Breaking a habit is hard. You get up at 6 a.m. every morning for 2 years and one morning the alarm does not got off and you still wake up at 6 a.m. Your body sends a notice about when it is time for the regular evening scotch or the 10 a.m. coffee. It takes force of will not to turn in the usual way when driving to work so that you may make a purchase before continuing along the normal route. Habits or addictions run deeply.

Sampling is a habit. Having heard about the need for it, its goodness and efficiency, most of us have quickly turned to the means for doing it - a dozen rules, methods, nuances and peculiar perspectives, and nearly magical results - "take 114, not one more or less!" We have sampled. We believe in it. It is part of science. Science is good. So is sampling. We are convinced, members of the faith, followers ... but we suspect it may be something with the label "Habit Forming" washed off. I think we need to re-attach the label. Because the implications of "habit-forming are not always considered beneficent, people in the natural resource field may need a replacement. Things need fixing! They are "broke." Repairs may not work. A replacement is needed.

I wander out into a murky, foggy land, with harm-doing creatures likely behind every boulder, past every wall, over every hillock. They are the spirits of prideful mathematicians, logicians, and a gross of people suffixing the word "statistical" or "sampling" to everything. I have no "truck" with them; I joust not from truck. I have a viewpoint - nay, a whole viewscape. I want to share it because it is fun; challenging in a recreational sort of way; easily poo-pooed; but potentially a threat and maybe even the basis for a whole new way of doing business in the wildland world. I doubt that I am really suggesting anything new. The older I get, the more I see duplications. I occasionally notice that I repeat the beauty of the classics: I have ideas and feelings over 2,000 years old! How satisfying! How leveling! How embarrassing to think I am original. I remember with glee (but I forget the author) that educated people are those who are not "excited by discovering the trite." I risk being called trite; I hope not; I am fearful enough even to ask for pre-nuptial agreements in what should be a writer-reader marriage.

I am a product of the Sputnik revolution. After Sputnik, only things scientific were good. I, with others, was in a world-class race and even though few knew how to run or what to ride, the race was on. Our efforts were to catch up to and get ahead of the Russians who must have been better in science to have gotten up a satellite first. This chapter is not about space flight or science itself, but its alternative. It is about means to suppress the inferno of multi-nation-wide emphasis on induction ignited on the day of the launch.

Science was good. Basic science especially good. Fundamental science worthy. Applied science tolerable. And all deserved financial support. Who spends money on things bad? ... so it must all be good. I benefited in various ways through student grants, university grants, research grants, publications, books, conferences, schools and in hundreds of ways about as unnoticed, unappreciated, and as important as sewer lines.

Science, I learned, was a many-splendored thing but reductionists fettered out the splendor and claimed that all it was was induction and deduction - but allowed that there could be continual conversation between the two, a dialogue, or, pretty fancy, a dialectic. So the inductive-deductive dialectic was born, later to be replaced by someone who had not read history and felt compelled to add a paper to the philosophical grounds. It arose as the "hypothetico-deductive," hardly a major leap forward.

Philosophers, early on over-awed by science, struggled to balance on a stepping stone in the fast-moving stream of science. They were not heard over the rushing roar of the stream. They seemed to whisper, "We have had the answer for a long time, centuries!" There are lots of ways to know. One way is by induction. Another way is by deduction. I know. I can show you. Let me show you, at least tell you.

An old, nearly-senile man stood beside a blazing house. All of his worldly possessions were ablaze. The fire trucks arrived too late. The total loss was clear. They began spraying water on nearby houses. The old man, in tears, screamed, "hand me the hose. I see the place!" I suspect the philosophers saw the place. There were other factors involved. They know well their subject, epistemology, or as some of them preferred: criteriology (Chapter 16). Not logic, not what you know (as if discussing some storehouse of knowledge) but how you know. How does anyone know anything, really! I know because by daddy told me (authority is the epistemological base); I know because I saw it (sensory base); I know because we have agreed on this language and the conditions and definitions match perfectly, I know this is X (contextual); I know Z will run down hill (coherence is the base).

It turns out that there are many ways to know ... anything. Induction is one way, deduction another. They "both together licked the platter clean" as goes the nursery rhyme. The other ways got lost, played down, ignored. Philosophy itself lost ground. I am protesting; I want a reformation; I have a thesis ... to the doors!

If you don't do an experiment, you're probably not doing science. Ho hum. Oh my! Sleepiness will allow induction to gain excessive power. On guard! The dialectic is needed; at least firmness is needed between the two (never mind, for now, the disgraceful lack of attention given to all of the other epistemological bases - all recognized for their strengths as well as criticized for their weaknesses).

I think describing natural stuff is science. Measuring birds and reporting the measures is good science. There are no hypotheses! I think listing all of the plants found in a watershed is science. It is as descriptive of part of a system as bird wing length is descriptive of a bird system. Derogatory use of "inventory" does not change the reality of its scientific nature. Measuring, listing, and comparing flow rates among 100 streams is science. It is easily classified as an effort to find the distribution (mathematical) of a wildland parameter (in this case flow rate such as expressed as cubic feet per second, cfs, which in its awesome way means that if you have a stream with a mere 2 cfs, that silvery streak is sending over 300,000 gallons of water past you every day). Knowing what the least of these is doing, the greatest, and the median can have profound consequences to fish, fisherpeople, and fiddler crabs.

My mental tears pool for the taxonomists. When did their lofty work become a lesser science? It was lost in the flood of post-sputnik money. Only molecular biology was good (never mind the name of the organism from which the molecule came was unknown). People who need to know nature have lost ground. Even to begin to save the parts, you must have a name for the parts.

