1. Structural Equation Modeling for Ecologists Using R
John Kitchener Sakaluk
University of Victoria
Department of Psychology

2. About These Workshop Materials

3. Why Are We Using R Because it is your favourite price (free)
Because it is increasingly popular
Because it can do virtually everything
EFA
CFA
SEM
IRT
LCA
Because it is becoming more user friendly
Because it is reproducible
Because it makes beautiful visualizations

4. Today’s Agenda Orienting you to R (importing data, using packages)
Why SEM, as an Ecologist?
Fundamentals of CFA
Advanced CFA
SEM and Multi-Group SEM

5. An Orientation to R/R-Studio

6. Orienting You to R: Anatomy of an R Script
Operator that tells R to save output from function (on right of =) into object (named on left of =)
name = function(options)
Each function has flexibility; we need to specify how we want it to perform a certain task
(e.g., for t-test, level of alpha?, one-tailed or two-tailed?, using what variables/data?, etc.,)
We store information (e.g., data, output, plots) in objects—we need to give objects a name
Need to specify a function--what are we trying to do/create?
(e.g., import data, perform an EFA, create a plot of some sort)

7. Orienting You to R: Anatomy of an R Script (example)
Save the output of the read.csv function (data importing) into a new object called SSSS.dat
Eco.dat= read.csv(file.choose())
There are a number of ways to “point” read.csv to a data file (e.g., an awful-looking file path).
The file.choose() option pulls up a navigation menu to make selecting data file *super* easy
The (arbitrary) name for our data file object; we will use it to refer to our data in later commands
The read.csv function is used to import data that is in a .csv file

8. Orienting You to R: Installing/Calling External Packages
install.packages (“package_name”)
Will automatically find/download/install external package on your computer.
Only need to do once, and then subsequent times will update.
library(package_name)
Tells R to make functions from an external package available for use.
Need to do this every time R restarts.

9. Check Out Section (1) of Script
See how comments/headings in R help to organize your code
Learn how to install/call packages, and request citation info to give developers credit
Import example data OR your own data

10. Why Use SEM?

11. Why Consider Using SEM? Multiple outcome variables
Modeling (vs. making) assumptions
Model comparison/constraint testing
Latent variables*

12. What Is a Latent Variable?: The Elephant and Blind Men Analogy
“It was six men of Indostan to learning much inclined, who went to see the Elephant (though all of them were blind) that each by observation might satisfy his mind … And so the men of Indostan disputed loud and long, each in his own opinion exceeding stiff and strong, though each was partly in the right And all were in the wrong”
John Godfrey Saxe

13. “Construct Space” of the Elephant+
4 legs
Grey
Trunk
Large
Tusks
Tail
Herbivore

14. A Hypothetical Latent Model of the Elephant Construct
Ψ11 = *1
λ11 = .92
λ51 = .35
Trunk
4 Legs
Trunk
Tail
Large …
.15
.88 𝜃1 𝜃5

15. Leaving the Analogical World: Confirmatory Factor Analysis
Abiotic
Stress
Ψ11 = *1
λ11 = .92
λ51 = .35
“Measurement Model”
Drought
Wind
Speed
Soil
Flooding
Temp.
Radiation …
.15
.88 𝜃1 𝜃5

16. Leaving the Analogical World: Structural Equation Modeling
“Structural Model”
Ψ11 = *1
Abiotic
Stress
Species
Count
???

17. Leaving the Analogical World: Structural Equation Modeling
Ψ11 = *1
Abiotic
Stress
Biodiversity
Ψ22 = *1
???

18. Grace et al. (2010)

19. Considerations for Ecologists Modeling Latent Variables (LVs)
Requires multiple indicators of the LVs
I.e., Collection of data from more observed variables
Why bother?
Theoretical precision
Increased statistical power for inferential tests of structural parameters
How do LVs facilitate this?

20. A Brief Foray into Classic Test Theory
Observed
Score
Variance
“True Score”
Variance for
LVs A, B, C…
(A) (B) (C)
Error Variance
(E) = +
Should covary with
other indicators of A, B, C…
Should not covary
with anything

21. Latent (Common) Factors Represent Shared Variance
Factor (A)
Shared Variance between
A1-A5 A1 A2 A3 A4 A5
B + C + E
B + C + E
B + C + E
B + C + E
B + C + E
Variance in A1-A5
unique to
B and C, plus error

22. Three Broad Types of Latent Variable Analysis
Exploratory Factor Analysis
You have observed variables, but need help building theory of construct measurement
Confirmatory Factor Analysis
You have observed variables and theory of construct measurement, and need to test its empirical support
Structural Equation Modeling
You have empirically supported theory of construct measurement, and wish to test theories of structural relations between constructs

23. “All models are wrong, but some are useful”
--George Box (1978), Statistician

