set.seed(4197)
xx <- rep(seq(0.5, 40, length.out = 20), each = 3)
Vm <- 20; K <- 8; cv <- 0.22
mu <- Vm * xx / (K + xx)
y <- mu + rnorm(length(xx), 0, cv * mu) # SD grows with the mean
d <- data.frame(x = xx, y = y)
f_ols <- nls(y ~ Vm * x / (K + x), data = d, start = list(Vm = 15, K = 5))Checking a nonlinear model
R
nonlinear regression
model checking
ecology tutorial
Checking an nls fit in R: spotting non-constant variance in the residuals, why Wald intervals under-cover, and why extrapolation is a choice of form.
A nonlinear fit that converged, and whose parameters are identified, is still not finished. The same three checks that matter for a linear model matter here, with nonlinear twists: are the residuals well behaved, are the intervals honest, and does the fit mean what you want it to mean outside the range you observed. This post, the last in the series, works through each, and ends on the check that no amount of diagnostics can pass for you.
Residuals: look for a funnel
Ordinary nls minimises the plain sum of squared residuals, which assumes the scatter around the curve is the same size everywhere. Ecological measurements often break that assumption: counts and positive quantities usually scatter more when the mean is larger. We simulate a saturating curve whose noise grows with the mean, fit it the ordinary way, and plot residuals against fitted values.
The residual plot shows a clear funnel: the spread widens as the fitted value grows. That is the signature of non-constant variance, and it means the ordinary fit is giving too much weight to the noisy high end, dragging the estimates around and reporting standard errors that assume a constancy that is not there.
Fix it by weighting
The fix is to weight each point by the inverse of its variance, so noisy points count for less. When the standard deviation is roughly proportional to the mean, that means weights near the inverse squared fitted value. Since the weights depend on the fit, we iterate: fit, form weights from the fitted values, refit, a few times.
w <- 1 / fitted(f_ols)^2
f_w <- nls(y ~ Vm * x / (K + x), data = d, start = coef(f_ols), weights = w)
for (i in 1:3) {
w <- 1 / fitted(f_w)^2
f_w <- nls(y ~ Vm * x / (K + x), data = d, start = coef(f_w), weights = w)
}The weighted fit pulls the asymptote estimate from 21.8 back to 20.5, nearer the true 20, and the half-saturation from 10.7 to 8.7, nearer the true 8. Its standard error for the asymptote is 1.37 against the ordinary 2.3: not just different but earned, because it reflects the real error structure. Standardising the weighted residuals removes the funnel.
library(ggplot2)
theme_te <- function(base_size = 12) {
theme_minimal(base_size = base_size) +
theme(plot.background = element_rect(fill = "#f5f4ee", colour = NA),
panel.background = element_rect(fill = "#f5f4ee", colour = NA),
panel.grid.minor = element_blank(),
panel.grid.major = element_line(colour = "#dad9ca", linewidth = 0.3),
axis.title = element_text(colour = "#2c3a31"),
axis.text = element_text(colour = "#46604a"),
strip.text = element_text(colour = "#16241d"),
plot.subtitle = element_text(colour = "#5d6b61"),
legend.position = "none")
}
rd <- rbind(
data.frame(fitted = fitted(f_ols), resid = resid(f_ols),
panel = "ordinary: raw residuals"),
data.frame(fitted = fitted(f_w), resid = resid(f_w) * sqrt(w),
panel = "weighted: standardised residuals"))
ggplot(rd, aes(fitted, resid)) +
geom_hline(yintercept = 0, colour = "#dad9ca") +
geom_point(colour = "#275139", alpha = 0.7, size = 1.6) +
facet_wrap(~ panel, scales = "free_y") +
labs(x = "fitted value", y = "residual", subtitle =
"The funnel on the left is non-constant variance") +
theme_te()
Intervals: profile beats Wald
Earlier posts flagged that the symmetric interval built from a standard error can be over-optimistic. Here is the check. We simulate the saturating curve many times with constant variance this time, and for each fit we record whether the true half-saturation falls inside the Wald interval and inside the profile interval. Over many runs an honest ninety-five per cent interval should contain the truth about ninety-five per cent of the time.
