Chapter 2 · Three Views of One Model: Notation, Equations, and Graphs

A student fits the same multilevel model in three programs and receives three outputs that look nothing alike. One program reports “level-1” and “level-2” coefficients, another reports “fixed effects” and “random effects,” and a third reports a covariance matrix called $V$ . The student concludes that the programs fit different models. They do not. Each program shows one of three equivalent ways of writing the same model: as a set of equations, one for each level (the hierarchical view); as a single equation with a fixed part and a random part (the mixed view); and as a statement about the mean and covariance of the outcome (the marginal view). This chapter introduces the notation used throughout the book, shows how the three views follow from one another, and adds a fourth, graphical view: the Unified Model Graph. Being able to move among the four views is the most useful skill for reading multilevel models, and every later chapter relies on it.

2.1Notation#

Table 2.1 lists the symbols used in this book. The notation follows Snijders & Bosker (2012) and Raudenbush & Bryk (2002) closely, with one simplification: random variables are written in lower case ( $y_{ij}$ , $u_{0 j}$ , $e_{ij}$ ) so that the same letters can be used in the equations, in the diagrams, and in the R code. The last column of the table gives the symbols used in other books and in software output, so that readers can move between sources.

Table 2.1 Notation Used Throughout the Book

Symbol	Meaning	Other common notation
Indices and sample sizes
$i$ , $j$	Level-1 unit (pupil, occasion), level-2 unit (school, person)	$i$ and $j$ reversed in some texts
$n_{j}$ , $J$ , $N$	Units in cluster $j$ ; number of clusters; $N = \sum_{j} n_{j}$	$N$ for clusters in S&B
Variables
$y_{ij}$	Outcome of unit $i$ in cluster $j$	$Y_{ij}$ (S&B, R&B)
$x_{ij}$ , $x_{p ij}$	Level-1 predictor(s), $p = 1, \dots, P$	$X_{ij}$
$z_{j}$ , $z_{q j}$	Level-2 predictor(s), $q = 1, \dots, Q$	$W_{j}$ (R&B), $Z_{j}$
Coefficients and random effects
$β_{0 j}$ , $β_{1 j}$	Intercept and slope of cluster $j$ (level-1 coefficients)	$β_{0 j}$ , $β_{1 j}$
$γ_{00}, γ_{01}, γ_{10}, \dots$	Fixed effects: first subscript names the level-1 coefficient, second the level-2 predictor	$γ_{00}, \dots$
$u_{0 j}$ , $u_{1 j}$	Level-2 random effects (cluster deviations)	$U_{0 j}$ (S&B), $u_{0 j}$ (R&B)
$e_{ij}$	Level-1 residual	$R_{ij}$ (S&B), $r_{ij}$ (R&B), $ε_{ij}$
Variance components
$τ_{0}^{2}$ , $τ_{1}^{2}$	Variances of $u_{0 j}$ and $u_{1 j}$	$τ_{00}$ , $τ_{11}$ (R&B, HLM)
$τ_{01}$	Covariance of $u_{0 j}$ and $u_{1 j}$	$τ_{01}$
$σ^{2}$	Variance of $e_{ij}$	$σ^{2}$
$ρ_{I}$	Intraclass correlation, $τ_{0}^{2} / (τ_{0}^{2} + σ^{2})$	ICC, $ρ$
Matrices for cluster $j$
$y_{j}$ , $e_{j}$	$n_{j} \times 1$ outcome and residual vectors
$X_{j}$ , $Z_{j}$	Design matrices of the fixed and the random part
$γ$ , $u_{j}$	Vector of fixed effects; vector of random effects of cluster $j$	$β$ , $b_{j}$ (statistics)
$T$ , $V_{j}$	Covariance matrix of $u_{j}$ ; marginal covariance matrix of $y_{j}$	$G$ , $Σ_{j}$ (SAS: $G$ , $V$ )

Note. S&B = Snijders & Bosker (2012); R&B = Raudenbush & Bryk (2002). Bold symbols denote vectors and matrices.

Two conventions deserve attention from the start. First, the subscript $i$ always refers to the level-1 unit and $j$ to the level-2 unit, so $y_{ij}$ is the outcome of unit $i$ in cluster $j$ . In longitudinal data (Part V) the level-1 unit is a measurement occasion and the level-2 unit is a person; the subscripts keep the same meaning. Second, the fixed effects $γ$ carry two subscripts. The first says which level-1 coefficient is being modeled ( $0$ for the intercept, $1$ for the first slope, and so on), and the second says which level-2 predictor it multiplies ( $0$ for the constant). Thus $γ_{00}$ is the overall intercept, $γ_{10}$ is the average slope of $x$ , $γ_{01}$ is the effect of the level-2 predictor $z$ on the intercepts, and $γ_{11}$ is the effect of $z$ on the slopes (a cross-level interaction).

