NMF-RE: Mixed-Effects Modeling with nmfkc

Introduction

This vignette demonstrates NMF-RE (Non-negative Matrix Factorization with Random Effects), a mixed-effects extension of NMF implemented in the nmfkc package.

Why Mixed Effects?

Standard NMF with covariates models the data as: \[Y \approx X \Theta A\]

where $\Theta A$ captures systematic (fixed) effects of covariates $A$ on latent scores. However, in many applications — such as longitudinal studies, panel data, or clustered observations — individuals exhibit unit-specific deviations that cannot be explained by covariates alone.

NMF-RE addresses this by adding random effects $U$: \[Y = X(\Theta A + U) + \mathcal{E}\]

$Y$ ($P \times N$): Non-negative observation matrix (e.g., repeated measurements over $P$ time points for $N$ subjects).
$X$ ($P \times Q$): Non-negative basis matrix learned from data.
$\Theta$ ($Q \times K$): Coefficient matrix for covariate effects (denoted C in the package output).
$A$ ($K \times N$): Covariate matrix (e.g., intercept, sex, treatment).
$U$ ($Q \times N$): Random effects matrix capturing individual deviations.
$\mathcal{E}$: Residual noise with variance $\sigma^2$.

The random effects follow $\mathrm{Var}(\mathrm{vec}(U)) = \tau^2 I$, and the penalty $\lambda = \sigma^2 / \tau^2$ determines the degree of shrinkage. The marginal model is \[\boldsymbol y_n \sim N\!\big(X\Theta\boldsymbol a_n,\; \tau^2 X X^\top + \sigma^2 I\big).\]

Key Features

EM/ECM estimation: An outer–inner ECM algorithm alternates a ridge/BLUP step for $U$, a semi-NMF step for the basis $X$, and a fixed-effect update for $\Theta$ (inner loop, fixed $\lambda$), with EM M-steps for $(\sigma^2, \tau^2)$ (outer loop). The variance components are estimated from the data — there is no tuning parameter to set.
Sign-free or non-negative coefficients via C.signed (see below).
Wild bootstrap inference: Standard errors, z-values, p-values, and confidence intervals for $\Theta$ without repeated constrained re-optimization.
Trace-based ICC: Quantifies the proportion of variance attributable to random effects.

Sign convention: `C.signed`

The single argument C.signed selects the whole estimation scheme:

C.signed = TRUE (default, recommended): the coefficients $\Theta$ are treated as real-valued (sign-free), updated by exact least squares. Inference uses a two-sided test (interior null $H_0: \Theta_{qk} = 0$). This is the natural choice when covariate effects can be positive or negative.
C.signed = FALSE: $\Theta$ is constrained to be non-negative, updated by a multiplicative update, with a one-sided test (boundary null $H_0: \Theta_{qk} = 0$ vs $H_1: \Theta_{qk} > 0$). Use this for compositional or intensity-type scores.

In both cases the basis $X$ is always non-negative.

1. Data: Orthodont (Longitudinal Growth)

We use the Orthodont dataset from the nlme package: orthodontic distance measurements for 27 children at ages 8, 10, 12, and 14. The covariate of interest is sex (Male/Female).

library(nmfkc)
library(nlme)
#> Warning: package 'nlme' was built under R version 4.4.3

data(Orthodont)
head(Orthodont)
#> Grouped Data: distance ~ age | Subject
#>   distance age Subject  Sex
#> 1     26.0   8     M01 Male
#> 2     25.0  10     M01 Male
#> 3     29.0  12     M01 Male
#> 4     31.0  14     M01 Male
#> 5     21.5   8     M02 Male
#> 6     22.5  10     M02 Male

1.1 Prepare the Observation Matrix Y

Each column of $Y$ is a subject, each row is a time point (age). Since there are 4 ages and 27 subjects, $Y$ is $4 \times 27$.

