What method should I use to study population level growth and individual level growth?

Question

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I have a panel dataset which contains diameter measurement data for 88 trees obtained from a mapped stand in Duke Forest. The description of variables is:

Each tree is indexed by a unique ID (ID), and the diameter of a given year (yr) is the variable cm(cm). For each year, three variables of weather information are also available: annual precipitation (annualprec), average summer Palmer Drought Severity Index (summerpdsi), and average winter temperature (wintertemp).

Now I want to study both stand (population) level growth and individual level growth. I currently use fixed effect and random effects model. These two models allow me to investigate the effects of the 3 weather variables. But my main focus should be stand level growth and individual level growth. I don't know what method I should use to attain this goal.

The dataset can be seen here: http://www2.stat.duke.edu/~lm186/data/diamdata.txt

The description of this data set is:

Tree growth provides essential information about forest ecology. One common method to estimate tree growth is based on repeated tape measurements of the diameter of the same tree, and the diameter increment is the difference between the current and previous measurement. The dataset “diamdata.txt” contains diameter measurement data for 88 trees obtained from a mapped stand in Duke Forest. The stand was established in 1991 for the purpose of studying forest dynamics. The measurements are made at breast height marked by a nail that holds a tag indicating the identifying tree number. Diameter censuses were conducted at intervals of one to four years starting in 1993. Each year, some trees died and were removed from the census, and some new trees were planted and added to the census, resulting in different numbers of trees measured in each census and different numbers of measurements for each tree. Each tree is indexed by a unique ID (ID), and the diameter of a given year (year) is the variable cm. For each year, three variables of weather information are also available: annual precipitation (annualprec), average summer (Jun. - Sep.) Palmer Drought Severity Index (PDSI) (summerpdsi), and average winter (Jan. - Mar.) temperature (wintertemp). Explore and analyze the data to infer about the pattern of tree growth over time. In this regard, we may be interested in learning about both stand (population) level growth and individual level growth. Write a report of up to three pages regarding tree growth based on your analysis that is understandable and useful to ecologists. Details of key statistical methods or models should be given.

Answer 1

There are several ways to evaluate data on individuals (trees, here) over time, discussed for example in Chapter 7 of Frank Harrell's course notes . A simple fixed-effects model isn't appropriate, as that wouldn't take into account the correlations of changes within trees. This data set illustrates strengths, weaknesses, and limitations of different approaches to modeling longitudinal data. As you are studying these issues, consider trying several different approaches.

You evidently have chosen a mixed model with the 3 weather variables as fixed effects, and handled the intra-tree correlations by using tree ID as a random effect (I suspect just as random intercepts). To that extent, you've already started to evaluate the associations of those variables with growth at the stand level, because there is only 1 stand to evaluate. Coefficients associating growth with fixed-effect weather variables will thus represent associations at the stand level. There's an alternate approach that you might consider, generalized least squares , to modeling data over time like these. Here that would give marginal estimates of covariate effects for this stand, while ignoring tree-level effects. Harrell's notes illustrate that approach.

It's not clear how you have incorporated time into the model, if you've modeled the diameter values (or some transformation, like a log) directly as a function of time, or evaluated the change per year in diameter as a function of the weather variables. Harrell discusses different ways to incorporate time; a flexible spline fit for time might be a good choice. It's important to make sure that your combination of time and weather variables makes sense in a causal direction. You want to make sure that the data are formatted in a way that the weather variables are measured just before or during the periods of growth, not after. For example, make sure the wintertemp values are for the winter prior to measurements.

Evaluating individual-level growth can be tricky with data like these. Look at your tree-by-tree data first, with plots of diameter against year. You have no more than 7 observations on any one tree, which greatly limits your ability to evaluate the effects of any predictors on a tree-by-tree basis. You also have several small trees evidently planted in 2007 or 2008, while most have data dating back many more years. Think about whether those more recently planted trees should be modeled differently from the others.

You could consider using modeling the distribution of responses to fixed effects among trees by incorporating random slopes, not just intercepts, into your mixed model. Playing with your data suggest that incorporating some random slopes is possible here, without running into singular-fit problems, particularly if you log-transform your diameter values first.

You'll have to apply your understanding of the subject matter (which I don't have) to describe the results in a way "that is understandable and useful to ecologists."