Statistical Principles in DendrochronologyStatistical Principles in Dendrochronology

1. Statistical distributions

• Why are we interested in “average” growing conditions over time?

• Average = SIGNAL. Means we must shoot for an average or mean when we sample.

• Suggests we also must know the variability about this mean.

• Which means we must be familiar with statistical distributions, which are defined by mean and variance:

• e.g., the normal distribution, the t-distribution, the z-distribution, the Weibull distribution

• population

• samples are drawn

• uncertainty = sampling error = noise

• maximize signal (= average), minimize noise

• be aware of sampling bias: examples?

• easy access

• physical limitations (altitude, health)

• low budget

• downright laziness!

1. Statistical distributions

• samples are drawnfrom a population

• descriptive statistics arecalculated (e.g. mean, median,mode, standard deviation,minimum, maximum,range)

• frequency distributionis calculated

1. Statistical distributions

a. Sample statistics have distributions.

b. These are normally distributed (considers both mean and variance).

c. As one increases sample size, our sample statistic approaches the population statistic.

Example: from a population of five trees, we can only sample three. For the year 1842, the five trees had the following ring widths:

0.50 0.75 1.00 1.50 2.00

population mean = ?

average of all sample means = ?

2. Central Limit Theorem

population mean = 1.15

(0.50+0.75+1.00)/3 = 0.75 (0.50+0.75+1.50)/3 = 0.92(0.50+0.75+2.00)/3 = 1.08(0.50+1.00+1.50)/3 = 1.00(0.50+1.00+2.00)/3 = 1.17(0.50+1.50+2.00)/3 = 1.33(0.75+1.00+1.50)/3 = 1.08(0.75+1.50+2.00)/3 = 1.42(1.00+1.50+2.00)/3 = 1.50

average of all sample means = 1.14 (rounding error here)

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2. Central Limit Theorem

Sample size means everything! The more samples one collects, the closer one obtains information on the population itself!

a. Average conditions become more prominent.

b. The variability about the mean becomes less prominent.

c. Notice relationship with S/N ratio! Signal increases while noise decreases!

2. Central Limit Theorem

• A procedure for selecting events from a population

• Pilot sample (or pretest)

• Simple random sample

• random number generators are handy for x/y selection

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3. Sampling Design

• Systematic random sample

• select k-th individual from gridded population

• lay out a line = transect, sample individual nearest the pre-selected point

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3. Sampling Design

• Stratified random sample

• population is layered into strata and then we conduct random or systematic sampling within each cell

3. Sampling Design

• Stratified, systematic, unaligned = point sampling

• Hybrid technique, favored among geographers

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3. Sampling Design

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3. Sampling Design

• Stratified, systematic, unaligned = point sampling

• Hybrid technique, favored among geographers

•Transect = line sampling, but must have a random component! (How can this be accomplished?)

•Many variations:

• Sample all individuals along the transect (row 1)

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• Sample quadrats along the transect (row 2)

• Sample all individuals within a belt (row 3)

3. Sampling Design

•Targeted sampling = non-random sampling

•Is this a legitimate technique?

•It is often necessary because of:

• Time constraints

• Budget constraints

• Lack of field labor

• Physical limitations of field labor

• Topographic limitations

•Advantages?

• Maximize information with minimum resources

• Target areas where samples are known to exist

• Less time needed and less money wasted

3. Sampling Design

•Targeted sampling = non-random sampling

•Used in practically all types of dendro research: fire history, climate reconstruction, insect outbreak studies, …

3. Sampling Design

• Specifically sample only trees that have best record of fire scars. (dots = trees, circles = trees collected with fire scars, X’s = fire scars, but not sampled = poor record.)

• What issues must we consider? Topography, slope, aspect, hydrology, tree density: all affect susceptibility to scarring by fire.

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3. Sampling Design

Shallow slope area Valley bottom Steep slope area

• Complete inventory is possible

• Sample all trees that have fire scars, regardless of number of scars or quality of preservation, but …

• Not very efficient (time, money, labor)

• Benefits are considerable, though.

3. Sampling Design