Y <- matrix(Orthodont$distance, nrow = 4, ncol = 27)
colnames(Y) <- paste0("S", 1:27)
rownames(Y) <- paste("Age", c(8, 10, 12, 14))
Y[, 1:6]
#>        S1   S2   S3   S4   S5   S6
#> Age 8  26 21.5 23.0 25.5 20.0 24.5
#> Age 10 25 22.5 22.5 27.5 23.5 25.5
#> Age 12 29 23.0 24.0 26.5 22.5 27.0
#> Age 14 31 26.5 27.5 27.0 26.0 28.5

1.2 Prepare the Covariate Matrix A

We create a $2 \times 27$ covariate matrix with an intercept row and a binary male indicator.

male <- ifelse(Orthodont$Sex[seq(1, 108, 4)] == "Male", 1, 0)
A <- rbind(intercept = 1, male = male)
A[, 1:6]
#>           [,1] [,2] [,3] [,4] [,5] [,6]
#> intercept    1    1    1    1    1    1
#> male         1    1    1    1    1    1

2. Fitting the NMF-RE Model

We fit the model with rank = 1 (a single latent growth trend). No tuning parameter is required: the variance components $(\sigma^2, \tau^2)$, and hence the penalty $\lambda$, are estimated by the EM/ECM algorithm. We keep the default C.signed = TRUE, since the male effect on growth could in principle have either sign. The prefix argument labels the basis.

res <- nmfre(Y, A, rank = 1, prefix = "Trend")

2.1 Model Summary (fit)

nmfre() performs optimization only. summary() on a freshly fitted object shows the variance components, fit statistics, and the $\mathrm{df}_U$ diagnostic; standard errors and p-values for $\Theta$ are added in a separate inference step (Section 2.2), mirroring the nmfkc() / nmfkc.inference() split.

summary(res)
#> NMF-RE: Y(4,27) = X(4,1) [C(1,2) A + U(1,27)]
#> Iterations: 41 (converged, epsilon = 1e-05)
#> R-squared (cor^2):    0.8246 (XB+blup), 0.4167 (XB)
#> R-squared (uncentered):     0.9974 (XB+blup), 0.9915 (XB)
#> R-squared (centered): 0.7594 (XB+blup), 0.2134 (XB)
#> 
#> Variance components:
#>   sigma2 = 1.933  (residual)
#>   tau2   = 48.19  (random effect)
#>   lambda = 0.04011  (sigma2 / tau2)
#>   ICC    = 0.6100  (tau2*tr(X'X) / (tau2*tr(X'X) + sigma2*P))
#>   dfU    = 23.28
#> 
#> Coefficients (Theta): run nmfre.inference(fit, Y, A) for SE / p-values.

Variance components:

$\sigma^2$: Residual variance (measurement noise).
$\tau^2$: Random effect variance (individual heterogeneity).
$\lambda = \sigma^2/\tau^2$: The shrinkage penalty on $U$, derived from the estimated variances.
ICC: Proportion of variance from random effects. Higher ICC means stronger individual differences relative to noise.
$\mathrm{df}_U$: The effective degrees of freedom consumed by the random effects, reported purely as a complexity diagnostic (the saturation ratio $\mathrm{df}_U/(NQ)$ is shown by res$dfU.frac). No cap is imposed.

cat("sigma^2 =", round(res$sigma2, 4), "\n")
#> sigma^2 = 1.9328
cat("tau^2   =", round(res$tau2, 4), "\n")
#> tau^2   = 48.1908
cat("lambda  =", round(res$lambda, 5), "\n")
#> lambda  = 0.04011
cat("dfU     =", round(res$dfU, 2),
    "  dfU/(NQ) =", round(res$dfU.frac, 4), "\n")
#> dfU     = 23.28   dfU/(NQ) = 0.8622

R-squared:

$R^2$ (XB+blup): Fit including both fixed and random effects.
$R^2$ (XB): Fit from fixed effects only — the gap shows how much the random effects improve the fit.

