Why Numerical Scale Matters
A GLMM optimizer works with finite-precision numbers. Even when the model and data are unchanged, very large or very small parameter values can make its calculations unreliable: tiny objective changes may be rounded away, and gradient or curvature calculations may be poorly scaled. The resulting warning is evidence to investigate, not proof that the statistical model is wrong or that the reported estimate is invalid.
The key diagnostic is controlled: express the same model in different, well-defined numerical coordinates, refit it, and compare the resulting fit after translating estimates back to a common scale. If important diagnostics change substantially, the original calculation may have been scale-sensitive. If the warning persists, rescaling alone does not resolve the concern; a genuinely weakly identified or nearly flat direction remains possible.
Rounding, Gradients, and Curvature
In standard double-precision arithmetic, the relative spacing between representable numbers near 1 is about \(2.22\times10^{-16}\). The absolute spacing increases with the magnitude of the number. Thus, subtracting two large, nearly equal log-likelihood values can lose meaningful digits. A finite difference may then be dominated by rounding rather than by the change in likelihood it was meant to measure.
Let \(\ell(\theta)\) denote the log likelihood as a function of a parameter vector \(\theta\). For a single parameter, a central finite-difference approximation to its derivative is
If \(h\) is too large, the approximation has truncation error; if \(h\) is too small, rounding in the two evaluations can overwhelm their difference. Therefore, changing parameter scales can affect numerical gradients and Hessians even though it does not change the underlying likelihood surface.
The coordinate transformation makes this precise. Write \(\theta=\theta_0+D u\), where \(D\) is a nonsingular diagonal matrix of scale factors and \(u\) is a vector of new coordinates. Define the same objective in the new coordinates by \(\tilde{\ell}(u)=\ell(\theta_0+D u)\). The chain rule gives:
These identities explain why numerical gradient sizes and Hessian eigenvalues depend on the coordinate scale. They also show what rescaling cannot do: because \(D\) is nonsingular, it does not change whether the exact Hessian has a zero-curvature direction. It may improve numerical representation of a direction, but it cannot create information the data do not contain.
Worked Example: A Small Likelihood Change at a Large Objective Value
Suppose a hypothetical optimizer evaluates log likelihoods near \(-12{,}345.678901\), and two candidate points differ in log likelihood by \(1.0\times10^{-13}\). Is that difference reliably represented in double precision?
Near \(12{,}345\), numbers lie between \(2^{13}=8{,}192\) and \(2^{14}=16{,}384\). The spacing between adjacent double-precision numbers in this range is approximately
The proposed likelihood difference, \(1.0\times10^{-13}\), is smaller than that spacing. Subtracting the two rounded objective values may therefore produce no difference or a difference altered by rounding. A numerical stopping rule based on such a subtraction may not be informative at this scale.
This calculation does not establish that the likelihood is flat. It shows that the reported objective values may not resolve the change. Subtracting a constant from already rounded values cannot recover lost digits. A more informative check is whether the software can evaluate a suitably centered objective, use more accurate derivatives, or produce stable results under a controlled parameter rescaling. Those checks must still preserve the same model and be compared on a common scale.
Rescaling Without Changing the Model
A common controlled transformation changes the units of a continuous predictor. Suppose a GLMM’s linear predictor contains \(\beta x\), where \(x\) is measured in meters. Define \(x^*=x/1000\), now measured in kilometers, and \(\beta^*=1000\beta\). Then \(\beta^*x^*=\beta x\) exactly. The fixed-effects contribution to the linear predictor is unchanged. If the same predictor enters other parts of the model, such as a random-slope term, those parameters must also be transformed consistently.
More generally, a transformation is controlled only if it preserves all relevant model quantities, including the linear predictor and any covariance structure affected by the transformation. Merely changing a parameter value, fixing it, or changing the random-effects structure is not a rescaling of the same model.
Worked Example: Rescaling a Predictor in a Binary GLMM
Consider a hypothetical binary GLMM with a logit link, fixed intercept \(\beta_0=-3\), a fixed distance effect \(\beta=0.002\) per meter, and a group-specific random intercept \(b_j\). For an observation in a group with \(b_j=0.2\) and distance \(x=1{,}000\) meters, the conditional linear predictor and success probability are
Now use distance in kilometers: \(x^*=1{,}000/1{,}000=1\), and set \(\beta^*=1{,}000(0.002)=2\). The transformed calculation is
For every observation, \(\beta^*x^*=\beta x\), so the conditional likelihood given the random effects is unchanged. The random-intercept distribution is unchanged because this transformation affects only the fixed distance term. Integrating over that same random-intercept distribution therefore also gives the same marginal likelihood. The fitted distance coefficient must be mapped back by \(\beta=\beta^*/1{,}000\); its original unit is log-odds per meter.
