Biostatistics — 01. Course Content. Theme 1. Basic Concepts of Biostatistics

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D. Shabanov, M.A. Kravchenko. Statistical Analysis of Data in Zoology and Ecology

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Course Content. Topic 1. Basic Concepts of Biostatistics

Topic 2. Using the Statistica Program

Biostatistics-01

Biostatistics-02

Content Topic 1. Basic Concepts of Biostatistics 1.1. What is Biostatistics and Why is it Needed 1.2. Probability 1.3. Population and Sample 1.4. What is Significance? A Humorous Example 1.5. Statistical Significance; Null and Alternative Hypotheses 1.6. Attributes 1.7. Distributions, Statistics, and Parameters 1.8. Parametric and Non-Parametric Statistical Methods and Tests Topic 2. Using the Statistica Program 2.1. Why Statistica? 2.2. The Statistica Program 2.3. Structure of a Statistica Data Table 2.4. Operations with Selected Cells 2.5. Working with Rows and Columns 2.6. Variable Specifications 2.7. Numeric and Text Forms of Data 2.8. Formulas for Data Recalculation Topic 3. Data Visualization (Using the Example of Green Frog Description Results) 3.1. Description of the Pelophylax_example.sta Sample File 3.2. Histograms: Example of Graph Construction 3.3. Graph Editing 3.4. Scatter Plots and Regression Lines Topic 4. Comparing Samples 4.1. In What Situations Might Comparing Samples Be Necessary? 4.2. Comparing Samples Using Student's t-Test 4.3. Using Fisher's Test for Comparing Samples 4.4. Box Plots in the t-Test Module 4.5. The Problem of Multiple Comparisons 4.6. Experiment on Obtaining False "Significant" Differences in Multiple Comparisons 4.7. Corrections for Multiple Comparisons 4.8. Non-Parametric Analogues of Parametric Methods 4.9. Mann-Whitney U-Test 4.10. Sign Test for Paired Comparisons 4.11. Kruskal-Wallis Rank Sum Analysis of Variance Topic 5. Brief Introduction to Analysis of Variance 5.1. What is Analysis of Variance? 5.2. Test for Normality of Distribution 5.3. One-Way Analysis of Variance: Manual Calculations 5.4. One-Way ANOVA in the Statistica Package 5.5. ANOVA and Student's and Fisher's Tests: Which is Better? 5.6. Two One-Way Analyses of Variance: Manual Calculation 5.7. Two-Way Analysis of Variance: Manual Calculation 5.8. Two-Way Analysis Using ANOVA in the Statistica Package Topic 6. Comparing Distributions 6.1. Examples of Problems Requiring Distribution Comparison 6.2. Determining Association of Qualitative Attributes Using Crosstabulation 6.3. Comparing Distributions Using the Non-Parametric Statistics Module Topic 7. Association Between Attributes Topic 8. Cluster Analysis 8.1. Essence of Cluster Analysis 8.2. Example of Performing Cluster Analysis "Step by Step" 8.3. Fundamental Limitations and Drawbacks of Cluster Analysis Topic 9. Principal Component Analysis 9.1. Essence of the Method (Using a Simple Example) 9.2. Transition to Initial Data with a Large Number of Measurements Topic 10. Discriminant Analysis 10.1. Purpose and Basic Logic of Discriminant Analysis 10.2. Example of Performing Discriminant Analysis: Morphometric Attributes of Frogs 10.3. Searching for More Effective Ways of Separating Groups Topic 11. Some Methods Specific to Zoology and Ecology 11.1. Fluctuating Asymmetry Analysis 11.2. Example of Fluctuating Asymmetry Data Analysis Program of the Major Practical Course "Statistical Analysis of Data in Zoology and Ecology" Additional Materials: M.A. Ghazali. Statistical Methods in Zoology. Materials from Open Lectures Delivered at the I.I. Schmalhausen Institute of Zoology in the 2017/2018 Academic Year for First-Year Graduate Students Lecture 1. Statistical Research Lecture 2. Statistical Estimates Lecture 3. Elementary Tests Lecture 4. Linear Models Lecture 5. Multivariate Statistics (part) Presentations D. Shabanov (2006). Lies, Damned Lies and... D. Shabanov (2009). Clusters, Clades and the Chimera of Objectivity K.P. Vorobiev (2008). "Format of Modern Journal Publication Based on Clinical Study Results. Part 4. Biostatistics" Promising Topics for Course Expansion: Using Microsoft Access DBMS for Creating Zoological Databases. Topic 1. Basic Concepts of Biostatistics 1.1. What is Biostatistics and Why is it Needed