2.2The Hierarchical View#

The hierarchical view writes one equation for the level-1 units within each cluster and one equation for each cluster-specific coefficient. It is the most natural way to state a multilevel theory, because it places each variable at its own level. For a model with one level-1 predictor $x_{ij}$ and one level-2 predictor $z_{j}$ , the level-1 model is an ordinary regression within cluster $j$ :

y_{ij} = β_{0 j} + β_{1 j} x_{ij} + e_{ij}, e_{ij} \sim N (0, σ^{2}) . (2.1)

Each cluster has its own intercept $β_{0 j}$ and its own slope $β_{1 j}$ . The residual $e_{ij}$ is the deviation of unit $i$ from its cluster’s regression line, and $σ^{2}$ is the within-cluster residual variance, assumed to be the same in every cluster.

The level-2 model treats the cluster-specific coefficients as outcomes and relates them to cluster characteristics:

β_{0 j} β_{1 j} = γ_{00} + γ_{01} z_{j} + u_{0 j}, = γ_{10} + γ_{11} z_{j} + u_{1 j}, (2.2) (2.3)

with

(u_{0 j} u_{1 j}) \sim N ((00), T), T = (τ_{0}^{2} τ_{01} τ_{01} τ_{1}^{2}), (2.4)

and the random effects $u_{0 j}, u_{1 j}$ independent of the residuals $e_{ij}$ and independent across clusters. In words: a cluster’s intercept is predicted from $z_{j}$ , and $u_{0 j}$ is what is left over; a cluster’s slope is also predicted from $z_{j}$ , and $u_{1 j}$ is what is left over. The variances $τ_{0}^{2}$ and $τ_{1}^{2}$ describe how much intercepts and slopes vary across clusters after $z_{j}$ is taken into account, and the covariance $τ_{01}$ describes whether clusters with high intercepts tend to have high or low slopes.

The hierarchical view makes the substantive meaning of each parameter clear. It also shows immediately where a level-2 variable can enter (only in the level-2 equations) and what a cross-level interaction is (a level-2 variable in the equation for a slope). Its weakness is that it hides what the model says about the outcome itself: Equations (2.1) to (2.3) do not show directly how $y_{ij}$ depends on $z_{j}$ , or how two outcomes in the same cluster are related.

2.3The Mixed View#

Substituting the level-2 equations into the level-1 equation gives a single equation:

y_{ij} = (γ_{00} + γ_{01} z_{j} + u_{0 j}) + (γ_{10} + γ_{11} z_{j} + u_{1 j}) x_{ij} + e_{ij} = fixed part γ_{00} + γ_{10} x_{ij} + γ_{01} z_{j} + γ_{11} z_{j} x_{ij} + random part u_{0 j} + u_{1 j} x_{ij} + e_{ij} . (2.5)

This is the linear mixed model. The fixed part contains the regression coefficients $γ$ , which are the same for all clusters; the random part contains the random effects $u_{0 j}$ , $u_{1 j}$ , which vary across clusters, and the residual $e_{ij}$ . The mixed view shows three things that the hierarchical view hides. The cross-level interaction appears as an ordinary product term, $γ_{11} z_{j} x_{ij}$ . The level-2 predictor appears as an ordinary predictor of the outcome, $γ_{01} z_{j}$ . Finally, the error term is no longer a single residual but a sum, $u_{0 j} + u_{1 j} x_{ij} + e_{ij}$ , whose random-effect terms $u_{0 j}$ and $u_{1 j}$ are shared by all units of cluster $j$ . This composite error is the formal source of the dependence discussed in Chapter 1.

The mixed view is also the form in which models are written in most software. In the R package lme4 (Bates et al., 2015), for example, the model of Equation (2.5) is written y ~ x * z + (x | school): the fixed part outside the parentheses and the random part inside them, with the grouping factor after the vertical bar.

2.3.1The mixed model in matrix form#

For computation and for later chapters it helps to stack the $n_{j}$ equations of cluster $j$ in vectors and matrices (Laird & Ware, 1982):

y_{j} = X_{j} γ + Z_{j} u_{j} + e_{j}, u_{j} \sim N (0, T), e_{j} \sim N (0, σ^{2} I_{n_{j}}) . (2.6)

Here $y_{j}$ is the $n_{j} \times 1$ vector of outcomes, $X_{j}$ is the design matrix of the fixed part, $γ$ is the vector of fixed effects, $Z_{j}$ is the design matrix of the random part, $u_{j}$ is the vector of random effects, and $e_{j}$ is the vector of level-1 residuals. For a cluster with three units the model of Equation (2.5) reads

y_{j} y_{1 j} y_{2 j} y_{3 j} = X_{j} 111 x_{1 j} x_{2 j} x_{3 j} z_{j} z_{j} z_{j} z_{j} x_{1 j} z_{j} x_{2 j} z_{j} x_{3 j} γ γ_{00} γ_{10} γ_{01} γ_{11} + Z_{j} 111 x_{1 j} x_{2 j} x_{3 j} u_{j} (u_{0 j} u_{1 j}) + e_{j} e_{1 j} e_{2 j} e_{3 j} .

The columns of $Z_{j}$ are the variables that have random coefficients (here the constant and $x$ ); they are usually a subset of the columns of $X_{j}$ . In a model with only a random intercept, $Z_{j} = 1$ , a column of ones.

