Chapitre 02

Random sampling

This chapter demonstrates that uniform random sampling is a mathematical requirement in stereology to eliminate any selection bias and ensure the impartiality of measurements. It presents systematic random sampling (SUR) as a more effective method than pure random sampling, as it ensures a more homogeneous spatial coverage of the object. Finally, it introduces random geometry and isotropy, where geometric probes (points, lines) enable complex spatial properties to be estimated through simple counts.

Sampling

Chapter 2 explores the mathematical necessity of random sampling to ensure the impartiality of stereological analyses. It defines uniform randomness (UR), whereby every point in space has an equal probability of being selected, thereby eliminating selection bias.
The text distinguishes between pure random sampling and systematic random sampling (SRS). SRS is more efficient as it covers the space uniformly and reduces the variance of the estimates. It details the practical generation of SRS points in 1D, 2D and 3D using number generators or grids. The chapter introduces stochastic geometry, using geometric probes (points, lines) applied to objects. These probes enable the estimation of lengths, areas and volumes using simple counting rules.
A key concept discussed is isotropy – that is, uniform random orientation – which is essential for the unbiased estimation of areas and lengths. Finally, it points out that accuracy increases with the number of points in a square-root progression. This fundamental chapter lays the theoretical foundations necessary for understanding the validity of the following methods

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Course materials

STERO-CH01-PDF-EN.pdf - 28Mo

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Contents of this chapter

;
Mandatory randomness

This is necessary to eliminate systematic bias and ensure that samples are statistically representative.
;
Uniform random (UR)

Every member of the population must have an equal probability of being selected in order to ensure impartiality.
;
Effectiveness of the SUR

Random systematic sampling is simpler to carry out and often reduces the variability of estimates compared with simple random sampling
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Estimation using probes

Properties such as area or length are estimated by multiplying the number of hits from geometric probes by a grid constant
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Isotropy

All orientations must be equally likely in order to avoid bias when measuring spatial characteristics that depend on orientation

Assessment quiz

Welcome to your Chapter-2

If you measure an area on an image magnified by a factor of M, how do you work out the actual area?

By subtracting the MMM factor from the total sum of the areas.

By multiplying the measured area by M2M^2M2.

By dividing the measured area by M2.

By dividing the area measured by MMM.

Why is systematic random sampling (SRS) generally preferred to pure random sampling (PRS) in practice?

It provides more uniform coverage of the space, thereby reducing the variance in the estimates.

It allows you to measure just one section and use that to work out all the others.

It completely eliminates the need for a random starting point.

It is the only method for calculating an error coefficient.

What is the fundamental relationship used to estimate the volume fraction Vv by counting particles?

Vv=(2×I=/L

Vv=Pp

Vv=Q−×A

Vv=P(total)/P(componant)×T

What is the fundamental formula in Cavalieri’s method for estimating the volume V of an object?

V= ∑A / T

V=T×∑A

V=π×T×Amax

V=T×T×∑A

What major problem does the ‘disector’ solve compared with traditional 2D counting?

The need to measure the diameter of each cell manually.

The bias linked to particle size, whereby larger particles are more likely to be cut off.

The inability to see transparent structures under a microscope.

Tissue deformation during chemical fixation.

What is the ‘central sampling paradox’ referred to by Stuart?

A purely random sample will always result in a visually uniform distribution of points.

This property emphasises that the randomness lies in the selection mechanism and not in the final appearance of the sample.

The larger the sample size, the greater the systematic bias inherent in the method.

It is impossible to tell whether a sample has been drawn at random from a uniform distribution simply by examining it once it has been extracted.

When using a grid of crosses to estimate an area, what is the conventional counting rule set out in the source material?

We count all the crosses that touch the edge of the object, even if only partially.

One in every two crosses is counted to compensate for the border effect.

The cross is counted if a specific point (e.g. the top-right corner of the intersection) falls within the object.

To ensure in-plane isotropy when rotating a linear probe, what range of angles is sufficient?

From 0 to 360 degrees.

From −180 to +180 degrees.

From 0 to 90 degrees.

From 0 to 180 degrees.

To estimate the surface density Sv using linear probes, which formula should be used?

Sv=2×Qa

Sv=∑A×T

Sv=2×I/L

Sv=2π×I/L

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