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ARBOR Project – Tree Survival Location The experimental site extends for approximately 1 mile along the eastern bank of the floodplain of the White River from 10th Street to New York Street in Marion County, Indiana (see maps). It is bounded to the north by the junction of Fall Creek and the White River, to the south by White River State Park, to the west by the White River, and to the east by an Army Corp of Engineers Flood Control Levee adjacent to the IUPUI campus. We have divided the site into a northern and southern section each including approximately 4.5 acres of floodplain. The northern section of the study site is bounded to the west by an existing forested wetland and curved sand bar complex. The southern section is bounded to the west directly by the White River. The Experiment There are three commonly used methods for floodplain or bottomland restoration in the Midwest. One of the questions we're interested in evaluating is which method, if any, works best. Each method was used in two 1-acre plots; we have also kept unplanted control plots for comparison. In both the north and south sections, four 1-acre areas were planted as follows (numbers = total trees planted in the north & south areas):
Each treatment method was planted in both the northern and southern sections. Plots were randomly placed in each section and individual trees were planted randomly within each treatment plot. Additionally, equal numbers of trees and an even distribution of tree species were planted in each treatment plot. Trees included 12 native species that naturally occupy the edges of rivers and streams (= riparian) and species adapted to the environmental conditions of the Midwest, generally, and to the Tipton Till Plain, specifically. They exclude extremely rare or habitat-restricted species, including rock elm and blue ash, and the American elm. The latter was formerly an important canopy species on many Indiana floodplains but Dutch elm disease now routinely kills them before they reach canopy height. Therefore, it was also excluded from the planting list. How Well Are the Trees Surviving? Restoring a section of the White River’s floodplain to its natural state is one goal of the ARBOR Project but we're also attempting to determine the best method of accomplishing the restoration. One way to evaluate the success or failure of our efforts is to examine how the trees are 1) surviving as time progresses within a single treatment plot (strategy) and, 2) whether or not different planting strategies enhance a species’ chances of survival. We want to statistically evaluate tree survival (= quantitative assessment) and not just look at raw numbers (or percentages) of surviving trees and make a value judgment (= qualitative assessment). How should this be accomplished? The chi-square test: This type of data (counts) is ideally suited for a statistical test called the chi-square test. The results of the test let you know the degree of confidence you can have in accepting or rejecting a hypothesis. Typically, the hypothesis being tested with chi square is whether or not two separate samples are different enough, in some characteristic or aspect of their behavior, that we can consider them to be different statistically. Here, for example, we’re interested in the survival of species of trees within each planting area as time passes and in survival patterns related to different planting strategies. Stating this another way, do particular trees tend to survive better than others, and do one or more methods of planting increase a species chances for survival over other methods or practices? How does chi square do this? Basically, the chi square test of statistical significance is a series of simple mathematical formulas that compare the observed frequencies of some phenomenon (in our sample) with the frequencies we would expect if there were no relationship at all between the two variables in the larger (sampled) population. This hypothesis of no difference is called the null hypothesis. Chi square tests our actual observations against the null hypothesis and assesses whether the actual results are different enough to overcome a certain probability that they could be due to sampling error. For example, our null hypotheses are that 1) there is no difference in the ability of trees to survive as time passes, and 2) differences in tree survivorship between areas or plotting strategies do not exist. An example: Let’s run a chi square test on data taken from Sokal and Rohlf (1987 p. 307) where we will test the effect of an antiserum on bacteria infecting populations of mice. The setup of the test will be similar to that for our tree data. 1. A sample of 111 mice was administered a dose of bacteria and was then divided into 2 groups: 57 mice that also received a dose of antiserum and a control group of 54 that did not. 2. After the disease had run its course, 38 dead mice & 73 survivors were counted. Thirteen of the dead mice had received the antiserum while 25 received only bacteria. 3. The question of interest is did the antiserum protect the mice so that there were proportionately more survivors in that group than in the control group?
Within the table, you can see the number of dead & surviving mice while the sums, called marginal totals, indicate the total number of mice that display any one property. For example, 57 mice were treated with antiserum while 38 mice did not survive the experiment. Next, we need to determine the expected numbers of living and dead mice based on the null hypothesis that the proportions do not differ between treated and untreated mice. To do this, we use the marginal totals from the table of observed frequencies of living & dead mice. Let’s calculate them and then display those values in a table similar to the one above: 1. Bacteria + antiserum/Dead: (38 x 57)/111 = 20 2. Bacteria + antiserum//Alive: (73 x 57)/111 = 37 3. Bacteria only/Dead: (38 x 54)/111 = 18 4. Bacteria only /Alive: (73 x 54)/111 = 36 Expected frequencies:
We now can compare the observed results against the results expected if the null hypothesis were true. However, we need some way to determine whether the null hypothesis can be rejected and, if it can be rejected, what level of confidence we have that we're not making a mistake. To do this we must measure the size of the difference between each pair of observed and expected frequencies for cells in our contingency table. To accomplish this, we calculate the difference between the observed and expected frequency in each cell, square that difference, and then divide that product by the difference itself. The formula for chi square is: Χ2 = (O - E)2/E Squaring each difference eliminates negative values because some will be larger and others smaller than the expected value. If we didn't do that, the positive and negative differences across the entire table would add up to zero. Dividing the squared difference by the expected frequency removes the expected frequency from the equation, so that the remaining measures of observed-expected difference are comparable across all cells. For our immunology example, the individual chi square calculations are: 1. Bacteria + antiserum/Dead: (13 - 20)2/20 = 2.45 2. Bacteria + antiserum//Alive: (44 - 37)2/37 = 1.32 3. Bacteria only/Dead: (25 - 18)2/18 = 2.72 4. Bacteria only /Alive: (29 - 36)2/36 = 1.36 Sum (å) = 7.85 The Total Chi Square Value = Χ2 = å (O - E)2/E = 7.85. Is this value significant,
statistically? We need to know how much larger than 0 (the absolute
chi square value of the null hypothesis) our calculated chi square
value must be before we can confidently reject the null hypothesis.