2.4The Marginal View#

The random effects $u_{0 j}$ and $u_{1 j}$ are not observed. What is observed is the outcome, and the model makes a statement about its distribution once the random effects are integrated out. This is the marginal model. From Equation (2.6), and because $u_{j}$ and $e_{j}$ have mean zero and are independent,

E (y_{j}) Var (y_{j}) = X_{j} γ, = V_{j} = Z_{j} T Z_{j}^{'} + σ^{2} I_{n_{j}}, (2.7) (2.8)

and outcomes in different clusters are uncorrelated. The marginal mean is the fixed part of the mixed model; the marginal covariance matrix $V_{j}$ collects everything the random part implies about variances and covariances. Because clusters are independent, the covariance matrix of all $N$ outcomes is block diagonal, with one block $V_{j}$ for each cluster.

2.4.1Random intercept models: compound symmetry#

With a random intercept only, $Z_{j} = 1$ and $T = τ_{0}^{2}$ , so $V_{j} = τ_{0}^{2} 1 1^{'} + σ^{2} I$ . Every outcome has the same variance, $τ_{0}^{2} + σ^{2}$ , and every pair of outcomes in the same cluster has the same covariance, $τ_{0}^{2}$ , whatever the values of the predictors. This pattern is called compound symmetry. The correlation between any two units of the same cluster is the intraclass correlation of Chapter 1, now computed after the predictors in the fixed part have been taken into account (a residual or conditional ICC).

2.4.2Random slope models: the variance function#

With a random slope the picture changes. Writing out Equation (2.8) for single elements gives

Var (y_{ij}) Cov (y_{ij}, y_{i^{'} j}) = τ_{0}^{2} + 2 τ_{01} x_{ij} + τ_{1}^{2} x_{ij}^{2} + σ^{2}, = τ_{0}^{2} + τ_{01} (x_{ij} + x_{i^{'} j}) + τ_{1}^{2} x_{ij} x_{i^{'} j}, i \neq = i^{'} . (2.9) (2.10)

The variance of the outcome is now a quadratic function of $x$ , the variance function, and the covariance between two units of the same cluster depends on their values of $x$ . A random slope therefore makes the marginal model heteroscedastic by construction: at values of $x$ where the cluster regression lines fan out, the outcome varies more; where they come closest together, it varies less. The minimum of the variance function lies at $x = - τ_{01} / τ_{1}^{2}$ . Chapter 6 develops these results in detail.

Technical detail Deriving the marginal variance and covariance

From the mixed model, the random part of $y_{ij}$ is $u_{0 j} + u_{1 j} x_{ij} + e_{ij}$ . Its variance is $Var (u_{0 j}) + x_{ij}^{2} Var (u_{1 j}) + 2 x_{ij} Cov (u_{0 j}, u_{1 j}) + Var (e_{ij}) = τ_{0}^{2} + τ_{1}^{2} x_{ij}^{2} + 2 τ_{01} x_{ij} + σ^{2}$ , because $e_{ij}$ is independent of the random effects. For two units $i \neq = i^{'}$ of the same cluster the residuals are independent, and only the shared random effects contribute: $Cov (u_{0 j} + u_{1 j} x_{ij}, u_{0 j} + u_{1 j} x_{i^{'} j}) = τ_{0}^{2} + τ_{01} (x_{ij} + x_{i^{'} j}) + τ_{1}^{2} x_{ij} x_{i^{'} j}$ . In matrix form these are the elements of $Z_{j} T Z_{j}^{'} + σ^{2} I$ . Why is the marginal model the one that software estimates? Because $u_{j}$ is unobserved, the likelihood of the data must be the density of $y_{j}$ with $u_{j}$ integrated out, $f (y_{j}) = \int f (y_{j} ∣ u_{j}) f (u_{j}) d u_{j}$ , and for normal $u_{j}$ and $e_{j}$ this integral is exactly the normal density with mean (2.7) and covariance (2.8). Chapter 9 builds the maximum likelihood and REML estimators on this fact.

Figure 2.1 makes Equation (2.8) concrete. It shows the model-implied correlation matrix of mathematics scores for nine NELS:88 pupils, three from each of three schools, chosen to have the lowest, median, and highest homework time in their school. The estimates come from the models fitted in Section 2.5. Under the random intercept model (panel a) the matrix has the compound-symmetry pattern: blocks of equal correlations (.23) within schools and zeros between schools. Under the random intercept and slope model (panel b) the within-school correlations depend on homework. Two pupils with similar and fairly high homework in the same school correlate strongly (.59 and .78 in schools 2 and 3), but a pupil who does no homework and one who does much homework are negatively correlated in the same school ( $- .42$ , $- .50$ ). The negative values reflect the crossing of school regression lines seen in Figure 1.3 of Chapter 1: in schools where pupils without homework do well, pupils with much homework do relatively poorly, and vice versa.

