Sampling Populations Ideal situation - Perfect knowledge Not possible in many cases - Size & cost...
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Transcript of Sampling Populations Ideal situation - Perfect knowledge Not possible in many cases - Size & cost...
Sampling Populations
• Ideal situation
- Perfect knowledge
• Not possible in many cases
- Size & cost
• Not necessary
- appropriate subset adequate estimates
• Sampling
- A representative subset
Sampling Concepts
• Sampling unit
- The smallest sub-division of the population
• Sampling error
- Sampling error as the sample size
• Sampling bias
- systematic tendency
Steps in Sampling
1. Definition of the population
- Any inferences that population
2. Construction of a sampling frame
This involves identifying all the individual sampling units
within a population in order that the sample can be drawn
from them
Steps in Sampling Cont.
3. Selection of a sampling design
- Critical decision
4. Specification of information to be collected
- What data we will collect and how
5. Collection of the data
Sampling designs
• Non-probability designs
- Not concerned with being representative
• Probability designs
- Aim to representative of the population
Non-probability Sampling Designs
• Volunteer sampling
- Self-selecting
- Convenient
- Rarely representative
• Quota sampling - Fulfilling counts of sub-groups
• Convenience sampling
- Availability/accessibility
• Judgmental or purposive sampling
- Preconceived notions
Probability Sampling Designs
• Random sampling
• Systematic sampling
• Stratified sampling
• Sampled locations in close proximity are likely to
have similar characteristics, thus they are
unlikely to be independent
Tobler’s Law and Independence
Everything is related to everything else, but near things are more related than distant things.
• Point Pattern Analysis
Location information
Point data
• Geographic Patterns in Areal Data
Attribute values
Polygon representations
Spatial Patterns
Point Pattern Analysis
Regular Random Clustered
1. The Quadrat Method
2. Nearest Neighbor Analysis
Point Pattern Analysis
1. Divide a study region into m cells of equal size
2. Find the mean number of points per cell
3. Find the variance of the number of points per cell (s2)
the Quadrat Method
(xi – x)2i=1
i=m
m - 1s =
where xi is the number of points in cell i
4. Calculate the variance to mean ratio (VMR):
the Quadrat Method
VMR = s2
x
VMR < 1 Regular (uniform)
VMR = 1 Random
VMR > 1 Clustered
5. Interpret VMR
the Quadrat Method
6. Interpret the variance to mean ratio (VMR)
2 =(m - 1) s2
x= (m - 1) * VMR
comparing the test stat. to critical values from the 2 distribution with df = (m - 1)
Quadrat Method Example
• Quadrat size
• Too small empty cells
• Too large miss patterns that occur within a single cell
• Suggested optimal sizes
• either 2 points per cell (McIntosh, 1950)
• or 1.6 points/cell (Bailey and Gatrell, 1995)
The Effect of Quadrat Size
• An alternative approach
- the distance between any given point and its nearest neighbor
• The average distance between neighboring points (RO):
2. Nearest Neighbor Analysis
diRO = i = 1
n
n
• Expected distance:
The Nearest Neighbor Statistic
RE =2
1 where is the number of points per unit area
• Nearest neighbor statistic (R):
R =RO
RE
=1/ (2
x where x is the average observed distance di
• Values of R:
• 0 all points are coincident
• 1 a random pattern• 2.1491 a perfectly uniform pattern
• Through the examination of many random point patterns, the variance of the mean distances between neighbors has been found to be:
Interpreting the Nearest Neighbor Statistic
V [RE] = 4 - 4n
where n is the number of points
• Test statistic:
Interpreting the Nearest Neighbor Statistic
V [RE]Ztest = RO - RE
(4 - 4n=
RO - RE
= 3.826 (RO - RE) n
• Standard normal distribution
Nearest Neighbor Analysis Example
• Observed mean distance (RO):
RO = (1 + 1 + 2 + 3 + 3 + 3) / 6 = 13 /6 = 2.167
• Expected mean distance (RE):
RE = 1/(2) = 1/(26/42]) = 1.323
and use these values to calculate the nearest neighbor statistic (R):
R = RO / RE = 2.167/1.323 = 1.638
• Because R is greater than 1, this suggest the points are somewhat uniformly spaced
Nearest Neighbor Analysis Example
Z-test for the Nearest Neighbor Statistic Example
• Research question: Is the point pattern random?
1. H0: RO ~ RE Point pattern is approximately random)
2. HA: RO RE (Pattern is uniform or clustered)
3. Select = 0.05, two-tailed because of H0
4. We have already calculated RO and RE, and together with the sample size (n = 6) and the number of points per unit area ( = 6/24), we can calculate the test statistic:
Ztest = 3.826 (RO - RE) n
= 3.826 (2.167 - 1.323) *
Z-test for the Nearest Neighbor Statistic Example
5. For an = 0.05 and a two-tailed test, Zcrit=1.96
6. Ztest > Zcrit , therefore we reject H0 and accept HA, finding that the point pattern is significantly different from a random point pattern; more specifically it tends towards a uniform pattern because it exceeds the positive Zcrit value