Why is the history of stereology so muddled?

I am often surprised at how people get the history of stereology wrong. Is it really that hard to get it right?

Recently an old article has been passed around the internet like some hoax letter. It makes many of the old mistakes. It tries to give Cavalieri’s work a modern twist by assigning work to him that isn’t done for at least a century. It gets the date of Buffon’s work wrong by 44 years. It assigns the date to the end of his life instead of the early years. Even Delesse is given credit for something he never did.

A while back I was looking into an issue with point counting. I never really knew where it came from although there are reports it was invented by a geologist and then reinvented independently by another geologist. Turns out that biologists invented point counting, not geologists. In fact, it was the rage for many years before the geologists learned about it.

What’s interesting is that many articles and books get it right. Many get it wrong.  The dividing line is 1979. Anyone want to venture which famous book on stereology got it wrong first? Here’s a little juicy tidbit. The geologist erroneously given the honor of inventing point counting in 1930 gave an address in 1938 in which he laments not making use of the the new and efficient method of point counting.

It is also fair to ask if all of this matters? Well it does. People place references in articles to justify their methods. Many articles use references to articles that do not use the stereological methods they employ. Does that sound like one of those health hoax letters that list all sorts of articles as references only to find out that the articles are not about the subject matter? That’s what it looks like, doesn’t it?

The history of stereology was muddled in 1979. The errors are not being corrected, but rather they are being increased by the recent rehash of an old article that is flawed in many, many ways.

Are designed based stereological methods unbiased?

There is a common misunderstanding that design based methods are by definition unbiased. That is simply not true. There are those that have said to me, “It’s design based because I designed this idea. It is designed and therefore unbiased.” I think that latter comment tells you that being a designed based method is not necessarily unbiased. I realize that the person making the comment was unclear about the meaning of design based, but that’s the way it happened.

In a design based approach it is is possible to show whether or not a method is unbiased. It doesn’t mean that biased can’t creep in if the method cannot be implemented properly, but at least there is the hope that the biases introduced during the implementation of the method are not overwhelming.

You might ask yourself if bias if really all that bad. Does it really matter if a method used in stereology has bias? What is that doing to the result? If the amount of bias is small relative to the value being determined, then it might not be bad if the method is biased.

Suppose a method had a bias estimated to be less than 5% and the data showed a 20% difference between control and experimental, then the 5% is not important. The method would be a reasonable method if it saved work.

Unfortunately, biases are difficult to determine. Showing that the bias is less than a certain magnitude is usually impossible.

That is why design based methods that are unbiased are favored. If the method can be shown to have zero bias, then the issue is how close to the mathematical ideal is the implementation.

Typical errors in online stereology information

Recently an old article of sorts has been spread around the internet. This article makes a number of claims that are not grounded in truth.

Consider the following:

In 1637, Bonaventura Cavalieri, a student of Galileo Galilei in Florence during the high Italian Renaissance, showed that the mean volume of a population of non-classically shaped objects could be estimated accurately from the sum of areas on the cut surfaces of the objects (right). The Cavalieri Principle provides the basis for the volume estimation of biological structures from their areas on tissue sections.

It’s true that Cavalieri did publish something in 1637 and he was associated with Galileo. Most of the rest is quite wrong. For instance Cavalieri was not a student. He was a professor at the same university as Galileo. Galileo and Cavalieri had somewhat of an academic problem since Galileo had his own ideas on the subject called infinitesimals. Cavalieri’s ideas did not agree with Galileo.

The real problem here is the math. The statement here about the work of Cavalieri is completely wrong. What Cavalieri’s theorem does is to compare the volumes or areas of two objects of equal height. Let’s consider only volumes here. Two objects are the same height if two parallel planes touch the tops and bottoms of both objects. Cavalieri goes on to prove that if any parallel plane in between the top and bottom planes intersects the two objects and the areas on the 2 planes are the same area, then the volumes are the same.

The reason that Cavalieri’s name is associated with the estimation or measurement of volumes is due to the same type of misuse of Cavalieri’s work. In 1902, two geologists mistakenly connected Cavalieri’s theorem to volume estimation of crystals observed under the microscope. The name stuck. It’s just a misnomer.

Here is another mistake from the same online article.

In 1777, Count George Leclerc Buffon presented the Needle Problem to the Royal Academy of Sciences in Paris, France. The Needle Problem supplies the probability theory for current approaches to estimate the surface area and length of biological objects in an unbiased (accurate) manner.

The records of the RAS, the Royal Academy of Science, show that Buffon gave his presentation in 1733. His name was not Buffon. That’s another mistake in this online article. The name Buffon was a nickname. It comes from a town his ancestors bought. They owned the town and collected taxes for the King. His name was George LeClerc. None of my research shows he was a count, i.e. French royalty. That is yet another mistake.

