Mitosis Data Analysis: Testing Statistical Hypotheses By Dana Krempels, Ph.D. and Steven Green, Ph.D.

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1 Mitosis Data Analysis: Testing Statistical Hypotheses By Dana Krempels, Ph.D. and Steven Green, Ph.D. The number of cells in various stages of mitosis in your treatment and control onions are your raw data (singular = datum). This chapter will guide you through statistical analysis. I. Data, Parameters, and Statistics: Quick Review Recall that data can be of three basic types: 1. Attribute data. o descriptive, "either/or" measurements o usually describe the presence or absence of a particular attribute o such data have no specific sequence, so are considered unordered 2. Discrete numerical data. o biological observations counted as integers (whole numbers) o have a specific sequence, and so are considered ordered o do not describe physical attributes of the things being counted. 3. Continuous numerical data. o fall along a numerical continuum o limit of resolution is the precision of collecting methods and instruments o generally fall along a normal (Gaussian) distribution Data measurements usually are distributed over a range of values. Measures of the tendency of measurements to occur near the center of the range: mean - the average measurement median - the measurement located at the exact center of the range mode - the most frequent measurement in the range Measurements of dispersion around the mean: range - the difference between the largest and smallest values variance - spread of data points from their mean standard deviation dispersion of data points around the mean (square root of variance) Parameters and Statistics If you were able to measure the height of every adult male Homo sapiens who ever existed, and then calculate a mean, median, mode, range, variance and standard deviation from your measurements, those values would be known as parameters. Parameters represent the actual values calculated from measuring every member of a population of interest. Mitosis Data Analysis - 1

2 Obviously, collecting data from every member of a population of interest is difficult, if not impossible. Randomly sampling a subset of the population allows calculation of an estimate of a parameter known as a statistic. Parameters are written as Greek symbols. Statistics are written as Roman letters equivalent to their Greek counterparts. For example The standard deviation of an entire population is written as σ. The standard deviation for a subset of an entire population is written as s. The variance of an entire population is written as σ 2. The variance for a subset of an entire population is written as s 2. and so on. II. From Raw Data to Mitotic Index Your mitotic cell counts comprise a survey of the number of different stages of mitosis in your two populations. You counted mitotic cells in 8 treatment and 8 control roots, and then calculated a Mitotic Index (M) for each sample. M = n m /N n m = the number of mitotic cells in the sample N = the total number of cells counted in the sample. Enter your M values in a table like the one shown below. Provide an appropriate table legend. Table. Treatment Sample # Mitotic Index (M) Control Sample # Mitotic Index (M) Enter your M values for any mitotic phase you are comparing between treatment and control in a table similar to the one above. Probability The probability that an observed result is due to some factor other than chance is known as alpha (α). By convention, α is usually set at 0.05, or 5%, which means that there is a 95% probability that a particular outcome is due to some factor other than random chance. 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. Mitosis Data Analysis - 2

3 (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. We will use an α level of 0.05.) In scientific endeavors, statistical significance has a highly specific definition. The difference between an observed and expected result is said to be statistically significant if and only if: Under the assumption that there is no true difference, the probability that the observed difference would be at least as large as that actually seen is less than or equal to a (5%; 0.05). Conversely, under the assumption that there is no true difference, the probability that the observed difference would be smaller than that actually seen is greater than 95% (0.95). A probability distribution assigns a relative probability to any possible outcome. III. Applying a Statistical Test to Your Mitotic Indices Your mitotic indices are ordinal, non-parametric data not distributed along a normal curve. A non-parametric statistical test, the Mann-Whitney U is appropriate for this type of data. The Mann Whitney U test measures the degree of overlap between two sets of data that can be ranked (i.e., put in order of ascending values). large overlap means no significant difference between your populations o fail to reject the null hypothesis small (5% or less) overlap means a significant difference between your populations o reject your null hypothesis. Non-parametric test for two samples: Mann-Whitney U The Mann-Whitney test allows the investigator (you) to compare your two cell populations without assuming that your Mitotic Index values are normally distributed. The Mann-Whitney U does have its rules. For this test to be appropriate: You must be comparing two random, independent samples (treatment & control) The measurements (Mitotic Indices) should be ordinal No two measurements should have exactly the same value o (though we can deal with ties in a way that will be explained shortly) 1. State your null and alternative hypotheses. H o : H A : Example: H o : There is no difference in the ranks of Mitotic Indices (M) between meristematic cells in an onion treated with aqueous trifluralin and an onion treated with plain water. H A : There is a difference in the ranks of Mitotic Indices (M) between meristematic cells in an onion treated with aqueous trifluralin and an onion treated with plain water. 2. State the significance level (to be compared to α, 0.05) required to reject H o. This is typically a probability value (P) of < Rank your Mitotic Indices from smallest to largest in a table Note which index came from which population of cells (Treatment or Control). Mitosis Data Analysis - 3