24. CFA Basic Principles

25. What Is the Goal of CFA?
Parsimonious, yet sufficient, representation of our data
Specifically, observed variances/covariances
Model too simple? Lose valuable information…
Model too complex? Too nuanced to be helpful

26. Anatomy of a CFA Path Diagram
Ψ 12
Ψ 11
Ψ 22
Factor 1
Factor 2
Not shown:
𝛂 = Latent Means
𝞃 = Item Intercepts
Will describe later
𝜆 11
𝜆 21
𝜆 31
𝜆 42
𝜆 52
𝜆 62
Var1
Var2
Var3
Var4
Var5
Var6
𝜃 11
𝜃 22
𝜃 33
𝜃 44
𝜃 55
𝜃 66
UF 1
UF 2
UF 3
UF 4
UF 5
UF 6

27. Measurement Model: Factor Loadings
Represent the direction and strength of association between Factor and Item
Factors typically cause items—not the other way around*
Glorified regression slopes
When standardized and squared = % of item variance explained by factor (i.e., communality [h])
Determined by shared variance between items
Factor 1
𝜆 11
𝜆 21
𝜆 31
Item 1
Item 2
Item 3
UF 1
UF 2
UF 3

28. Measurement Model: Unique Factors
Also called: residual variances, error-variances, or uniquenesses (standardized)
Represent random error variance and other (not- modeled) construct variance
Factors that explain more variance in items will have smaller unique factors
Item 1
Item 2
Item 3
𝜃 11
𝜃 22
𝜃 33
UF 1
UF 2
UF 3

29. Structural Model: Latent Variances/Covariances
between Factor 1 and 2.
Variance of Factor 1
Variance of Factor 2
Ψ 12
Ψ 11
Ψ 22
Factor 1
Factor 2

30. Why Am I Telling You All of This?
What your model says the observed variances/covariances should be (i.e., model-implied)
Σ=Λ Ψ Λ ′ +Θ
Comparing
these is how
we appraise
model fit!!! S
What your actual variances/covariances are (i.e., observed)

31. Model-Implied Variances/Covariances
Σ: Model-Implied Var/Covar Matrix
Ψ 11
Factor 1 X1 X2 X3
𝜆 11 * Ψ 11 * 𝜆 𝜃 11
𝜆 11 * Ψ 11 * 𝜆 21
𝜆 21 * Ψ 11 * 𝜆 21 + 𝜃 22
𝜆 11 * Ψ 11 * 𝜆 31
𝜆 21 * Ψ 11 * 𝜆 31
𝜆 31 * Ψ 11 * 𝜆 31 + 𝜃 33
𝜆 11
𝜆 21
𝜆 31 X1 X2 X3
𝜃 11
𝜃 22
𝜃 33
UF 1
UF 2
UF 3

32. Model Fit: S vs. Σ S: Observed (Co)Variances
Σ: Model-implied (Co)Variances
Item1
Item2
Item3
Item4
Item5
Item6
Var1 --
Cov12
Var2
Cov13
Cov32
Var3
Cov14
Cov24
Cov34
Var4
Cov15
Cov25
Cov35
Cov45
Var5
Cov16
Cov26
Cov36
Cov46
Cov56
Var6
Item1
Item2
Item3
Item4
Item5
Item6
Var1 --
Cov12
Var2
Cov13
Cov32
Var3
Var4
Cov45
Var5
Cov46
Cov56
Var6

33. Evaluating Model Fit: The χ2 test
The “original” model fit index—all else calculated from it
Tests H0 of perfect-fitting model
i.e., 𝑆= Σ
Not especially informative
H0 virtually always rejected at typical levels of n
We don’t expect 𝑆= Σ—all models are wrong!

34. Evaluating Model Fit: Absolute Indexes
Standardized root mean residual (SRMR)
Average standardized residual from 𝑆 𝑣𝑠.Σ
Root means square error of approximation (RMSEA)
Amount of misfit per df of model
Can calculate 90% CI to test null of close fit
Worst
Fit
Our
Model
Perfect
Fit
Absolute
Indexes
> .10 (poor); (mediocre); (acceptable); (close); .00 (perfect)

35. Evaluating Model Fit: Relative Indexes
Compare our model to “null” model
Reasonable “worst-fitting” model
Worst
Fit
Our
Model
Perfect
Fit
Relative
Indexes

36. Evaluating Model Fit: Relative Indexes
Σ: “Null” Model
Σ: Our Model
Item1
Item2
Item3
Item4
Item5
Item6
Var1 --
Var2
Var3
Var4
Var5
Var6
Item1
Item2
Item3
Item4
Item5
Item6
Var1 --
Cov12
Var2
Cov13
Cov32
Var3
Var4
Cov45
Var5
Cov46
Cov56
Var6
Compare each to S to get χ2 and df for each