If you look online you will see that the date 1777 is common. How did this mistake of 44 years come about? Part of it is poor research by the authors. The other part is a lack of understanding of the date. The person that discovered the work of Buffon was Crofton. He was looking over some books and came across an odd entry in a book by Buffon. In 1777 Buffon published a book on Natural History and added a number of odds and ends to the book. One of the add-ons was a paper he had written years earlier to get into the RAS. That is what the records of the RAS show. Buffon was known for adding all sorts of off topic entries into the backs of his books. Crofton recognized the importance of this work and mentioned it and the date of publication, which was 1777.

Today the date of publication is sometimes given erroneously as the date when Buffon presented the work.

In 1815 LaPlace corrected some of Buffon’s work from his 1733 paper. LaPlace did not name the person who’s work he was correcting. Another little item is that in 1736 Buffon did write his second article on the idea. Finally, in 1780 Buffon died. He had been in ill health and was not developing new ideas in the last few years of his life.

Another mistake is the following:

In 1847, the French mining engineer and geologist, Auguste Delesse, demonstrated that the expected value for the volume of an object varies in directly proportion to the observed area on a random section cut through the object. The Delesse Principle provides the basis for accurate and efficient estimation of object and regions volumes by point counting.

Delesse had a good idea. He did not demonstrate or prove. He developed an idea to estimate volume fraction. He never proved it worked. The method was so difficult to implement that the idea was for all intensive purposes never done. A major misleading suggestion is that the method of Delesse involved point counting.

The article is about as accurate in its content as articles about 2012, and claims that the ancient Egyptians lit their temples with electric bulbs. There is some semblance of truth, but the details are a mess of mistakes.

How can anyone suggest that Delesse had anything to do with point counting, or expected values?

How can anyone suggest that the Cavalieri principle deals with one object?

A fair assumption is that the other details provided are just as unreliable as the other claims. It ‘s largely made up and done with a bit of bravado.

Early and Late Recognition

Each observer is different when it comes to doing stereological research. The reason for this is the way that the researchers interpret what they see. One of the basic issues in recognition is deciding whether or not something is a part of the population being counted. A cell or structure seen in the microscope is something to count or something not to count. After a decision is reached about whether or not something should be counted, comes the issue about the intersection of the probes and the item of interest.

The easiest decision is whether or not an object is intersected by a line. A counting frame is composed of 2 types of lines. The green or dashed lines are the inclusion lines and the red or solid lines are the exclusion lines. Touching a line seems to be a simple rule. In fact, there are many cases that becomes gray issues. Whatever decision is made in one place has to be used at all times. The decision that a certain fuzzy condition is a touch or not needs to be used in all similar conditions. The same conditions need to be used for both the inclusion and exclusion lines. If the rules are not applied in a consistent manner then a bias is introduced. This bias may be quite small if the counting frame is small. The bias becomes more pronounced as the counting frame decreases in size.

The most difficult decision is the decision about the z-position of an object. Some observers tend to be early recognizers. The tendency is to see something in focus before others. The idea is that the researcher is focusing through the material and decides that something is in focus before other people. A late recognizer is someone that decides later than other people that an object is in focus. Near the top of the optical disector an early recognizer might say that a cell is not counted, while a later recognizer might decide that the cell is indeed counted. Problems occur when late and early recognition are used consistently across the height of the disector.

If someone uses early recognition at the top of an optical disector and late recognition near the bottom of the optical disector it is as if the optical disector is smaller than it is. The result is that fewer counts are made and the population is underestimated. The opposite can happen as well if someone uses late recognition at the top and early recognition at the bottom, then the result is that more cells are accepted and an over projection occurs.

It does not matter if a person is an early or late recognizer. What matters is being consistent in the application of counting rules regardless of the position where counts are made.

First effort at Computing the Sampling Variance

The two basic things we want out of stereology are the right answer and some idea of how well the work was done.

The second issue is obtaining some measure of how well the sampling went. When sampling is performed there is always some variation between samples. This spread of sample results gives us uncertainty about the answer. So how good is the estimate? That depends on two factors. One is the method that is used and the other is the thing being sampled. The same method applied to two different objects gives two different spreads. There are statistical measures that quantify the spread. These are said to be measures of dispersion.

One of the most important of these is the variance. Another measure of dispersion is the standard deviation. There is a statistical measure called the coefficient of variance. This is a relative measure of the dispersion. These values talk about the sample. The measure called the coefficient of error talks about the method being used.