Statistical analysis of biological research results allows solving several types of problems: 1. Clearly presenting the results of describing the diversity of studied objects; 2. Reasonably (with a certain probability of error) accepting or rejecting hypotheses about the existence of patterns reflected in the variation of the studied variable; 3. Detecting hidden patterns concealed in the variation of the studied data. One should not think that there exists some special biological statistics fundamentally different from mathematical statistics in general. However, the variability of biological objects has certain peculiarities distinguishing them, for example, from the variability of financial indicators or results of technological processes in production. This leads to the fact that the set of methods used in biology differs from that in other fields of statistical application. Furthermore, one should remember that statistical research in biology is not an end in itself: it is subordinated to the tasks of biological research and cannot be fully interpreted outside the biological problem being studied. However, not only data analysis should be subordinated to the logic of biological research; the research itself should be constructed with consideration of future analysis. The collection of empirical data and the setup of experiments should in advance consider how the analysis of the obtained data will be organized. Thus, although the application of statistics in biology cannot be completely separated from mathematical statistics as such or from the branches of biology studied using certain methods, it still constitutes a special field of science, a special complex of problems and methods for their solution. For this field, one can use the term proposed in 1899 by Francis Galton — biometry. Since the term "biometry" has been appropriated by specialists in identity verification based on individual characteristics, in many cases it is simpler to use the term biostatistics. The objects studied by biology have a high level of uniqueness. Almost any biological phenomenon exhibits both general patterns and the influence of special circumstances, often related to the uniqueness of biosystems. This means that for biological research, methods that allow seeing general patterns manifesting themselves through the variability of particular manifestations are very important. Perhaps this is why biologists have made a major contribution to the development of statistics as a whole. The works of Francis Galton, Karl Pearson, and Ronald Fisher constitute an important part not only of biostatistics but of mathematical statistics as a whole. 1.2. Probability Statistically repeatable events can be studied. For example, we blindly select rabbits from a box. Rabbits can be black or white. Each selection is an elementary event. A person puts their hand into the box opening and grabs some rabbit... Can we find out which rabbit they grabbed? No (if there are no other sources of obtaining information and no other factors influencing the result). Can we find out what the ratio of black to white rabbits in the box is? Also no.

As soon as the rabbit is pulled out, we not only learn its color. We can learn something about the composition of rabbits in the box. For example, if a white rabbit is pulled out, we can state that there was at least one white rabbit in the box. A little... However, if 10 rabbits are pulled out sequentially, based on the composition of the group of rabbits gathering at the feet of the person pulling them out, we can make a more detailed assumption about the composition of rabbits in the box. These predictions are based on the phenomenon of probability manifesting in regular, repeatable events. Probability is a numerical measure of the possibility of an event. Probability 1 means the event will certainly occur, and probability 0 means it is impossible. Suppose there are 50 white and 50 black rabbits in the box. What is the probability of randomly selecting a white rabbit in a single selection? Out of the total number of possible outcomes (100), 50 correspond to this condition, so the probability is 50/100 = 1/2 = 0.5. Should we consider the variant that, for example, the hand pulled out of the box had no rabbits or, say, two? In real life — yes, but in its simplified model, to which the apparatus of probability theory basics can be applied — we may not need to. The cases when a person did not pull out any rabbit or pulled out two at once do not correspond to the conditions of a single selection. However, if the reader of this text put their hand into a real box filled with dodging and kicking rabbits, the probability that they would pull out nothing could not be ignored. And what is the probability of selecting two rabbits of the same color? It might seem like 0.5, though in reality it is less. After selecting a rabbit of a certain color, the probability of selecting a second one of the same color is 49/99 versus 50/99. Thus, the probability of selecting two rabbits of the same color is 49/99 = 0.4949..., and two white rabbits — 0.24747...

