# Biodiversity Data Analysis: Testing Statistical Hypotheses By Joanna Weremijewicz, Simeon Yurek, Steven Green, Ph. D. and Dana Krempels, Ph. D.

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1 Biodiversity Data Analysis: Testing Statistical Hypotheses By Joanna Weremijewicz, Simeon Yurek, Steven Green, Ph. D. and Dana Krempels, Ph. D. In biological science, investigators often collect biological observations that can be tabulated as numerical facts, also known as data (singular = datum). Biological research can yield several different types of data. Important measurements include counts (frequency) and those that describe characteristics (length, mass, etc.). Data from a sample are often used to calculate estimates of the average values of the population of interest (mean, mode, and median) and others describing the dispersion around those values (range, variance, and standard deviation). I. Data, Parameters, and Statistics: A Review Recall that data can be of three basic types: 1. Attribute data. These are descriptive, "either-or" measurements, and usually describe the presence or absence of a particular attribute. The presence or absence of a genetic trait ("freckles" or "no freckles") or the type of genetic trait (type A, B, AB or o blood) are examples. Because such data have no specific sequence, they are considered unordered. 2. Discrete numerical data. These correspond to biological observations counted as integers (whole numbers). The number of leaves on each member of a group of plants, the number of breaths per minute in a group of newborns or the number of beetles per square meter of forest floor are all examples of discrete numerical data. These data are ordered, but do not describe physical attributes of the things being counted. 3. Continuous numerical data. These are data that fall along a numerical continuum. The limit of resolution of such data is the accuracy of the methods and instruments used to collect them. Examples are tail length, brain volume, percent body fat...anything that varies on a continuous scale. Rates (such as decomposition of hydrogen peroxide per minute or uptake of oxygen during respiration over the course of an hour) are also numerical continuous data. (Figure 1). (Continuous numerical data generally fall along a normal (Gaussian) distribution. This distribution is a function indicating the probability that a data point will fall between any two real numbers.) When an investigator collects numerical data from a group of subjects, s/he must determine how and with what frequency the data vary. For example, if one wished to study the distribution of shoe size in the human population, one might measure the shoe size of a sample of the human population (say, 50 individuals) and graph the numbers with "shoe size" on the x-axis and "number of individuals" on the y-axis. The resulting figure shows the frequency distribution of the data, a representation of how often a particular data point occurs at a given measurement. Biodiversity Data Analysis 1

3 Your team should have counted at least 10 samples from each of your two habitats, and can now calculate one Menhinick s index (D value) for each sample. Tabulate your D values here: Sample # D habitat1 D habitat So what do we do with these indices? You may have an intuitive sense that they will allow you to determine whether your two sampled habitats overlap in their degrees of biodiversity. But science isn t about intuition. Statistics and statistical tests are used to test whether the results of an experiment are significantly different from the null hypothesis prediction. What is meant by "significant?" For that matter, what is meant by "expected" results? To answer these questions, we must consider the matter of probability. B. Probability The significance level (also known as alpha (α)) for a given study is set by the investigator before the analysis is begun. Alpha is defined as the probability of mistakenly rejecting a null hypothesis that is true (Type I error). By convention, α is usually set at 0.05 (5%). The probability that an observed result is due to some factor other than chance is known as P. The result of a statistical test is a statistic. For example, the student s t test yields a t statistic, the Chi-square test yields a X2 statistic, and the Mann-Whitney U test yields a U statistic. Every value of a particular statistic is associated with a particular P value. If the P value associated with a calculated statistic (e.g., the U statistic you will calculate with the Mann- Whitney test, to be described below) is 0.05, this means that there is only a 5% chance that the rejection of the null hypothesis will be incorrect. A P value of less than 0.05 means that there is an even lower chance of a Type 1 error. (For example, a P value of 0.01 means that there is only a 1% chance that the results are due to chance, and not to the factor you are examining.) In essence, α is a cut off value that defines the area(s) in a probability distribution where a particular value is unlikely to fall. In some studies, a more rigorous α of 0.01 (1%) is required to reject the null hypothesis, and in some others, a more lenient α of 0.1 (10%) is allowed for rejection of the null hypothesis. For our study of biodiversity, you will use an α level of The term "significant" as used in every day conversation is not the same as the statistical meaning of the word. In scientific endeavors, significance has a highly specific and important definition. Every time you read the word "significant" in this lab manual, know that we refer to the following scientifically accepted standard: Biodiversity Data Analysis 3

