Is it true that ...

Changing the prism factor from plot to plot in a stand in order to "optimize" the number of trees per plot gives just as reliable and unbiased an answer as using the same prism factor within a stand?

In short – NO. There is a slight tendency toward bias, and it is probably less efficient.

The Logic

The "optimum" prism size works just like it does in fixed plots. Larger plots are less variable (smaller coefficient of variation, CV) and so you need fewer of them. Larger plots have more tress, so they are more expensive. So, as plot size increases the number of plots goes down and the effort goes up, and at some point there is a kind of "minimum cost" size.

For old growth in the Northwest they decided many years ago that this would be around ¼ acre and adopted that size widely. Nobody changed the plot size at each sample point – they were looking for the right one on the average.

The same thing happens in Variable Plot sampling (since a smaller angle just implies a larger plot around each tree). It is more general to talk about average tree count (the result) than about the angle (BAF) itself. The right average tree count for this "minimum cost" situation was about 4-8 trees when the research was done, and it has proven true whenever we have retested it with field data. This has been true with old growth or second growth.

There is a second reason not to get a high tree count – you start missing trees (personal error), particularly fallen trees, and it makes any "edge effect" problems worse. Above 10 trees seems to be where these problems begin to occur – with experienced people, not just new cruisers.

Switching Prisms

Would you be better off getting 4-8 trees on every sample point? No. There are two main reasons. First, there is a slight bias when you do this. The largest effect is bumping up the answers which would have been "zero plots" (always a positive bias). If you leave "zero plots" as they are, and make large prism switches (like 40 vs. 80) rather than small adjustments (50 vs. 55) then this bias is probably small enough to tolerate – but you never know for sure. Why do it?

The second reason is that it is not more efficient. We have tested this in a few situations, and the CV of the system with the prism switching routine was more variable than using a constant BAF.

So – the situation is:

• It can be somewhat biased under practical conditions.
• It has no statistical advantage.

In addition, the chance of messing up the paperwork and writing down the wrong prism is pretty high, and it is nice to use the same prism all the time so you get a feel for which trees to check.

Checking the (4 to 8 tree) rule of thumb yourself:

To do this you need to get coefficients of variation (CV) of tree count for a cruise using various basal area factors. This is easily done in two ways: (1) Measure the diameters and distances to all trees at each point and use various plot radius factors to simulate tree counts for different BAF prisms, or (2) use a range of prisms to get tree counts at many points. Use the tree counts from the various prisms to compute the CV [CV = Standard Deviation of tree count / Average tree count] and then plot the CV against the average tree count and smooth out the curve. This will tell you how CV relates to average tree count.

Next, use the CV to calculate how many sample points would be needed for a fixed sampling error [n = (CV/SE)2]. Last, work out the cost of doing that number of trees on that many plots. Plot the cost against the average number of trees, and see where it is minimized. Or, you can just go along with the conventional wisdom.

All of the calculations are easily done on most calculators with statistics functions or in a computer spreadsheet. Let us know if you try it, particularly if you get a surprising result.

History of the problem

For those interested in the history of this problem, it runs something like this – the exact details of disasters are always rather muddied, for obvious reasons:

The 4-8 rule has been published in books on sampling for some time. The standard, Bell and Dilworth Variable Plot Sampling book, has had it for years. The USFS and at least one company decided that if the range was good, getting an exact number was even better, and settled on 7. Field crews were told to get 7 every time, and switched prisms to do so.

Eventually, people got nervous, and the first widely published account was ["Selection of basal area factor in point sampling" by Wensel et al, Journal of Forestry, 1980, volume 78, pages 83-84]. They showed a bias, using field data, when very small adjustments were made (the worse case for possible bias). No theoretical basis here – just field data.

A second publication ["Ensuring an adequate sample at each location in point sampling", Schreuder et al, Forest Science, 1981, Volume 27, pages 567-573] showed a mathematical bias, but entirely based on an assumption that was wrong (and, in fact, impossible).

The last chapter in the saga appears to have been ["Changing angle gauges in variable plot sampling: Is there a bias under ordinary conditions?", Iles & Wilson, Canadian Journal of Forest Research, 1988, Volume 18, pages 768-773]. This showed that switching prisms could be unbiased, under a very reasonable sounding assumption which most people would believe. At the same time, good field data showed that this assumption did not seem to be true under field conditions, and led to some bias. This is probably the most complete discussion of the problem.