2.2 Inference on $\Theta$ with `nmfre.inference()`

To obtain standard errors, z-values, p-values, and confidence intervals for $\Theta$, pass the fit to nmfre.inference() (sandwich SE + wild bootstrap, conditional on the estimated $\hat X, \hat U$). It returns the same object with the inference fields added, so summary() now prints the coefficient table.

res <- nmfre.inference(res, Y, A, wild.B = 1000)
summary(res)
#> NMF-RE: Y(4,27) = X(4,1) [C(1,2) A + U(1,27)]
#> Iterations: 41 (converged, epsilon = 1e-05)
#> R-squared (cor^2):    0.8246 (XB+blup), 0.4167 (XB)
#> R-squared (uncentered):     0.9974 (XB+blup), 0.9915 (XB)
#> R-squared (centered): 0.7594 (XB+blup), 0.2134 (XB)
#> 
#> Variance components:
#>   sigma2 = 1.933  (residual)
#>   tau2   = 48.19  (random effect)
#>   lambda = 0.04011  (sigma2 / tau2)
#>   ICC    = 0.6100  (tau2*tr(X'X) / (tau2*tr(X'X) + sigma2*P))
#>   dfU    = 23.28
#> 
#> Coefficients:
#>                   Estimate Std. Error (Boot) z value Pr(>|z|) 
#> intercept:Trend1    90.506      2.472  2.524   36.62   <2e-16 ***
#>      male:Trend1     9.423      3.057  3.070    3.08 0.002052 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> C (= Theta) update: sign-free (real-valued); p-values two.sided

Coefficient table ($\Theta$):

Estimate: Effect of each covariate on the latent trend.
Std. Error: Analytic (sandwich) standard error; the (Boot) column is the wild-bootstrap standard error.
Pr(>|z|): With the default C.signed = TRUE, this is a two-sided p-value ($H_0: \Theta_{qk} = 0$). Both the intercept and the male effect are detected as statistically significant.

The sidedness follows the fit: sign-free $\Theta$ (C.signed = TRUE) uses a two-sided test, while the non-negative variant (C.signed = FALSE) uses a one-sided (boundary) test.

3. Visualization

We plot the growth curves to see how the model captures both population-level trends (fixed effects) and individual deviations (random effects).

age <- c(8, 10, 12, 14)

plot(age, res$XB[, 1], type = "n", ylim = range(Y),
     xlab = "Age (years)", ylab = "Distance (mm)",
     main = "Orthodont: NMF-RE Growth Curves")

# Plot observed data points
for (j in 1:27) {
  pch_j <- ifelse(male[j] == 1, 4, 1)
  points(age, Y[, j], pch = pch_j, col = "gray60")
}

# Plot individual predictions (fixed + random effects)
for (j in 1:27) {
  lines(age, res$XB.blup[, j], col = "steelblue", lty = 3, lwd = 0.8)
}

# Plot population-level fixed effects (two lines: male and female)
for (j in 1:27) {
  lines(age, res$XB[, j], col = "red", lwd = 2)
}

legend("topleft",
  legend = c("Fixed effect (male/female)", "Fixed + Random (BLUP)",
             "Male (observed)", "Female (observed)"),
  lwd = c(2, 1, NA, NA), lty = c(1, 3, NA, NA),
  pch = c(NA, NA, 4, 1),
  col = c("red", "steelblue", "gray60", "gray60"),
  cex = 0.85)

Interpreting the Plot

Red solid lines: Population-level fitted values from fixed effects ($X \Theta A$). There are two distinct curves — one for males (higher) and one for females (lower) — reflecting the sex effect in $\Theta$.
Blue dashed lines: Individual-level predictions ($X(\Theta A + U)$). Each line is shifted from the population curve by the subject’s random effect $U$, capturing personal growth trajectories.
Gray points: Observed measurements. Crosses ($\times$) for males, circles ($\circ$) for females.

4. Examining Learned Components

4.1 Basis Matrix X

The basis $X$ represents the latent temporal pattern shared across all subjects.

cat("Basis X (temporal pattern):\n")
#> Basis X (temporal pattern):
print(round(res$X, 4))
#>        Trend1
#> Age 8  0.2308
#> Age 10 0.2408
#> Age 12 0.2567
#> Age 14 0.2717

Since rank = 1, there is one basis vector. Its shape shows how the latent factor manifests across the four ages. The column is normalized to sum to 1.

4.2 Coefficient Matrix $\Theta$

$\Theta$ (the C element) maps covariates to latent scores:

cat("Coefficient matrix (Theta):\n")
#> Coefficient matrix (Theta):
print(round(res$C, 4))
#>        intercept   male
#> Trend1   90.5058 9.4227

intercept: Baseline level of the latent trend for females.
male: Additional contribution for males. A significant value indicates that males have higher orthodontic distances on average.