Figure 2.1: Model-Implied Correlations of Mathematics Scores for Nine Pupils in Three NELS:88 Schools — Figure 2.1 Model-Implied Correlations of Mathematics Scores for Nine Pupils in Three NELS:88 Schools

Figure 2.2 shows the corresponding variance functions. Under the random intercept model the variance of mathematics scores is the same (92.6) at every value of homework. Under the random intercept and slope model it is 115.7 for pupils with no homework, falls to a minimum of 72.8 between the categories “less than one hour” and “one hour,” and rises steeply for pupils with much homework. The dotted line marks the within-school part $σ^{2}$ , which is constant; everything above it comes from differences between schools, and under the random slope model the between-school share depends on the value of the predictor.

Figure 2.2: Model-Implied Variance of Mathematics as a Function of Homework Time (NELS:88) — Figure 2.2 Model-Implied Variance of Mathematics as a Function of Homework Time (NELS:88)

2.5Moving Among the Three Views: A Worked Example#

This section fits three models to the NELS:88 data of Chapter 1 (519 pupils in 23 schools) and writes each of them in the three views, using the estimates. The outcome is mathematics achievement, and the level-1 predictor is the homework category $h_{ij}$ (0 = none, 1 = less than one hour, 2 = one hour, and so on up to 7 = ten hours or more per week). The models are the empty model (M0), a random intercept model with homework (M1), and a random intercept and slope model with homework (M2). Table 2.2 shows the estimates, obtained by restricted maximum likelihood (REML; Chapter 9) with lme4.

Table 2.2 Estimates for Three Models of Mathematics Achievement (NELS:88, 23 Schools, 519 Pupils)

Parameter	Symbol	Empty (M0)	Random intercept (M1)	Random intercept and slope (M2)
Fixed effects
Intercept	$γ_{00}$	50.76 (1.15)	46.36 (1.16)	46.33 (1.76)
Homework	$γ_{10}$		2.40 (0.28)	1.98 (0.93)
Variance components
Intercept variance	$τ_{0}^{2}$	26.12	21.34	62.42
Slope variance	$τ_{1}^{2}$			17.73
Intercept–slope covariance	$τ_{01}$			$-$ 27.59
Level-1 variance	$σ^{2}$	81.24	71.28	53.29
REML deviance	$- 2 ℓ_{R}$	3,798.7	3,729.3	3,635.6
Parameters	$p$	3	4	6

Note. Estimates are REML estimates; standard errors in parentheses. $p$ = number of estimated parameters (fixed effects and variance components). REML deviances can be compared only between models with the same fixed part (here M1 and M2; see Chapter 11).

library(lme4)
nels <- read.csv("nels88_school23.csv")
M0 <- lmer(math ~ 1 + (1 | school), data = nels)                   # empty model
M1 <- lmer(math ~ homework + (1 | school), data = nels)            # random intercept
M2 <- lmer(math ~ homework + (homework | school), data = nels)     # random intercept and slope
summary(M2)       # fixed effects, variance components, correlation of random effects
VarCorr(M2)$school  # the matrix T; sigma(M2)^2 gives sigma^2

Each view answers different questions. The hierarchical view says what happens in a typical school and how schools differ. In the average school, the expected mathematics score increases by about 2 points per homework category, but schools differ a great deal: the standard deviation of the slopes is 4.2, so in a school one standard deviation above the average the slope is about $1.98 + 4.21 = 6.2$ , and in a school one standard deviation below it is about $- 2.2$ . If the slopes are normally distributed, about a third of schools in the population ( $Φ (- 1.98/4.21) \approx .32$ ) have a negative slope. The strong negative correlation between intercepts and slopes ( $- .83$ ) says that schools whose pupils do well without homework tend to have flat or negative homework slopes, which matches the pattern of the two schools in Figure 1.3.

The mixed view says what the software estimates as fixed effects (46.33 and 1.98) and which terms are random. Comparing M1 and M2 in Table 2.2 shows something that will recur in Chapter 6: when the slope is allowed to vary, the estimated average slope changes (from 2.40 to 1.98), and its standard error becomes more than three times larger (from 0.28 to 0.93). The random intercept model treats the homework effect as the same in every school and therefore overstates the precision of the average effect.

The marginal view says what the model implies about the observed scores: the population-average regression line and the variances and correlations shown in Figures 2.1 and 2.2. It also shows how the variance components make up the total variance. In the empty model the total variance is $26.1 + 81.2 = 107.3$ , of which 24% lies between schools. Adding homework as a fixed effect (M1) reduces both components ( $τ_{0}^{2}$ to 21.3 and $σ^{2}$ to 71.3). Allowing the slope to vary (M2) moves variance out of $σ^{2}$ (to 53.3) and into the school-level part, which now depends on homework. How to summarize such changes as “explained variance” is taken up in Chapter 11.

2.6Drawing the Model: The Unified Model Graph#

Equations are precise but hard to scan; path diagrams are easy to scan but, in their usual form, cannot show which coefficients vary across clusters. This book draws every model as a Unified Model Graph (UMG), a formally specified graphical grammar in which a well-formed diagram corresponds to one statistical model (Yu, 2026a). The R package umg (Yu, 2026b) builds, checks, and renders such diagrams. The grammar answers three questions about every quantity in a model and uses one visual channel for each answer.

