Glossary
Key terms used across the tutorials
This glossary collects the key terms used across the tutorial series. Terms are linked from individual tutorials on first use. Where a term has a Greek symbol, the symbol is shown in the notation key.
A
assignment
Storing a value in a named R object with the <- operator. For example, x <- 5 assigns the value 5 to the object x.
ANOVA
Analysis of variance. A generalisation of the t-test to three or more groups. Tests whether the means of the groups are all equal. In R, fitted with aov().
B
bootstrap
A resampling method used to estimate the sampling distribution of a statistic — most commonly a regression coefficient or an indirect effect — by repeatedly sampling with replacement from the observed data. In lavaan, requested with se = "boot" and bootstrap = N. Bootstrap confidence intervals are preferred over the Sobel test for mediation analysis because the product of two normally distributed coefficients is not itself normally distributed.
BCa
Bias-corrected and accelerated. A refinement of the bootstrap confidence interval that corrects for both bias and skewness in the bootstrap distribution. Requested in lavaan with parameterEstimates(fit, boot.ci.type = "bca.simple").
C
CFA
Confirmatory factor analysis. A statistical technique for testing whether a hypothesised factor structure — which observed variables load on which latent factors — fits the observed covariance matrix. Fitted in lavaan with cfa().
common method variance
Variance shared between observed variables due to the measurement method (e.g. all items measured on the same Likert scale in the same questionnaire) rather than due to the latent construct. Can be modelled as residual covariances between items in CFA.
configural invariance
The weakest level of measurement invariance: the same factor structure (the same indicators loading on the same latent factors) holds in every group, but all parameters are freely estimated in each group. Also called pattern invariance. A prerequisite for testing stronger forms of invariance. See Tutorial 17.
constraining parameters
Forcing two or more model parameters to take the same value. In lavaan, achieved by labelling parameters with the same identifier — for example, y ~ a*x1 and z ~ a*x2 constrains the two regression coefficients to be equal.
convergence
The iterative estimation algorithm has found a solution — the difference between parameter estimates across successive iterations has fallen below a tolerance threshold. If a model does not converge, the parameter estimates are not trustworthy. lavaan prints a warning when convergence fails.
D
degrees of freedom
The number of independent pieces of information available to estimate model parameters. In SEM, df = q(q+1)/2 - t, where q is the number of observed variables and t is the number of free parameters. A model with df = 0 is just-identified; df > 0 is overidentified; df < 0 is underidentified and cannot be estimated.
direct effect
In a mediation model, the effect of the independent variable \(X\) on the outcome \(Y\) that does not pass through the mediator \(M\). Denoted \(c'\). In lavaan, estimated directly as a regression path.
disturbance
The residual variance of a latent dependent variable — the part of the latent variable’s variance not explained by its predictors in the structural model. Denoted \(\zeta\) (zeta).
E
endogenous
A variable that is predicted by at least one other variable in the model — a dependent variable. Contrast with exogenous.
estimation
The process of finding the values of a model’s free parameters that best reproduce the observed covariance matrix. The default estimator in lavaan is maximum likelihood (ML); the robust variant MLR is recommended for non-normal data.
exogenous
A variable that is not predicted by any other variable in the model — an independent variable. Exogenous variables have variances and covariances as model parameters but no structural equations predicting them.
F
factor loading
The regression coefficient linking a latent variable to one of its observed indicators. Denoted \(\lambda\) (lambda). In lavaan, specified with the =~ operator. The first loading of each factor is usually fixed to 1 (the marker-variable method) to set the latent variable’s scale.
FIML
Full information maximum likelihood. An approach to missing data that uses all available information in the dataset — every participant contributes to whichever parameters their observed values inform. The default missing-data treatment in lavaan. Preferred over listwise deletion because it is unbiased under MCAR and MAR.
fit indices
Statistics that measure how well a model reproduces the observed covariance matrix. The canonical set used in this series is \(\chi^2\), RMSEA, CFI, TLI, and SRMR. See Hu and Bentler (1999) for the cutoff criteria.
fixed parameter
A model parameter whose value is set by the researcher rather than estimated from the data. Common examples: a factor loading fixed to 1 (marker variable), a residual covariance fixed to 0 (default). In lavaan, fixed with * followed by a number — e.g. f =~ 1*x1.
free parameter
A model parameter whose value is estimated from the data. Most parameters in a SEM are free.
H
Heywood case
An impossible parameter estimate — most commonly a negative variance or a standardised factor loading above 1. Usually indicates either a misspecified model, an underidentified model, or a sample that is too small. lavaan prints a warning when it detects a Heywood case.
