Lecture 1—Wednesday, January 11, 2006

What was covered?

• Overview of course
• Six steps of ecological data analysis
• Types of probability models
• Some probability basics

Terminology Defined

• discrete distribution (discrete random variable)
• continuous distribution (continuous random variable)
• probability function
• probability mass function
• probability density function
• expectation
• variance

Overview of Course

• Our goal: Learn how to analyze data that typically arise in ecology.
• What distinguishes ecological data from other data?
  1. Typically require probability models other than the normal distribution.
  2. Observations of interest are often not obtained independently so that observations are correlated. This arises when the same unit is observed more than once or the observations are obtained in clusters.
  3. Ecological data can possess a very complex structure, often hierarchical.
  4. Most ecological data are observational rather than experimental. Thus there is a need to control statistically that which could not be controlled for by design.
• Good news: there are now readily available statistical tools for handling the complexities listed above.
• Bad news: understanding the tools is not easy. There is a lot of theory to master. While it's easy to fit complicated models, it is also easy to obtain nonsense. In short, you need to know what you're doing.
• The basic problem we're faced with in this course is how to go from only a little bit of statistical knowledge to being able to analyze the most complicated kinds of data sets imaginable. I feel we have two options.
  1. Option 1. Learn to swim by playing in the shallow end of the pool and moving into deeper water only gradually. Using this approach we would start with what you know and gradually build up your knowledge base. While this seems attractive, we have only 16 weeks. We may end up learning a lot but in the end we could run out of time before ever getting to the complexities inherent in ecological data.
  2. Option 2. Learn to swim by jumping in the deep end. In this approach we would grapple with the complexities from the get-go. Perhaps we might ignore some of the complexity initially, but not for long. We would try to analyze the data the way it needs to be done and fill in the details as we go. The down side is that we would end up using methods that at least initially you know nothing about.
• I've elected to try Option 2. On Tuesdays we'll look at data in all of its ecological beauty (but statistical ugliness) and then spend MWF trying to make sense of what we've seen and what we did on Tuesday. In short, on Tuesdays we drown and during the rest of the week we try to learn from our drowning experience.

Some comments

• I view this as your course. I don't have an agenda. I will not introduce any ideas solely for your edification. Everything we do I can guarantee from my experience in data analysis should eventually prove useful to you in your careers.
• I realize that you probably view statistics as a necessary evil, as opposed to a thing of beauty, so I would like to make this experience as palatable as possible. If you have data you need to get analyzed I welcome you to submit it to me for use during this semester. We can analyze it as part of the weekly computer sessions, or as homework, or in exams. I have plenty of interesting data to work with but I don't mind tailoring things to your needs. There is no better way to learn statistics than by analyzing data you actually care about.
• Software
  • At UNC we have the luxury of having all the major, and many minor, statistical packages available for free or at nominal cost. SAS, Splus, SPSS, Stata, Matlab, MLwiN, and Lisrel are just a few of the packages available.
  • The problem is that when you leave UNC you probably will no longer have access to these. Therefore I think it makes sense to learn a statistical package that has all the capabilities that these do yet you will be able take it with you when you leave. Such a package is R.
• R
  • R is free and is available from the R project site, http/www.r-project.org.
  • R runs on all major computer operating systems, including Windows and Mac OSX.
  • R is a comprehensive, state of the art statistical package written by statisticians. It is modular in that additional statistical routines are constantly being made available as add-on packages. So if a capability you desire is not currently available, it probably soon will be.
  • R can be used to make publication quality graphics.
  • The down side is that R is not easy to use and learn. We will spend each Tuesday learning how to use R.
• Textbook
  • The optional textbook for the course is: Crawley, Michael J. 2002. Statistical Computing: An Introduction to Data Analysis Using S-Plus. Wiley, New York.
    • It's a great book but I made it optional because I think it's too expensive.
    • The bookstore did not order copies of the book so if you want it you'll need to special order it. I recommend checking out some of the book price comparison sites on the web, because there are some great bargains available for this book. See the course description page for more details.
    • S-Plus and R are both based on the same S language, so almost all the S-Plus code included in Crawley's book will work in R too.
    • If you own this book you should begin by reading the first 11 chapters. These chapters review elementary statistics and explain how to implement basic statistical methods in S-Plus (or R).
  • Another good textbook choice is the two volume series by Julian Faraway: Faraway, Julian J. 2004. Linear Models with R. CRC Press, Boca Raton, FL. and Faraway, Julian J. 2005. Extending the Linear Model with R: Generalized Linear, Mixed Effects, and Nonparametric Regression Models. CRC Press, Boca Raton, FL. We will cover a lot of what's in the second book.

