ESCI 407/507: Forest Ecology

Spring 2024

Last updated: 5/7/24

Lab #4: Introduction to Quadrat Sampling (Capture the Flag!)
 
 Note for 2002: add some discussion in here to point out that we are attempting to evaluate sources of BIAS associated with quadrat-based sampling.

Introduction: It is frequently necessary n ecological research to obtain information about the numerical characteristics of natural population. Knowledge of numerical density (number of individuals per unit area) of one or more populations may be required even before a field study can be planned or a statistical experiment can be formulated.

Complete enumeration of natural populations is seldom practical or possible. Even if complete counts were possible, this information would actually be no more useful in most cases than information obtained through proper sampling of the population. Thus, ecologists rely heavily upon sampling procedures to estimate the numerical characteristics of populations. Some knowledge of and first-hand experience with sampling theory and practice are fundamental to research planning, execution and interpretation.

We begin by distinguishing between the sample itself and the population from which it is drawn. A sample can be counted, measured and otherwise described more or less exactly. Inferences drawn about the population, however, are estimates made on the basis of sample information. The central question asked in all sampling programs is "how accurately and how precisely do the samples describe or represent the original population?"
 

Objectives: This laboratory exercise is designed as a "tools" lesson in field ecology, to provide first-hand familiarity with some of the principal considerations involved in quadrat-based sampling of plant population density and spatial distribution. To simplify things we will practice the sampling technique using different colored flags instead of plants. Different colors represent different plant species. Particular aspects of sampling that will be considered in this lab exercise include:

    1. the effect of quadrat size (area) on estimates of population density,
    2. the effect of sample size (number of quadrats) on estimates of population density, and
    3. the effect of quadrat size (area) on the detection of spatial pattern.


Methods and Procedures: We will be sampling flags in a 25 by 25 meter area using three different sized nested square quadrats (0.25 m2, 1 m2 and 4 m2). Each group will be provided with a plastic frame for the 0.25 m2 and 1 m2 quadrats. The 4 m2 quadrat is created by flipping the 1 m2 quadrat. Population density will be estimated from the data for each size quadrat. Density estimates will be adjusted to units of number of flags per 1 m2 to facilitate statistical comparisons. We will work in groups of two or three people. Each group will locate and inventory as many randomly placed sample points as time allows (try for at least 20).
 

Randomization Procedure: For each replicate, "blindly" select a starting point in the random number table (these will be handed out in lab). From this first number, move across the table to the right until you:

    1. find the first two digit pair that is 23 or under. This will be the X-coordinate;
    2. find the second two digit pair of 23 or under for your Y-coordinate.

Example: 8820 6656 3384 4813 6244 2517 3596 0356

The coordinates of your first sample point are, x=20, y=13. You will find this point in the field and place the corner of your nested quadrats at this location (see diagram). The counting procedure will be described below. When you are finished collecting data at this first sampling point, you will continue moving through the random numbers table until you locate the next pair of numbers that are between 0 and 23. The coordinates of the second sampling point are, x=17, y=03.
 

Sampling Procedure: At each sample point the group will count and record the number of individuals for each color of flag contained within each of the three nested quadrats. The number of individuals in the smallest quadrat will be counted and the data recorded (make your own data sheet like the one illustrated below). Then count and record the number of individuals in the 1 m2 quadrat, including those that were counted in the smallest quadrat. Finally, the total number of individuals in the largest (4 m2) quadrat will be counted and recorded (see sample below).

Data Sheet:
 

 

0.25 m2

 

 

1 m2

 

 

 4 m2

 

Red

Yellow

Blue

Red 

Yellow

Blue

 Red

Yellow

Blue

1

0

1

2

1

3

3

5

6

Data Compilation: For this and subsequent labs, all groups will compile their data in a common digital format so that I can produce one complete data set. For some labs, we will combine data from the Monday and Tuesday labs into a single data set for the entire class. In other cases (such as for this lab) we will have separate data sets for the Monday and Tuesday labs. The desired format will be explained for each week. For this week's lab, you will take your field data and enter it into Excel in the same format as the field data table: column#1 is the replicate #, col. 2-5 are counts for the four colors of flags in the 0.25 m2 quadrat, col. 6-9 are counts for the four colors of flags in the 1 m2 quadrat and col. 10-13 are counts for the four colors of flags in the 4 m2 quadrat. Label the columns so we can be sure of the organization. You will need to get a copy of everyone else's data (just the data collected on either Monday and Tuesday). We will compile these data and get them back to you so we can work on the analysis in lab.