Induction is doing experiments. You get an idea, you develop a hypothesis, you do an experiment, collect data and analyze it, draw a conclusion, and (if in my camp), present or publish the conclusion so everyone can learn or criticize the effort so that "science improves." I suggest a revision of that phrase to: so knowledge is increased and the process of gaining knowledge is improved. More on this later. High school kids remember: "induction means going from the particular to the general." The "general" is a conclusion (which I call "Q") and from this I can usually suggest that, for example, if 10 rats are representative, then I can say "Q" about most rats or even all rats. The reasoning is that knowledge-workers seek generalizations, things useful for daily living, the two major ones being explaining and predicting. "Explaining" is said to take care of the past and present. (Some people add "understanding" so that research money flows to present-related projects since some people oppose studies of history.) "Understanding" is a mere part of "explaining." Inductive studies and reasoning based on them have been powerful. Inductivists seek evidence and there is evidence the induction works (the pragmatists' argument, not the inductivists'). It is helpful, often very powerful.

Deduction is said to be equal to induction in the scientific "approach." Objective observer - scientist certified - seems to have the patch over the wrong eye. A dialectic, a conversation, between induction and deduction is muted and often does not occur. Deduction is the process of modeling. It takes or more observations of generalities, then uses them to estimate or approximate in some way a specific condition. Philosophers or epistemologists call deduction "correspondence," an easy relationship to understand. A great model will give great answers; a perfect match with nature. This is pure fantasy ... but it sounds reasonable. For small, tight, invariant systems (a few exist) a good match is possible. A heart beat models well. Heart beats over a day model well. Output from a forest model matches a forest only occasionally.

Because of these rare matches or close fits of one curve to another, then the model must be "bad," so we go back to inductive procedures to get more data because the first ones collected may be wrong and we tinker with the model because it might be improved. (Perhaps this is the so-called dialectic.) The resulting image: my grandchildren "dancing" their gummy-bears on jello.

The situation grows progressively worse and I feel almost compelled to write about it for the sake of the entire scientific enterprise but especially for my grandchildren. There is too much insistence on induction; insufficient and poorly informed work in deduction; unbalanced reporting, partially because of "inductive" editorial influences; excessive pressure to publish, thus excessive publications, mostly pieces-parts. There is insufficient amalgamation of the results of the other epistemological bases. Insufficient attention to the reason for it all - not fun, not "just because" as asserted by the so-called basic scientist - but to provide inputs to a knowledge-base to allow people to comprehend and reduce risks of their decisions.

If I buy a computer model for a forest off the shelf, enter the required data, and run the model ,am I doing science? Perhaps I am predicting. I use the results to harvest the specified trees in the forest and replant . I am the manager, the designer and operator of the present and future system. I care little whether someone calls me a scientist. I'll claim "scientific management" for public relations benefits and feel not one pang of prevarication. Three years later the interest rate on money will shift greatly and the decision that I made to cut the amount of wood will be wrong. Is the model wrong? Perhaps I should have a model to predict interest rate change ... or disease ... or insect ... or forest fire ... or the vandal ... or the cows that got loose and ate my seedlings. That damned no-good model! The inductivist calmly claims: you should have collected more data; you should have verified your model. You should always verify and validate ( the parallel equivalent of spit and expectorate). If a model is worth anything it will be big, complex, inclusive because they are the very things for which we need models. We can do controlled experiments on the tacky pieces-parts. If a model is good it will be as complicated as all get-out and because it will simulate unique systems. There will only be one data base (as uni-) and thus there will unlikely be a repeat performance in another even closely-similar system. The odds of a complete match are very low. The small probability of getting a close fit was known by the modeler from the beginning. Why should we expect two very different, complex systems to perform the same? Some people do. Some people reject deductive work because it does not meet the rules of inductive science.

As a wildland manager, I would rather go to the woods armed with a model, any model, than without one. I use word-models like "the more moist, the better the site, and the better the tree growth." I know that is untrue when things are too wet, when other site factors are at work, and when the trees are very old. A simple word model takes more word models to become real or useful. By the time you engage an entire stand, all within a landscape, there are 50 non-straight-line, often conflicting relations going on within your head - at least the head of the forester working on our land (all of us public citizens, corporate stockholders or grandparents). A model may not be a bad idea.

I would rather go to the field with a model than with daily data from the past year. One year of data in a wildland is one point of a graph. A wildland needs 200 points. The chance of any year being exactly like last year has to be approximately 1 x 10-10. Yes, dear, there is a chance. Wildland systems are unique. This awareness suggests the need for emphases on maximums and minimums, on approximations, on median values. My point, and I make the emphasis because I have been misunderstood: the wildland manager will be more right over a longer period when using a comprehensive regional model than when assuming that a detailed short-term data set "represents" an area. With model in hand, I shall never delay a project for a year, even 3 years, to collect more data for which my inductivist crowd would clamor. I will settle for a delay of 50 years, but rarely will anyone else.

A year, you see, is a sample. Inductivists live and breath samples. Foresters - all wildland managers - are educated (and have been for 75 years) as scientists. You "cruise" a forest. This means to observe, and sample, and evaluate it. Samples are important but wildland workers need a new awareness about them. The concepts here are not new; the awareness, it seems to me, is new, at least a refurbishment - a slip-cover over an old comfortable chair by the fireplace. The reader will probably forget, but I'll not repeat: the following is about the wildlands and "nature," not industrial sampling or quality control.