2.6.1Three questions about every quantity#

Figure 2.3(a) shows the vertices. Is the quantity observed? Observed variables are shaded; unobserved (latent) quantities, including random effects, are white. Is it continuous or categorical? Continuous quantities are circles; categorical quantities (a binary response, a latent class) are squares. What role does it play? A random variable has a single border; a fixed unknown parameter, the target of frequentist estimation, is a diamond; a known constant is a triangle; and a quantity computed deterministically from others, such as a cluster mean $\overset{x}{ˉ}_{j}$ , has a double border.

Figure 2.3: The Lexicon of the Unified Model Graph — Figure 2.3 The Lexicon of the Unified Model Graph

Figure 2.3(b) shows the edges. A single-headed arrow is a dependence: the quantity at the head depends on the quantity at the tail, as in a regression. A curved double-headed arrow is a covariance, a symmetric association with no direction. A dashed arrow into a double-bordered vertex is a deterministic assignment. An arrow with an open head from a latent class to a set of parameters is a mixing edge, used for mixture models (Chapter 22). Figure 2.3(c) shows two shorthand marks, a loop for the variance of a random effect and a short arrow for a residual variance, and the plates introduced in Chapter 1.

2.6.2The core models as UMGs#

Figure 2.4 draws three models: the empty model M0 (panel a), a random intercept model with a pupil predictor $x_{ij}$ and a school predictor $z_{j}$ (panel b; without $z_{j}$ and $γ_{01}$ it is M1), and the random intercept and slope model M2 (panel c). In every panel the pupil plate ( $i$ ) is nested in the school plate ( $j$ ), observed pupil variables are shaded circles inside the pupil plate, the observed school variable $z_{j}$ (panel b) is a shaded circle inside the school plate but outside the pupil plate, the random coefficients ( $β_{0 j}$ in panels a and b, $β_{0 j}$ and $β_{1 j}$ in panel c) are white circles in the school plate, and the fixed effects $γ$ are diamonds outside all plates. In panel (b) the slope is fixed, so its diamond $γ_{10}$ points directly at $y_{ij}$ . Reading panel (c) from top to bottom reproduces the hierarchical view: the population parameters $γ_{00}$ and $γ_{10}$ generate the school coefficients $β_{0 j}$ and $β_{1 j}$ , which vary across schools with variances $τ_{0}^{2}$ and $τ_{1}^{2}$ and covariance $τ_{01}$ ; the school coefficients and the pupil’s predictor $x_{ij}$ (homework, written $h_{ij}$ in the worked example) generate the pupil’s score $y_{ij}$ , with residual variance $σ^{2}$ .

Figure 2.4: The Empty, Random Intercept, and Random Intercept and Slope Models as Unified Model Graphs — Figure 2.4 The Empty, Random Intercept, and Random Intercept and Slope Models as Unified Model Graphs

A UMG is not only a picture. Every vertex with incoming arrows has a distribution given the vertices that point to it, and the joint distribution of the random quantities in the diagram, given the parameters and observed predictors, is the product of these conditional distributions, replicated as the plates indicate. For panel (c) this product is

j = 1 \prod J [p (β_{0 j}, β_{1 j} ∣ γ_{00}, γ_{10}, T) i = 1 \prod n_{j} p (y_{ij} ∣ β_{0 j}, β_{1 j}, x_{ij}, σ^{2})], (2.11)

which is exactly the hierarchical model of Equations (2.1) to (2.4) without $z_{j}$ . Integrating the white (unobserved) random coefficients out of Equation (2.11) gives the marginal model. In this sense the diagram carries all three views: the arrows give the hierarchical equations, the white vertices mark the random part of the mixed model, and removing them by integration gives the marginal model.

2.7Parameter Promotion: Fixed and Random as Two Positions#

The most useful single idea in the UMG grammar is parameter promotion (Yu, 2026a). Figure 2.5 shows one coefficient, the effect of $x$ on $y$ , in four positions. In panel (a) the coefficient $γ_{10}$ is a diamond outside both plates: one fixed slope for all schools. In panel (b) the diamond has been moved into the school plate and has become a white circle $β_{1 j}$ with a distribution across schools, $β_{1 j} \sim N (γ_{10}, τ_{1}^{2})$ ; the old diamond remains as the mean of that distribution. This single move is the difference between a fixed and a random effect. In panel (c) the coefficient has been given a prior distribution with known constants (the triangle), which is how Bayesian estimation treats every parameter (Chapter 13). In panel (d) a latent class vertex selects one of several class-specific versions of the coefficient through a mixing edge, which is how mixture models work (Chapter 22).

Figure 2.5: Parameter Promotion: One Coefficient in Four Positions — Figure 2.5 Parameter Promotion: One Coefficient in Four Positions

The statistical content of this idea is not new. Random effects are latent variables (Curran, 2003; Mehta & Neale, 2005; Skrondal & Rabe-Hesketh, 2004), a prior turns a parameter into a random variable, and a mixture switches parameters with a latent class. What the diagrams add is that the four cases look different on the page while the move between them is a single relocation. Readers who have understood panel (b) have understood what a random effect is, and they will recognize the same move when it reappears for random intercepts (Chapter 5), random slopes (Chapter 6), Bayesian estimation (Chapter 13), growth mixtures (Chapter 22), and latent variables in multilevel structural equation models (Chapter 29).

