Lecture 38—Monday, March 27, 2006

What was covered?

• Examples of structured data in ecology
  • Repeated measures experiment
  • Meta-analysis of individual regression models
  • Nonlinear growth curve analysis
  • Spatially clustered data
  • Spatially clustered repeated measures data

Terminology defined

• structured data

Background

• Whether we're dealing with an ecological experiment or with observational data, structured data are pretty much the norm in ecology. It's rare to find ecological data that do not exhibit at least some structure. By structured data I'm referring to a situation where the observations making up a data set are not all equivalent. There are two distinct ways of thinking about this lack of equivalence.
  • In structured data sets observations are heterogeneous. They occur in groups such that the observations within a group exhibit less variability than exists for observations coming from different groups.
  • Said differently in structured data sets observations possess a correlation structure. They occur in groups such that the observations within a group are more similar to one another than they are to observations coming from different groups. The upstart is that observations arising from the same group are correlated.
• It is often the case that the "structure" in structured data exhibits a hierarchical form. Often observations are members of groups that are nested in larger groups that in turn may be nested in still larger groups. Because of this structured data are also called hierarchical or multilevel data.
• We spend this lecture examining examples of structured data in ecology some of which we'll then begin to formally analyze next time. The analysis of structured data will occupy our time for the remainder of this course. It is crucial that you learn to recognize structure in data and become comfortable analyzing it. The most common statistical error I encounter in my consulting work is a researcher treating structured data as if they were independent data. This immediately raises the specter of pseudo-replication and calls into question the conclusions of the analysis.

Example 1—A repeated measures experiment

• A 2 × 2 factorial design was employed to study the factors affecting daily caterpillar movements on a host plant. The factors of interest were two randomly assigned treatments: the amount of lighting, sun versus shade (2 levels), and temperature (2 levels). Together these yield a total of four treatment combinations. Each of the four treatment combinations was replicated 12 times and the entire experiment was repeated on two additional days for a total of three days.
• The experiment was carried out as follows. Four caterpillars were placed in a terrarium containing a single plant and the terrarium was maintained at one of the four treatment combinations for the duration of the study. At six pre-determined time periods the locations of the four caterpillars was ascertained, i.e., how many of them were present on top of a leaf versus how many of them were found below a leaf.
• The experiment yielded a total of 864 = 48 × 6 × 3 observations.
  • 48 observations were made at each time period because the 4 treatment combinations were each replicated 12 times.
  • There were six time periods.
  • The entire experiment was repeated on three separate days.
• Research question: How does lighting and temperature affect the behavior of caterpillars? Does the behavior evolve over time and is it consistent across days?
• Statistical analysis—logistic regression
  • Observe that the response at each realization of the experiment is a grouped binary random variable. If we call "caterpillar on top of leaf" a "success" our response is the number of successes out of four trials. (Some of the caterpillars died during the course of the experiment so in truth the total number of trials was not always four.)
  • Given the nature of the response logistic regression is a sensible method of analysis. The linear predictor consists of sun/shade as a factor, temperature as a factor, and the interaction between the two. In addition day (as a factor) and time period (treated either as a continuous variate or as an ordinal factor) were included as covariates.
• Why are these structured data?
  • The 864 observations making up this study are not all equivalent. The observations come in packets of six because six observations were made on the same caterpillar group. Observations coming from the same group of caterpillars might be expected to be more similar than observations coming from different groups of caterpillars.
  • The study described here is generally referred to as a repeated measures design. A repeated measures design is a classic example of structured data.

Example 2—A meta-analysis of individual regression models

• Data were obtained on the larval development times of 74 marine invertebrate and vertebrate species. The data were taken from 65 published studies in which the pelagic larval duration (time spent in the larval state) was experimentally determined by rearing larvae at various ambient temperatures. Only studies reporting the duration time for at least two separate temperatures were chosen for the analysis.
• It was assumed that something analogous to an Arrhenius power law is a reasonable model of pelagic larval duration time (PLD) as a function of temperature (T).

• To simplify model fitting with available software this model was linearized by log-transforming both sides.

As the analysis proceeded it was discovered that a quadratic version of this model provided a better fit to the data.

• Research question: Does a single model of one of the above forms adequately represent all 74 species or do we need many different relationships? For instance, in the quadratic model above, do the same choices of β₀, β₁, and β₂ work for all species, or do we need different values for one or more of these parameters for different species?
• Why are these data structured?
  • The data come in packets, each packet corresponding to the set of temperature-development time pairs obtained for a single species in a given study.
  • The separate species correspond roughly to separate published studies. Because different researchers use different protocols we would expect observations coming from the same study (species) to be more similar to each other than to observations coming from different studies.
  • Once again the data are heterogeneous. Observations from the same study are expected to be correlated but independent of observations coming from different studies.
• Statistical analysis—linear regression.
  • What would it mean in this case if we ignored the structure? If we ignored the species label and just fit a single model to all the data we would be violating one of the primary assumptions of regression, namely that the (x, y) data pairs are mutually independent. Furthermore it's not clear how doing this would directly answer the research question. Fitting a single model does not tell us whether a single model is appropriate.
  • At the other extreme we could fit 74 separate models, one to each species.
    • Once again this doesn't tell us whether we actually need to fit 74 models or whether we could get by with a single model.
    • Fitting 74 separate models would clearly overfit the data. This is most obviously seen for those species that provide only two data points.
  • As we'll see a better approach is to take an intermediate strategy in which we fit a single model while still allowing individual species trajectories to vary randomly about this single model if need be. By examining the variability of the individual trajectories about a common population trajectory we can determine whether a single model is appropriate.

