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Statistics for Wildlifers: How much and what kind?

Recommended quantitative education for wildlife students

We now turn to the question of how much quantitative training wildlife biologists should have. In a nutshell, we recommend taking as many statistics courses as one can. Because statistics is a broad and diverse subject, an understanding of all topics, especially at a detailed level, is virtually impossible. We believe that good introductory courses—which include basic principles of probability theory, followed by courses in linear models, sampling, experimental design and analysis, and regression—are a must for wildlife biologists, particularly those involved in research. We suggest the following coursework in statistics and mathematics be completed by the end of a masters program.

Introductory probability and statistics
This course provides both definitions for many terms such as "population," "sample," and "statistic," and formulas for a number of commonly used statistics. The course often is offered at two levels, one that requires calculus as a prerequisite, and one that does not.

Probability and mathematical statistics
This course, in part, offers a more advanced treatment of most of the topics covered in the introductory course. It provides a deeper understanding of statistics by presenting the theory underlying the methods. This course includes the principles of probability theory, random variables, basic statistical distributions, and estimation and testing. This is usually a 2-semester sequence that requires first-year college calculus.

Theory of linear models
Building and interpreting models, for either explanatory or predictive purposes, are vital activities in many scientific and management endeavors. While the exact functional relationship between 2 or more variables is rarely known, linear equations often provide useful approximations. In a 2-semester course, the student will learn the connection between regression analysis and the analysis of variance, and gain a foundation for better understanding of those and related topics. Generally, a good knowledge of mathematical statistics (see above) and linear algebra (see below) is required to fully appreciate the theory and analysis of linear models.

Sampling methodology
Although introductory statistics courses emphasize simple random sampling, ecologists generally use more complex sampling schemes, either because of constraints in how data can be collected or because more sophisticated methods yield more accurate estimates. A one-semester course in sampling introduces students to concepts such as stratified, systematic, and cluster designs. Other topics introduced are ratio estimates, estimating proportions, determining appropriate sample sizes, and possibly topics specific to biologists such as adaptive cluster sampling, capture-recapture methods, and line-intercept methods.

Experimental design
Careful experimentation offers the quickest and most trustworthy way to understand a system. Indeed, wildlife management is most usefully viewed as experimentation (Caughley and Sinclair 1994, White 2001). Unfortunately, most ecological studies are far more complicated than the experimental designs covered in introductory statistics courses (Hilborn and Mangel 1997), and usually involve some form of a nested, split-plot, or repeated-measures design. A one-semester course in experimental design offers wildlife biologists insight into designing and analyzing complex experiments, as well as into the important concepts of blocking, confounding, and determining sample units. Sample size considerations, methods of reporting results, and limitations of experimental studies typically are covered as well.

Applied regression analysis
Regression is often applied to observational data, which are typical of field biology, where controlled experimentation too often is precluded. Such studies provide weaker inferences than manipulative experiments, but at a minimum generate hypotheses to test further. Most regression courses cover a broad range of topics, such as plotting data, fitting linear models to data, examining assumptions of such fits, and assessing the usefulness of fitted models.

Additional Statistics Courses
Other courses, such as multivariate analysis, survival analysis, categorical data analysis, time series, spatial statistics, and stochastic processes, are useful, especially for research biologists. The choice of courses to take depends on the individual needs of the student. A student planning to study population dynamics might benefit more from a course in survival analysis, whereas one intending to examine how animals use their habitats might gain more from a class in spatial statistics. We also recommend a course in quantitative ecological methods, which covers topics specific to ecologists.

Because statistics relies greatly on mathematics, an ideal training program would also involve preparatory mathematics. We recommend the following courses. These offer valuable training even if they were not prerequisites for further coursework in statistics. As Nowak and May (2000:preface) stated, "Mathematics is no more, but no less, than a way of thinking clearly."

White (2001) argued forcefully that wildlife majors should take calculus; he emphasized the value of such training for logical and critical thinking. A calculus course also will provide the student with a set of useful analytical tools for solving real problems. Can calculus techniques be meaningful to wildlife biologists, who on an average day do not concern themselves with the trajectory of rockets or bouncing balls? Such examples are typical of many calculus textbooks, which are geared towards majors in mathematics, engineering, or physical sciences. Are these topics relevant? Yes. Formulating a problem as a functional relationship (i.e., a mathematical equation or model) is a necessary first step toward solving the problem. This skill generally is invoked in pre-calculus courses but is not thoroughly exercised until a yearlong calculus class. Although a biologist may not need to determine the trajectory of a rocket, he or she may well want to determine the trajectory of a population. Further, some of the same mathematical techniques are used for both problems.

Other advanced mathematics
The following additional training would be useful, especially for students oriented toward research. Often, equations or models come in pairs or other multiples, with the need for simultaneously solving for 2 or more variables. For example, a predator-prey dynamic model might yield one equation for the predator species and another for the prey species, both of which need to be true at the same time. Linear algebra offers techniques to model and solve such systems of equations. Like calculus, linear algebra offers a deeper understanding of statistical concepts and methods (see above), especially statistical modeling techniques such as linear models and multivariate techniques.

Most biological processes involve changes through time. Therefore, mathematical equations or models describing such processes involve describing the rate of change of a variable as a function of other variables. Animal or plant population growth, as one example, can usually be expressed as an equation, with the growth rate being a function of time and possibly other variables. Such formulations lead to differential equations. Although differential equations are introduced in a first-year calculus sequence, the detailed study of differential equations generally is included in the fourth or fifth semester of a calculus sequence.

We also advocate that students of wildlife biology take at least one course in computer programming. This coursework should involve writing programs in languages such as Fortran or C, not just using canned packages such as SAS or Excel. The goal is not only to learn the programming language itself, but also to practice defining and solving problems using algorithms, which would benefit students leaning toward either research or management. Additionally, we recommend other courses that emphasize analytic thinking and logic.

Individuals vary in their mathematical inclination. While we encourage everyone to study as much mathematics and statistics as possible, we recognize that not everyone will master calculus and more advanced topics. Some wildlifers will be able to gain a deep appreciation for statistics, others will not. For wildlife students who cannot complete the mathematics coursework outlined above, we recommend they take as many mathematics classes as possible. That should include at least calculus for nonmajors. For statistics coursework they should take at a minimum the introductory probability and statistics course mentioned above. They also can take courses in sampling methods, experimental design, and applied regression analysis, which sometimes are offered for students lacking a calculus background.

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