37. Evaluating Model Fit: Relative Indexes
Compare our model to “null” model
Reasonable “worst-fitting” model
Recommended:
Tucker-Lewis Index (TLI)/Non- Normed Fit Index (NNFI)
Comparative Fit Index (CFI)
Worst
Fit
Our
Model
Perfect
Fit
Relative
Indexes
< .85(poor); (mediocre); (acceptable); (close); 1.00 (perfect)

38. Recommendations for Evaluating Model Fit
Hu & Bentler (1999)
Recommend two-index evaluation strategy: χ2 + 1 absolute index + 1 relative index
Take note when similar indexes radically diverge

39. But There’s A Problem or Two…
Scale-setting: Latent variables are “unobservable”–how do we come to understand their scale? We need a reference point of some kind.
Identification: Many unknowns to solve for. We need to ensure the equations for the model are solvable.

40. Scale-Setting and Identification Methods
Fix an estimate for every factor to a particular meaningful value; defines latent scale, and makes equations solvable
“Marker-variable”: fix a loading for each factor to 1 (the default of most SEM software)
Privileges marker-variable as “gold-standard”, introduces problems later—best avoided
“Fixed-factor”: fix latent variance of each factor to 1
Standardizes the latent variable—should be your go-to

41. Scale-Setting Impacts Model-Implied Variances/Covariances
Σ: Model-Implied Var/Covar Matrix
Ψ 11
Factor 1 X1 X2 X3
𝜆 11 * Ψ 11 * 𝜆 𝜃 11
𝜆 11 * Ψ 11 * 𝜆 21
𝜆 21 * Ψ 11 * 𝜆 21 + 𝜃 22
𝜆 11 * Ψ 11 * 𝜆 31
𝜆 21 * Ψ 11 * 𝜆 31
𝜆 31 * Ψ 11 * 𝜆 31 + 𝜃 33
𝜆 11
𝜆 21
𝜆 31 X1 X2 X3
𝜃 11
𝜃 22
𝜃 33
UF 1
UF 2
UF 3

42. Levels of Identification
Under-identification
#of parameters to estimate > # of known var/covars
Model fit cannot be estimated
Just-identification
#of parameters to estimate = # of known var/covars
Model fit is meaningless/artificially good
Over-identification
#of parameters to estimate < # of known var/covars
Model fit is meaningful

43. Critical Commentary on Grace SEM Examples
There is nothing remotely
“latent” about this.
Single-indicator factors
require ridiculous #’s of
fixed parameters to render
them identified. This is a
glorified path-analysis*,
nothing more. Of course it
fits well—it’s artificial!
*Path analysis is totally cool; just call a spade a spade

44. Identification with Suboptimal “Latent” Variables
Ψ 11
Factor 1 ∗1
Factor 1 ∗1
∗𝜆 11
∗𝜆 11 X1 X1 X2 ∗0
𝜃 11
𝜃 11
UF 1
UF 1
UF 2

45. CFA in lavaan() (Section (2) of Script)
Save CFA model syntax in an R object
Fit CFA model and specify scale-setting method; save output in a new object
Request summary output from CFA object

46. Advanced CFA

47. If Your CFA Model Fit Is Unacceptably Bad…
Tread carefully!
Any model revisions are now exploratory
You will be tempted to justify *anything* for good fit
Replication is a must
Software will produce “mod indexes” on request
What model changes in current sample would improve model fit the most

48. On the Abuse of Correlated Error Variances in Post-Hoc CFA Model Revision
Arguably most common post-hoc modification made to improve model fit
Often times, theoretically indefensible
And when they are, more often than not, should have been predicted from the start

49. Example of Defensible/Predictable Correlated Error Variances
Affect
Cog
Oral
PVI
Anal
Oral
PVI
Anal
UF 1
UF 2
UF 3
UF 4
UF 5
UF 6

50. A More Likely (…Probably) Example for Ecologists: Time
Abiotic Stress (T1)
Abiotic Stress (T2)
X1.1
X2.1
X3.1
X1.2
X2.2
X3.2
UF 1
UF 2
UF 3
UF 4
UF 5
UF 6

51. Evaluating Measurement Generalizability via Invariance Testing
Eventually, you/others may wish to compare groups/time points on structural parameters
Means
Variances
Covariances/correlations
Regression slopes
Such comparisons are only valid if construct(s) being measured is the same for all groups/time points
Assumption still applies even if latent variables not being analyzed (e.g., using a generic t-test)

52. Group-Based Model Constraints at a Glance: Parent Model
Ecosystem 1
Ecosystem 2 ∗1 ∗1
Abiotic
Stress
Abiotic
Stress 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 X1 X2 X3 X1 X2 X3
𝜃 11
𝜃 22
𝜃 33
UF 1
UF 2
UF 3
UF 1
UF 2
UF 3