4.3 Random Effects U

$U$ captures individual deviations. Subjects with positive $U$ values grow faster than the population average; negative values indicate slower growth.

barplot(res$U[1, ], names.arg = colnames(Y),
        las = 2, cex.names = 0.7,
        col = ifelse(male == 1, "steelblue", "salmon"),
        main = "Random Effects (U) by Subject",
        ylab = "Random effect value")
legend("topright", fill = c("steelblue", "salmon"),
       legend = c("Male", "Female"), cex = 0.85)

5. Model Diagnostics

5.1 Convergence

Check that the algorithm converged properly:

cat("Converged:", res$converged, "\n")
#> Converged: TRUE
cat("Iterations:", res$iter, "\n")
#> Iterations: 41
cat("Stop reason:", res$stop.reason, "\n")
#> Stop reason: outer_lambda

plot() shows the marginal negative log-likelihood $\ell(X,\Theta,\sigma^2,\tau^2)$ (random effects integrated out), which the ECM algorithm decreases monotonically. Note that the fixed-$\lambda$ penalized objective (res$objfunc.iter) is not monotone across outer iterations — it jumps each time $\lambda = \sigma^2/\tau^2$ is updated — so it is unsuitable for illustrating convergence.

plot(res, main = "Convergence (marginal NLL)")

5.2 Residual Analysis

Compare the fitted values against the original data:

residuals <- Y - res$XB.blup
cat("Mean absolute residual (BLUP):", round(mean(abs(residuals)), 4), "\n")
#> Mean absolute residual (BLUP): 0.8799
cat("Mean absolute residual (fixed):", round(mean(abs(Y - res$XB)), 4), "\n")
#> Mean absolute residual (fixed): 1.759

# Fitted vs Observed
plot(as.vector(Y), as.vector(res$XB.blup),
     xlab = "Observed", ylab = "Fitted (BLUP)",
     main = "Observed vs Fitted", pch = 16, col = "steelblue")
abline(0, 1, col = "red", lwd = 2)

6. Comparison: With and Without Random Effects

To appreciate the value of random effects, compare NMF-RE with a standard NMF covariate model (no random effects).

# Standard NMF with covariates (no random effects)
res_fixed <- nmfkc(Y, A = A, rank = 1)
#> Y(4,27)~X(4,1)C(1,2)A(2,27)=XB(1,27)...0sec

cat("=== Standard NMF (fixed effects only) ===\n")
#> === Standard NMF (fixed effects only) ===
cat("R-squared:", round(1 - sum((Y - res_fixed$XB)^2) / sum((Y - mean(Y))^2), 4), "\n\n")
#> R-squared: 0.4161

cat("=== NMF-RE (fixed + random effects) ===\n")
#> === NMF-RE (fixed + random effects) ===
cat("R-squared (XB):      ", round(res$r.squared.fixed, 4), "\n")
#> R-squared (XB):       0.4167
cat("R-squared (XB+blup): ", round(res$r.squared, 4), "\n")
#> R-squared (XB+blup):  0.8246
cat("ICC:                 ", round(res$ICC, 4), "\n")
#> ICC:                  0.61

The improvement from fixed-only $R^2$ to BLUP $R^2$ quantifies the contribution of individual random effects.

7. Why optimization and inference are separated

Because nmfre() does not run the bootstrap, fitting is cheap — useful when the model is fitted many times (rank selection, cross-validation). Run nmfre.inference() only on the final model, and re-run it (without re-fitting) to try different settings such as the number of bootstrap replicates, confidence level, or p-value sidedness:

# coefficient table from the inference object built in Section 2.2
res$coefficients[, c("Basis", "Covariate", "Estimate", "SE", "p_value")]
#>    Basis Covariate  Estimate       SE     p_value
#> 1 Trend1 intercept 90.505774 2.471719 0.000000000
#> 2 Trend1      male  9.422697 3.056691 0.002051691

Visualization with `nmfkc.DOT()`

The inference result is compatible with nmfkc.DOT() for graph visualization. Edges are filtered by significance level and decorated with stars:

dot <- nmfkc.DOT(res, type = "YXA", sig.level = 0.05)
plot(dot)

