Undergraduate biology labs often explore the techniques of data collection but neglect the statistical framework necessary to express findings. Students can be confused about how to use their statistical knowledge to address specific biological questions. Growth in the area of observational ecology requires that students gain experience in sampling design and the scope of inference relevant to observational studies. I developed a laboratory-based guided inquiry that illustrates these concepts by comparing ponderosa pine (Pinus ponderosa) trees in northeastern Washington State. This approach presents a hands-on experience whereby students apply the statistics they learn in the classroom to a field-based investigation, giving students an appreciation of the design and interpretation of observational studies in ecology.
Recent calls have been made for an increased emphasis on quantitative skill development in the sciences, both in this journal (Leonard, 2010) and from the science community (National Research Council, 2009; AAAS, 2011). A part of developing these skills is the coordination of lecture-based statistical theory, hands-on data-collection techniques, and the use of statistics to express findings. In the ecological sciences, research is often conducted through observational studies instead of the controlled experiments that prevail in many other disciplines. Observational studies require careful attention to the conclusions drawn from statistical results. I developed a laboratory-based guided inquiry whereby students frame a research question, select and measure samples, calculate statistics, and present results based on their investigation.
Scientific studies can be categorized as either randomized experiments, whereby each experimental unit is assigned randomly to a treatment group, or as observational studies, in which experimental units are measured under the conditions in which they occur (Ramsey & Schafer, 1997). With randomized experiments, one can infer that the assignment to a treatment group caused measured differences. In observational studies, differences between groups may be attributed to many factors, therefore limiting this inference to one of association rather than causality. Observational studies are often employed in ecological research because of logistical or ethical challenges associated with manipulating organisms or ecosystems such that they can be randomly assigned to treatment groups (Sagarin & Pauchard, 2010).
Two statistical tests used in observational research are the two-sample t-test and simple linear regression (Ramsey & Schafer, 1997). The two-sample t-test establishes whether samples come from two populations that differ from one another in some property. With randomized experiments, treatment can be assumed to be the cause of such differences, but for observational investigations, while a t-test can establish whether two groups differ, it cannot indicate what the cause of such differences might be.
Simple linear regression (SLR) is used to evaluate the significance of a relationship between two properties that are measured within a single population. For SLR, investigations are observational when treatments are not assigned to sample units but are observed factors pertinent to a particular sample unit. In observational studies, SLR can be used to evaluate correlation between the two properties (A is related to B), but not causality (A is caused by B). Two other potential expressions of causality could be true for the collected samples. First, it may be that causality is reversed – that is, that B is caused by A rather than being the causal factor responsible for changes in A. Second, it may be that A and B are correlated because they are both related to a third, unmeasured process. This scenario is also commonly experienced because of the challenges associated with accounting for all the potential factors that influence the dynamics of complex living systems.
Students should understand the nuances associated with observational investigations as they report on their research; however, these concepts are difficult to convey in the abstract. When statistical techniques are studied in the context of relevant applications, students can make connections between abstract conceptual content and relevant applications. I introduced the two statistical procedures described above in an undergraduate biology course, Ecological Measures (BI 304). The intent of the course is to expose students to techniques for collecting and analyzing ecological data. The lab reinforces statistical concepts presented in lecture through a hands-on activity that illustrates the value of statistics in describing ecosystems. Students measure ponderosa pine trees (Pinus ponderosa) in two areas near Whitworth University’s campus in northeastern Washington State. Ponderosa pines are prevalent near campus and are easily measured using annual growth rings. A variety of species could be used, depending on availability at a specific site. Gymnosperms (conifers) such as pine, fir, or cedar trees work well because of seasonal variations in tracheid morphology. Among angiosperms, monocots do not produce clear annual rings, but dicots do. Ring-porous dicots (e.g., oak and ash) produce the clearest rings, whereas diffuse-porous species (e.g., maple, birch, and aspen) do not produce rings that are as clear (Speer, 2010).
The determination of tree diameter and age are fundamental measures in the field of forest ecology because they pertain to tree dominance and economic value. Tree diameter is typically measured at 1.3 m, referred to as “diameter at breast height” (DBH), with the use of a tape that is calibrated to measure tree diameter or with tree calipers (Avery & Burkhart, 2002). Tree age can be approximated by counting tree rings acquired with the use of an increment borer, which extracts a core of a tree’s trunk, providing a record of tree rings (Speer, 2010). Tree rings are formed by seasonal variations in the morphology of vascular cells formed in the cambium of gymnosperm and dicotyledonous angiosperm trees. Techniques can be used to ensure that the core does not contain “false” or missing rings, but those approaches are beyond the scope of this lab, so we assume that trees put on a single ring per year.
In the lab described below, students select trees to sample for both diameter and age for two different areas. They use a two-sample t-test to compare the diameters of trees in the two areas, and SLR to assess the relationship between tree age and diameter.