A useful numerical check is to compare the mapped estimates, objective values, and convergence diagnostics from both parameterizations. A different warning after rescaling is a reason to investigate numerical sensitivity, not a license to select whichever fit is more convenient.
A Controlled Diagnostic Workflow
Rescaling is informative only when the comparison is designed to isolate scale. First record the original model, estimates, objective value, convergence messages, and any available gradient or Hessian diagnostics. Then choose a transformation with an exact inverse, refit the same model, map estimates back, and compare. The earlier material on scaling predictors and convergence tolerances is useful background; here the goal is specifically to test sensitivity to finite numerical resolution.
Look for parameters with very different magnitudes, predictors with extreme units, or objective changes close to floating-point resolution. Record which diagnostic prompted concern.
Write down the forward and inverse mappings, and verify algebraically that the model’s linear predictor and affected covariance terms are preserved.
Use the same data, model, likelihood target, and relevant fitting settings. Translate parameter estimates to the original units before comparing them.
Check the mapped estimates and objective, and examine available gradient or curvature diagnostics on appropriately scaled coordinates. Decide whether the result is stable enough to support the intended use.
An optimizer’s raw gradient norm is coordinate-dependent: the chain rule shows that it changes under rescaling. An absolute gradient tolerance can therefore look better merely because the coordinate units changed. Interpret that diagnostic with the optimizer’s parameter scaling and stopping rule in mind, and do not treat a smaller raw gradient alone as proof of a more accurate fit.
Worked Example: Interpreting a Rescaling Check
Suppose a hypothetical GLMM has a continuous predictor recorded in meters. In the original fit, the optimizer reports a convergence warning and a poorly conditioned numerical Hessian. After expressing the predictor in kilometers, the same fitting procedure reports no warning. The mapped intercept and slope differ from the original estimates by less than \(0.1\%\), and the two log-likelihood values differ by \(2\times10^{-8}\).
The exact slope transformation is \(\beta^*=1{,}000\beta\), so the original-scale slope from the second fit is \(\beta^*/1{,}000\). The small mapped-estimate differences and close objective values support agreement of the fitted solutions to the displayed precision. The changed warning and Hessian diagnostic indicate that the original numerical calculation was sensitive to parameterization.
This is not a proof that the Hessian warning was harmless. The Hessian diagnostic is numerical, and its behavior depends on the parameter coordinates. A careful report would retain the warning, document the rescaling and comparison, and check that the intended estimates and uncertainty calculations are stable under the transformation. If mapped estimates or substantive conclusions change materially, do not present the warning-free fit as an automatic correction.
What Rescaling Can and Cannot Diagnose
The main alternative checks answer different questions. Trying another optimizer can show whether the numerical result depends on a particular algorithm, but it does not by itself identify a scale problem. Changing a convergence tolerance changes the acceptance criterion, not the underlying precision. A more accurate derivative calculation can reduce finite-difference error, but cannot resolve a likelihood that genuinely contains little information in a direction. These approaches may complement rescaling; they are not interchangeable.
A scale-sensitive warning may be reduced by choosing more suitable units or parameter coordinates. A warning that persists across reasonable, equivalent parameterizations deserves further investigation. The earlier material on flat likelihood directions and profile likelihood as a boundary diagnostic helps distinguish weak information or boundary behavior from a purely numerical issue; a rescaling check does not replace those diagnostics.
- Do not alter the model silently. Changing a variance constraint, removing a random effect, or fixing a parameter is not controlled rescaling.
- Do not compare raw coefficients with different units. Map estimates back before deciding whether they agree.
- Do not equate a warning-free fit with a validated fit. Check the objective, mapped estimates, and relevant diagnostics.
- Do not assume rescaling repairs true non-identifiability. An exact change of coordinates preserves the information in the data.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A predictor is changed from meters to kilometers. What must happen to its fixed slope so that its contribution to the linear predictor is unchanged?
- Why can a raw gradient norm not be compared directly across two parameterizations?
- Near an objective value of magnitude \(10^4\), is a change of \(10^{-14}\) comfortably larger than double-precision spacing?
- If a warning disappears after an exact rescaling, what does that establish—and what does it not establish?
Solutions. (1) If \(x^*=x/1{,}000\), use \(\beta^*=1{,}000\beta\), since \(\beta^*x^*=\beta x\). (2) The chain rule transforms the gradient by the coordinate scale factors, so its numerical magnitude depends on the units. (3) No. The spacing near \(10^4\) is about \(1.82\times10^{-12}\), which is larger than \(10^{-14}\). (4) It shows that the diagnostic is sensitive to parameterization and suggests numerical scale may be involved. It does not prove the fit is well identified, the model is appropriate, or the warning is harmless.