I
identification
A model is identified if a unique solution exists for every free parameter. Necessary conditions include positive degrees of freedom (the t-rule) and a set scale for every latent variable. A model that is not identified cannot be meaningfully estimated.
indirect effect
In a mediation model, the effect of the independent variable \(X\) on the outcome \(Y\) that operates through the mediator \(M\). Equal to the product \(ab\) of the two paths \(X \rightarrow M\) and \(M \rightarrow Y\). In lavaan, defined with the := operator — e.g. indirect := a * b.
indicator
An observed variable that measures a latent variable. For example, pa1 is an indicator of the latent factor “positive affect”.
J
just-identified
A model with zero degrees of freedom — the number of free parameters equals the number of non-redundant elements in the covariance matrix. Such a model fits perfectly by construction and cannot be rejected by the data. Also called a saturated model.
L
latent variable
An unobserved variable that is inferred from its effects on observed indicators. Represented by circles in path diagrams. Examples in this series: “positive affect”, “job control”, “creativity”.
listwise deletion
Removing any participant with a missing value on any variable in the model. Reduces sample size, reduces power, and can bias parameter estimates if the missingness is not completely random. We request FIML instead by passing missing = "ml" to cfa() or sem().
loadings
See factor loading.
M
marker variable
A method for setting the scale of a latent variable by fixing one of its factor loadings to 1 (and freely estimating the latent variance). lavaan’s default. The alternative is std.lv = TRUE, which fixes the latent variance to 1 and freely estimates all loadings.
MAR
Missing at random. The probability of a missing value on variable \(Y\) is unrelated to the value of \(Y\), conditional on other variables in the analysis. FIML gives unbiased estimates under MAR.
MCAR
Missing completely at random. The probability of a missing value is unrelated to any variable in the dataset — missingness is purely random. The strongest missing-data assumption; FIML and listwise deletion both give unbiased estimates under MCAR, but FIML is more powerful.
measurement error
The portion of an observed variable’s variance that is not due to the latent construct — a combination of random error and item-specific systematic variance. Denoted \(\theta\) (theta). Modelled as the residual variance of each indicator in CFA.
measurement invariance
The property that the same measurement model — the same items measuring the same latent factors, with the same relationships between items and factors — holds in every group. Tested as a four-level hierarchy: configural, metric (equal loadings), scalar (equal loadings + intercepts), and strict (equal loadings + intercepts + residuals). A prerequisite for comparing latent means or structural paths across groups. See Tutorial 17.
measurement model
The part of a SEM that specifies how latent variables are measured by observed indicators — that is, the CFA portion of the model. Contrast with the structural model.
mediation
A relationship in which an independent variable \(X\) influences an outcome \(Y\) through an intermediate variable \(M\) (the mediator). The total effect of \(X\) on \(Y\) is decomposed into a direct effect (\(c'\)) and an indirect effect (\(ab\)). See Tutorials 11 and 12.
metric invariance
The second level of measurement invariance: factor loadings are equal across groups, in addition to the same factor structure (configural invariance). Also called weak invariance. Required before comparing structural paths or factor variances across groups. Tested in lavaan with group.equal = "loadings". See Tutorial 17.
missingness
The pattern of missing values in a dataset. Distinguished into MCAR, MAR, and MNAR (missing not at random).
MNAR
Missing not at random. The probability of a missing value depends on the unobserved value itself — for example, people with very high salaries are more likely to refuse to report them. FIML is biased under MNAR; no standard method fully addresses MNAR without strong assumptions.
MLR
Maximum likelihood with robust standard errors (Satorra–Bentler). The default estimator we recommend throughout this series. Robust to non-normality. Requested in lavaan with estimator = "MLR".
modification indices
Statistics that estimate how much the model’s \(\chi^2\) would improve if a currently-fixed parameter were freely estimated. A value above 3.84 suggests a parameter worth considering for freeing (\(\alpha = .05\)). Should be used cautiously — they are data-driven and can lead to overfitting. In lavaan: modindices(fit, minimum.value = 3.84, sort. = TRUE).
multigroup SEM
A structural equation model fitted simultaneously in two or more groups, with parameters progressively constrained equal across groups. Used both for testing measurement invariance (on the measurement model) and for testing moderation by a categorical variable (on the structural paths). Specified in lavaan with the group argument and group.equal / group.partial. See Tutorials 16 and 17.
O
observed variable
A variable that is directly measured — a survey item, a test score, a demographic. Represented by squares or rectangles in path diagrams. Also called a manifest variable.
overidentified
A model with positive degrees of freedom — fewer free parameters than non-redundant elements in the covariance matrix. Such a model can be empirically tested against the data. The goal of SEM is to specify an overidentified model that fits well.