If we do this manually, we’d have to look at a table of chi square
values, choose a level of significance, and look up the appropriate
value for a test with 1 degree of freedom (for a 2 x 2 table). 1. If we did that, we’d discover that the tabled value of chi square for 1 degree of freedom and a significance level of 0.05 (we’d be willing to be wrong 1 time in 20 or correct 95% of the time) would be = 3.841. 2. Our calculated value of 7.85 exceeds the tabled value by a large margin. 3. This means that we cannot accept the null hypothesis of no difference in the survival of mice. The antiserum DOES make a difference! The statistical programs that are currently available for personal computers do all of this for you: 1. they automatically calculate the expected frequencies for your table and determine the probability of getting the chi square value that was calculated. 2. This means that you don’t have to worry about degrees of freedom or looking values up in a table for comparison! 3. If you’re interested in the details of the assumptions for chi square tests, making calculations, the concept behind the number of degrees of freedom, and confidence levels, you can go to Dr. Jeff Connor-Linton’s web site at Georgetown University to read about them and examine his examples. OK, What About the Trees at the ARBOR Site? The Data: Here is what the data look like for October of 2001 and March of 2002 (see tabs at bottom of spreadsheet). Notice that we’ve recorded the number of trees that were planted in each area and have been counting, at regular intervals of time, the number of trees that have perished. Survival Within an Area: First, let’s take a look at the survival pattern of Red Maples from Area 1 (containerized trees) that were planted in October of 2000. 1. Notice that in the Fall Growing Season (October 2001) 18 of the 21 trees that were planted still survived. Survivors had decreased to 11 by March 2002 (Fall-Spring Season). 2. Go to Dr. Connor-Linton’s Web Chi Square Calculator. Using this web site, you can calculate chi square values interactively by simply entering frequencies of living and dead trees into an electronic contingency table. 3. Begin by entering the dimensions of the table, which are 2 rows x 2 columns, and press the Generate Table button. 4. Double click in the Table Name box & enter Red Maple Survival – Area 1. 5. Double click in the Col 1 box & enter Alive, press the Tab Key & enter Dead in Col 2. 6. Press the Tab Key again & enter October 2001 in the Row 1 box, tab again & enter 18 in the alive column, and tab once more & enter 3 in the dead column. 7. Press the Tab Key & enter March 2002 in the Row 2 box, tab again & enter 11 in the alive column and, finally, tab once more & enter 10 in the dead column. 8. Press the Calculate Chi Square Key to perform the calculations. a. The normal radio button is the default and produces the contingency table, degrees of freedom, chi square value, the probability that you’d get a value of that magnitude assuming no difference between categories, and whether or not that value is significant. b. If you choose the verbose radio button, you’ll receive all of the output for the normal button plus calculations of the expected frequencies for each cell in the table and calculations of probabilities. 9. What can you conclude from these results? Next, perform a similar test for Sycamores from Area 2 (bare-root, random planting). a. What do your results indicate? b. Are Sycamores surviving as well as the Red Maples? Differences in Survivorship Between Areas: This time, we’ll use data for Swamp White Oak: 1. Let’s begin by comparing different planting schemes within the same general area of study – the northern area.
2. Next, let’s compare the same planting style in a northern and a southern area, again using data for Swamp White Oak from the March 2002 survey.
References Sokal, R. R. and F. J. Rohlf. 1987. Introduction to Biostatistics. W. H. Freeman and Co., New York, 363 p. Chi Square Tutorial, J. Connor-Linton, Department of Linguistics, Georgetown University (www.georgetown.edu/cball/webtools/web_chi_tut.html; accessed 5/2/02) Practical Stats: A collection of useful guides to the application of data analytic procedures in the social sciences. Compiled by Kristopher J. Preacher and Derek D. Rucker, The Ohio State University (quantrm2.psy.ohio-state.edu/kris/index.htm; accessed 5/2/02) Useful General Link on Statistics from BigChalk at www.bigchalk.com/cgi-bin/WebObjects/WOPortal.woa/964/wa/BCPageDA/sec~CAB~10003~MATH~ (accessed 5/2/02). How to Identify plants: www.biologie.uni-hamburg.de/b-online/e02/02.htm (accessed 5/3/02). U.S. Department of Agriculture, Plant Database - an excellent resource for identifications (with images): plants.usda.gov/
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