1.3. Population and Sample A population is the actual or hypothetical set of all objects with the technology of studying a category. In most cases, it is impossible to study the entire population, and researchers work with samples (empirical sets, sample sets) – groups of objects obtained from the population. The size of the population is determined by the research task (and can significantly change when it is reformulated). Comparing the growth of boys and girls in a group studying biometrics can be a study of this specific group (thereby falling under the entire population), a study of students of a specific university (the population is at least finite and legally defined). Students in general or people in general (in the latter cases, the population, at least hypothetically, turns out to be potentially infinite). A significant paradox of statistics is that the researcher works with samples, but studies the populations from which these samples were obtained. Can a sample be used to judge a population that is significantly larger than the sample? To some extent, yes. However, it is clear that not every sample reflects the composition of the population from which it was obtained. Can a sample be taken to judge the variability of human growth from among students? No, because this sample will include people of predominantly young age who wanted to get higher education and were able to enter the relevant university. Such a sample is biased. To obtain a completely random sample, the process of its formation should be organized in such a way that any object in the population has an equal probability of being included in the sample. In most cases, such selection is practically impossible. Nevertheless, only representative samples should be used to study the population, the deviations from randomness in their formation of which cannot lead to significant bias of the sample. The non-randomness of sample formation, which biologists work with, is one of the constant (and completely irremediable) problems in biological research. Imagine that we need not to take black and white rabbits out of a box, but to determine their ratio in a particular habitat. How to do this? For example, go out into the field and count the rabbits of each color that cross the researcher's path. However, white rabbits will be more noticeable on black plowed land, and black rabbits after snowfall. Perhaps it is worth not relying on the researcher's eyesight and catching rabbits with traps? However, if white rabbits are albinos, they may have poorer eyesight than black ones and fall into traps more often. The sample of rabbits observed during a route count, and the sample of rabbits caught in traps, are not fully representative for assessing the population of rabbits inhabiting the study area. Now imagine that a zoologist is trying to assess the composition of a population of agile lizards. He visited the habitat of this population on a cloudy, windy day, preceded by several days of rain. In such weather, only young individuals and pregnant (carrying mature eggs) females (those individuals experiencing particularly strong hunger) came to the surface to forage. The researcher collected several individuals that seemed "typical" to him, as well as a few more specimens that interested him with their unusualness. During further analysis, he will judge the properties of the studied population (lizards of this population) based on the properties of the sample he has. Unfortunately, no statistical analysis methods can completely correct the bias of such a sample.

1.4. What is Significance? A Humorous Example{ "translated_text": "Let us consider a humorous example. A well-known magic trick involves a magician pulling a rabbit out of a hat (Fig. 1.4.1). Where does the rabbit extracted by the magician come from? It is unknown… One can imagine that the hat is an “entry point” to some analogue of a box of rabbits, similar to the one we used to discuss the concept of probability. The procedure of extracting rabbits from the hat can be compared to obtaining a sample from a population. The sample consists of the extracted rabbits (possibly one, possibly more, drawn one after another), and the population is the rabbits in that “magical space” from which they are extracted.\n\n[IMG_2]\n\nFig. 1.4.1. What can we assert about the “magical space” from which the magician extracted the rabbit (i.e., what can we learn about the population from the sample we obtained in this case)? There was at least one white rabbit there…\n\nSuppose the magician pulls rabbits out blindly: whatever his hand grasps when inserted into the hat, he pulls out. Inserting his hand into one hat, he pulled out a white rabbit, and inserting it into another — a black one (Fig. 1.4.2).\n\n[IMG_3]\n\nFig. 1.4.2. Another rabbit, a black one, appeared from the other hat… In the space accessed by the right hat, there was at least one black rabbit. Do the two hats lead to the same space, or to different ones (in other words, are the two samples drawn from the same population or from different populations)?\n\nWe do not have sufficient grounds to choose between these options. It is possible that the two samples are drawn from the same population, which contains both black and white rabbits, or they may come from different populations.\n\nCan we determine whether the two samples corresponding to the two hats are drawn from the same population based on the composition of rabbits in each? Sometimes the data we obtain are useless for choosing between mutually exclusive possibilities, while other times they can form the basis for reasoned assumptions (Fig. 1.4.3).