There are no "wow" experiences in the wildlands for the experienced person. In the city they are frequent. "Wow, that sure looked heavy!" (a styrofoam "cement" building block). "Wow, this is heavy! " (iron in a magnetic field). At Peculiar Manor, there are sunsets and dodged snake bites that produce "wow," but the inflection is different. Even with a little experience, I know how long it takes to walk a mile, the weight of rocks, the height of most trees, where 4.5 feet is from the ground, how fast water flows, how fast birds fly, and which flowers grow how fast. There are a world of pleasant observations to be made but few "wows!". Now that 70 percent of the nation's people are urban, I expect great insecurity and more wow experiences by people in the wildlands. I cannot even imagine why so many urban youth would make a career decision to do wildland work, given such limited experience. They have no hero, no mentor. They have vicarious TV experience and few hiking-camping experience. TV viewing seems sufficient to make a career choice. Such irrationality! Why would we be surprised to see such irrationality crop up in their career work? My irrationality! I learned that executives are often the children of executives. They know the language (contextual epistemology), know the people, are connected, have information, are familiar with the system, have a hero and mentors. No wonder! In the wildland areas there is little of this and what there is, I perceive to be declining. There is trouble ahead. Solutions may lie in new, intensive, practical, experience-based, decision-requiring education .

Claims of an experienced people not withstanding, it is possible to estimate well almost everything in the wildlands. "How tall is that tree?" Answer: 10 feet. "Don't be silly! It is 65 feet tall!" We said estimate well. Here is part of the problem. It is another epistemological issue. How will we know when we are estimating well? Just any old estimate will not do.

My view is that until things shape up in the wildland professions, until landowners become more sensitively attuned to what they need and can get from the land, until the quantitatively and economically illiterate college graduates of the U. S. (themselves only 35 % of high school graduates) become more perceptive about public land management and management of lands affecting their drinking water, almost any estimate will do. The time suggested, needed, it seems to me, is that equivalent to teaching a pig to sing.

Inductive scientists based on a peculiar logic emanating from early work in agriculture claim that for most studies, they can tolerate being wrong in their estimate no more than about 1 time in 20. For example, we may observe or estimate A and B and then ask: Is A greater than B? We'll do 200 experiments and draw conclusions. A is greater than B. We made 10 wrong decisions (A was equal to or less than B). Who really cared? How much money was on the line? How many peoples' lives were at stake? How many jobs? How many species lost? Acres ruined for 100 years? In some agricultural work, is may be that greater confidence is need, say a mistake no more than 1 time in 100. Statisticians would express the latter as "alpha is 0.01" or the probability of an erroneous conclusion is 0.01.

I am attempting here to awaken old concepts important for modern wildland management. I am not arguing against sampling, only for a return to its fundamentals. The first fundamental is that you can never achieve certainty. Knowledge with a probability of 1.0 is impossible. "That tree will die! That's certain!" Probably, but even of that I cannot be certain. The standard of 1.0 is excessive. I think 1 in 10,000 is not a bad standard. I call it S. It is a mistake in 1 day out of about 2 life times. The probabilities I compute for any wildland event should be compared to this number or some other one well decided so I get a certainty ratio of :

C = P/S.

For old, asocial, short-term profit maximizers, S is probably 0.9 and C probably very high. Close enough is good enough! Do it!

Sampling by a rule-of-thumb is costly because resource and wildland managers have fat thumbs. Sampling, especially, the decision about how many samples to take, is based on many factors, but the ruling three are (1) how variable are the things being sampled, s2; (2) how confident do you have to be in your answers or conclusions, based on the sample, t2; and (3) how accurate must you be, the allowable errors d2. There's always confusion about the difference between (2) and (3) and there are constraints of many types, formal mathematical, budgets, time and others.

The smaller the sample needed, the better, because of costs. If the calculated sample-size-needed is 200 and each sample costs $200 to collect and to process ($40,000); and the available funds are $10,000, then what? In my view, the study should not be done. "We'll learn something even if we do not collect the proper samples" has been claimed. The alternative view is that the population will be misrepresented, faulty conclusions drawn, and even though cautiously worded, research "results" will be used (even though it is inappropriate to do so) because managers are hungry for "results," the needs are great, they have to show something for funds spent, and hidden assumptions about the prosperity of research are overlooked because of graciousness ("we are all limited"), hunger, lack of skepticism, lack of sufficient critical skills, insufficient time for critical work, etc.

The knowledge-building system needs to assure quality performance leading to effective use of results and this includes justifiable sample sizes leading to conclusions at pre-stated levels of confidence and pre-stated allowable error.

Figure 18.1. The sample size tetrahedron. The sample size, n, is located somewhere within the 3-dimensional space of the tetrahedron. All of the factors operate for the resource manager. Selecting an optimum point may be a false objective, given the multi-dimensional uncertainties.

In the wild, things are variable. They are not innately variable like potato chips. They are influenced by hundreds of things. Two oak tree seeds, acorns, exactly identical on a tree limb will be different in a few days after being on the ground due to algae, insects, drying and other factors. All of the factors cannot be known. Even position on the limb may have resulted in one acorn being larger than the other. They vary in weight. This is not mystical, magical, or the workings of evil woodland spirits. They vary. The more variable, the more samples it is said are needed. An expression of variability is statistical variance, usually symbolized as s2. The higher the variance, the more the samples needed. This is true for things equally clumped around an average or mean value. My observation is that things in the wildlands are not "normally distributed" or do not have a bell-shaped curve when frequency is plotted. These curves are frequently skewed, have a long tail off to one side or the other. The central value of interest is the median, not the average. I think a "variance" computed using the differences from a median (not the mean) may suggest more reasonable sample sizes than in the past for wildland work. When the required number is very large, the observer had better go elsewhere ... or home.