5 You must be comparing two random, independent samples (your two sites) The measurements (Menhinick s Indices, in our case) should be ordinal No two measurements should have exactly the same value (though we can deal with ties in a way that will be explained shortly). The Mann-Whitney U test allows the investigator to determine whether there is a significant difference between two sets of ordered/ranked data, such as those your team has collected in its biodiversity study. Here is a stepwise explanation and example of how to apply this test to your data. 1. State your null and alternative hypotheses. (You already have done this, right?) H o : H a : Example: H o : There is no difference in the ranks of species richness between a silted pond and a clear pond. H o : There is a difference in the ranks of species richness between a silted pond and a clear pond. 2. State the significance level (alpha, α) necessary to reject H o. This is typically P < Rank your Menhinick s Indices from smallest to largest in a table, noting which index came from which habitat. Example: Table 1 shows 18 (imaginary) values for Menhinick s Indices from the two ponds mentioned before, silted (S) and clear (C). Table 2 shows the values ranked and labeled by pond type. Table 1. Menhinick s Indices Table 2. Ranked Menhinick s Indices for silted and clear ponds D silted D clear Rank Ranked D values Habitat S S S S S S C S S C C C S C C C C C Notice in the ranked table that if two values are the same, then the rank each one receives is the average of the two ranks. For example, value nine appears twice, at rank 6 and 7. Add the two ranks and divide by two to get their mean: 13/2 = 6.5. Each value is assigned their same, mean rank whenever there is a tie. Biodiversity Data Analysis 5

6 4. Assign points to each ranked value. Each silted rank gets one point for every clear rank that appears below it. Every clear value gets one point for every silted value that appears below it. For example, the first rank, 2(s) has 9 clear values below it, so it gets 9 points. Value 9(c) has 3 silted values below it, so it gets 3 points. Table 3. Points assigned to ranked D values in silted and clear ponds. Rank Ranked D Habitat Points values 1 2 S S S S S S C S S C C C S C C C C C 0 5. Calculate a U statistic for each category by adding the points for each habitat. U silted = = 75 U clear = = 6 Your final U value is the smaller of these two values. In this example our U value is 6. In general, the lower the U value, the greater the difference between the two groups being tested. (For example, if none of the D values overlapped, the U value would be zero. That means there is a large difference between the two groups: they do not overlap at all.) 6. You are now ready to move to the final step, determining whether to reject or fail to reject your null hypothesis. (Proceed to Section IV.) A video explanation of the Mann-Whitney U test procedure can be viewed here: B. Non-parametric test for multiple samples: Kruskal Wallis test We told you not to. But some teams just have to go that extra mile. Biodiversity Data Analysis 6

8 Table 5. Critical values for the Mann-Whitney U statistic. Find the value that corresponds to the sample sizes of your two habitats. If your U value is smaller than that shown in the table, then there is less than 5% chance that the difference between your two habitats is due to chance alone. If your U value is smaller than the one shown in this table for your two sample sizes, reject your null hypothesis. If your U value is larger than that shown in the table, fail to reject your null hypothesis. (From The Open Door Web Site, Biodiversity Data Analysis 8

9 V. Project Completed. Is This the End? The study you are now completing is only the beginning of what could be a long-term research project to discover the various factors that affect biodiversity. The only thing you are determining now is whether or not there is a statistically significant difference between your two sample habitats. In other words, the research project you are now completing is a pilot Biodiversity Data Analysis 9

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