53. Group-Based Model Constraints at a Glance: Nested Model
Ecosystem 1
Ecosystem 2 ∗1 ∗1
Abiotic Stress
Abiotic Stress 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 X1 X2 X3 X1 X2 X3
𝜃 11
𝜃 22
𝜃 33
UF 1
UF 2
UF 3
UF 1
UF 2
UF 3

54. Comparing Nested Models
𝜒 𝑁𝑒𝑠𝑡𝑒𝑑 2 𝑑𝑓 𝑁𝑒𝑠𝑡𝑒𝑑 = ???
Test of perfect fit for the Nested Model:
Will always have larger chi-squared statistic
and df because it is simpler
Test of perfect fit for the Parent Model:
Will always have smaller chi-squared statistic
and df because it is more complex
𝜒 𝑃𝑎𝑟𝑒𝑛𝑡 2 𝑑𝑓 𝑃𝑎𝑟𝑒𝑛𝑡 = ???
Δ 𝜒 2 𝑑𝑓 𝑁𝑒𝑠𝑡𝑒𝑑 − 𝑑𝑓 𝑃𝑎𝑟𝑒𝑛𝑡 = 𝜒 𝑁𝑒𝑠𝑡𝑒𝑑 𝜒 𝑃𝑎𝑟𝑒𝑛𝑡 2
Tests whether the constraints oversimplify the model, resulting in significantly worse
model fit; null is that the more parsimonious nested model is “worth it”

55. What Level(s) of Invariance Needed for Valid Group Comparisons?
Invariance Level
What Constraint(s) Imposed?
Needed for Valid Group Comparisons of… 1
“Configural”/“Pattern”
Same # of factors, and same pattern of items loading onto factors
All structural parameters 2
“Weak”/ “Loading”/ “Metric”
1 + equivalent factor loadings
Variances, covariances, and regression slopes 3
“Strong”/“Intercept”
equivalent intercepts
Means

56. How to Evaluate Measurement Invariance?
Two strategies:
Same Δχ2 testing process
ns difference = invariance level supported
ΔCFI (see Cheung & Rensvold, 2002)
Invariance supported if ΔCFI < .01

57. In-Depth Theory Testing via CFA in R (section (3) in code)
Request mod indexes for CFA model
Use semTools() package for easy testing of measurement invariance

58. Structural Equation Modeling

59. From CFA to SEM Analytic focus on structural level of the model
Latent means, correlations, specifying regression pathways, etc.,
Major perk of SEM w/ latent variables: more statistical power
Bigger effects or less variability, depending on scale-setting

60. Traditional SEM: Fancy Multiple Regression Models∗1
Abiotic Stress
Light a
Biodiversity ∗1 ∗1
Disturbance b

61. Same Intuitive Constraint-Testing Approach∗1
Abiotic Stress
Light a
Biodiversity ∗1 ∗1
Disturbance a

62. Group Comparisons of Latent Means (e.g., latent t-test or ANOVA)
semTools() measurementInvariance() command tests this by default
Constrains all means to equality (omnibus test)
Nested in strong/intercept invariance model
Requires follow-up tests, if more than 2 groups

63. Group Comparisons of Latent (co)Variances, Correlations, and Regression Slopes
Can test predictions about group variances
Assumed values not needed for comparing latent means
Covariances/Correlations/Slopes
Akin to testing categorical X continuous interactions
Constrain 1. or 2. to equality*, nested within weak/loading invariance model
*Comparing group covariances requires “Phantom” variables if group variances are unequal

64. Example: Abiotic Stress --> Biodiversity (Two Ecosystems): Parent Model∗1
Ecosystem 1
Abiotic Stress
Biodiversity ∗1 a ∗1
Ecosystem 2
Abiotic Stress
Biodiversity ∗1 b

65. Example: Abiotic Stress --> Biodiversity (Two Ecosystems): Nested Model∗1
Ecosystem 1
Abiotic Stress
Biodiversity ∗1 a ∗1
Ecosystem 2
Abiotic Stress
Biodiversity ∗1 a

66. SEM Considerations Scale-setting method matters
#1 reason marker-variable sucks: biases results of significance testing of structural estimates involving the latent variable; USE FIXED-FACTOR!
Model complexity matters (especially with small samples)
Convergence/estimation problems common when too much is asked of smaller amounts of data

67. Structural Equation Modeling via R
Fit measurement model for all variables to-be analyzed
Specify latent regressions and test for constraint of equal predictive strength
Specify multiple group latent regressions and test for constraint of equal predictive strength between groups

68. Resources for You
See selected list of references for latent variable analysis
Most informed this talk
StackExchange and CrossValidated
Online Q&A communities for programming and stats
PsychMAP and Psychological Methods Discussion FB Groups
Twitter

69. Thank You!
And good luck!