Link to class datasets will be added below (note that links are here but not functional until I get everyone’s data):

Quadlab_wed2024.xlsx   Data now available

Quadlab_fri2024.xlsx    Data now available


 

Calculations:

  1. In order to statistically compare density estimates between quadrat sizes, we need to adjust each density estimate to a common unit of area (flags / m2). Retain the raw count data but prepare a duplicate worksheet with standardized (flags/m2) density values as well.
  2. Calculate descriptive statistics, based the adjusted (flags/m2) data, for each quadrat size. Compute the adjusted mean, variance, standard deviation, and standard error of the mean. Also calculate the 95% confidence interval around your estimate of the mean. These descriptive statistics should be included in Table 1 of your write up. The sample size should be stated in the table legend.  Something to add next year; have them calculate 95% confidence limits.  Is “true” value within CL bounds?  Also maybe calculate statistical power?
  3. Compute a statistical test of the equality of population means between the three quadrat sizes for each species using a one-way ANOVA. Run a separate ANOVA for each species. In these tests, the null hypothesis (Ho) is that, for each species, there is no difference between your estimates of the mean among the three quadrat sizes.  Rejection of this hypothesis would suggest that estimates of population density are dependent upon quadrat size.  This should not happen if your sample size is adequate.  Include the results of this analysis in Table 2 of your write up.  Note: See the discussion below about the Independence Assumption of ANOVA before doing this one.
  4. Calculate the simple variance-to-mean ratio for the unadjusted (raw count data) density estimate for each quadrat size, V=s2/mean. V is an index of dispersion. The variance and mean of a random distribution are equal, so that V=1 for a random distribution. Values of V greater than 1 indicate a departure from randomness in the direction of clumping, whereas values less than one indicate a uniform distribution. Also include this V-index in Table 1 of your write up.
  5. Graph the cumulative (adjusted) sample mean and variance for each quadrat size, recalculating the sample mean after each additional five sample increment in sample size. The X-coordinate in these graphs is the cumulative number of quadrats included and the Y-axis is the cumulative sample statistic. You will be producing 8 graphs: one for each "species" showing the cumulative mean density (4 graphs) and one for each "species" showing the cumulative variance (4 more graphs). Each graph will include three lines; one line for each quadrat size. These graphs will provide information about the "adequacy" of your sample size. With a small sample size, your estimate of the mean and variance is not very reliable. Initially (with just a few samples) your estimate of the mean and variance will jump around quite a bit. As you approach an "adequate" sample size, your estimate of the mean and variance will stabilize. The point at which this stabilization occurs is likely to occur at a different point for each quadrat size. This stabilization will also occur at a lower sample size for uniformly distributed "species" than for those with a clumped distribution.


The Independence Assumption of ANOVA: You will be using Analysis of Variance (ANOVA) to analyze your data. As you recall from your Biostatistics class, one of the fundamental assumptions of ANOVA is that all observations are independent of one another.  In our case, since our smaller quadrats are nested within the larger quadrats, any given estimate of population size based on the 4 m2 quadrat is correlated with the estimate of population size based on the two smaller quadrats that are nested within it.  For this reason an ANOVA using all of our data violates the independence assumption and any conclusions based on this sort of analysis are not reliable.  We could solve this dilemma in two ways.  One approach would involve a change in the way we collected data.  We could have split the class into three groups and have one group collect data using only the smallest quadrat, have another group use only the medium quadrat and have the third group use only the largest quadrat.  This would be the preferred approach but it would have been much more time consuming.  The second approach (and the one I want you to use), is to split our existing data set into three groups.  For the first 1/3rd of our cases, use only the data collected using the smallest quadrat.  For the second 1/3rd of our cases, use only the data collected using the medium quadrat.  Finally, for the last 1/3rd of the cases, use only the data collected using the largest quadrat.  This will insure that each of our observations are independent of one another and the results of our ANOVA will be valid. Splitting the data set will only be necessary for the ANOVA!