4 Example: Table 1 shows 16 (imaginary) MI from treatment (T) and control (C) onion root tips. Table 2 shows the values ranked and labeled by population. Table 1. Mitotic Indices for Table 2. Ranked Mitotic Indices treatment and control root tips Note tied values in blue. Sample # M treatment M control Rank Ranked M values Cell Population T T T T T C T T T C C C C C C C 4. Assign points to each ranked value (see Table 3): Each treatment rank gets one point for every control rank that appears below it. Every control value gets one point for every treatment value that appears below it. For example, the first value, 0.10 (T) has 8 Control values below it, so it gets 8 points. Value 10 (C) has 3 Treatment values below it, so it gets 3 points. Tied values split the sum of their points. For example: o Rank 5 (0.35) has 8 points o Rank 6(0.35) has 3 points o = 11 o Each rank gets half of 11, or 5.5 Table 3. Points assigned to ranked M values in Treatment and Control onion cell populations. (example) Tied values split their total points equally. Rank Ranked M values Cell population Points T T T T T 8 à C 3 à T T T 7 à C 0 à C C C C C C 0 Mitosis Data Analysis - 4

5 5. Calculate a U statistic for each category by adding the points for each cell population. U treatment = = 55 U control = = 9 Your U statistic is the smaller of these two values. In the imaginary example our U value is 9. The lower the U value, the greater the difference between the two groups being compared. (For example, if none of the M values overlapped, the U value would be zero.) IV. Critical values for non-parametric statistics We have defined our significance level (α) as This implies: a true null hypothesis will be rejected only 5% of the time a false null hypothesis will be rejected 95% of the time if the P value obtained from your data is less than or equal to A critical value of a statistic (e.g., Mann-Whitney U) is the value associated with a significance level less than or equal to α. (We are using the traditional value of α, 0.05.) Critical values for the Mann-Whitney U statistic (at different sample sizes) are shown in Table 4. In the previous imaginary example, treatment and control groups with 8 samples each, a critical value of 23 is required for rejection of the null hypothesis. The Mann Whitney U statistic of 9 is far lower than this cut-off value. This means there is very little overlap between the two populations (they are significantly different). The null hypothesis is rejected. V. Experimental Error vs. Human Error Your team made sure that all factors except one the chemical used on one population of onion root tips were exactly the same for Treatment and Control groups. But did you get exactly the same number of mitotic cells in each sample? Probably not. What might account for slightly different results among samples? Slight variation in results in carefully run trials is known as experimental variability or experimental error. This natural variability is NOT the same as variability caused by actual mistakes in experimental technique (human error). DO NOT CITE HUMAN ERROR AS A REASON FOR UNEXPECTED RESULTS IN YOUR EXPERIMENT! TO DO SO IS UNPROFESSIONAL. If you make accidental mistakes that could affect your results, you should re-do the experiment, not simply explain away those mistakes as human error. Citing human error as a good reason for your results is about as good as saying, Oops! We are terrible at science. But we don t really care enough to do it right. NEVER include human error in discussions of experimental variability. Experimental error mistakes! When contemplating your results, your fellow scientists will assume you have done your experiments as carefully as possible, and have minimized inaccuracies due to human error. Mitosis Data Analysis - 5

6 VI. Project Completed. Is This the End? The study you are now completing is only the beginning of what could be a longterm research project to discover the various factors that direct and affect mitosis. The research project you are now completing is a pilot study. It establishes an observable fact (that there is or is not a difference in chromosome migration between cells treated or not treated to disable microtubules). Eventually, your findings should lead to more detailed investigation. Science is not a one-project endeavor. Every new piece of valid information can be seen as opening a new doorway to discovery of the most intimate mechanisms of life. Table 4. Critical values for the Mann-Whitney U statistic. Find the value that corresponds to the sample sizes (8 and 8) of your two cell populations. 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 cell populations is due to chance. If your U value is smaller than the one shown in this table, reject your null hypothesis. If your U value is larger than that shown in the table, fail to reject your null hypothesis. Mitosis Data Analysis - 6