Confidence is expressed by an alpha level (described above) and shows up in the sampling equation as a value of "t" from a statistics table. I hate being wrong but I am now very experienced at it and it is rarely fatal. How confident? remains a problem. I have seen scientists at work for over 40 years in the various wildland fields. Most like to work at the "95-percent level" of confidence (alpha = 0.05 or a chance of being wrong no more than 1 time in 20). I've seen 10,000 experiments done. There must have been about 500 mistakes; really bad errors in concluding whether these were real differences. I have not seen the mistakes - but I would certainly have noticed 1 or 2; the rest could have been cunningly hidden. I think the criterion of 95 percent is much too conservative. Too many samples are being required by classical equations relating to sample size to confidence. I have suggested lowering confidence. I think that rarely more than 1 time in 5 do we truly experience job-costing, promotion-losing, profit-foregoing, ego-impairing, or embarrassing errors. Very rarely will anyone have a baseline or standard for comparison. Rarely will objectives be stated well enough so that success can be measured and taken to court with confidence. The consequences of making an improper sample size estimate can be profound. Other things being equal, say an allowable error not of 5 percent, but 10 percent, then the sample size needed when the confidence level is 95 percent compared to 80 percent is 83 vs. 17. Samples cost big bucks - collecting, analyzing, storing, retrieving, data processing, and publishing. A reduction, a simple shift from 95 to 80 needed in results, can have enormous consequences in understanding the wildlands and making massive, large-area, long-effect decisions.

Being aware that nature works on the basis of winning most of the time, presumably a rule no more elegant than "about 51 percent of the time," it is reasonable to use the sample size relationship above and compute a sample size when the median is about 50, the variance (s2) 80, and the acceptable error about plus or minus 10 percent, as:

n = s²t²/d²

n = (80)(.90)/(50x0.10)² = 2.3

It doesn't take much information (3 samples) to win over the long run. If the variance was decreased to about 53 (because we are not interested in the outlier observations), the number of samples would decrease to 2. In the wildlands we've done a lot more sampling than we have needed to to stay in the game and win more than we lose. It turns out that as long as d² is equal to or greater than the variance s², then as long as the needed confidence is no more than 25 percent, one or two samples is usually sufficient.

The third piece of the sample-size equation is d². It is the mean (I suggest, however, using the median) plus or minus a proportion, then squared. If I say "it is OK if my estimate is within about an inch of the true diameter of the tree" then when I am pondering the fatness of a 20-inch diameter tree, then we are talking about accuracy to within 5 percent.

Consider another situation. How many squirrels were killed by hunters last year? Who wants to know? What will they do with the numbers? How dangerous to squirrels, forests, or people is being a little wrong? The answer to the question of harvest accuracy will probably be "within 20 to 30 percent." It sounds gross. Why not? There is almost no situation in which more precise information can be used given the complexity of the squirrel system, the forest, and the hunters (Lee and Giles 1982) When the variance is 50, the mean 50, and the desired confidence level 80 percent, the difference between wanting accuracy at 5 percent and that at 15 percent is 14 samples vs. two samples. This above discussion is based on fairly conventional sampling concepts but all grounded in an inductive philosophy. I suggest an alternative view which I call natural heuristic. Heuristic usually means "discovering" or an eclectic approach to things; not using a singular formal namable algorithm, but a kind of neo-pragmatism. I use "natural" to suggest excluding technological sampling quality control (but it probably includes psychological studies). Natural heuristics means discovering or gaining knowledge by any means possible, then storing it, and then beginning use of it and revising it. It is a systems approach to gaining useful knowledge. Science is just one way to gain.

The "light-bulb" experience is well known. Sitting, silent, by the fireplace contemplating. All of a sudden there is a new combination, a step on a pathway never before seen, a turning point. This is genuine discovery, as real as finding a new species of plant, seeing a chemical reaction for the first time. Research is done to gain knowledge. "Basic research" is said to build the knowledge base, to put a finding on a shelf or disks where it can be retrieved. It seeks a truth without a predictable, even expected use. It is a search for a truth, just because. "Applied research" is a quest for an answer to a particular question; a solution likely to be used when found. There is a mission and the wildland scholar seeks answers of any type or form, new or old, to achieve the mission.

Natural heuristics is employed by wildland managers to discover incrementally, cost-effectively, knowledge needed to achieve stated system objectives. It sounds like "applied research" but it isn't. First, it is bounded by the essential that at least something must be known about every part of a system.. Far too much has been devoted to gaining knowledge about certain parts of the natural resource system (none to other parts). Then, based on objectives and after sensitivity analyses, the manager determines the greatest need for knowledge about some topic (call it C). This knowledge, once obtained, will increase human explanatory power over the system performance (perhaps "prediction" but most likely the likely location of the system in the condition. The strategy includes asking a competent set of researchers how they prepare to find C. A reasonable and low-cost project is selected. [This reduces the waste of ad hoc proposal writing, reduces writing time, directs the thoughts of managers and their staffs and utilizes past investments in ideas, experience, facilities, and equipment.] It is not "basic research" in any classical sense, but "extra" work or findings of a more fundamental nature, rewarded with bonus payments, travel sponsorship, graduate stipends and assistants. Incentives work well within research societies as elsewhere.