2.8The Data Displays a Model Implies#

Every element of a UMG implies a display of the data that would show whether that element is needed (Yu, 2026a). A plate implies small multiples: one panel per cluster, so that the within-cluster relation can be seen in each. A random coefficient implies a display of the cluster-specific estimates of that coefficient: for a random slope, a “spaghetti” plot of cluster regression lines. A covariance edge between two random coefficients implies a scatterplot of one set of cluster-specific estimates against the other. A random slope also implies a variance function, as in Figure 2.2. Read forward, a diagram is a model to be fitted; read backward, it is a checklist of displays to look at before and after fitting.

Figure 2.6 shows the three displays implied by panel (c) of Figure 2.4 for the NELS:88 data. The small multiples (panel a) show that the within-school relation between homework and mathematics is roughly linear but differs clearly from school to school. The school-specific least-squares lines (panel b) show the spread of intercepts and slopes around the pooled line, which motivates the random intercept and the random slope. The scatterplot of school intercepts against school slopes (panel c) shows a strong negative association (the correlation of the 23 pairs of least-squares estimates is $- .84$ ), which is what the covariance edge $τ_{01}$ represents and what the model estimates, net of sampling error, as $- .83$ . Chapter 4 develops these displays systematically, and the function umg_eda_scaffold() in the umg package generates them from a diagram and a data set.

Figure 2.6: Data Displays Implied by the Random Intercept and Slope Model (NELS:88) — Figure 2.6 Data Displays Implied by the Random Intercept and Slope Model (NELS:88)

2.9What Software Needs to Be Told#

Every program for multilevel models asks for the same four pieces of information, although in different ways: (1) the outcome and the fixed part (which predictors, which interactions); (2) the grouping variable that defines the clusters; (3) the random part (which coefficients vary across clusters); and (4) the structure of the covariance matrix $T$ of the random effects (whether the random intercept and slope may be correlated). Most programs also allow a covariance structure to be specified for the level-1 residuals, which becomes important for longitudinal data (Chapter 20), and all of them require an estimation method, set explicitly or by default (Chapter 9).

Programs written around the mixed view (R, SAS, SPSS, Stata, JASP) take a single model statement with a fixed part and a random part. Programs written around the hierarchical view (HLM) take a level-1 equation and level-2 equations. Programs written around latent variables (Mplus) take a within-cluster model and a between-cluster model in which the random coefficients appear as latent variables, which is exactly the picture in Figure 2.4(c). The software crosswalk at the end of this section gives the syntax for model M2 in each program, and Table 2.3 shows where each variance component appears in their output.

Table 2.3 Where the Variance Components of Model M2 Appear in Program Output

Program	$τ_{0}^{2}$ , $τ_{1}^{2}$	$τ_{01}$	$σ^{2}$
R (`lme4`)	`school (Intercept)` and `homework` under Variance; standard deviations also printed	printed as a correlation (Corr), not a covariance	`Residual`
SAS (`MIXED`)	`UN(1,1)`, `UN(2,2)` in Covariance Parameter Estimates	`UN(2,1)`	`Residual`
SPSS (`MIXED`)	`UN (1,1)`, `UN (2,2)`	`UN (2,1)`	`Residual`
Stata (`mixed`)	`var(_cons)`, `var(homework)`	`cov(homework,_cons)`	`var(Residual)`
Mplus	between-level variances of `MATH` and `S`	`WITH` statement for `MATH` and `S` (between level)	within-level residual variance of `MATH`
HLM 8	$τ_{00}$ and $τ_{11}$ in the $τ$ matrix	$τ_{01}$	$σ^{2}$

Note. Labels are those printed for the model in the crosswalk box below; they can change with the order of terms in the model statement and between program versions. $τ_{0}^{2}$ = intercept variance; $τ_{1}^{2}$ = slope variance; $τ_{01}$ = intercept–slope covariance; $σ^{2}$ = level-1 residual variance.

Pitfall Default covariance structures differ between programs

In lme4, a term such as (homework | school) estimates the variances of the random intercept and slope and their covariance. In SAS PROC MIXED, SPSS MIXED, and Stata mixed, the default structure for several random effects is diagonal: variances are estimated but the covariance $τ_{01}$ is fixed at zero unless an unstructured matrix is requested (type=un, COVTYPE(UN), covariance(unstructured)). In Mplus the covariance between a random intercept and a random slope is fixed at zero by default and is estimated only when a WITH statement requests it. Fitting “the same model” with different defaults is a common reason why two programs disagree. Also check the estimation method: lme4, SAS, SPSS, and HLM (for two-level models) use REML by default for linear models, whereas Stata mixed uses maximum likelihood unless reml is specified, and Mplus uses maximum likelihood (MLR by default for multilevel models).

Software crosswalk The random intercept and slope model M2

Outcome math, level-1 predictor homework, clusters school; unstructured $T$ ; REML where available. All programs should reproduce the M2 column of Table 2.2 (Mplus gives maximum likelihood estimates, which differ slightly, and so does Stata if reml is omitted).