P
parcel
An aggregate — usually a mean or sum — of two or more items, used as a single indicator of a latent variable. Parcelling reduces the number of indicators per factor, which can simplify a model and improve fit, at the cost of losing item-level information. Controversial; see Tutorial 08.
parameter
A quantity in a model that is either estimated from the data (free) or set by the researcher (fixed). In SEM, parameters include factor loadings, regression coefficients, variances, and covariances.
partial invariance
A relaxed form of measurement invariance in which most, but not all, parameters of a given type are equal across groups — for example, all but one or two factor loadings. As long as each factor retains at least two invariant indicators (the marker plus one other), the factor’s scale remains comparable across groups and subsequent comparisons are interpretable. In lavaan, requested with group.partial. See Tutorial 17.
path analysis
A structural equation model with observed variables only — no latent variables. A system of regression equations. Fitted in lavaan with sem(). See Tutorial 09.
psi
R
residual covariance
A covariance between the residuals of two observed variables. Indicates that the two variables share variance not explained by their common latent factor(s) — often due to common method variance or a shared occasion. In lavaan, specified with ~~ — e.g. x1 ~~ x2.
RMSEA
Root mean square error of approximation. A badness-of-fit index (lower is better). Values below .05 indicate close fit, .05–.08 acceptable fit, above .10 poor fit. Reported with a 90% confidence interval; the upper bound of the CI should be below .08.
S
saturated model
See just-identified.
scalar invariance
The third level of measurement invariance: item intercepts are equal across groups, in addition to equal loadings (metric invariance). Also called strong invariance. Required before comparing latent means across groups, because equal intercepts fix the origin of each factor’s scale. Tested in lavaan with group.equal = c("loadings", "intercepts"). See Tutorial 17.
second-order factor
A latent variable whose indicators are themselves latent variables. For example, “burnout” might be a second-order factor measured by the first-order factors “emotional exhaustion” and “cynicism”. See Tutorial 08.
SEM
Structural equation modelling. A framework for testing models that combine a measurement model (latent variables measured by observed indicators) with a structural model (regression paths among latent and observed variables). Fitted in lavaan with sem().
simple slopes
In moderated regression, the slope of \(Y\) on \(X\) at a specific value of the moderator \(W\). Usually computed at \(W = \pm 1\) SD from its mean. See Tutorial 16.
Sobel test
A test of the indirect effect \(ab\) based on assuming the product of two normally distributed coefficients is itself normally distributed — which it is not. Superseded by bootstrap confidence intervals. Included here for historical context; not used in this series.
SRMR
Standardised root mean square residual. A badness-of-fit index (lower is better). The average of the standardised residuals between the observed and model-implied covariance matrices. Values below .08 indicate good fit.
standardised solution
Parameter estimates expressed in standard deviation units, so that loadings and path coefficients are comparable across variables on different scales. In lavaan: summary(fit, standardized = TRUE) or standardizedSolution(fit). The std.all column fully standardises both observed and latent variables.
strict invariance
The strongest level of measurement invariance: item residual (measurement-error) variances are equal across groups, in addition to equal loadings and intercepts (scalar invariance). Rarely required for substantive conclusions — scalar invariance is enough for comparing latent means — but tested for completeness. Tested in lavaan with group.equal = c("loadings", "intercepts", "residuals"). See Tutorial 17.
structural model
The part of a SEM that specifies the regression relationships among variables — which variables predict which. Contrast with the measurement model.
T
t-rule
A necessary condition for identification: the number of free parameters \(t\) must be less than or equal to the number of non-redundant elements in the observed covariance matrix, \(q(q+1)/2\), where \(q\) is the number of observed variables. The difference is the degrees of freedom.
theta
See measurement error.
total effect
In a mediation model, the sum of the direct and indirect effects: \(c' + ab\). The total effect of \(X\) on \(Y\) ignoring the mediator.
tibble
A modern data frame used by the haven and tidyverse packages. Prints more neatly than a base data.frame and never converts strings to factors. Behaves like a regular data frame for the purposes of this series.
U
underidentified
A model with negative degrees of freedom — more free parameters than the data can support. Cannot be estimated. The remedy is to fix or constrain parameters, not to collect more data.
V
vector
The fundamental data structure in R: a sequence of values of the same type (numeric, character, logical, integer). Created with c(). See Tutorial 01.
latent variance
The variance of a latent variable — a model parameter in CFA and SEM. Denoted \(\psi\) (psi). In the marker-variable method, the latent variance is freely estimated; with std.lv = TRUE, it is fixed to 1.