Random number table (azimuths 0–359°; distances 0–30 m)
50-m tape (one per team of three or four students)
Surveyor’s flagging (one roll per team)
Compass (one per team)
Increment borer (one per two teams)
DBH tape (one per team)
Core mounts (one per student)
Sanding blocks (one per student)
Sandpaper – coarse through fine (no. 60–400)
Computers with statistical software (one per team)
Stereo dissecting scope (optional, one per student)
This lab accompanies lecture instruction related to the appropriate selection of samples and to the use of both the two-sample t-test and SLR. The lab takes place over two 3-hour lab periods, with an additional 20-minute preparation time between lab periods.
Framing the Question
At the beginning of the first lab period, students work together in groups of three or four to frame research hypotheses. The instructor prompts students to discuss which factors might result in differences in tree size. Factors that influence plant productivity, and therefore tree size, include age, water and nutrient availability, solar insolation, and soil characteristics (Chapin et al., 2002). On the basis of this discussion, the instructor asks students to make predictions and frame hypotheses about differences in trees at two sites and about the size of trees in relation to tree age. The instructor guides students in the formulation of a research hypothesis regarding tree growth, using that hypothesis to predict how trees in different circumstances might differ in size, and crafting a testable hypothesis about trees in the specific sites under consideration. At our campus, two study areas are identified and discussed. One area is irrigated and treated with fertilizers and herbicides, while the other area is unmanaged. Students typically hypothesize that trees in the managed area will have a larger average diameter and that there is a positive relationship between tree age and tree diameter, but they acknowledge that other factors influence tree diameter.
Collecting the Data
During the second part of the first lab period, groups of students are assigned to one of the two study areas. A starting location is assigned, and students are given random number tables of distances (1–30 m) and azimuths. Students use a 50-m tape and a compass to navigate to a selected location within the study area. If the location is outside of the study area, the next pair of random numbers in the sequence is used. Students select the closest ponderosa pine to the established location and measure the tree’s diameter.
During the process of selecting trees, one tree per student is flagged for coring when an increment borer becomes available. The timing of diameter measurements and tree coring works well with one increment borer per two teams. Increment borers come in a variety of lengths and bore diameters. A 4.3-mm diameter bit works well, and a 45-cm-long borer is appropriate for our large trees. The instructor moves between teams and guides the students through the extraction of a core from the tree by inserting the bore as close to the ground as the handle of the borer will allow and perpendicular to the trunk of the tree (Figure 1). The borer is pressed into the bark and turned clockwise until it extends to the middle of the tree. An extractor spoon is inserted into the borer, a half-counterclockwise turn breaks the core off, and the extractor spoon is used to extract the core (Figure 1). The borer is removed from the tree by turning counterclockwise. Coniferous trees sustain little damage in this process because they quickly compartmentalize the area around the injury. The impact of coring on hardwoods is typically minimal as well, although it varies by species (Grissino-Mayer, 2003). Students store their cores in straws for transport back to the lab.
At the lab, cores are removed from the straws and the instructor demonstrates how to gently tape cores onto mounts with masking tape. They should be allowed to dry for a few days before they are glued to the core mount. Tape should be tight enough to hold the core in place and keep it from warping, but not so tight that it constrains the contraction of the core, lest the core break as it dries. Core mounts consist of a wooden stick that is at least as long as the longest anticipated core (I choose to use 450 × 20 × 20 mm mounts), with a 6-mm semicircular groove down the middle to accommodate the core (Figure 1). Hardwood core mounts (I use oak) work well, since they are not as prone to being damaged in the process of surfacing the cores. Mine are made at our woodshop, but they can also be ordered. Grissino-Mayer’s website (http://web.utk.edu/~grissino/supplies.htm) on dendrochronology research has information on ordering or making a variety of dendrochronology supplies.
Analyzing Diameter Differences
Collection of diameter data takes about an hour and a half, allowing students to perform their first statistical test during the remainder of the period. Students record their data in a common spreadsheet that is then distributed to the class by the instructor. The spreadsheet consists of four columns: tree ID, site ID, tree diameter, and number of rings (Table 1). During this first lab period, the “number of rings” column will be left blank. Students have data for tree diameters for a number of trees equal to 20× the number of teams in the class. The instructor then guides the students through the process of analyzing the diameter data from the two areas.
|Tree ID .||Site ID .||Diameter (cm) .||Number of Rings .|
|Tree ID .||Site ID .||Diameter (cm) .||Number of Rings .|
Many statistical and some spreadsheet software packages can be used to calculate appropriate statistics. The specifics of the procedures will vary by software package. The instructor should become familiar with these procedures and write clear instructions for students to analyze the diameter data. Both high school and undergraduate students can use software to calculate the mean and standard deviation for the two samples and make observations about which area shows larger trees and by how much, in relation to the standard deviation. Most applications will produce box-and-whisker plots, which can be used for a visual interpretation of the differences between the two areas (Figure 2). Undergraduate students are given instructions on how to perform a t-test to statistically compare the two samples. Because they are not paired observations, a two-sample t-test should be specified, selecting the option to assume equal variances. Students can interpret the resulting P value in relation to a predetermined significance threshold, usually such that a P value of <0.05 ( < 0.05) indicates a statistically significant difference between populations. It typically takes students some time to come to a clear understanding of the meaning of the P value; therefore, the instructor should prompt them to interpret the P value they obtain from the statistical analysis in terms of the hypotheses they originally posed.