\n\n[IMG_4]\n\nFig. 1.4.3. There is more information available to make a decision…\n\nIf we assume that the magician accesses two different rabbit populations via the two hats, we need no additional assumptions. If both samples are drawn from the same population, we must assume that a less probable scenario has occurred.\n\nWhat is the probability that the magician obtains two white rabbits in one sample and two black rabbits in the other, if he draws them from the same population? What ratio of white to black rabbits can we expect in the population (if it is a single population)? We know for certain that both white and black rabbits are present there, and we can estimate their ratio from the combined sample (in this case, we assume that the differences between rabbits from different hats are due solely to chance). The most probable scenario is an equal number of white and black rabbits, as this exactly matches the overall sample obtained.\n\nWhen the magician first inserts his hand into one of the hats, he pulls out a rabbit of some color. The scenario shown in Fig. 1.4.3 occurs if a rabbit of the same color is pulled from that hat (i.e., an event with a probability we estimated as ½), and two rabbits of the other color are pulled from the other hat in succession (i.e., two independent events, each with a probability of ½). Thus, such a distribution as shown in the figure would occur in only 1 out of 8 trials if the space is shared. It is more probable to assume that the rabbit populations are different, although of course we do not have sufficient grounds to reject the assumption that the rabbits are drawn from a single common population…\n\nHowever, more complex cases are possible (Fig. 1.4.4)…\n\n[IMG_5]\n\nFig. 1.4.4. There is even more information, but the probability calculation is not as trivial as in the previous case.\n\nThe overall ratio of white to black rabbits remains the same. The probability that one sample has a ratio of 1:3 and the other 3:1 (ignoring the order in which rabbits were extracted in each sample) is the same as in the previous example: given the sample sizes, the outcome shown in the figure occurs in 1 out of 8 cases.\n\nHow would our estimates change if the samples were larger and the differences between them more pronounced (Fig. 1.4.5)?\n\n[IMG_6]\n\nFig. 1.4.5. In this case, the question of whether the magician inserts his hands into the same “magical space” via different hats, or into different spaces, can be answered with high probability: different spaces. If the space were shared, a split of 10 rabbits of one color in one sample and 10 rabbits of the other color in the other would occur in only 1 out of 524,288 trials.\n\nIf the rabbit samples obtained via the two hats differ strongly (10 white rabbits from one hat, 10 black from the other), the magician is almost certainly pulling rabbits from different populations. The probability of such an outcome is 1 divided by 2^19. By rejecting the assumption of such an improbable event, we are almost certainly not making a mistake. Thus, we can accept that in the case shown in Fig. 1.4.5, the hats lead to different spaces. However, it is important to remember: we have not proven the difference between the spaces accessed by the hats, but only obtained grounds to assume their difference with high probability.\n\nOne might assume that if the samples do not differ in composition (Fig. 1.4.6), we could make the opposite decision and assume that both hats are portals to the same location. However, such a decision would be incorrect. We have only established that the assumption of an equal ratio of white to black rabbits in the populations accessed by the right and left hats in Fig. 1.4.6 is quite probable based on the comparison of rabbits extracted from them. But we have no grounds to choose between the assumptions that these are two different spaces with identical composition, or a single shared space.\n\n[IMG_7]\n\nFig. 1.4.6. Is the observed pattern in this case consistent with the assumption that the magician inserts his hands into different “magical spaces” via the right and left hats? It is entirely consistent!\n\nThus, the case in Fig. 1.4.5 provides grounds for a definite conclusion, while the case in Fig. 1.4.6 does not! This reflects a general pattern: when comparing two samples, we can sometimes prove that they are drawn from different populations (i.e., justify that the opposite conclusion is extremely improbable), but we cannot prove that they are drawn from the same population! However, we can justify that the difference between the populations from which the two samples are drawn does not exceed a certain level, with a given probability…\n\nIn the sample comparison case we considered in this example, the probability that the samples are drawn from the same population and the differences between them are due to chance is called the statistical significance of the hypothesis that the populations from which the samples are drawn are different. In other cases (e.g., when studying the correlation between the variation of two traits), statistical significance is defined analogously — it is the probability that the observed effect is due to chance. To summarize: the significance level is the probability that the observed result is merely a product of chance in sample formation.\n\nWhat does the phrase “statistically significant result” mean? It means that the random occurrence of this result is very improbable, and we have every reason to consider the result non-random, reflecting the characteristics of the object of our study.