Running ANOVA in Excel: There are a number of statistics packages available on the various university computers.  Feel free to use any package you are familiar with.  I will provide instructions here for doing ANOVA using Excel.  Unfortunately, Excel is rather fussy about the arrangement of your data so you will have to do some cutting and pasting to prepare.  Let's assume you have a total of 120 observations arranged on an Excel worksheet as described above.  You will need to set up a new worksheet arranged like this:
 

0.25 m2

1 m2

4 m2

0.25 m2

1 m2

4 m2

0.25 m2

1 m2

4 m2

Red

Red

Red

Yellow

Yellow

Yellow

Blue

Blue

Blue

Data for Obs. #1-40

Data for Obs. #41-80

Data for Obs. #81-120

Data for Obs. #1-40

Data for Obs. #41-80

Data for Obs. #81-120

Data for Obs. #1-40

Data for Obs. #41-80

Data for Obs. #81-120

All the data should be in #flags/m2 , not the raw count data.  Go to Data - Data Analysis - ANOVA: single factor.  Click on OK and then point to the three columns of data for the Red flags.  Click on OK and you should get some output that looks like this:

Anova: Single Factor
SUMMARY
 

Groups

Count

Sum

Average

Variance 

Column 1

 40

0 

0 

0 

Column 2

40 

0

0

0

Column 3

40

16

0.4

1.476923077

 
ANOVA
 

Source of Variation

SS

df 

MS

F 

P-value

F crit

Between Groups

4.266666667

2

2.133333333

4.333333333

0.015293163

3.073765242

Within Groups

57.6

117 

0.492307692 

 

 

 

 

 

 

 

 

 

 

Total

61.86666667

119 

 

 

 

 

 
Naturally, the values in your output tables will be different than those in this example.  At this point, It is important to do a reality check!  In the Summary table above, the average values for "column 1, column 2 and column 3" represent the average values for the red flags based on the data for the 0.25 m2, 1 m2 and 4 m2 quadrats, respectively.  Go back to your worksheet and calculate the averages for these columns and make sure that you get the same value as given on the ANOVA output page.  If you don't get the same values, you've done something wrong.  Note that these average values will not be the same as the average values you calculate using the full data set (the values you will report in your table 1).  The key thing to look at in your ANOVA output is the P-value.  A P-value of less than 0.05 indicates that we should reject our null hypothesis.  What does this mean?

You will need to run the ANOVA four times; once for each color of flags.

Things to think about and discuss in your report:

  1. How closely do your estimates of flag density match the “true” flag density? 
  2. Did the adjusted density estimates differ substantially with quadrat size? How were the differences related to distribution pattern, as shown by the V ratio?
  3. Was the sample size used in our ANOVA large enough?  Take a look at your figures (mean vs. cumulative sample size plots) to assess this issue.
  4. Did the sample mean "stabilize" for each quadrat with increasing sample size (i.e., with increasing number of quadrats)? Did one quadrat size show stabilization earlier than the others?  Did the “optimum” quadrat size vary among species?  If so, was this related to the species distribution pattern as shown by the V ratio?
  5. Did all three quadrat sizes indicate the same pattern of intrapopulation dispersion, as indicated by the V ratio?
  6. What effects might differences in quadrat shape have on density estimates for this population? (hint: what if each flag had a "canopy?")
  7. Under what conditions might you choose to use a sampling procedure other than simple random sampling?
  8. Outline the various factors you might consider in selecting a quadrat size in a real plant population study. How would you determine the appropriate number of samples and the appropriate quadrat size?

Due Date for Lab Reports: Lab reports are due by 1:00 PM on the day of your lab during the week of May 13. As on all lab reports, there is a 5% per day penalty for turning in your report late. Be sure to check the Lab Index page for information on the proper format, organization and grading criteria for your lab reports.

Quadlab_tues2017.xlsx

quadlab_wed2017.xlsx

quadlab_monday2012.xlsx

quadlab_tuesday2012.xlsx

quadlab_mon2011.xlsx

quadlab_tues2011.xlsx

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