Interim Prep Time
Between labs, the cores (after they are dry) should be glued into mounts, such that they are oriented in the same way as they were in the tree (i.e., with vertical grain; Figure 1). This can be determined by looking at the end of the core. Glue should be applied evenly along the length of the mount, but used sparingly, and excess glue should be cleaned off. Students either have a tendency to use too much glue, making surfacing difficult, or they apply a spotty distribution of glue, making some portions of the core come apart when sanded. Masking tape can be used to hold the cores in place while the glue dries. Cores that have broken as they dried can be realigned before gluing them in place. It is helpful for the instructor to demonstrate the process of aligning, gluing, and taping the core with a sample core and with diagrams of core grain drawn on the whiteboard. Glued cores should be allowed to dry for ~24 hours before the surfacing procedure that occurs in the next lab block.
Surfacing Cores & Counting Rings
During the first part of the second lab period, after the glue has dried, cores should be surfaced using a sanding block and sandpaper, starting with coarse grit (no. 60) and ending with fine (no. 400). Cores should be sanded until they are flush with the core mount (Figure 1). Students often stop short of sanding the core all the way flush, which makes for a rounded surface that is more difficult to interpret. Sanding nearly flush with the coarsest paper before moving to a finer grit makes it easier to finish the cores flush with the mount. A sample that has been done well also helps students gauge their progress. Rings can then be counted, starting from the bark and counting toward the pith of the tree. The number of rings is recorded by each student in the common class spreadsheet (Table 1). Optionally, a dissecting scope may be useful in counting rings when they are very close together.
Analyzing Size–Age Relationships
After counting the rings, undergraduate students can analyze the data using SLR during the latter portion of the second lab period. Using appropriate statistical software, the independent variable will be specified as the number of rings, and the dependent variable will be set as tree diameter. The SLR will result in a P value that can be used to determine the likelihood that there is a relationship between the age of a tree and its size, given the data that were collected (typically at α < 0.05). Both undergraduate and high school students can create a plot of the data by producing a scatterplot with a least-squares line fit to the data. Undergraduates should record the coefficients and P value for the SLR and the R2 value for the fit (Figure 2). The instructor should again query students about the meaning of the P value, in particular the difference between P values for the two-sample t-test and for the SLR procedure; in SLR, the P value indicates the likelihood that the slope of the best-fit line is actually zero, given the sample collected. Students should also differentiate between the P value and the R2 value, which is simply an indication of the degree to which the best-fit line describes the variability in the sample data and does not indicate a measure of statistical significance.
Students summarize their work in a formal lab report that includes introduction, methods, results, discussion, and conclusion sections. Although the lab work is done in teams, each student is responsible for writing their own report. Students are not instructed regarding the specific content of the report; rather, they are given a template that includes the purpose of each section and are required to describe their research using the format specified. The instructor should describe the intent of each section of the report, paying particular attention to the way in which the different sections should work together. The introduction describes the importance and objectives of the research and sets out the hypotheses. The methods section describes the study area, details the data collection, and outlines the analysis. The methods section assumes an audience that is familiar with the techniques used, but it records sufficient detail that the study could be repeated. The results section highlights the results of the statistical procedures, including professionally presented figures. The discussion section puts the results into context, most importantly discussing the nature of the results in relation to the restrictions of an observational study. Students must verbalize results such that they do not make claims of causality but still discuss the degree of support that their results convey to the predictions and hypotheses that they set out in the introduction. The instructor should guide a conversation that asks students to evaluate how their methods might influence the validity of their results. For example, astute students discern that while the random number table results in an adequately randomized sample, it does not sample the entire area. Students might then discuss alternatives to this shortcoming. Finally, the conclusion section summarizes the student’s findings in relation to the hypotheses posed in the introduction. Assessment of the report focuses on the degree to which each section aligns with the intent of the report. Emphasis is placed on a clear and concise description of the methods and the professional presentation and accurate interpretation of statistical results.
The use of easily collected data with straightforward interpretation provides the reiteration of statistical concepts introduced during lecture. The lab described is relatively inexpensive, making it accessible to most institutions. The value of this approach is in presenting a hands-on experience whereby students can apply the statistics they learn in the classroom to a relevant field-based investigation. This gives students an appreciation of the design and interpretation of observational studies in ecology.