1 Hypothesis generation
8.2 Exploring hypothesis space
8.3 The shadow’s cause revisited
8.4 Obtaining the undirected dependency graph
8.5 The undirected dependency graph algorithm
8.6 Interpreting the undirected dependency graph
Directed versus undirected paths
8.7 Orienting edges in the undirected dependency graph using unshielded colliders assuming an acyclic causal structure
8.8 Orientation algorithm using unshielded colliders
8.9 Orienting edges in the undirected dependency graph using definite discriminating paths
8.10 The Causal Inference algorithm
8.12 Detecting latent variables
8.13 Vanishing Tetrad algorithm
8.14 Separating the message from the noise
8.15 The Causal Inference algorithm and sampling error
8.16 The Vanishing Tetrad algorithm and sampling variation
8.18 Orienting edges in the undirected dependency graph without assuming an acyclic causal structure
8.19 The Cyclic Causal Discovery algorithm
Cause and Correlation in Biology
：A User's Guide to Path Analysis, Structural Equations and Causal Inference
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