\n\n1.5. Statistical Significance; Null and Alternative Hypotheses\n\nTo formalize such logical choices, it is customary to formulate two hypotheses, between which a choice must be made during statistical research.\n\nThe null hypothesis (H0) states that there are no differences between the populations from which the samples are drawn (and the difference between the samples is a consequence of chance during their formation).\n\nThe alternative hypothesis (H1) states that the differences between the samples reflect differences between the populations from which they are obtained.\n\nIt is impossible to unambiguously choose between these options, and there is always a possibility of error. Based on the available data on sample composition, one must estimate the probability that the null and alternative hypotheses are true, and choose the optimal decision. Statistical tests — rules that allow such choices — are used for this selection.\n\nNull and alternative hypotheses can be two-tailed (only the fact of a difference between the populations from which the samples are drawn matters) or one-tailed (e.g., it is important that a specific treatment increases the value of a trait; the trait value is higher in the treated population). For example, when determining whether sex affects tail length, we may consider both cases where females have longer tails than males and where they have shorter tails as examples of such an effect. When determining whether a new drug “works”, cases where it promotes recovery and where it hinders recovery are considered completely different. The alternative hypothesis should specifically state that the drug promotes recovery. Thus, two-tailed tests should be used in the first case, and one-tailed tests in the second.\n\nThe significance level is the probability that we consider a difference significant (accept the alternative hypothesis) when it is actually random. The significance level can be defined as the probability of committing a Type I error by accepting the alternative hypothesis when the null hypothesis is actually true. A Type II error is accepting the null hypothesis when the alternative is true. Type I errors are usually more serious. The probability of a Type I error is denoted as α; the probability of a Type II error is denoted as β. Accordingly, the power of a test is defined as 1 — β.\n\nExamples of incorrect use of the terms “reliability” and “significance” are frequently observed. The concept of “statistical significance” (or simply “significance”) has a clear mathematical interpretation. The statistical significance of a given result (e.g., detection of a difference between data groups or a correlation between two variables) is the low probability of its random occurrence. The statement “two samples differ statistically significantly” means that the probability they are drawn from the same population is so low that it can be considered proven that they are drawn from different populations. “Reliability” is a much broader concept, used in a wide range of fields (from jurisprudence to philosophy) and lacking a mathematical definition. It is used to denote well-founded, evidence-based knowledge. The statement “the conclusions of the dissertation are reliable” means they are justified by the logic of the study design and presentation. Remember: reliable conclusions are based on statistically significant results! Incidentally, with improper experimental design or interpretation errors, unreliable conclusions may reference numerous statistically significant phenomena…\n\nThe vast majority of sources simply refer to “significance level”. This is by no means an error, and such usage is entirely acceptable. However, since this text is intended for educational use, its author will strive to use the full formulation in all cases: the concept of “statistical significance”; this makes it easier to remind readers of its statistical nature.\n\n1.6. Traits\n\nWhen describing study objects, researchers record the values of various traits — characteristics by which the compared objects may differ from each other. Traits can have different natures.\n\nTable 1.6.1. Trait Categories\n\nTrait Categories\n\n\nExpressed as\n\n\nExample\n\n\n\nQuantitative\n\n\nMetric (continuous, measured)\n\n\nNumber from a continuous series\n\n\nFrog body length\n\n\n\nMeristic (discrete, countable)\n\n\nInteger\n\n\nNumber of stripes on the shin\n\n\n\nRanked (ordinal)\n\n\nInteger (rank), where the difference between ranks is not a measure of the difference between the objects themselves\n\n\nRank of forelimb digit length (1 – longest, 2 – next longest, etc.)\n\n\n\nQualitative (attributive)\n\n\nMultistate (nominal, polytomous)\n\n\nSpecific quality from a given set\n\n\nDorsal color\n\n\n\nBinary (dichotomous)\n\n\nOne of two possible states (present – absent)\n\n\nPresence of a dorsomedial stripe\n\n\nTraits from different groups differ in their properties. For example, an individual with 4 stripes on the shin differs from an individual with 3 stripes by the same number of stripes as it differs from an individual with 2 stripes. At the same time, for individuals that differ in the rank of the first digit length on the forelimbs, it is impossible to say by how much the digit of an individual with rank 4 is shorter than that of an individual with rank 3, and the difference between individuals with ranks 4 and 3 cannot be compared to the difference between individuals with ranks 3 and 2.\n\nThus, traits are characteristics by which objects can be compared to each other. The result of describing an individual by a given trait is called the value of that trait, or simply a value. When working with computer software, what is recorded in a separate cell of a data table is most simply referred to as a “value” (although other variants exist, e.g., “datum”).