NMF-RE: Mixed-Effects Modeling with nmfkc"

In nmfkc: Non-Negative Matrix Factorization with Kernel Covariates

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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)

data(Orthodont)
head(Orthodont)

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]

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]

---

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)

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$sigma2, 4), "\n")
cat("tau^2   =", round(res$tau2, 4), "\n")
cat("lambda  =", round(res$lambda, 5), "\n")
cat("dfU     =", round(res$dfU, 2),
    "  dfU/(NQ) =", round(res$dfU.frac, 4), "\n")

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)

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")
print(round(res$X, 4))

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")
print(round(res$C, 4))

• 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$converged, "\n")
cat("Iterations:", res$iter, "\n")
cat("Stop reason:", res$stop.reason, "\n")

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

# 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)

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

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

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")]

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$C >= 0), "\n")
cat("P-value side   :", res_nn$C.p.side, "\n")
res_nn$coefficients[, c("Basis", "Covariate", "Estimate", "SE", "p_value")]

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

nmfkc documentation built on July 14, 2026, 1:07 a.m.