A variety of discovery efforts are used from library searches for a coefficient, a comprehensive review with a bibliography, to a preliminary study to establish an estimate of the variance of a population, to full-scale, long-range research with an elaborate sampling scheme. The strategy includes finding, assembling, and organizing past work (including oral reports of experts); collecting (e.g., new insects in an inventory); describing and measuring; describing distributions in time and space; testing hypotheses; and modeling. It includes writing theory. It includes developing techniques, "discovering a better way" in the situation.

Storing results for ease of access and retrieval is essential. To fail to do so is some pseudo-practice not to be rewarded, certainly discouraged.

Each researcher answers for him or her self: is the question in a request for a proposal (RFP) appropriate for me to answer? If I find an answer, will society be wiser and more constructive with the answer? Can I produce peripheral knowledge, perhaps as important as the contractual question in the time and for the resources available and still meet the contractual obligations?

By no means opposed to basic research, the strategy is regional and mission-oriented, and voluntary for the scholar (s) contacted. Basic knowledge is judged to be of parallel and equal value, conditional upon the contractual knowledge being gained. Credit for creating the contractual conditions for the discoverable moment is required. The beauty and grandeur of such discovery need celebration. The end is to have a fund of useful knowledge, of increasing confidence levels improved by direct, as well as ancillary, studies and a demonstration of power from using the knowledge base.

The pathways to quantifying anything in the wildlands in a rationally robust fashion (Giles et al. 1993) are to use many schemes in a discovery mode. Each is used to reject or shore-up each other. All create the "ballpark," the constrained space in which numerical truth probably lies. It is perfectly true that no one scheme within the following scheme will yield truth or an acceptable chunk of knowledge. They are to be used in combinations, within a system, to build tentative knowledge. Even I find myself critical of a procedure when working with it. I risk presenting them; each can be readily criticized. Each, however, has led to advanced work. The points to be remembered: There is no more time, no more money, no more taxonomists, and human populations are increasing and resources are being consumed faster than being replaced or produced. This concept of natural heuristics is one set, a solution. I do not like it, but it is better than the alternatives I perceive.

Schema 1: The Power of Three

An experienced person takes three samples believed to be representative (say 11, 16, 13) and puts them in rank order (11, 13, 16). If both side numbers are no more than 25 percent different than the center number (13 plus or minus 3.25) as they are here, the person grins and runs with the result. He or she has gotten control of the system! Use the center value! You can be wrong, but the sun may not rise.

If numbers were 11, 26, and 13, they violate the 25 percent rule. Then compute the median by the following: ((high value) + 4 . center value) + (low value))/6 or (( 26 + (4 . 13) + 11)/6 = 14.8). Use it. The variance is probably ((high-low)/6)² or ((26-11)/6)² or 6.25.

In this procedure the innate synthetic capability of the human mind is acknowledged and used. We do not surrender to numbers, equations or computers. We capitalize on experience and we sense "representative."

Schema 2: Standard Error of the Median

Use the standard error of the median (for example as estimated above) as computed by:

s_m = 1.25s/(N^0.5)

When N, the total population observed, is 3, then an estimate

s_m = 0.72s

and s_m can be used in expressing how much each of the three numbers contributes, on average, to the variation in the population. The amount is about 25 percent more when the population is not bell-shaped but is skewed (as it seems to me to be throughout nature).

Schema 3: Variance to Mean

I do not know why more people do not use the ratio of the variance (s²) to the mean or median. When that ratio is large (greater than 3.0), you probably can't pick representative numbers or even get a large enough sample size to feel confident about any decision. When it is very small (less than 0.8), take a few samples. A few will do. These observations are based on the general concept than when things in nature are distributed as the Poisson, that is the variance equals the mean, they are random. When the variance is large (thus the ratio large), there is evidence of clumped conditions (like animals near a den, dropping near barns, and seedling near trees). When the variance is very small, things are uniformly distributed (like trees in an orchard or the weight of bunkhouse biscuits).

Schema 4: Cast the Rank Order

Take a small sample, then plot the points in rank order on a graph, high to low. Is it convex, concave, or linear? If concave left, limits are likely. If linear, cast the line to the maximum and minimum reasonable or known limits. There are answer to a variety of questions where the lines cross. If convex left, ponder the implications of zero and infinite limits.

Re-do the above, using the log (x+1), and plot the numbers. Linear results are likely. Study the meaning of the intercepts.

Schema 5: Use Ranges

The high and low values are probably more likely to be of use to wildland managers than an average. With a median, they are especially useful in describing a population or set of observations. Study the statistics of range.

Schema 6: Create Iterative Ranges

Hydrologists plot peak flows of a stream. If any peak flow depth of a river exceeds any past flow depth, then it is plotted; otherwise the data are ignored. A curve is developed that eventually flattens. The longer there is no change, the flatter the curve. The rate of change in the knowledge of the maximum depth becomes zero, but not exactly, for there may yet occur a storm producing a depth never yet seen.

This procedure can be used in sampling. Starting in the center of a page, plot incrementally lower and high values taken in each sample. The results are lines cast toward both a high and a low value. These are estimates of the range bounds. Progressive sampling eventually reaches a point when a higher or lower value is unlikely to be found. The probability can be expressed as the frequency of changes among all samples entered. The median value of wildland things is not likely to be exactly half-way between these two extremes.

Schema 7: The One-Third Guess I cannot imagine an experienced person missing a guessed value by more than one-third of its actual value.

How old was she? How heavy was he? How tall was the shrub layer? What was the diameter? Really! More than one-third off?