Possibly the first coefficient of error calculation used was the binomial distribution formula. This was used for point counting in the 1930s and right into the 1960s and possibly even today. The problem with this method was known right from the start. Quite a few rules for sampling were added in the 1930s, 1940s, and 1950s to account for the mathematics of the binomial distribution.

Any mathematical formula is based on the assumption or ground rules used to derive a formula. If a formula is based on the assumption that samples are taken from an infinite set, then applying the sample to a population of 100 things is likely to give you the wrong answer. If the formula is based on the notion that all samples are independent, then using the formula when doing systematic sampling is likely to give you the wrong answer. It is very important to know what ground rules must be met to make a formula provide the correct answer. None of us would be foolish enough to use F=ma and substitute a temperature into the formula. But sampling is so complex that we might apply a formula without realizing that the formula was not applicable.

Point counting in the early years was quickly applied to studies under the microscope. Although it had been developed for large scale studies of land management, by the early 1930s geologists were doing point counting under the microscope. The binomial distribution formula was latched onto as a means of describing the precision of the result.

The problem with using this formula was recognized right away. The sampling was done in a systematic manner, but the formula is very clear that the results must be independent. Geologists doing modal analysis of rocks identify minerals through the microscope by producing thin sections of the rocks and analyzing them using a polarizing microscope. The view under the microscope is of crystals in the rocks revealed in different colors. Suppose you were sampling and several points fell inside of the same crystal under the microscope. Are these independent results?

The sampling was in a systematic manner because it was cheap to build a device that supported systematic sampling. Previously devices relied on the Rosiwal method. Those devices were also cheap to build, but their use was tedious. The newer point counting devices produced sampling speeds of 1500 points per hour. Compare that to modern computerized systems and you see that the older mechanical systems with automatic totals were pretty darn good.

By the 1960s the discussion of the binomial distribution formula was again raised. The underlying mathematics was yet again brought to light and a serious discussion of the limitations and requirements were openly discussed in the literature.

Eventually the matter was addressed in more theoretical terms. Instead of justifying the use of a particular formula, work was done to see what needed to be done. The use of the binomial distribution formula dropped as better methods were developed. These newer methods dropped the independent requirement. The results that are obtained now are in keeping with the nature of the sampling that is being done.

Who needs to use the Proportionator

Anyone doing stereology and counting cells needs the proportionator.

• If you use your stereological equipment more than once a year you need the proportionator.
• If you have tens or hundreds or more tissue samples to process  you need the proportionator.
• If you find the the CE is often too large then  you need the proportionator
• If you have plans to increase your research you need the proportionator
• If you are moving to a larger experiment with more lab animals you need the proportionator

The proportionator is the stereological tool that makes more effort easier to handle. It is the only tool today that makes it possible to obtain the best results with the least effort. Weibel is quoted as saying “Do more, less well.” The fractionator makes it possible to “Do better, less effort.”

The quadrat

The quadrat is sampling shape. It is often thought of as a square or rectangular frame used in the field to sample an area. This is the way the quadrat appears in geographical, ecological, or forestry studies. But the fact is that the quadrat is not necessarily a 2-dimensional sampling shape.

Before looking into the more general definition of a quadrat lets consider the 2-dimensional quadrat. That an area. Typically, an area is a square or rectangle. There’s a simple reason for that. It’s easy to make a quadrat of that shape. A few boards and fasteners and the quadrat is done.  Early on it was realized that there existed a problem with quadrats. The problem was realized as early as probably the late 1890s when Clements and Pounds published there article on the quadrat. One of the early solutions was the proposal to use disc shaped quadrats. This meant that the quadrat minimized the edge effect. In fact, it didn’t. The problem was that the sampling rules associated with curved quadrats were too complex to use effectively.

You’d think that the square quadrat would have been the choice. It enclosed a large area for the perimeter. That didn’t happen. It was learned that the long thin quadrats such as the belt or strip quadrats were more effective sampling methods.

The shapes of quadrats was a hot topic in a number of disciplines from range management, to forestry, to ecology, to grassland studies. In all of this of course the assumption was that a quadrat was a 2-dimensional shape.

A quadrat is geometric sampling shape. It can be any dimension from 0 to 3 dimensions. These are points, lines, areas, and volumes. Even though you might think of a quadrat as a square frame, a quadrat can be any sampling geometric shape that is needed to get the work done and done efficiently.

Measures of abundance

One of the basic concerns of field work is to develop a sense of how much is out there. In a recent presentation I encountered a fascinating study in which the researchers were studying a shore bird called the Wilson’s plover. This bird is a beach nester that is not too common, but not endangered.