Six steps of ecological data analysis

• The process of model building that we'll use in this course will typically adhere to the following six steps. I assume in this that the data of interest have already been collected and sensible ecological hypotheses about the data have been formulated.
  1. Choose a probability generating mechanism for your data. This involves choosing a sensible probability distribution and thinking about how that distribution might vary with respect to various features in your data. Together all of this entails the probability model for your data.
  2. Formulate your ecological hypothesis in terms of the parameters that occur in the probability model. This in turn will involve the introduction of additional parameters.
  3. Incorporate any data structure into your probability model. By the term data structure I mean observational heterogeneity, i.e., any lack of equivalence that exists between sets of observations. Such structure might include one or more of the following:
     • clustering, e.g., you are interested in the characteristics of individual plants but you obtained your data via transects or quadrats in which a whole group of individuals are obtained at once,
     • repeated measures, i.e., the same individual, quadrat, or transect is resampled at different times, and
     • hierarchical structure (nesting), i.e., individuals are contained in quadrats, quadrats are located in forest tracts where the forest tracts are a sample of available tracts.
  4. This step and the next may be done implicitly (by software) or explicitly (by you). Construct the loglikelihood function for your data based on the chosen probability model.
  5. Maximize the loglikelihood function with respect to the parameters introduced in step 2.
  6. Assess fit.
     • Relative fit. Compare your model to other models. These models may differ in which variables are included, in how they are parameterized, or even in which probability model was assumed as the data generating mechanism. Typically such comparisons are made using significance testing or via some information-based criterion, such as AIC.
     • Absolute fit. Compare your model to the raw data. This usually involves goodness-of-fit tests, such as the Pearson chi-squared statistic or deviance-based measures, and residual analysis. Residual analysis allows you to look for model misspecification and the violation of assumptions in choosing the probability model that you did.

Common Probability Models Used in Ecology

• The following overhead summarizes the relationships between the different probability models commonly used in ecology. Overhead 1
• The primary distinction made in this fiture is the difference between discrete and continuous distributions.
  • Discrete distributions apply to random variables (the name we use in statistics for the quantities we actually measure on our observations) whose set of possible values is finite or countable. Technically countable means that a random variable's potential values can be put in a one-to-one correspondence with the set of positive integers. So a countablyl discrete random variable can have an infinite number of possible values, but infinite only in this special sense.
  • Continuous distributions apply to random variables that can take any value in a continuum, i.e., between any two possible values one can always find a third no matter which two values you choose. Due to the resolution issues of measuring devices, continuous random variables don't exist in practice, but we still tend to think of random variables of being continuous at least in principle. Height, weight, biomass, etc. are all examples of continuous random variables.
  • An example of a discrete random variable would be "abundance", the number of individuals of a species found in a particular sample unit, or species richness. Technically we can think of abundance or richness as being potentially infinite. Practically speaking of course each of these can only be finite. In truth of course we don't know what that maximum value is, so philosophically no damage is done by treating these quantities as being potentially infinite.
  • A discrete random can under certain circumstances be treated as if it were continuous.
  • One useful ecological distinction that is mad eamong continuous random variables is whether negative values are possible. Distributions appropriate for positive-only random variables include the lognormal and gamma distributions. In principle this distinction only becomes important when the distribution of the random variable in question seems to "bunch up" near zero resulting in a skewed distribution with a long upper tail.
  • Discrete random variables include the Bernoulli, binomial, and multinomial (all with a finite number of outcomes), and the Poisson and negative binomial distributions (each with potentially an infinite number of possible outcomes).

A Little Bit of Probability Theory

• The function that defines a probability distribution is called a probability mass function, for discrete distributions, and a probability density function, for continuous distributions. Both are used to calculate probabilities but in different ways.
  • For discrete random variables the probability mass function gives the probabilities directly, i.e., .
  • For X a continuous random variable the situation is more complicated. The probability of any single value is zero; nonzero probabilities arise only with respect to intervals. To obtain these probabilities we need to integrate the density function over the interval. For example,

where the approximate equality in the last step only holds for very small intervals over which the density function is approximately constant. Notice the difference. For density functions we obtain probabilities by multiplying the density function by a length. For mass functions, the function itself yields a probability.

• Formally, a probability function (discrete or continuous) is any non-negative function whose range lies in the interval 0 to 1 such that if we sum (integrate) over all of its values (over its domain) we obtain 1.
• Probability density and mass functions typically require that the specification of auxiliary variables called parameters in order to completely characterize the distribution.
• In addition we often use the mean and variance of probability distributions as further characterizations.
• The mean of a probability distribution , or alternatively, the mean of the random variable X whose probability distribution is is called its expectation and is denoted . Here E is refered to as the expectation operator. It is defined differently for continuous and discrete random variables.
  • Discrete random variable, finite:
  • Discrete random variable, countable:
  • Continuous random variable:

  In the last definition we assume when x is outside the natural domain of the function. For example, if X is non-negative then on the interval .

• Observe that in the first formula (where X is discrete and finite) if all the values of X are equally likely, then for all i and the formula reduces to the usual formula for the sample mean.
• The variance of a probability distribution , or alternatively, the variance of the random variable X whose probability distribution is is defined as follows.

where in the last step I follow the convention of using the symbol to denote .

• Using properties of the expectation operator that follow from its definition (but are not developed here) it can be shown that an equivalent expression for the variance is the following.

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