Design-based stereology and model-based stereology differ in what justifies the step from sampled sections to a three-dimensional estimate. Design-based methods rely on a controlled probability sampling procedure. Model-based methods rely on a mathematical description of the structure being measured. Both can produce useful estimates, but their validity rests on different conditions.
The practical question is not which label sounds more rigorous. It is whether your study can defend its sampling design, its structural assumptions, or both. A method that works for nearly spherical particles may fail when treatment changes their shape. A design-based estimator can also fail when investigators select convenient fields rather than follow the sampling plan.
This distinction belongs within the broader fundamentals of stereology: define the quantity of interest, establish how observations enter the sample, and use an estimator that matches those observations.
What Makes a Method Design-Based?
In design-based inference, the specimen is treated as fixed. Its cells, pores, vessels or grains already have their locations and geometry. Randomness enters through the investigator’s sampling procedure: which sections are selected, where counting frames are placed, and, when required, how probes are oriented.
The estimator uses those known sampling probabilities to make an inference about the defined specimen or region. It does not need to assume that cells are spherical or that every part of an organ contains the same density of structures.
This does not mean every sampling unit must always have an equal probability of selection. Unequal probabilities can be valid when they are known and correctly accounted for. The requirement is a defensible relationship between the sampling procedure and the estimator, developed formally in Davy and Miles’s sampling theory for spatial specimens.
For a simple hypothetical counting design, suppose every target object has a known inclusion probability of 1/100. If 240 objects are counted, multiplying by 100 gives an estimated total of 24,000. The arithmetic is straightforward. Establishing that each object really had the stated opportunity to enter the count is the demanding part.
What Makes a Method Model-Based?
In model-based inference, a mathematical model describes properties of the structure. Those properties allow measurements from sections to support statements about three dimensions. Depending on the method, the model may concern particle shape, size distribution, spatial arrangement or orientation.
There are two usages worth separating. In biological microscopy, “model-based” often describes geometric corrections that assume a particular particle shape or dimension. In mathematical stereology, the term also covers random structure models, such as a model whose statistical properties remain unchanged when its position shifts. Such a model need not assume spherical particles.
The distinction matters because model-based does not automatically mean biased. An estimator may be unbiased under a valid model. The concern is whether the model describes the material well enough for the intended inference, and whether departures from it would change the answer.
Research on anisotropic sampling in stereology, for example, develops model-based estimation for stationary fiber and surface processes. That is a broader statistical approach than simply treating every cell as a sphere.
Design-Based vs Model-Based: Practical Differences
| Question | Design-based approach | Model-based approach |
|---|---|---|
| Where does randomness enter? | Through the sampling design applied to the specimen. | Through a model of the structure, potentially alongside random sampling. |
| What supports the inference? | Known selection probabilities and a matching estimator. | Valid structural assumptions and a matching estimator. |
| Must particle shape be assumed? | Not for shape-independent estimators of quantities such as total number. | Some methods require shape assumptions; others do not. |
| What threatens validity? | Departures from the sampling or measurement protocol. | Model mismatch, as well as sampling and measurement errors. |
| Does collecting more data remove bias? | No, if the implemented design or measurement is biased. | No, if the model is wrong for the intended inference. |
These categories describe the basis of inference, not the microscope. A two-dimensional image can contribute to a design-based volume estimate. A three-dimensional image can still be analyzed using unsupported shape assumptions. “2D versus 3D” is therefore an unreliable substitute for “model-based versus design-based.”
How the Difference Changes an Actual Measurement
Estimating Volume Without Assuming Shape
Consider an irregular tissue region. One approach measures a few dimensions and treats the region as an ellipsoid. Its estimated volume depends on how well that geometric model fits the actual boundary.
A Cavalieri estimate instead uses areas measured on parallel sections at a known spacing, with a uniform random start and coverage of the entire region. For spacing T and section areas Ai, the estimate is T multiplied by the sum of the sampled areas. It does not require an ellipsoid, a sphere or a smooth outline. The original research on systematic sampling efficiency and Cavalieri volume estimation addresses this sampling approach.
For an illustrative calculation, suppose the sampled areas sum to 36 mm2 and the section spacing is 0.5 mm. The volume estimate is 18 mm3. That result follows from the sampling arrangement and area measurements, not from matching the specimen to an ideal solid.
The random start is not optional decoration. Taking regularly spaced sections from an arbitrarily chosen starting position is not the same probability design. For implementation details, see the Cavalieri principle for volume estimation.
Estimating Cell Number Without Counting Profiles as Cells
A cell profile is an intersection between a cell and a section. It is not necessarily one whole cell. A larger cell can intersect more section planes than a smaller one, so profile frequency can change even when total cell number stays constant.