If you can get a value of within one-third "for free" (no sampling), how much more will it cost to get an accuracy of 10? If we take our sampling equation and use a mean of 50, variance of 50, confidence of 80 percent, then we realize that one sample is needed to achieve 33 percent accuracy. In Table 18.1 it can be seen that many more samples are needed to provide higher levels of accuracy.

Schema 8: Parametric Simulation

Make three best estimates: highest, lowest, and most likely. Run an analysis three times (usually a simple computer program) describing some wildland activity in watersheds, wildlife, or wood yields to see what the consequences are to the system (other variables held constant) when these three values are used. This is a type of sensitivity analysis. If the decision is not likely to change, given the range of estimates, there is no further justification for further sampling. If there are differences, further quantification may be needed, but usually only of one of the estimates.

Schema 9: Iterative Improvement

Progressively increase the data set, weighing each entry by time or units of effort since the first sample. The latest entries are probably most relevant, best collected, etc. Use a model; as each new value is collected, re-run the model. When the model results no longer change, stop sampling.

In the wildlands, certain things occur infrequently, say once a year. Around Peculiar Manor, animals "hide" for obvious reasons. People are in the woods infrequently. To see nature you have to be a trapper, logger, fisherman, or farmer - out there every day. Sampling for things in the wildlands is silly (except for trees and big rocks). There is always possible the accumulation of information, one observation after the other. Each observation (sensory

Table 18.1. Samples needed when the population has an estimated mean of 50, variance of 50, confidence level of 80 percent, and the distribution is normal or bell shaped.
Percent Accuracy 33 25 20 15 10 5	Samples Needed <1 or 0.20 0.35 0.55 0.98 2.22 8.90

epistemology) can be flawed; faulty reasons or causes ascribed. Educated observers, however, are the best than we can do. Pasteur's "nature favors the prepared mind" does not say to reject all observations until we have some counted number, then pronounce the truth of the event. State the observation, then stand back, ready to be reviled, then announce revision or confirmation.

Schema 10: Dynamic Sliding Mean

As data are collected, drop the earliest collected value (if time-related in any way) and add the new data, computing statistics of the field with two changes in the data but a constant number of items in the analysis (e.g., a set of data which are the logarithm of annual harvest plus 1.0 over the past 10 years used to get a trend in harvest).

Schema 11: The Prior

Almost everything in the wildlands is related to its condition previously. Things are sequential. The prior condition is the major determinant of whatever you are observing. Many statistical procedures require "independence." The very nature of things in ecological systems is that they are not independent. Use a regression analysis where possible. Lacking any information, guess the prior condition.

Schema 12: Just Ask

It is unwise to go to the woods alone. It is a hazardous place. It is unwise to go to the woods with a novice. When an estimate is needed, ask your colleague. Pride goeth before a large sampling bill.

Schema 13: Enough?

Sample 5 to 10 items, whatever you can get easily and quickly and at low cost and low risk (as in: "I felt lucky to get 6!"). Put them in rank order of magnitude, low to high. Assign each a number (1 to the total, say 1 to 6). Plot (or regress) the log(x+1) of the observations against the same for the numbers themselves. The numbers are "samples" so the plot or equation shows the value of the observation at zero - the minimum value you are likely to get. Then by inserting the estimate of the maximum value, you can see how many more samples will be needed to reach it. Not having a clue about the maximum, note whether the curve bends to the right. If so, you can estimate likely values you will encounter if you double, triple, or quadruple the sample size. If it turns upward, you are in big trouble. In this case, even with a few samples, the numbers are likely to be "all over the board" (whatever that means).

Schema 14: One Sample

Take 1 sample. Assume with complete certainty that this value obtained is the median for the thing observed. Difficult, this must be done with complete commitment. Also assume that this value plus 9 percent more (multiply it by 1.09) is the upper range limit. Next, assume this value minus 6 percent (multiply it by 0.94) is the lower range limit. Then write a report as if these values were true. This is the principle of "starting at the end." The odds are very great that you won't know what to do with the results of sampling because the context is unclear; your boss has not clarified objectives; you had no clear idea of the confidence needed; you were unsure of the accuracy and suspected you were likely to be ridiculed for a decision about it; and you were unsure of the readers (who might think that "30 samples" or "10 percent" are rules to live by).

If problems arise in writing, get clarification. No more wasted money or human time on sampling! If no problems arise, go and sample, then come back and over-write the preliminary numbers. It is not "practice;" this writing is not wasted effort.

One discovery will be that in most cases all that was needed is an answer to "more or less?", "which was higher or lower?", "use or not?", "cut or not?", "burn or not?" These are questions for which the concept of accuracy influencing sample size almost disappears and the concept of sampling from a binomial distribution with help from a statistician can become useful.

Schema 15: Banish Outliers

When conventional sampling is used to compute a variance which is then used to compute sample size, by all means drop the outliers. You're looking, usually for central tendencies, not trying to describe precisely a mathematical distribution of the occurrence of observations. In the wildlands, decisions are made for-the-long-run, on average, usual, and the expected. There will be extreme events - forest fires, storms, etc. They cannot be planned - central tendency is used and that is usually expressed as the median. Adjusting central tendency for the extreme values does not improve decisions. They occur or do not; there is no gain in saying a mean is 3.2 when all observations group around 3.0, just because a giant of 10 units appeared and overloaded the mean to suggest a size larger than 3.0. Extremes do occur, but over the long run, inflating (or deflating) the mean because of some extreme event or condition will rarely serve the manager well.

