|
Tree Ordinance Guidelines
Determining sample size for dot grid estimates
If you are using the dot
grid method to assess tree canopy cover, how many dots do you need to
count? Unfortunately, there is no single answer to this question, but you can
calculate the minimum sample size of
dots required for a given application if you have some basic information about
the population. Several basic principles apply when determining the necessary
sample size. First, the reliability of the canopy cover estimate will increase
as the dot density increases, but the increase in statistical
power begins to plateau at high sample sizes. This effect is evident
when power is plotted against sample size (Graph 3-1).
Larger sample sizes are needed when making comparisons between similar canopy
cover levels (e.g., comparing tree cover changes over time due to natural
mortality) than when comparing widely different canopy levels (e.g., comparing
tree cover in residential and industrial areas) (Graph 3-2).
Also, the sample size needed to detect a difference of a given magnitude (e.g.,
10%) increases as the percent cover approaches 50% (Graph 3-3).
The upshot of this is that almost any application will require a
count of at least 300-400 dots. If you need higher precision or if you need to
differentiate between levels of canopy that are close to 50%, the minimum dot
count will be closer to 500, and higher numbers would be preferable.
Graph 3-1. Power, confidence, and sample size
The graph below shows the general shape of a sample size power
curve. All of the curves shown on this page are based on the formula for a test
comparing two proportions (p1 and p2). Power is
plotted on the vertical axis and sample size is on the horizontal axis. You can
see from the graph that power increases with increasing sample size, but the
slope of the curve decreases progressively as sample size increases, that is,
you reach a point of diminishing returns. For example, at alpha (or
confidence level)
=0.05, sample size needs to be increased by about 200 to increase the power
from 0.6 to 0.7, but it needs to be increased by about 450 to increase the
power from 0.8 to 0.9. The graph below shows curves for two different levels of
alpha. For a given level of power, a larger sample size is needed to obtain a
higher confidence level.
Graph 3-2. Sample sizes for detecting differences of
various magnitudes
As illustrated in the graph below, sample size must increase to
detect relatively small differences at a given level of
power and confidence. In the
example shown, at alpha=0.05 and power=0.8, a sample size of about 170 dots
will suffice for detecting the difference between 30% and 45% canopy cover, but
a sample size of 1400 dots is needed to detect a real difference between 30%
and 35% canopy cover. Looking at this effect another way, at a sample size of
400 dots, a statistical difference between 30% and 45% canopy will be detected
more than 99 times out of 100; a difference between 30% and 40% canopy will be
detected about 85 times out of 100, but a difference between 30% and 35% canopy
will only be detected about 33 times out of 100. 
Graph 3-3. Required sample sizes increase as proportions
approach 0.5
In the graph below, the magnitude of the difference to be detected
is the same for all three curves (0.1). You can see that progressively larger
sample sizes are needed to obtain a given level of power
as the proportions approach 0.5. This relationship is symmetrical around the
center of the range (0.5). Thus, the curve for the pair p1=0.9 and p2=0.8 is
the same as the curve for the pair p1=0.1 and p2=0.2. 
|