Stats

Low variance of the response variable

Step-by-step statistics solution: Low variance of the response variable

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1. What the student is asking (in plain language)

The student is fitting a Gaussian mixed‑effects model with a spatial random field using spaMM.
The dependent variable is a log‑transformed morphological ratio (logSMI). Its raw values vary only a little (range ≈ 0.35–0.52, variance ≈ 5.7 × 10⁻⁴).

When the model is run, spaMM prints the warning

The variance of the response is low, which may lead to imperfect estimation of variance parameters.
Perhaps rescale the response?

The student wants to know:

• Why does low variance cause a problem?
• What does “rescale the response” mean in practice?
• What concrete steps can be taken to remove the warning (or decide it can be ignored)?

Below is a step‑by‑step worked solution that explains the issue, shows how to diagnose it, and gives practical R code for an appropriate rescaling (or alternative strategies).

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2. Why low variance of the response can be a problem

Concept	What happens in a Gaussian mixed model
Likelihood	For a normal model the log‑likelihood contains the term ‑(1/2) * Σ (y_i‑μ_i)² / σ². If all y are very close together, the residual sum of squares is tiny.
Numerical precision	R (and most C/Fortran libraries used under the hood) works with double‑precision numbers (~15 decimal digits). When the residual variance σ² is on the order of 10⁻⁶ or smaller, the computation of ` (y‑μ)² / σ²` can lose precision, leading to flat or poorly behaved likelihood surfaces for the variance components (the random‑effect variances).
Variance‑component estimation	spaMM estimates the random‑effect variance τ by maximizing the (penalised) likelihood. If the data provide almost no information about the scale of the response, the likelihood is nearly flat in τ and σ², so the optimiser may stop with a warning and produce unreliable standard errors.
Interpretation	The warning does not say the model is “wrong”; it merely says that, because the response variation is so tiny, the algorithm may have trouble separating the residual variance from the random‑effect variance. If the random‑effect variance really is (close to) zero, the warning is harmless – the model will simply estimate a variance near zero. If you do expect a non‑zero spatial effect, you need to give the optimiser a better numeric scale.

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3. What “rescaling the response” actually means

Rescaling = applying a linear transformation that changes the units of the response but does not change the underlying statistical model.

Two common linear rescalings are:

1. Standardisation (z‑scoring)
   [ y^{*}= \frac{y-\bar y}{\text{sd}(y)} ] After this transformation the variance of the response is exactly 1.

2. Multiplying by a constant (e.g., 1 000)
   [ y^{*}=c\; y\qquad (c\gg 1) ] This simply stretches the scale so that the residual variance becomes of order 1–10⁰ rather than 10⁻⁶.

Both transformations keep the shape of the Gaussian likelihood unchanged; only the numeric values that the optimiser works with are altered. After fitting, you can back‑transform any predictions or variance estimates to the original scale.

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4. Step‑by‑step solution

Step 1 – Inspect the data and compute the variance

summary(subset_fulldata$logSMI)
var_logSMI <- var(subset_fulldata$logSMI)
var_logSMI
# [1] 5.7e-06   # (example; actual value may differ)

If the variance is < 10⁻⁴, the warning is likely to appear.

Step 2 – Decide whether the warning is serious

• Fit a null model (no random effects) and look at the estimated residual variance:

null_mod <- fitme(logSMI ~ Sex + temperature,
                  data = subset_fulldata,
                  family = gaussian())
summary(null_mod)$sigma2          # residual variance estimate

• If the residual variance is already tiny (≈ 5 × 10⁻⁶) and the spatial variance estimate in the full model is exactly zero (or a very small number with a huge SE), the data simply contain no spatial signal. In that case you may keep the model as‑is and ignore the warning.

• If you expect a spatial component (e.g., strong spatial autocorrelation in raw measurements) and the estimate is non‑zero but imprecise, proceed to rescaling.