What is an outlier? As an extreme standard, it is any number farther away from the median than the value of the median itself. For example, if the median is 72, then an outlier is a number greater than 144. There are almost no negative numbers in the wildlands; plenty of zeros. Not to fit a curve through the zero point on a graph is to ignore reality. All samplers in the wildlands probably start with two free observations, namely the zero value (e.g., if no water then no plants; if no food, no animals) and an estimated projected maximum as in Figure 18.2).

Schema 16: Parsimony

In the past scientists have tried to live up to their mathematical teachers - to develop their elegance, universality, and parsimony. As we have learned, the greater the parsimony, the

Figure 18.2. In Schema 15, outliers are discarded but a maximum is estimated and a zero initial value can usually be assumed.

less the accuracy, but parsimony reduced costs. We now have one cost-reducer for the wildlands, the computer. We no longer need to strive for parsimony (except as a fundamental imperative, to find the rules) and can collect and fit data to our heart's content, painlessly (compared to pencil-paper, even electric calculator work of the past). To eschew parsimony - heresy! So be it. It may be an improved pathway to knowledge for the pragmatist, another epistemological route.

Schema 17: Area Proportional

It is given, that in linear systems, we can break up complicated systems into their simple parts and analyze them independently (the superposition principle). By knowledge of the parts, we can learn what we need to know about the complete system. If we are sure of this linearity (and the evidence in publications is that it is the law) we may be able to concentrate our attention on a part of this curve (or place) and not waste time wandering around the edges.

A new meaning has been brought to the idea of "area-proportional sampling." A form of stratification, "area-proportional" simply allocates the number of samples in proportion to the area present of each cover type, geological grouping, or other ways of interest in which the area is mapped. The new meaning has been brought by the technology called (improperly) GIS or "geographic information systems." By segregating potential sampling areas on the basis of slope, aspect, elevation, land form, geology, vegetation cover (e.g., from satellites), hundreds (even thousands) of distinctively different areas can be demarked. On these grounds I think each square meter of Earth is unique. I recommend selecting areas for sampling proportional to the sub-areas available and near roads or trails. This last "map layer" of roads is applied after the above work is done. The chance for bias because of the many layers becomes so small that bias is not worth discussing. The idea of random sampling in the wildlands is preposterous. The costs are exorbitant, the rules rarely followed, the orientation of staff doing the work limited, and the odds of finding the same spot (or having enough money in later years) for paired- or sequential-comparisons are very low. The "causes" for variation are removed by using multiple-factor map layers. Using the road for much (not all) sampling simply acknowledges the realities of modern society: no time, no money, no skill in orientation, high safety risks. A person in a car, with a map can usually orient well. With GPS instrumentation on a road or trail, there is little excuse for not gaining the needed fit between map, field reality, and the reasons for sampling.

Greenfield (1974) noted that "epidemiological experiments are difficult to carry out and when stratified to reveal the single or specific multiple pollutant effect, quickly reduce even a large body of data to an insufficient sample." The same condition besets the wildlife sampler. Even with resources, the computed adequate sample cannot be gotten. It may be reasonable to isolate a population (not a sample), say 17 cells or pixels in a map that, by definition of 13 map layers, are unique in a map. Then study them! They are the population of interest, sampling is not needed.

Schema 18: Equifinality

As a boy I was more interested in "skinned cats" than in the wisdom of my grandfather's oft-used phrase that "there's more than one way to skin a cat." I could not imagine why there were so many such events or that a saying would have emerged. Unquestioning, I waited, for I had heard the non-answer enough times: "you'll learn one of these days." I think I have learned, and I want to share knowledge of equifinality because it has provided me many new insights and a pattern for some explanations.

Perhaps equifinality is only a word for some old ideas. I do not know where or when I first heard it but it has taken at least 20 years to grow from its seed bed. Ludwig Von Bertalanffy, though not the first, used it in General Systems Theory in 1968. It means, simply, that there can be several pathways to the same destination.

Machines rarely exhibit equifinality. For them there is one pathway. There may be several end states from a machine, the good products and the rejects, and these are obviously not the same. Wild animal and other natural resource systems often exhibit equifinality. The same animal populations or the same forest growth will come from a wide range of conditions. There is rarely a unique procedure or pathway to any single outcome.

A "well-loved child" growing up and leaving home is an end state of a family. Given the usual differences in children in a family, each is treated quite differently, even in the same environment. The result is the same - a full awareness of vast parental love. Crop systems often exhibit equifinality. A lot of water and a little nitrogen on the same site will produce the same crop yield that a little water and a lot of nitrogen will produce.

Given about 50 factors working even in the most simple game animal or furbearer population model, it is easy to see how the same population can occur year after year even when conditions change conspicuously. Harnessing these observations has been my difficulty, answering "so what?" my quest.

Suppose after elaborate field research and statistical analyses, a satisfactory model results. It is:

YIELD = 1.63 + 2.01 (STUFF) + 0.83 (OTHER STUFF)

It turns out that if a manager needs 1100 units of yield, then he or she had better add 546 units of stuff and 0.01 units of other stuff or 0.01 units of stuff and 1323 units of other stuff ... or any combination of the pair between these extremes! There are lots of ways to produce yield ... or to skin a cat.

This observation allows me to return to the field plots. My view is that every square meter of land is unique. That results in every forest stand in the world, at least at Peculiar Manor, being unique, and given stand history (the permutations of events), then it is truly amazing (at least to me) that so many stands or other natural communities appear so similar. The apparent end states are approximately the same after an enormous array of pathways have been taken. As a result of these observations, I find analyses leading to regression models of the sort shown above to be overly heroic. Many heroes are stupid. To seek one equation as if it were the singular expression of a single, non-varying system implies a theory of life at odds with observations, thus refutable. Alternative theories exist and some of us are eager to hear them and look for them. Generalizations are less important than they once were because resource managers can use computers to work quickly with the ungeneralized data. Managers, nevertheless, do seek similarities and tendencies, and to generalize. In the future, we should be more particular, more discriminating, placing smaller and smaller pieces of knowledge into models to do the work of generalizing for us.