8. Non-negative Coefficients (`C.signed = FALSE`)

When the latent scores are compositional or intensity-type and the covariate effects must be non-negative, set C.signed = FALSE. The basis is then updated by a positive-part multiplicative rule and inference switches to a one-sided (boundary) test.

res_nn <- nmfre(Y, A, rank = 1, prefix = "Trend", C.signed = FALSE)
res_nn <- nmfre.inference(res_nn, Y, A, wild.B = 500)
cat("All Theta >= 0 :", all(res_nn$C >= 0), "\n")
#> All Theta >= 0 : TRUE
cat("P-value side   :", res_nn$C.p.side, "\n")
#> P-value side   : one.sided
res_nn$coefficients[, c("Basis", "Covariate", "Estimate", "SE", "p_value")]
#>    Basis Covariate  Estimate       SE       p_value
#> 1 Trend1 intercept 90.433728 2.460764 5.757441e-296
#> 2 Trend1      male  9.543617 3.043123  8.559770e-04

For the Orthodont data the male effect is positive, so the sign-free and non-negative fits agree on its direction; the difference matters when an effect is genuinely negative (the non-negative variant clips it to zero).

Summary

NMF-RE provides a principled way to model individual heterogeneity in NMF:

Step	Function	Purpose
1	`nmfre()`	Fit the mixed-effects model (optimization only; variances estimated by EM/ECM)
2	`nmfre.inference()`	Standard errors, p-values, and CIs for $\Theta$
3	`summary()`	Examine variance components and the coefficient table
4	`nmfkc.DOT()`	Visualize with significance stars

When to use NMF-RE:

Longitudinal or repeated-measures data where subjects differ systematically.
Panel data with unit-level heterogeneity.
Any NMF application where fixed covariates alone leave structured residual variation.

Choosing C.signed: keep the default TRUE (sign-free, two-sided) when covariate effects can be positive or negative; use FALSE (non-negative, one-sided) for compositional or intensity scores.

For more details on the underlying methodology, see:

Satoh, K. (2026). Wild Bootstrap Inference for Non-Negative Matrix Factorization with Random Effects. arXiv:2603.01468. https://arxiv.org/abs/2603.01468

Introduction

Why Mixed Effects?

Key Features

Sign convention: `C.signed`

1. Data: Orthodont (Longitudinal Growth)

1.1 Prepare the Observation Matrix Y

1.2 Prepare the Covariate Matrix A

2. Fitting the NMF-RE Model

2.1 Model Summary (fit)

2.2 Inference on \(\Theta\) with `nmfre.inference()`

3. Visualization

Interpreting the Plot

4. Examining Learned Components

4.1 Basis Matrix X

4.2 Coefficient Matrix \(\Theta\)

4.3 Random Effects U

5. Model Diagnostics

5.1 Convergence

5.2 Residual Analysis

6. Comparison: With and Without Random Effects

7. Why optimization and inference are separated

Visualization with `nmfkc.DOT()`

8. Non-negative Coefficients (`C.signed = FALSE`)

Summary

NMF-RE: Mixed-Effects Modeling with nmfkc

Introduction

Why Mixed Effects?

Key Features

Sign convention: C.signed

1. Data: Orthodont (Longitudinal Growth)

1.1 Prepare the Observation Matrix Y

1.2 Prepare the Covariate Matrix A

2. Fitting the NMF-RE Model

2.1 Model Summary (fit)

2.2 Inference on \(\Theta\) with nmfre.inference()

3. Visualization

Interpreting the Plot

4. Examining Learned Components

4.1 Basis Matrix X

4.2 Coefficient Matrix \(\Theta\)

4.3 Random Effects U

5. Model Diagnostics

5.1 Convergence

5.2 Residual Analysis

6. Comparison: With and Without Random Effects

7. Why optimization and inference are separated

Visualization with nmfkc.DOT()

8. Non-negative Coefficients (C.signed = FALSE)

Summary

Sign convention: `C.signed`

2.2 Inference on \(\Theta\) with `nmfre.inference()`

Visualization with `nmfkc.DOT()`

8. Non-negative Coefficients (`C.signed = FALSE`)