Archive for September, 2009

Sampling Trees

September 25, 2009

One of the interesting examples of PPS, probability proportional to sampling, is the use of the angle gauge. This is a single device. It has an arm with a gap at the end of it. The other end of the arm is brought up to the eye. The gap forms an angle.  At a sampling site the person doing t he sampling simply turns in a circle and counts how many trees are larger than the observed angle.  If the tree fits inside of the gap, then the tree is too small to count. The goal in this type of sampling is to estimate the total volume of wood in the forest.

Think about it. A small tree can be very close and  not counted, yet a large tree some distance away is counted. Does that make sense? Does it appear that this is biased sampling? It turns out that the sampling is not biased.

What is clear is that the trees are sampled with different probabilities. The difference is probability is due to a characteristic of the tree. In this case the characteristic is a physical characteristic – the diameter of the tree.

The importance of this sampling technique is that the person doing the sampling is doing a small amount of work relative to a typical quadrat sampling technique.

A quadrat is a sampling site.  It is often a rectangular area. Stopping laying out a rectangle at place in the forest is often slow and not without its complications. There are all sorts of impediments from bushes, cliffs, mud, and water courses to whatever that make it hard to lay out a quadrat.

With the angle gauge there is no quadrat layout step. There are no tree diameter measurements to take. Its turn, count, and go. What a sweet advance. What a huge gain in efficiency!

The proportionator also uses PPS to increase the efficiency of counting. Anyone looking at a vast number of slides to process can only hope that a better way of counting is just around the corner. Now it is here.

Count less, count better, and use the proportionator.

Sampling with equal probability

September 22, 2009

I left off with an issue that is likely to confuse many people. I appear to say yes and no to the equal probability of sampling issue.

The difference is in what infers what. If cells are sampled with equal probability, then it is true that the sampling method could lead to an unbiased answer. There may other factors that prevent the estimate from being unbiased, but the equal probability of sampling is a good start to obtaining an unbiased result.

On the other hand, if cells are not sampled with equal probability it does not necessarily mean that the results are biased.

There are a number of ways in which sampling with unequal probability leads to a biased result. A well known way is profile counting. That is what most people still do today when they count “cells.” In fact, no cells are counted, just profiles seen under the microscope. The thin sections reveal slices through cells. These slices of the cell are called profiles. A number of ways have been conceived to deall with profiles vs cells.
1. Don’t do anything
2. Skip sections to avoid double counting
3. Abercrombie
4. Floderus
5. Rose-Rohrlich

None of these methods leads to an unbiased result.

The proportionator does lead to an unbiased result and makes use of an interesting form of PPS.


September 21, 2009

One of the more recent developments in stereology is the proportionator. This method is likely to be one of the most sought after methods, not because it is new or flashy, but because it makes stereological work easier.

The most common stereological work in the biological sciences is counting cells.  In studies from toxicology to the neurosciences the basic question question is how many cells or structures are there. The first unbiased stereological technique was the physical disector. The next was the optical fractionator.  Now there is the proportionator. just as the optical fractionator reduces the work over the physical disector so does the proportionator reduce the work over the optical fractionator.

If a study requires only a week of counting then you may decide to skip learning about the proportionator. But, if the study requires hundreds or thousands of hours of counting then avoiding the proportionator is a bad idea.

The goal in all three methods
1. physical disector

2. optical fractionator

3. proportionator

is to obtain the correct answer. It is also important that the amount of work is not excessive.  The optical fractionator reduces the work by avoiding the needed to align two sections. The optical fractionator works in situations where the cells are well dispersed such as counting neurons. The optical fractionator fails in many dense tissues such as the liver where it is not possible to focus down through the tissue and obtain optical sections. The proportionator reduces the work by reducing the amount of counting that has to be done.

The basic idea behind the proportionator is that sampling does not always have to be with equal probability. The notion that all cells have to be sampled with equal probability to obtain an unbiased result is incorrect. What is correct is that if cells are sampled with equal probability, then an unbiased estimate is possible.

Stereology – The unknown science

September 17, 2009

Stereology is a rather poorly known science that has been around for quite a while. The purpose of stereology is to develop methods that provide estimates of the size of things. Common measures of size are volume, length, surface, and the numbers of things.

In this blog we’ll explore both recent and innovative finds as well as cover interesting aspects of older issues.