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Neither mixed models nor permutation tests are a universal winner for spatial case–control analysis. A mixed model represents structure such as grouping or replication through random effects; a permutation test evaluates a specified null by rearranging data in ways that must preserve the study design. They may answer different questions, so choose by the estimand, sampling process, dependence structure, and output you need—not by method label alone.
Start by defining the question your analysis must answer
Spatial case–control data can support several distinct kinds of inference. Before choosing a method, define the outcome, how cases and controls were sampled, the spatial support of the observations, and the scientific target.
- Association: Does case status vary with location or with spatial covariates?
- Risk-surface estimation: Where does a smoothed geographic pattern of case status or risk appear?
- Global clustering: Is the overall spatial pattern more clustered than expected under a stated null?
- Local cluster detection: Is there an unusually concentrated area, perhaps around a prespecified focus?
A smoothed risk map, a global clustering test, and a local cluster statistic are not interchangeable outputs. A method suited to one target does not automatically answer the others.
What each approach represents
| Dimension | Mixed model | Permutation test |
|---|---|---|
| Core idea | Model structured variation using fixed and random effects; spatial random effects can represent spatially structured variation. | Build a null reference distribution by rearranging observations or labels according to a defined randomization scheme. |
| Especially relevant design feature | Repeated or replicated spatial units, clusters, or other grouping that should be represented explicitly. | A defensible null under which the permitted rearrangements preserve the sampling design and relevant dependencies. |
| Typical inferential output | Model-based estimates and tests for effects represented in the model. | A test result relative to the null distribution generated by the chosen rearrangements. |
| Main design risk | Spatial confounding can complicate interpretation of fixed effects when spatial covariates overlap with spatial random effects. | Invalid exchangeability: unrestricted shuffling can break the dependence or constraints that the null must preserve. |
This comparison is about the logic of the methods, not a claim that they estimate the same quantity. A random-effects model and a permutation test can be complementary in some analyses, but one does not automatically substitute for the other.
#1 Best Overall
When a mixed model is a plausible choice
Use the design’s grouping or replication when it matters
A mixed model is a reasonable candidate when observations come from replicated spatial point patterns or other grouped sampling units, and the analysis needs to represent variation among those units. Bell and Grunwald’s 2004 work develops mixed models for replicated spatial point patterns using maximum pseudolikelihood and generalized linear mixed modeling, and compares fixed- and mixed-effect formulations. That evidence supports mixed models for that kind of structure; it does not establish a general preference for mixed models in every case–control study.
Interpret fixed effects carefully when spatial random effects are present
Spatial confounding arises when a smooth covariate pattern aligns with spatial random effects. In that situation, the estimated fixed-effect association can depend on how the model separates covariate and spatial structure. Restricted spatial regression is one approach discussed in the literature, but it is not a universally established fix. Report the model structure and treat fixed-effect interpretation as conditional on those choices.
Rank #2
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When permutation inference is a plausible choice
Make the null and rearrangement explicit
A permutation test is useful when you can state a defensible null and specify exactly what is randomized and what remains fixed. For example, a 2006 population-based case–control mapping study used a generalized additive model (GAM) with a bivariate spatial smoother. It tested whether case status depended on location by comparing model deviances with and without the spatial smoothing term. The investigators conditioned on the case and control counts, randomized locations, and refit the model for each permutation. They used 999 permutations in that particular analysis; that is a study-specific implementation, not a general minimum or recommendation.
That example illustrates one conditional randomization design. It is not a universal recipe: the rearrangement must follow from how the data were sampled and from the null hypothesis being tested.
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Check exchangeability before shuffling
Permutation validity depends on exchangeability: under the null, the observations being rearranged must be interchangeable under the proposed scheme. Spatial correlation, repeated measures, and grouping can violate that assumption. FSL’s permutation documentation warns that correlated data can break exchangeability and describes blocks for accommodating some repeated-measures designs. Blocks do not make every spatial permutation valid; their suitability depends on the study design and null.
A study of spatial random-shift methods also documents a setting in which a procedure that disrupted spatial correlation produced liberal tests. The practical lesson is to avoid unrestricted shuffling when it destroys dependence that should remain under the null. State the randomization unit, constraints, and rationale alongside the p-value.
What comparative performance evidence does—and does not—show
Published performance results are conditional on the simulated or sampled design, the alternative pattern, and the performance measure. One simulation compared permutation-based GAM approaches with a spatial scan statistic, not with mixed models. The scan statistic had the highest power for the study’s circular-cluster scenario, while GAM methods performed better for its point-source and line-source scenarios. GAM sensitivity exceeded the scan statistic’s in all three simulated cases.
Those results show why alternative geometry can matter; they do not establish that permutation-based GAMs always outperform mixed models, or that either method is best for spatial case–control analysis generally. The compared methods and scenarios must be named whenever those findings are used.
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A practical decision sequence
- Specify the estimand. Decide whether you need an association estimate, a smoothed geographic risk surface, a global clustering test, or local cluster detection.
- Write down the sampling design. Record what was sampled, whether case and control counts were fixed, and which observations or labels could plausibly be randomized under the null.
- Represent replication and grouping. If the design includes replicated patterns, repeated units, or clusters whose variation matters, consider whether random effects are needed to represent that structure.
- Define any permutation scheme before interpreting its p-value. Specify what is moved or relabeled, what is held fixed, and why those rearrangements are valid under the stated null.
- Assess dependence and exchangeability. Check whether spatial correlation or repeated measurements rule out unrestricted shuffling; use design-appropriate restrictions only when justified.
- Review spatial confounding if using spatial random effects. Consider whether smooth covariates overlap with the spatial effect and how that affects fixed-effect interpretation.
- Match performance claims to the evidence. Identify the target method, data-generating conditions, alternative geometry, and performance measure; do not turn a study-specific comparison into a general ranking.
Keep nearby spatial methods distinct
Case–control risk mapping, point-process intensity modeling, and cluster detection address related but distinct questions. A bivariate spatial smoother can describe how case status varies geographically under a specified model. Point-process methods focus on the spatial distribution or intensity of events. Scan statistics and other clustering methods assess global or local departures from a null pattern. Choose and describe the method in terms of the question it answers, rather than treating all spatial analyses as competing ways to produce the same map or test.
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