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					FOR 220 Aerial Photo
Interpretation and
Forest Measurements

Lecture 4

Sampling Designs




Avery and Burkhart,
Chapter 3
       Why/Why Not Randomly Select Samples?


       We will discuss several sampling designs in this lecture

       • Most statistical methods assume simple random sampling
       was used.

       • Sampling methods, however, will vary depending on the
       objectives of the survey, the nature of the population being
       sampled, and prior information about the population being
       sampled.




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Why Not Measure Everything?

      Complete Enumeration (Census)

         Measure every feature of interest.
         The result is a highly accurate description of the population.

         Drawbacks:
               Only viable with small populations.
               Only cost-effective with high-valued features.




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Frame


      Sampling Frame
     The list of all possible units that might be drawn in a sample.

     • Developing a reliable frame may be difficult (e.g., campgrounds)

     • Access


             How is a sampling frame different than a population?



FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Designs


         Sampling Design


              The method of selecting [non-overlapping]
              sample units to be included in a sample.




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Designs


                Sampling Designs Covered Today:

                         1. Simple Random Sampling

                         2. Systematic Sampling

                         3. Stratified Random Sampling




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling

      Assumptions:
     Every possible combination of sampling units has an equal
     and independent chance of being selected.

     The selection of a particular unit to be sampled is not
     influenced by the other units that have been selected or will
     be selected.

     Samples are either chosen with replacement or without
     replacement.


FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling

      With and without replacement?
     Why worry about replacement?

     Small populations – samples may “remove” considerable
     portion. Removes “independence”


                                                              N 12
                                                              n -6
                                                                 5

FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling


        Design:




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling
     We use the familiar equations for estimating common statistics for the population

       Estimate the population                                        x        
       mean:                                                       x 
                                                                       n        
                                                                                 
                                                                                


       Estimate the variance of
                                                               2           
                                                                   x 2   x 2 / n
                                                              s 
                                                                                         
       individual values:                                        
                                                                         n 1            
                                                                                          




       Compute the coefficient
                                                                   CV  
                                                                             s
       of variation:                                                         
                                                                            x



FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling

      Compute the standard error:

                                   s2                              s2  N  n 
                              SE                             SE            
                                                                   n  N 
                                                       OR
                                   n                                         


                         (with replacement, or                (without replacement and
                         infinite population)                 from a finite population)

      Compute confidence limits:
                                                                t = t statistic from table,
                                  95% CI  x  t SE             determined by degrees of
                                                                freedom (n-1).

    In general, it is safe to use: t = 2.0 for small n (n < 30), or t = 1.96 for large n (n > 30)
    if you do not have access to a t-table.

FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling

      Finite population term:
                                                                  N n
              N = Population size                                     
              n = sample size                                     N 

                 s2  N  n 
            SE      N 
                                                 Standard error (without replacement
                 n                               and from a finite population)




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling


       How many samples to take?

          • Sample size should be statistically and practically
          efficient.

          • Enough sample units should be measured to obtain
          the desired level of precision (no more, no less).




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling

     Determine sample size (with replacement, or infinite population)

     Knowing the standard deviation and the desired                             2
                                                                       t s 
     objective (e.g., to be within xxx board feet per acre)       n       
     we can calculate the necessary minimum sample size.               E 
                                                                 Stein’s Formula

     Example:

     s = 2000 board feet / acre                                             2
                                                                  2 (2000) 
     t = 2 (assuming a 95% confidence interval)
                                                                  500  64
                                                              n           
     E = want to be within +/- 500 board feet / acre                       

        (E must be in same units as std. deviation)


FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling

      Determine sample size (with replacement, or infinite population):

       Knowing the coefficient of variation and the                            2
                                                                   t CV   
       desired objective (e.g., error is expected to be        n 
                                                                   A      
                                                                           
       within x % of the value of the mean)                               

    Example:

                CV = 30%
                t = 2 (assuming a 95% confidence                       2
                                                                  2 (30) 
                    interval)                                     5  144
                                                              n         
                                                                         
                A = we want the allowable error to
                    be within +/- 5% of the mean
             (A is expressed as a percentage of the mean)

FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design I: Simple Random Sampling


       How many samples to take?

          • How do we know variance in population before we
          sample it?!?!




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design II: Systematic Sampling

      Assumptions:

        The initial sampling unit is randomly selected or
        established on the ground. All other sample units are
        spaced at uniform intervals throughout the area sampled.

        Sampling units are easy to locate.

        Sampling units appear to be representative of an area.




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design II: Systematic Sampling

      Design:
                              A Grid Scheme is most common




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design II: Systematic Sampling

      Arguments:
      For:
                  Regular spacing of sample units may yield efficient
                  estimates of populations under certain conditions.


      *** Against:
                  Accuracy of population estimates can be low if there is
                  periodic or cyclic variation inherent in the population.



FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design II: Systematic Sampling

       Arguments:
       For:
                   There is no practical alternative to assuming that
                   populations are distributed in a random order across
                   the landscape.

       Against:
                   Simple random sampling statistical techniques can’t
                   logically be applied to a systematic design unless
                   populations are assumed to be randomly
                   distributed across the landscape.


FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design II: Systematic Sampling

        Summary:
     We can (and often do) use systematic sampling to obtain estimates about the
     mean of populations.

     When an objective, numerical statement of precision is required, however, it
     should be viewed as an approximation of the precision of the sampling effort.
     (i.e. 95% confidence intervals)

     Use formulas presented for simple random sampling, and where appropriate,
     use the “without replacement” variations of those equations (if sampling from
     a small population), otherwise use the normal SRS statistical techniques.




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design III: Stratified Random Sampling

      Assumptions:
     A population is subdivided into subpopulations of known sizes.

     A simple random sample of at least two units is drawn from each
     subpopulation.

     Why? To obtain a more precise estimate of the population mean. If the
     variation within a subpopulation is small in relation to the total
     population variance, the estimate of the population mean will be
     considerably more precise than a simple random sample of the same size.

     Why? To obtain an estimate of the resources within the subpopulations.



FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design III: Stratified Random Sampling

       Design:

      Simple random
      samples within 3
      strata




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design III: Stratified Random Sampling

      Estimate the overall population mean:

      Where:
                                                                      L        
      L = number of strata                                             Nh y h 
                                                              y st   h 1     
      Nh = total number of [area] units in strata h
      N = total number of [area] units in all strata                       N   
                                                                               
      << Essentially a weighted average >>                                     


     We often use area for strata units (acres, hectares) in natural resource applications




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design III: Stratified Random Sampling

        Estimate the overall population mean:

                   Example:            Strata      acres      mean dbh

                                       1           10         12.2
                                       2           12         31.6
                                       3           7          20.1




                       (10)(12.2)  (12)(31.6)  (7)(20.1) 
               y st                                         22.1 inches
                                       29                  



FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design III: Stratified Random Sampling

     Estimate the overall population standard error of the mean:
      1. First compute variance within each strata


                                        x h  x h 2 
                                sh                    
                                  2
                                         nh  1        
                                                       

    Where: nh = total number of [tree or plot] units sampled in strata h

                                                                   2              
                                                                        x h 2   x h 2 / nh
                                                                 sh  
                                                                                                   
                                                                                nh  1             
                                                                                                   
FOR 220 Aerial Photo Interpretation and Forest Measurements   Avery and Burkhart formula
       Sampling Design III: Stratified Random Sampling

     Estimate the overall population standard error of the mean:
     2. Second, compute population standard error of the mean


                                                                   1     L Nh 2 sh 2 
                                                          SE st   2                 
        (with replacement)
                                                                   N  h 1  nh       
                   OR


                                                      1  L  Nh sh  Nh  nh  
                                                                2  2
                                                    
      (without replacement)                 SE st   2           
                                                                             
                                                                              
                                                     N  h1  nh  Nh  
                                                                               

                         Remember: N = [area] units, n = [tree or plot] units
FOR 220 Aerial Photo Interpretation and Forest Measurements
       Sampling Design III: Stratified Random Sampling

     Estimate the overall population standard error of the mean:
           3. Compute confidence intervals if desired


        Estimated confidence intervals:


                                        y st  t SE
                                                              st




FOR 220 Aerial Photo Interpretation and Forest Measurements
  Sampling Design III: Stratified Random Sampling
                               Stratum (h)   Stratum Size ac. (N)       Stratum Mean
 Example: Mean
                                   1                   20                    43
                                   2                   40                    80
                                   3                   70                    27
                                   4                   60                    56
                                   5                   50                    63


      (20  43)  (40  80 )  (70  27 )  (60  56 )  (50  63)
yst                                                                51 .9
                    (20  40  70  60  50 )

                                               2

     yst   
             N  yh       h
                                                                                  4


              N
                                                                    5
                       h
                                                   3                              1
  Sampling Design III: Stratified Random Sampling

 Example: Standard Error
     Stratum (h)   Sample Size (n)   Stratum Size ac. (N)   Stratum Mean   Stratum s2
         1               10                  20                  43           285
         2               25                  40                  80           472
         3               30                  70                  27           295
         4               30                  60                  56           304
         5               25                  50                  63           387



         1  20 2  285 40 2  472 70 2  295 60 2  304 50 2  387 
SEst      
          2 
                                                                
       240  10             25         30         30         25    

   SEst  2.86                                                N h  sh
                                                                2    2
                                                   1
                                                          2 
                                          SEst            
                                                 ( N h )       nh
Sampling Design III: Stratified Random Sampling

Example: Confidence Interval
                                      Remember:
                                      Mean = 51.9
95 %CI  yst  SEst                   SE = 2.86
                                      Overall n > 30, t = 1.96