Step 3 – Rescale the response

Option A – Standardise (recommended for interpretability)

subset_fulldata$logSMI_z <- scale(subset_fulldata$logSMI,
                                  center = TRUE, scale = TRUE)  # returns a matrix
# keep it as a numeric vector
subset_fulldata$logSMI_z <- as.numeric(subset_fulldata$logSMI_z)

# Fit the model on the standardised response
modSMI_z <- fitme(logSMI_z ~ Sex + temperature +
                 Matern(1 | Long_km + Lat_km),
                 data   = subset_fulldata,
                 family = gaussian())
summary(modSMI_z)

The residual variance will now be ~1, and the random‑effect variances will be on a comparable scale, so the optimiser works comfortably.

Option B – Multiply by a large constant (quick fix)

subset_fulldata$logSMI_k <- 1e4 * subset_fulldata$logSMI   # stretch by 10 000

modSMI_k <- fitme(logSMI_k ~ Sex + temperature +
                 Matern(1 | Long_km + Lat_km),
                 data   = subset_fulldata,
                 family = gaussian())
summary(modSMI_k)

After fitting, divide the estimated residual variance by (10 000)² to obtain the variance on the original scale.

Step 4 – Check that the warning disappeared

# The warning is printed during fitting; if it does not appear, you are good.
# You can also verify that the Hessian is finite:
modSMI_z$opt$convergence   # should be 0 (successful)

Step 5 – Back‑transform variance components (if you need them on the original scale)

For the standardised model (logSMI_z):

• Let σ²_z be the residual variance reported by summary(modSMI_z).

• The original residual variance is

  [ \sigma^{2}{\text{orig}} = \sigma^{2}{z}\;\times\;(\text{sd}(y))^{2} ]

• The spatial variance component τ_z is back‑scaled analogously:

  [ \tau_{\text{orig}} = \tau_{z}\;\times\;(\text{sd}(y))^{2} ]

For the constant‑multiplied model (logSMI_k), divide by the square of the multiplier (e.g., (10^{8}) for 10 000).

sd_y <- sd(subset_fulldata$logSMI)
sigma2_orig <- summary(modSMI_z)$sigma2 * sd_y^2
tau_orig    <- summary(modSMI_z)$theta[1] * sd_y^2   # assuming first theta = spatial variance

Step 6 – Interpret the final model

• Fixed‑effect estimates are unchanged by linear rescaling (they are multiplied by the same constant, so the t‑values and p‑values stay the same).
• Random‑effect variances are now estimated with better numerical precision; confidence intervals are narrower, and the optimiser no longer warns.
• If the spatial variance remains essentially zero after rescaling, you can conclude that the data do not contain a detectable spatial signal.

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5. Final answer (concise)

The spaMM warning occurs because the response variance (≈ 5 × 10⁻⁶) is so small that the numerical optimisation struggles to separate residual variance from random‑effect variance.
The remedy is to rescale the response – either standardise it (subtract the mean and divide by the standard deviation) or multiply it by a large constant – and then refit the model. After rescaling the warning disappears, and variance‑component estimates can be back‑transformed to the original scale for reporting.

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6. Common mistakes & how to avoid them

Mistake	Why it’s wrong	Correct approach
Ignoring the warning completely	You may think the model is fine, but the estimated random‑effect variances can be numerically unreliable (e.g., huge standard errors, boundary estimates).	Check the size of the estimated variances and their SEs; if they are on the order of 10⁻⁸ or hit the lower bound (0), rescale first.
Re‑transforming the response with a non‑linear function (e.g., log again)	Non‑linear transforms change the model (different link), so the original interpretation is lost.	Use only linear rescaling (add/subtract a constant, multiply/divide by a constant).
Standardising without storing the scaling factors	You cannot convert the variance components back to the original units.	Save mean_y and sd_y (or the multiplier) before fitting; use them later to back‑transform.
Rescaling only the random‑effect terms	spaMM rescales the whole model internally; manually scaling the design matrix does not affect the variance‑component optimisation.	Scale the response variable itself; leave covariates untouched (they are already on a reasonable numeric scale).
Using a different family (e.g., binomial) because of low variance	The data are continuous and Gaussian; changing the family would model a completely different distribution.	Keep family = gaussian(); low variance is a numerical issue, not a distributional one.
Rounding the response to fewer decimal places to “increase variance”	Rounding reduces information and can actually make the problem worse.	Keep the original precision; only change the scale (multiply/divide).

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Original question: Low variance of the response variable on Cross Validated (Stats Stack Exchange), licensed CC BY-SA.

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