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 RandomClustered

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 (s 2 ) the Quadrat Method (x i – x) 2  i=1 i=m m - 1 s  = where x i is the number of points in cell i

4.Calculate the variance to mean ratio (VMR): the Quadrat Method VMR = s2s2 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) s 2 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 (R O ): 2. Nearest Neighbor Analysis  d i R O = i = 1 n n

Expected distance: The Nearest Neighbor Statistic R E = 2 1 where is the number of points per unit area Nearest neighbor statistic (R): R = RORO RERE = 1/ (2  x where x is the average observed distance d i

Values of R: 0  all points are coincident 1  a random pattern  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 [R E ] = 4 -  4n4n where n is the number of points

Test statistic: Interpreting the Nearest Neighbor Statistic V [R E ] Z test = R O - R E (4 -  4  n  = R O - R E = (R O - R E ) n Standard normal distribution

Nearest Neighbor Analysis Example

Observed mean distance (R O ): R O = ( ) / 6 = 13 /6 = Expected mean distance (R E ): R E = 1/(2  ) = 1/(2  6/42]) = and use these values to calculate the nearest neighbor statistic (R): R = R O / R E = 2.167/1.323 = 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.H 0 : R O ~ R E  Point pattern is approximately random) 2.H A : R O  R E (Pattern is uniform or clustered) 3.Select  = 0.05, two-tailed because of H 0 4.We have already calculated R O and R E, and together with the sample size (n = 6) and the number of points per unit area ( = 6/24), we can calculate the test statistic: Z test = (R O - R E )  n = ( ) *  

Z-test for the Nearest Neighbor Statistic Example 5.For an  = 0.05 and a two-tailed test, Z crit = Z test > Z crit, therefore we reject H 0 and accept H A, 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 Z crit value