1. ReduceObserved indicators form input and/or output latent variables.
2. RegressLatent and direct variables enter the selected regression engine.
3. EvaluateMetrics, predictions, residuals, paths, and optimization history are retained.
NDRLM is predictive and exploratory. A fitted path is not, by itself, evidence
of a causal effect. Its role as a data-driven benchmark for structural models
is developed in @Kosztyan2026NBSEM.
# Published latent-input example
The example follows the article's low-sample-size `mtcars` analysis. The ten
input indicators form two communities. The first groups `cyl`, `hp`, `qsec`,
`vs`, and `carb`; the second groups `disp`, `drat`, `wt`, `am`, and `gear`.
The two resulting input latent variables predict `mpg`.
```{r latent-input-fit}
X_car <- mtcars[, setdiff(names(mtcars), "mpg")]
Y_car <- mtcars["mpg"]
input_membership <- c(
cyl = 1, disp = 2, hp = 1, drat = 2, wt = 2,
qsec = 1, vs = 1, am = 2, gear = 2, carb = 1
)
fit_latent <- ndrlm(
Y_car, X_car,
latents = "in",
optimize = FALSE,
membership.X = input_membership,
regression = "lm",
seed = 42
)
fit_latent
summary(fit_latent)
```
The NDA object remains available inside the supervised model, so the
measurement structure is inspectable rather than hidden.
```{r latent-input-results}
data.frame(
indicator = colnames(fit_latent$NDAin$R),
community = unname(fit_latent$NDAin$membership)
)
round(fit_latent$NDAin$loadings, 3)
coef(fit_latent$fits[[1]])
fit_latent$fit_stats
```
The latent scores are computed only from their own community indicators. The
OLS coefficients then describe the fitted relationship between each input
latent variable and `mpg`.
# Predictions and residual diagnostics
```{r latent-prediction}
prediction <- predict(fit_latent, X_car)
diagnostic <- data.frame(
observed = Y_car$mpg,
fitted = fitted(fit_latent)$mpg,
residual = residuals(fit_latent)$mpg
)
head(prediction)
head(diagnostic)
```
```{r diagnostic-figure, fig.cap="Observed versus fitted fuel efficiency and residual diagnostic for the latent-input model."}
op <- par(mfrow = c(1, 2), mar = c(4, 4, 3, 1))
plot(diagnostic$observed, diagnostic$fitted,
pch = 19, col = "#0072B2",
xlab = "Observed mpg", ylab = "Fitted mpg",
main = "Observed versus fitted")
abline(0, 1, col = "grey45", lty = 2)
plot(diagnostic$fitted, diagnostic$residual,
pch = 19, col = "#009E73",
xlab = "Fitted mpg", ylab = "Residual",
main = "Residual pattern")
abline(h = 0, col = "grey45", lty = 2)
par(op)
```
These plots are descriptive in-sample diagnostics. A predictive study should
evaluate the complete reduction-and-regression pipeline using held-out data or
resampling.
# Input, output, and full latent models
| `latents` | Predictors used by regression | Responses used by regression |
|---|---|---|
| `"none"` | observed `X` | observed `Y` |
| `"in"` | input NDA scores | observed `Y` |
| `"out"` | observed `X` | output NDA scores |
| `"both"` | input NDA scores | output NDA scores |
The next executable example reduces both sides. Two observed response
indicators form one output latent variable. With `dircon = TRUE`, membership
zero would retain an excluded indicator as a direct observed variable.
```{r both-sides-fit}
X_both <- mtcars[, c("cyl", "disp", "hp", "wt", "qsec", "vs", "am", "gear")]
Y_both <- mtcars[, c("mpg", "drat")]
fit_both <- ndrlm(
Y_both, X_both,
latents = "both",
optimize = FALSE,
membership.X = c(1, 2, 1, 2, 1, 1, 2, 2),
membership.Y = c(1, 1),
regression = "lm",
seed = 42
)
fit_both$NDAin$membership
fit_both$NDAout$membership
fit_both$indep_names
fit_both$dep_names
head(fitted(fit_both))
```
# Regression engines with visible results
The enclosing NDRLM interface is the same for every engine. OLS provides
coefficient tests and intervals. Ridge stabilizes correlated predictors.
Elastic net combines shrinkage with coefficient selection. LOESS provides a
small-predictor nonlinear model but no linear-path significance tests.