A geometric correction attempts to account for this relationship using information such as section thickness and particle dimensions. That correction needs the appropriate dimension and a defensible model. A diameter measured in the section plane is not automatically a valid substitute for particle extent perpendicular to that plane.
The disector takes a different approach. It uses observations separated in depth and a defined counting event rather than treating every visible profile as a separate particle. Correct implementation permits number estimation without requiring a particle size or shape model, the central result of Sterio’s original disector paper.
Suppose a treatment enlarges nuclei without changing their number. A profile count could suggest an increase because the enlarged nuclei intersect more planes. A properly implemented disector avoids that size-dependent counting mechanism. It still requires reliable identification of the target objects and adherence to its counting rules.
Handling Structures with Preferred Orientations
Orientation is another point where the approaches separate. A structure can have a preferred direction without preventing design-based estimation. What matters is whether the combined sectioning and probe design meets the requirements of the chosen estimator.
For surface estimation, simply applying a convenient line grid to a convenient anatomical section is not enough. Suitable randomization can be introduced through section orientation, probe orientation, or an appropriate combination. Methods using vertical uniform random sections retain a chosen axis while applying the required sampling rules.
Do not extend this requirement indiscriminately. Cavalieri volume estimation does not require isotropic section orientation. The right question is “What randomization does this estimator need?”, not “Have all directions been randomized?”
When Is a Model-Based Method Defensible?
A model-based method deserves consideration when its assumptions are explicit, plausible for the material and open to testing. It becomes harder to defend when those assumptions are chosen mainly because the necessary sections are already available.
Validation should address the intended application. Agreement in untreated tissue does not establish validity after an intervention that changes particle size, shape or orientation. Nor does agreement in one anatomical region establish agreement in another.
A useful comparison comes from a mouse substantia nigra study that evaluated model-based and design-based neuron estimates against serial reconstruction. Neither estimate differed statistically from the reconstruction count in that experiment. The Baquet and colleagues comparison of neuron counting methods demonstrates that a model-based procedure can perform well under tested conditions. It does not establish universal equivalence, and absence of a statistically detectable difference is not proof that two methods are identical.
For a proposed model-based analysis, ask how the result changes across plausible alternative assumptions. If modest changes in the assumed particle dimension reverse the biological interpretation, the model uncertainty belongs in the results, not out of sight in the methods section.
A convenience sample also needs an honest label. Calling selected fields “representative” does not establish either a probability sample or a valid model.
Why Design-Based Does Not Mean Error-Free
A design-based estimator can be unbiased in theory while its laboratory implementation produces biased measurements. The statistical guarantee applies to the specified procedure under its conditions, not to every dataset produced by software bearing a stereology label.
Consider a region where some selected sections are damaged. If those sections are discarded and replaced with nearby attractive sections, the implemented selection procedure may no longer match the planned one. Likewise, a counting rule cannot recover cells that staining consistently fails to reveal.
Section deformation creates another problem. In optical methods, nominal cutting thickness may not describe the depth actually available for measurement after processing. Research introducing estimators for deformed tissue shows that changes in section depth can bias optical disector and fractionator measurements. Thickness assessment and the treatment of unusable section surfaces must therefore match the estimator.
Keep bias separate from precision. An unbiased estimate is not guaranteed to equal the true value in one specimen. Unbiasedness concerns its average behavior over repeated applications of the sampling design. Precision concerns how much those estimates vary.
Collecting more observations can reduce sampling variability without repairing a systematic error. Counting the wrong thing more diligently remains the wrong measurement. A coefficient of error is useful for assessing sampling precision, but it is not a certificate that selection, staining and object identification were correct. These distinctions are developed in bias, precision and coefficients of error.
Choosing an Approach for Your Study
Start with the endpoint, not the available counting tool. Total cell number, number per unit volume, volume fraction and mean particle volume answer different questions. Even a well-executed estimator can answer the wrong one.
For a hypothetical example, a region containing 100,000 cells in 10 mm3 has a numerical density of 10,000 cells/mm3. If its volume falls to 8 mm3 without cell loss, density rises to 12,500 cells/mm3. The increased density does not indicate increased cell number. Decide which quantity supports the biological claim before choosing how to estimate it.
Then assess what the available material permits. A complete region with documented sampling offers different options from a few archived sections with unknown selection history. If the original sampling cannot support a whole-region claim, narrow the claim or state the additional assumptions needed to make it.
Prefer a design-based approach when you can control sampling and want an estimate that does not depend on uncertain structural assumptions. Consider a model-based approach when the model addresses the intended endpoint and has relevant validation. Neither choice excuses poor preparation or unclear object definitions.
Finally, document the basis of inference so another researcher can assess it: the target quantity, reference region, selection procedure, estimator, measurement rules and remaining assumptions. The guide to reporting stereological methods and results covers that record. A defensible estimate needs more than a method name; it needs a traceable explanation of why the observations support the claim.