Statistical Principles in Dendrochronology

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Statistical Principles in Dendrochronology
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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

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1. Statistical distributions

• 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!

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1. Statistical distributions

• samples are drawnfrom a population
• descriptive statistics arecalculated (e.g. mean,
  median,mode, standard deviation,minimum,
  maximum,range)
• frequency distributionis calculated

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2. Central Limit Theorem
a. Sample statistics have distributions. b. Thes
e 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 ?
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2. Central Limit Theorem
population mean 1.15 (0.500.751.00)/3
0.75 (0.500.751.50)/3 0.92(0.500.752.00)/3
1.08(0.501.001.50)/3 1.00(0.501.002.00)
/3 1.17(0.501.502.00)/3 1.33(0.751.001.5
0)/3 1.08(0.751.502.00)/3
1.42(1.001.502.00)/3 1.50 average of all
sample means 1.14 (rounding error here)
0.50
0.75
1.00
1.50
2.00
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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!

• Average conditions become more prominent.
• The variability about the mean becomes less
  prominent.
• Notice relationship with S/N ratio! Signal
  increases while noise decreases!

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3. Sampling Design

• 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

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3. Sampling Design

• Stratified, systematic, unaligned point
  sampling
• Hybrid technique, favored among geographers

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3. Sampling Design

• Stratified, systematic, unaligned point
  sampling
• Hybrid technique, favored among geographers

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3. Sampling Design

• Transect line sampling, but must have a random
  component! (How can this be accomplished?)
• Many variations
• Sample all individuals along the transect (row
  1)

• Sample quadrats along the transect (row 2)

• Sample all individuals within a belt (row 3)

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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

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3. Sampling Design

• Targeted sampling non-random sampling
• Used in practically all types of dendro research
  fire history, climate reconstruction, insect
  outbreak studies,

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3. Sampling Design

• Specifically sample only trees that have best
  record of fire scars. (dots trees, circles
  trees collected with fire scars, Xs 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.

Shallow slope area Valley bottom
Steep slope area
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3. Sampling Design

• 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.