\n\n\n1.7. Distributions, Statistics, and Parameters\n\nA distribution is a function describing the probability of different values of a randomly varying quantity. The fact that a coin can land heads or tails with equal probability defines the distribution of coin toss outcomes.\n\nRandom variables (and their distributions) can be discrete or continuous. Quantitative and countable traits have discrete distributions, while metric traits have continuous distributions.\n\nSamples can be described by assuming that the distribution of values within them follows a law characteristic of the population from which the sample is drawn.\n\nSuppose a studied sample is characterized by measurement results. The sample mean can be calculated for the sample. If the sample is fully described, its mean can be determined with high precision. Based on the sample mean, one can infer the population mean from which the sample was drawn with a certain level of accuracy.\n\nMathematical quantities characterizing a sample are called statistics and are denoted by Latin letters; those characterizing a population are called parameters and are denoted by Greek letters.\n\nIn a typical biometrical study, sample statistics are used to infer the mathematical quantities characterizing the population — its parameters.\n\nTable 1.7.1. Most Common Statistics and Corresponding Population Parameters\n\nStatistics\n\n\nParameters\n\n\n\nSample size — n.\n\n\n\n\nSample mean — [IMG_8], [IMG_9]\n\n\nPopulation mean — [IMG_10]\n\n\n\nSample standard deviation — s; [IMG_11]\n\n\nPopulation standard deviation — [IMG_12]" }{"translated_text": "Arithmetic Mean (Mean) [IMG_13], where [IMG_14] is the arithmetic mean of the studied variable x; n is the number of elements in the sample; xi are individual values of variable x, from x1 to xn. Individual [IMG_15] obtained for different samples can be considered as sample estimates of the population mean [IMG_16] (the arithmetic mean of the population, including the entire set of objects represented by the studied sample).\nVariance, Standard Deviation (Variance). The standard deviation of the population could be calculated as [IMG_17], but such an estimate would require examining all elements of the population. In reality, this parameter is always determined for a specific sample, which is unlikely to include the rarest values and those most deviating from the mean. Therefore, the sample standard deviation, denoted as s2, must be calculated with a correction. The formula [IMG_18] is used for this purpose. The value df=n-1 is called the degrees of freedom. It can be considered that with a known [IMG_19], the values of all sample elements except the last one can be varied (i.e., their number, equal to n-1): when all other values and the mean are determined, the last value of the sample is unambiguously determined by these quantities.\nIn Russian, variance is often called dispersion (from Latin dispersio — scattering; hence the name analysis of variance). Sometimes it is stated that the term dispersion should be used only to denote the mere fact of scattering of individual values around the mean, and the described measure should be called variance, by analogy with English. Variance is the square of the Standard Deviation, denoted s and calculated, naturally, as [IMG_20].\nSometimes other statistics characterizing samples are also used. These include the range (difference between minimum and maximum values), the median (the value that is exactly in the middle of the ordered series of sample elements, so that half of the sample elements are less than this value and half are greater), the mode (the most numerous class of values in the sample), the mean absolute deviation, the geometric mean, etc. A good analysis of these and other statistics can be found here.\nAccording to the law of large numbers, originating from J. Bernoulli (1713) and proven by P.L. Chebyshev in the 19th century, as the sample size increases, sample statistics tend toward the population parameters. The smaller the sample, the more likely the deviation of sample statistics from population parameters.\nIf a metric characteristic is influenced by multiple random effects, it acquires a normal distribution. Graphically, this distribution is described by the normal curve, which is uniquely defined by just two parameters: [IMG_21] and [IMG_22].\nIn the normal distribution, the mean, median, and mode coincide. 99.7% of observed values in a normal distribution fall within [IMG_23] (the three-sigma rule).\nEmpirical distributions may resemble normal distributions, yet differ from them. The most common differences are skewness and kurtosis.\n\n1.8. Parametric and Nonparametric Statistical Methods and Tests\nStatistical tests (rules allowing one to make a choice between the null and alternative hypotheses) can be divided into parametric (those in whose procedure it is assumed that the compared samples are drawn from populations with a specific, most often normal, distribution) and nonparametric, distribution-free (requiring no assumptions about the nature of the distribution of the populations studied). Thus, if we do not know how the compared quantities are distributed, nonparametric methods can be used \"by default.\" However, most nonparametric methods have less power (1 — β, where β is the probability of \"missing\" a difference, accepting the null hypothesis when the alternative is true) than parametric methods (and this is natural, since parametric methods already \"know\" something about the distributions of the compared quantities)."}