(lme4)

M2 <- lmer(math ~ homework + (homework | school), data = nels, REML = TRUE)

(PROC MIXED)

SAS

proc mixed data=nels method=reml covtest;
  class school;
  model math = homework / solution;
  random intercept homework / subject=school type=un;
run;

(MIXED)

SPSS

MIXED math WITH homework
  /FIXED = homework | SSTYPE(3)
  /METHOD = REML
  /PRINT = SOLUTION TESTCOV
  /RANDOM = INTERCEPT homework | SUBJECT(school) COVTYPE(UN).

Stata

mixed math homework || school: homework, covariance(unstructured) reml

(DATA command as in the crosswalk of Chapter 1; the random slope s and the random intercept math are between-level latent variables, and math WITH s frees their covariance)

Mplus

VARIABLE: NAMES = school student sex race homework schtype ses pared math
                  classstr schsize urban geo minority ratio female white public;
          USEVARIABLES = math homework;
          CLUSTER = school;
          WITHIN = homework;
ANALYSIS: TYPE = TWOLEVEL RANDOM;
MODEL:    %WITHIN%
          s | math ON homework;
          %BETWEEN%
          math WITH s;

HLM 8. Level 1: MATH $= β_{0} + β_{1} (HOMEWORK) + r$ . Level 2: $β_{0} = γ_{00} + u_{0}$ ; $β_{1} = γ_{10} + u_{1}$ (activate the random effect $u_{1}$ by clicking it). HLM estimates the full $T$ matrix.

JASP (Mixed Models module). Linear Mixed Models: dependent variable math; fixed effect homework; random-effects grouping factor school; under Random effects, keep the random slope for homework and the Correlations option checked (both are checked by default).

2.10How to Read the Rest of This Book#

From now on every new model is presented in a “three views” box like the one in Section 2.5, drawn as a UMG, and accompanied by the data displays it implies. When a result is easier to see in one view, the text uses that view. Interpretation of coefficients and the theory behind a model are usually clearest in the hierarchical view. Software syntax, interactions, and the effect of centering (Chapter 7) are clearest in the mixed view. Estimation (Chapter 9), tests of variance components (Chapter 11), covariance structures for longitudinal data (Chapter 20), and comparisons with population-average models such as generalized estimating equations (Chapters 23 and 32) are clearest in the marginal view. The UMG is the view to return to when a model becomes complicated, because it shows at a glance which quantities vary at which level.

Chapter summary

The book writes $y_{ij}$ for the outcome of level-1 unit $i$ in cluster $j$ , $β_{0 j}, β_{1 j}$ for cluster-specific coefficients, $γ$ for fixed effects, $u_{0 j}, u_{1 j}$ for random effects with covariance matrix $T$ (elements $τ_{0}^{2}$ , $τ_{1}^{2}$ , $τ_{01}$ ), and $e_{ij}$ for level-1 residuals with variance $σ^{2}$ .
The hierarchical view writes a level-1 regression within clusters and level-2 regressions for its coefficients. The mixed view substitutes the level-2 equations into the level-1 equation, giving a fixed part and a random part; in matrix form, $y_{j} = X_{j} γ + Z_{j} u_{j} + e_{j}$ . The marginal view gives the mean $X_{j} γ$ and covariance $V_{j} = Z_{j} T Z_{j}^{'} + σ^{2} I$ of the outcome.
A random intercept implies compound symmetry; a random slope implies a quadratic variance function and covariances that depend on the predictor. In the NELS:88 example the random slope model implies correlations between pupils of the same school that range from strongly positive to negative.
The Unified Model Graph encodes observability by fill, support by shape, and role by border; fixed parameters are diamonds outside all plates, random effects are white circles inside cluster plates. A well-formed diagram corresponds to a factorization of the joint distribution.
A fixed effect becomes a random effect by moving its diamond into the cluster plate (parameter promotion); the same move with a prior or a latent class gives Bayesian and mixture models.
Each element of a model implies a data display: plates imply small multiples, random coefficients imply spaghetti plots, covariances imply scatterplots of cluster-specific estimates.
All programs need the fixed part, the grouping variable, the random part, and the covariance structure of the random effects. Default covariance structures and estimation methods differ between programs and must be set explicitly.

Key Terms#

compound symmetry
covariance matrix of random effects ( $T$ )
design matrix ( $X_{j}$ , $Z_{j}$ )
fixed effect
fixed part
hierarchical (level-specific) view
level-1 model
level-2 model
linear mixed model
marginal covariance matrix ( $V_{j}$ )
marginal model
model-implied data display
parameter promotion
plate
random effect
random part
slopes as outcomes
Unified Model Graph (UMG)
variance function

Exercises#

A. Concept checks#

For each statement, say whether it belongs to the hierarchical, the mixed, or the marginal view: (a) “Each school has its own regression of anxiety on workload.” (b) “The variance of anxiety increases with workload.” (c) “The model contains the product of school climate and workload.” (d) “Two teachers in the same school have correlated anxiety scores, and the correlation is larger when both have a high workload.” (e) “The slope of workload is predicted by school climate.” (f) “The error term is $u_{0 j} + u_{1 j} x_{ij} + e_{ij}$ .”
In Figure 2.4(b), explain why $z_{j}$ is drawn inside the school plate but outside the pupil plate, and why $γ_{01}$ is drawn outside both plates.
A colleague reports the correlation between the random intercept and slope as $- .83$ and concludes that “schools with high achievement benefit less from homework.” What additional information is needed before this interpretation can be judged? (Hint: see the pitfall on where $x = 0$ .)
Which of the following change when a level-1 predictor with a random slope is shifted by a constant: $γ_{00}$ , $γ_{10}$ , $τ_{0}^{2}$ , $τ_{1}^{2}$ , $τ_{01}$ , $σ^{2}$ , the deviance?