95%CI  51.9  1.96  2.86


95%CI  51.9  5.6
Sampling Design III: Stratified Random Sampling


 Minimum sample size – modified Stein’s formula:


     t s
       2   2
                               N  t   Nh  s
                                     2                2
  n     2               n                           h
      AE                            N  AE
                                         2        2


                     Here, N = Total N (N1 + N2 + …Nn)
                     AE = Allowable Error
       Example: Estimate Average Age of Campers:



                   28      41              27        27             29
                                                                         Complete enumeration:
                                                                         (Census)
                   32      35         26             29            23
                   45      63         38                                 Mean age = 35.95 years
                                                38                  46
                   19      31         41        56                  35
                   25      19         19        55


                                                          Shower
                                                                    48
                   26      25         22        47                  27
                   33        54                 56
                                      26                           32
                   42
                            28        58        48                 39




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Example: Estimate Average Age of Campers:


                                                                        Simple
                  28      41              27        27             29   Random
                                                                        Sampling:
                  32      35         26             29            23    n = 10
                  45      63         38        38                  46
                  19      31         41        56                  35   Average age =
                  25      19         19        55                       38.4  7.6 years

                                                         Shower
                                                                   48
                  26      25                                            30.8 to 46.0
                                     22        47                  27
                  33        54                 56
                                     26                           32
                  42
                           28        58        48                 39    True average = 35.95




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Example: Estimate Average Age of Campers:


                                                                        Simple
                  28      41              27        27             29   Random
                                                                        Sampling:
                  32      35         26             29            23    n = 10
                  45      63         38        38                  46
                  19      31         41        56                  35   Average age =
                  25      19         19        55                       40.2  8.5 years

                                                         Shower
                                                                   48
                  26      25         22        47                       31.7 to 48.7
                                                                   27
                  33        54                 56
                                     26                           32
                  42
                           28        58        48                 39    True average = 35.95




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Example: Estimate Average Age of Campers:


                                                                      Systematic
                                                  27
                28      41              27                       29   Sampling:
                32      35         26                                 n = 10
                                                  29            23
                45      63         38        38                  46
                19      31         41        56                  35   Average age =
                25      19         19        55                       37.7  8.0 years
                                                       Shower
                                                                 48
                26      25         22        47                       29.7 to 45.7
                                                                 27
                33        54                 56
                                   26                           32
                42
                         28        58        48                 39
                                                                      True average = 35.95




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Example: Estimate Average Age of Campers:

                               Every 4th camper

                                                                        Systematic
                                                    27
                  28      41              27                       29   Sampling:
                  32      35         26                                 n = 10
                                                    29            23
                  45      63         38        38                  46
                  19      31         41        56                       Average age =
                                                                   35
                  25      19         19                                 37.6  8.2 years
                                               55

                                                         Shower
                                                                   48   29.4 to 45.8
                  26      25         22        47                  27
                  33        54                 56
                                     26                           32
                  42
                           28        58        48                 39    True average = 35.95




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Example: Estimate Average Age of Campers:


                                                                     Systematic
                                                 27
               28      41              27                       29   Sampling:
               32      35         26                                 n = 15
                                                 29            23
               45      63         38        38                  46
               19      31         41        56                       Average age =
                                                                35
               25      19         19                                 38.5  5.9 years
                                            55
                                                      Shower
                                                                48   32.6 to 44.4
               26      25         22        47                  27
               33        54                 56
                                  26                           32
               42
                        28        58        48                 39
                                                                     True average = 35.95




FOR 220 Aerial Photo Interpretation and Forest Measurements
       Class Example: Timber Cruise


   Stand ~ 32 acres




                                                              Lets use plots and a
                                                              systematic sampling scheme
                                                              typical for a timber cruise

FOR 220 Aerial Photo Interpretation and Forest Measurements
       Class Example: Timber Cruise



   32 plots

   cruise lines                                                  #


                                                     #

   5 chains apart                        #
                                                                         #

                                 #
                                                             #



   4 chains                                      #
                                                                                 #

                                     #

   between plots                                         #
                                                                     #




                                             #
                                                                             #


                                                                 #


                                                     #




                                                         #

                                                                                     32 acres = 320 square chains
                                                                                     320/16plots = 2 ac. represented by each plot
                                                                                     2 acres = 20 square chains
                                                                                     Plot spacing: (5 x 4 = 20)
                                                                                     Randomize location of first plot
FOR 220 Aerial Photo Interpretation and Forest Measurements
       Class Example: Lab 2



   1/10 acre plots
   (37.2 ft radius)
                                                                 #


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FOR 220 Aerial Photo Interpretation and Forest Measurements
                                         What happened here?




FOR 220 Aerial Photo Interpretation and Forest Measurements
                                  What is this? What is the red “stuff”?




FOR 220 Aerial Photo Interpretation and Forest Measurements

				
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