```{r engine-comparison}
fit_ols <- fit_latent
if (requireNamespace("glmnet", quietly = TRUE)) {
fit_ridge <- ndrlm(
Y_car, X_car,
latents = "in", optimize = FALSE,
membership.X = input_membership,
regression = "ridge", lambda = 0.05,
seed = 42
)
fit_elastic <- ndrlm(
Y_car, X_car,
latents = "in", optimize = FALSE,
membership.X = input_membership,
regression = "elastic_net", alpha = 0.5, lambda = 0.05,
seed = 42
)
engine_results <- rbind(
OLS = fit_ols$fit_stats,
Ridge = fit_ridge$fit_stats,
Elastic_net = fit_elastic$fit_stats
)
engine_results
round(cbind(
ridge = fit_ridge$fits[[1]]$standardized_coefficients,
elastic_net = fit_elastic$fits[[1]]$standardized_coefficients
), 3)
} else {
cat("Install the suggested package 'glmnet' to run ridge and elastic net.\n")
}
```
```{r loess-fit}
fit_loess <- ndrlm(
Y_car,
X_car["wt"],
latents = "none",
regression = "loess",
loess_span = 0.80
)
head(predict(fit_loess, X_car["wt"]))
fit_loess$fit_stats
```
The displayed ridge and elastic-net values are standardized coefficients.
The path plots use the same scale by default, making linear path magnitudes
comparable across differently scaled predictors and responses. Prediction
metrics should still be compared on a common validation design when
prediction is the scientific goal.
# NSGA-II optimization and convergence history
When `optimize = TRUE`, NSGA-II tunes indicator weights and the four NDA
thresholds. This deliberately small run demonstrates the complete interface.
Larger populations, more generations, and repeated runs are appropriate for a
final analysis.
```{r optimization}
fit_optimized <- ndrlm(
Y_car, X_car,
latents = "in",
membership.X = input_membership,
optimize = TRUE,
target = "adj.r.square",
rel_weight = TRUE,
lower.bounds.x = rep(0.5, ncol(X_car)),
upper.bounds.x = rep(1.5, ncol(X_car)),
lower.bounds.latentx = c(0, 0, 0, 0),
upper.bounds.latentx = c(0.01, 0.01, 0.01, 0.01),
popsize = 4,
generations = 3,
restarts = 1,
regression = "lm",
seed = 42
)
fit_optimized$optimized
fit_optimized$fval
round(fit_optimized$NDAin_weight, 3)
tail(fit_optimized$NSGA_trace)
```
```{r convergence-figure, fig.cap="Best adjusted R-squared encountered during the compact NSGA-II run."}
trace <- fit_optimized$NSGA_trace
plot(
trace$evaluation,
trace$best_target,
type = "s", lwd = 2, col = "#0072B2",
xlab = "Objective evaluation",
ylab = "Best adjusted R-squared",
main = "NSGA-II search history"
)
points(trace$evaluation, trace$target_value,
pch = 16, cex = 0.65, col = grDevices::adjustcolor("grey35", 0.45))
lines(trace$evaluation, trace$best_target, lwd = 2, col = "#0072B2")
```
The grey points are evaluated candidates; the blue step line is the best
target value encountered so far. A short flat segment is not proof of global
convergence, but the trace makes search progress transparent and helps compare
longer runs and restarts.
# Static and interactive latent path networks
## Static latent path network
The both-sides model is used here because it shows the full four-layer
structure: observed inputs, two input latent variables, one output latent
variable, and observed outputs.
```{r ndrlm-static-network, fig.cap="Both-sides NDRLM graph. Rectangles are observed indicators, circles are latent variables, and path colour carries sign."}
path_network <- plot(
fit_both,
sig = 1,
interactive = FALSE,
show_edge_labels = TRUE,
shape = "automatic"
)
path_network
```
Observed variables are rectangles and latent variables are circles, following
standard structural-model notation. Each latent variable and its observed
indicators share one colour from the default light qualitative palette.
Measurement loadings are grey. Positive
regression paths are green, negative paths are red, and absolute magnitude is
mapped to width separately within loadings and structural paths. Structural
labels and widths use standardized coefficients for OLS, ridge, and elastic
net; loading labels retain their loading scale. Measurement
links between observed and latent variables are dashed; structural paths are
solid. Latent nodes are centred on the indicators assigned to them. Active
layers are packed without leaving blank columns, and the default fills are
slightly lightened and translucent. Edge labels use opaque backgrounds
displaced from their paths. Colours, line types, shapes,
and the minimum/maximum width can be overridden explicitly when a publication
requires another convention.
## Interactive latent path network
The interactive graph preserves the static layout and visual grammar. Nodes
remain movable, and edge labels stay readable on an opaque background while
the view is zoomed or rearranged.
```{r ndrlm-interactive-network}
interactive_path_network <- plot(
fit_both,
sig = 1,
interactive = TRUE,
show_edge_labels = TRUE,
shape = "automatic"
)
interactive_path_network$widget
```
# References
The references connect the original and generalized NDA methods to the package
implementation and the supervised network-based structural extension
[@Kosztyan2022; @Kosztyan2023; @Kosztyan2024; @Kosztyan2026Packages;
@Kosztyan2026NBSEM].
::: {#refs}
:::