Vm2 <- 20; K2 <- 12; sig <- 1.6
xc <- rep(c(1, 2, 3, 4, 6, 8, 11, 15), each = 2)
mu2 <- Vm2 * xc / (K2 + xc)
set.seed(20260812)
B <- 1200; wald_hit <- 0; prof_hit <- 0; n <- 0
for (b in seq_len(B)) {
yb <- mu2 + rnorm(length(xc), 0, sig)
fb <- try(nls(y ~ Vm * x / (K + x), data = data.frame(x = xc, y = yb),
start = list(Vm = 18, K = 10)), silent = TRUE)
if (inherits(fb, "try-error")) next
se <- summary(fb)$coefficients["K", "Std. Error"]
if (!is.finite(se)) next
n <- n + 1
wK <- coef(fb)["K"] + c(-1, 1) * 1.96 * se
wald_hit <- wald_hit + (K2 >= wK[1] && K2 <= wK[2])
pk <- try(suppressWarnings(confint(fb, "K")), silent = TRUE)
prof_hit <- prof_hit +
if (inherits(pk, "try-error") || !all(is.finite(pk))) 1 else (K2 >= pk[1] && K2 <= pk[2])
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Waiting for profiling to be done...wald_cov <- 100 * wald_hit / n
prof_cov <- 100 * prof_hit / nAcross 1199 fits the Wald interval contains the truth only 89.2 per cent of the time, well short of the nominal ninety-five, because the half-saturation has a skewed sampling distribution and the symmetric interval sits in the wrong place. The profile interval reaches 96.5 per cent, close to nominal. For any parameter that enters the model nonlinearly, prefer the profile interval from confint.
One check to skip: do not judge the fit by an R-squared. Spiess and Neumeyer (2010) show it is not a meaningful summary for nonlinear models. Read the residual plots, compare candidate models by AIC, and report the residual standard error.
The honest limit: extrapolation is a choice of form
Every check so far is about the range you observed. The deepest limit of a nonlinear model is that its behaviour outside that range is set by the functional form you chose, not by the data. To make this concrete, we generate data over a range that stops before the curve levels off, and fit two forms: a saturating one that bends to an asymptote, and an unbounded power curve.
set.seed(4297)
xr <- rep(seq(1, 12, length.out = 16), each = 3) # all below the half-saturation
yr <- 25 * xr / (15 + xr) + rnorm(length(xr), 0, 0.6)
dr <- data.frame(x = xr, y = yr)
mm <- nls(y ~ Vm * x / (K + x), data = dr, start = list(Vm = 20, K = 10))
pw <- nls(y ~ c * x^p, data = dr, start = list(c = 2, p = 0.8))
pred <- function(m, xnew) predict(m, data.frame(x = xnew))On the observed data the two forms are indistinguishable. Their AIC values differ by 1.2 and their residual standard errors are 0.61 and 0.6, so no goodness-of-fit statistic and no residual plot could tell them apart. Yet they say completely different things about what happens next.
gx <- data.frame(x = seq(1, 100, length.out = 400))
gx$saturating <- pred(mm, gx$x)
gx$power <- pred(pw, gx$x)
ggplot() +
annotate("rect", xmin = 1, xmax = 12, ymin = -Inf, ymax = Inf,
fill = "#93a87f", alpha = 0.14) +
geom_point(data = dr, aes(x, y), colour = "#16241d", alpha = 0.5, size = 1.4) +
geom_line(data = gx, aes(x, saturating), colour = "#275139", linewidth = 1) +
geom_line(data = gx, aes(x, power), colour = "#b5534e", linewidth = 1, linetype = "21") +
annotate("text", x = 70, y = pred(mm, 70), label = "saturating", colour = "#275139", vjust = 1.8) +
annotate("text", x = 70, y = pred(pw, 70), label = "power", colour = "#b5534e", vjust = -1) +
labs(x = "x", y = "y", subtitle =
"Shaded band is the observed range; the curves agree there") +
theme_te() + theme(legend.position = "none")
At the edge of the data the two predictions sit within 3 per cent of each other. Push well past the data, to an x of 40, and the power curve predicts 25.6 against the saturating 17.3; at an x of 100 the gap is a factor of 2.4. Nothing in the data chose between them. If you need the asymptote, or any prediction beyond the observed range, that number is coming from your choice of equation, so state the form as an assumption, fit more than one, and report how far apart they drift. A model that passes every check inside its range can still be a guess outside it.
Where this leaves the series
Across four posts the thread has been the same. Fit nonlinear models on their own scale rather than linearising them, get honest starting values and check that the parameters are identified, and read the fit sceptically: profile intervals, weighted residuals, and a clear head about extrapolation. The estimation is the easy part; the assumptions are where the ecology lives.
References
- Spiess and Neumeyer 2010 BMC Pharmacology 10:6 (10.1186/1471-2210-10-6)
- Bates and Watts 1988, Nonlinear Regression Analysis and Its Applications, Wiley (ISBN 978-0-471-81643-0)
- Seber and Wild 1989, Nonlinear Regression, Wiley (ISBN 978-0-471-61760-0)
- Carroll and Ruppert 1988, Transformation and Weighting in Regression, Chapman and Hall (ISBN 978-0-412-01421-4)
- Motulsky and Christopoulos 2004, Fitting Models to Biological Data Using Linear and Nonlinear Regression, Oxford University Press (ISBN 978-0-19-517179-2)
- Bolker 2008, Ecological Models and Data in R, Princeton University Press (ISBN 978-0-691-12522-0)