One of the foods of choice for the bird is the hermit crabs. The food source lives in burrows along the shore in muddy zones. The crabs dig burrows. A knowledge of the number of crabs in the nesting are used by the Wilson’s plover provides information about the available food supply. The measure of abundance in this case is the number of crabs and not the weight of the crabs.

The present sampling strategy is to toss a quadrat, a wooden square in this case, onto the mud flat where the crabs live and to count the number of crabs in the frame. Crab counting is hard. They move. They hide. The strategy used today is count multiple times and use the average number of crabs as the estimate for the selected quadrat.

A short discussion with the researchers demonstrated their knowledge of sampling and its issues, but they were not aware of 2 things.

1. They were not aware of counting frames
2. They were not aware of the efficiency of SRS

This isn’t surprising since stereology is a relatively unknown science.

The researchers will be doing more studies next year. They know that it is better to count burrows than it is to count crabs that run and hide. They are likely to use the counting frame and SRS in next year’s study.

A more general way to examine the nucleator and the circle

The nucleator is a local stereological probe.  Being a local probe means that the nucleator is based on the selection of a reference point. The probe, i.e. the geomtrical shape that is used to sample, is constructed relative to the reference point. A nucleator probe is a ray that starts at the reference points and projects outward. The nucleator is an unbiased probe because there is some essence of randomness to the sampling. That is accomplished by making the ray IUR. The abbreviation IUR stands for isotropic uniform random. That’s a fancy way of saying that all directions in space have the same chance of being chosen. On a flat surface a ray should have the same chance of choosing any of the directions from 0 to 360 degrees.

The measure of the nucleator is the positions along the ray where the ray crosses the boundaries of whatever is of interest. If the nucleator’s reference point is inside of the object of interest, and the ray goes out and never crosses the boundary again, then the measure is sort of like the length of the ray inside of the object.

In the case of the circle it is possible to come up with a formula that relates the circle and nucleator and the intercepts.

1. The circle is defined by its size which is its radius. That will be called r.
2. The nucleator is defined by its reference point and the orientation of the ray. These will be shown to be c and theta.

The first step is to work out the circle. The circle is defined to be centered at (0,0) and to have a radius r. A rather simple equation defines the point (x,y) that lie on the circle. Remember that a circle is the curve. A disc is the area inside of a circle.

The nucleator is a bit trickier. A circle is radially symmetric. No matter how it is rotated it is still the same. That allows us to simplify the choice of point inside of a circle. All points a distance c from the center of the circle are really the same. Therefore, all reference points inside of the circle a distance c from the center are essentially the same. Think about drawing a line through the reference point and the center of the circle. The line is a diameter of the circle. Use this line as the x-axis of a coordinate system.  For another point also a distance c from the center use a different coordinate system. The angle theta is the angle between the ray and the x-axis.

Knowing r, c, and theta it is possible to derive a formula that gives the intercept distance I.

The mean of this formula is found as follows:

Copyright of the author - click to view

It should be fairly obvious that this is not related to the circumference of the circle. Actually this result gives the correct value if c=0. But what if I squared was used instead of I? The mean of I squared is:

Copyright of the author - click to view

This result, r squared, is in units-squared. Length requires units. The square root of this result is proportional to the circumference of the circle.

The question is: Does this work for shapes other than the circle?

The answer is seen to be a resounding NO. All stereologists already know that perimeter is related to another measure, not the mean intercept length or the mean intercept squared, or on any variation of that except for a few trivial cases.

The special case of the nucleator and the circle

The daisy fails, but do things work for a circle

The 4-way nucleator, when clicked inside of a convex profile creates 4 lengths – one for each ray of the nucleator. In this case we examine the circle, the simplest of shapes.

A circle is simple, because it is isotropic. Rotate the circle and nothing changes. The circle looks the same. That makes many questions about the circle easy to answer.

Here an arbitrary point is selected inside of the circle. Four rays start at the arbitrary point and intercept the circle. Because the circle is isotropic it is possible to simplify the math and use rays parallel to the coordinate axes. Also, without loss of generality let’s select the arbitrary point from the first quadrat.

These two simplifications mean that the lengths show in the drawings, a, b, c, and d, can be used to identify the 4 intercepts as (0, a), (b, 0), (0, -c), and (-d, 0). The axes cross at the arbitrarily chosen point. That point has coordinates (0,0). The intercepts on the y-axis are at a and -c. The x-axis intercepts are at -d and b.

So the question is whether or not the information given here is capable of estimating the circumference of the circle? Yes. Let’s see how that is done.

<insert math>

In general, the nucleator cannot be used to estimate perimeter. There is at least 1 case in which the nucleator does work. Are there other cases? The answer to that is unfortunately no. No shape other than a circle has a perimeter that can be estimated by the nucleator  without bias.