B. Analytical exercises#

Consider the hierarchical model $y_{ij} = β_{0 j} + β_{1 j} x_{1 ij} + β_{2 j} x_{2 ij} + e_{ij}$ , with $β_{0 j} = γ_{00} + γ_{01} z_{j} + u_{0 j}$ , $β_{1 j} = γ_{10} + γ_{11} z_{j} + u_{1 j}$ , and $β_{2 j} = γ_{20}$ . (a) Write the mixed model and identify its fixed and random parts. (b) Write $X_{j}$ , $γ$ , and $Z_{j}$ for a cluster with three units. (c) Derive $V_{j}$ for this cluster. Solution →
Show that under a random intercept model $V_{j} = τ_{0}^{2} 1 1^{'} + σ^{2} I$ has compound symmetry, and find the correlation between any two units of the same cluster. Solution →
For the random slope model, show that the variance function Equation (2.9) has its minimum at $x = - τ_{01} / τ_{1}^{2}$ , and compute the minimum value. Using the M2 estimates, verify the values 115.7 (at $h = 0$ ) and 72.8 (at the minimum) reported in the text. Solution →
Show that if $x$ is replaced by $x^{*} = x - c$ , the random slope model with parameters $(τ_{0}^{2}, τ_{01}, τ_{1}^{2})$ is equivalent to a model with $τ_{1}^{* 2} = τ_{1}^{2}$ , $τ_{01}^{*} = τ_{01} + c τ_{1}^{2}$ , and $τ_{0}^{* 2} = τ_{0}^{2} + 2 c τ_{01} + c^{2} τ_{1}^{2}$ . Solution →

C. Computational exercises#

Fit M0, M1, and M2 to the NELS:88 data in R and reproduce Table 2.2. Then fit M2 in one other program from the crosswalk, once with the default covariance structure and once with an unstructured one, and explain the differences.
Using the M2 estimates, compute the model-implied covariance matrix $V_{j}$ for a school with four pupils whose homework categories are 0, 1, 3, and 6, and convert it to a correlation matrix. Which pairs of pupils are negatively correlated, and why? Solution →
Center homework at its grand mean and refit M2. Compare $\overset{τ}{^}_{0}^{2}$ , $\overset{τ}{^}_{01}$ , and the deviance with those in Table 2.2, and check the results against exercise B4.
Draw M2 as a UMG with the umg package, both by hand with umg_model() and with umg_from_lmer(), and run umg_validate() on each. Then produce the displays of Figure 2.6 with umg_eda_scaffold().

D. Simulation exercise#

Write a script (with the book’s header and block structure) that simulates 5,000 schools of four pupils each from M2, with the M2 estimates as true values and homework fixed at 0, 1, 3, and 6 within every school. Compute the empirical covariance matrix of the four scores across schools and compare it with the matrix you derived in exercise C2. How large must the number of simulated schools be for the two to agree to one decimal place? Solution →

E. Applied mini-project#

Choose a published article that reports a two-level model with at least one random slope. (a) Write the model in the three views, using the published estimates where they are reported. (b) Draw it as a UMG. (c) List what the article does not report that you needed for (a), for example the covariance of the random effects or the centering of the predictors. (d) Write a two-sentence comment to the authors about the missing information.

Lab#

References

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Curran, P. J. (2003). Have multilevel models been structural equation models all along? Multivariate Behavioral Research, 38(4), 529–569. https://doi.org/10.1207/s15327906mbr3804_5

Laird, N. M., & Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38(4), 963–974. https://doi.org/10.2307/2529876

Mehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations modeling. Psychological Methods, 10(3), 259–284. https://doi.org/10.1037/1082-989X.10.3.259

Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Sage.

Singer, J. D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 23(4), 323–355.

Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models. Chapman & Hall/CRC.

Snijders, T. A. B., & Bosker, R. J. (2012). Multilevel analysis: An introduction to basic and advanced multilevel modeling (2nd ed.). Sage.

Verbeke, G., & Molenberghs, G. (2000). Linear mixed models for longitudinal data. Springer. https://doi.org/10.1007/978-1-4419-0300-6

Yu, H.-T. (2026a). The Unified Model Graph: Drawing the statistical models of psychology [Unpublished manuscript].

Yu, H.-T. (2026b). umg: Unified Model Graphs for statistical models in psychology (Version 0.6.1) [Computer software]. https://github.com/hsiutingyu/umg