Most regression models are judged on the basis of the R-square value, a readily computed index. The closer the R-square value to 1.0, presumably the better the equation, the more the variability is accounted. I have seen vast amounts of field data scrapped because it "didn't show anything" (the R-square was, too low). Such a waste! Not at all hostile to regression analysis, perhaps overly fond of it, I find the concept of equifinality suggesting

1. enough samples (in each class, therefore in total) will rarely be available;
2. threshold and non-linear phenomena are at work behind every stump;
3. a low R-square is a reasonable hypothesis in the woods, because they are variable and no equation will ever accommodate a forest; and
4. alternative managerial modeling approaches (e.g., expert systems) may be more useful than regression analysis.

Another major area in which the consequences of understanding equifinality can influence decisions is in the never-never world of biodiversity. Biodiversity lurks around every pillar in conference halls. I have a computer program with 10 ways to compute diversity (which I now call variety because of the diverse definitions of diversity). I can change numbers (e.g., simulate stocking 50 animals of a rare species) and see what happens to the index. Invariably, changing the animals causes 5 of the indexes to increase, 5 to decrease! The frequently-used Shannon-Weiner index is notable for its ability to produce the same index from very different numbers. A community with 55 animals in each of 10 species has a diversity index of 0.23 ... just as does a 3-species community with 50, 100, and 400 animals in each species. The index is descriptive of an end state of a system, a condition. The details of estimating diversity are not at issue here. It can be comforting to know that there are several ways that it (whatever it is) can be achieved. It can be comforting to lawyers to know that the index sword has 2 edges that may cut in almost any way desired. The sparking edges or glittering points along the blade will be of little comfort to those claiming in court that diversity has not been achieved or maintained. There is a great amount of very difficult work that lies ahead in developing the concept of diversity as a system performance measure and its estimation. I suspect there are several characteristics of the desired end state of natural systems and that this desired condition it too loosely and too hastily expressed by words such as "diversity" and "biodiversity" that are now in the law. The characteristics describing "the good system" will be teased apart eventually. Once that is done, then systems for achieving these desirable natural systems that are good for people and other creatures, many and varied, can be developed.

Cyclic natural phenomena are obvious examples of equifinality in the same system. The same recurring population or economic numbers are not produced by the same phenomena. Presumably there is one system at work producing the undulations, but the alternative (and I believe more plausible) hypothesis is that there may be very different factors and conditions, maybe several phenomena producing the "curve." The end state, at points, A, B, C, and D in Figure 18.3 are identical. They are manifestations of system potentials, over-riding constraints, and they are probably unique combinations of usually over 300 conspicuous, generalized working factors in an average forest or North American wildland, such as that around Peculiar Manor. The potential relations (R) among this n=300 factor system is merely R = n (n-1), about 89,700. Ecologists are said to study relations. They may be irrational even to pretend to be able to or actually engage R relations in their decision processes, as well as to work with n species.

Figure 18.3 Conventional representations of harvest over time show equifinality at A, B, C, and D.

Infrequently seen is a graph such as that in Figure 18.3, which a picture of deer harvest as related to the harvest two years previous (often a strong inverse relationship). Equifinality in the system is seen in nearly identical harvests but these are a result of three very different harvests at B, C, and D.

Figure 18.4. Harvests in one year are likely related to phenomena in two years previously. Equifinality occurs at B, C, and D. The mean is shown as a dot at the center.

The cyclic or irruptive populations or the circular so-called "phase plane" appearance shown in Figure 18.4 can be combined to produce a picture in three dimensions (Figure 18.5) that can be very instructive. If an observer is careful or resistant to standard educational fare, he or she may loosen the bindings of the conventional wisdom of two-dimensional blackboard images.

Figure 18.5. Information is Figures 18.3 and 18.4 can be considered in three dimensions. The central tendency is shown in the dotted core. The system may never occur in this state.

As seen in Figure 18.5 there can be many states of systems that are working that produce the "coil." There are many crossings of projections, equifinal states. When managers quickly generalize about systems, they often use the average or the central value around which numbers clump. This shaded center-core (Figure 18.5) does not exist! No point on the curve showing final states of the system occurs along the shaded line at the center. Equifinality is descriptive of the ways that points on the curve are reached. It does not describe how the non-existent center is achieved.

The awesome reality is that forests and related natural resource systems are not three-, but n-dimensional. Knowledge of the center space, the "central tendency," is not likely to serve practical, responsible managers well in the future. What can serve is knowledge of the total, the many systems that produce measurable ends, many of which can closely achieve human needs.

What comes next (or first, or simultaneously) is clear thinking and articulating the wildland objectives - the desired end state.

Wrap-up

Not science, one means to an end, but the end, thus the means for building a knowledge base is what is needed. People know things. They know things that are untrue as well as true. There are many ways to know. There are few replacement procedures, how to undo the untrue, and that is a growth industry for those working in the wildlands of the future. The tasks ahead, as they have always been for people, is to discover, to use all approaches or schema possible (at least feasible), gain knowledge, store it precisely, and move it as rapidly and as closely as possible to truth, the high-probability statements, unattainable certainty.