Example 3—Nonlinear growth curve analysis

• Research question: Is there selection for a sex ratio bias in egg order for egg-laying birds?
• It is known that androgens accumulate in avian egg yolk after the eggs are formed in the ovary. Androgens are known to affect the growth and mass of nestlings, either enhancing or inhibiting it depending on the species. This suggests that egg order can affect the future success of nestlings.
• The role of yolk androgens in the American kestrel was examined by manipulating androgen levels in the first-laid egg to mimic the dosage that occurs in later laid eggs. Specifically the yolk androgen levels in the first-laid eggs of American kestrels were artificially elevated. The growth of hatchlings was then monitored over time and differences in growth rate and size of hatchlings for treated and untreated male and female eggs were examined. Each individual bird provided multiple size measurements over time.
• Why are these data structured?
  • Once again the data are heterogeneous. We have observations that come from the same bird as well as observations that come from different birds. We would expect observations made on the same bird to vary less than observations coming from different birds. Put another way, these data exhibit a correlation structure in that observations coming from the same bird are correlated.
• Statistical analysis—nonlinear growth curve analysis
  • Growth curve models were developed for individual birds as a function of their treatment status (treated or not) and gender (male or female).
  • While linear and polynomial models could have been used, the preferred growth curve models for birds are nonlinear ones. Popular choices include the logistic model (3 parameters), Gompertz model (3 parameters), and the von Bertalanffy model (4 parameters). The basic 3-parameter nonlinear growth curve model can be written as follows.

where f is the logistic or Gompertz function and ε denotes random error such that . Here are fitting parameters of the nonlinear function that could be allowed to be functions of treatment status or gender.

• The available data consist of the growth records of the nestlings from 18 first-laid eggs of which half were males and half were females. As many as six temporal observations were made per egg or as few as two. Observe that because the growth models being considered all have at least three parameters to be estimated, it is not possible to fit these models individually to nestlings that provided only two temporal observations. A similar issue arose in trying to fit separate regression models in Example 2.
• As in our first example this is another instance of a repeated measures analysis. Here though the focus is on developing a functional form for the growth of birds over time. The effect of treatment and gender on growth is determined by the manner in which these factors modify the estimated values of the parameters in our nonlinear growth function.

Example 4—Transects (spatially clustered data)

• Research problem: Develop a statistical technique to measure the width of an ecotone (the transition zone from one habitat type to another) for different vegetation types.
• Multiple transects were laid out extending from a forest edge into a forest. At various points along the transect richness and abundance values were recorded for various vegetation categories: herbs, woody plants, grasses, vines, etc.
• Why are these data structured?
  • The structure arises from the way in which the data were collected. Different transects were laid down at random. Thus it can be assumed that observations obtained from different transects will be uncorrelated.
  • On the other hand, observations made on the same transect would be expected to exhibit serial correlation based on the physical distance separating them. For instance, adjacent observations would be expected to be more similar than observations at the endpoints of the transect.
• Statistical analysis: Breakpoint Poisson regression. Breakpoint regression fits a piecewise regression model to data. The rationale for using this approach here is that richness/abundance should be approximately constant within a habitat, but undergo a change when the transect crosses an ecotone during the transition from one habitat type to another. Thus a graph of richness versus location on the transect might by approximated by two horizontal line segments connected by a line segment of nonzero slope. The endpoints of this latter line segment correspond to the change points of the breakpoint regression and define the boundaries of the ecotone.

Example 5—Spatially clustered repeated measures data

• Research question: How does riparian land use affect the abundance and richness of birds in the Sonoma Valley wine region of California?
• Land use was classified as natural, vineyard, or urban. Land owners of riparian land from each of these land use types with multiple sites occasionally selected from same land owner's holdings.
• The point count transect method was used to count birds in the riparian corridor of each selected site. The number of points varied from one to thirteen. A total of 18 different properties bordering nine different creeks were used in the study. Counts were done twice in the first year and then twice again in the following year. Vegetation variables were also recorded at each site at the first measurement period only.
• Statistical analysis: count regression (Poisson regression or some modification thereof) with vegetation variables and land use type as predictors.
• Why are these data structured? The observations are heterogeneous in that multiple observations were made at the same site but at different times. So we again have a repeated measures design. In addition some of the individual sites belonged to the same land owner. If a land owner engages in a consistent land use strategy, then it's conceivable that different sites coming from the same land owner would be more similar to each other than they are to different sites having different land owners.
• The data in this example possess two levels of nesting. We have temporal observations nested in sites but in addition sites that are then nested in land owner.

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