PCB 3043L General Ecology

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PCB 3043L - General Ecology

• Data Analysis

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OUTLINE

• Organizing an ecological study
• Basic sampling terminology
• Statistical analysis of data
• Why use statistics?
• Describing data
• Measures of central tendency
• Measures of spread
• Normal distributions
• Using Excel
• Producing tables
• Producing graphs
• Analyzing data
• Statistical tests
• T-Tests
• ANOVA
• Regression

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Organizing an ecological study

• What is the aim of the study?
• What is the main question being asked?
• What are your hypotheses?
• Collect data
• Summarize data in tables
• Present data graphically
• Statistically test your hypotheses
• Analyze the statistical results
• Present a conclusion to the proposed question

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Basic sampling terminology

• Variables
• Populations
• Samples
• Parameters
• Statistics

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What is a variable?

• Variable any defined characteristic that varies
  from one biological entity to another.
• Examples plant height, bird weight, human eye
  color, no. of tree species
• If an individual is selected randomly from a
  population, it may display a particular height,
  weight, etc.
• If several individuals are selected, their
  characteristics may be very similar or very
  different.

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What is a population?

• Population the entire collection of measurements
  of a variable of interest.
• Example if we are interested in the heights of
  pine trees in Everglades National Park (Plant
  height is our variable) then our population would
  consist of all the pine trees in Everglades
  National Park .

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What is a sample?

• Sample smaller groups or subsets of the
  population which are measured and used to
  estimate the distribution of the variable within
  the true population
• Example the heights of 100 pine trees in
  Everglades National Park may be used to estimate
  the heights of trees within the entire population
  (which actually consists of thousands of trees)

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What is a parameter?

• Parameter any calculated measure used to
  describe or characterize a population
• Example the average height of pine trees in
  Everglades National Park

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What is a statistic?

• Statistic an estimate of any population
  parameter
• Example the average height of a sample of 100
  pine trees in Everglades National Park

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Why use statistics?

• It is not always possible to obtain measures and
  calculate parameters of variables for the entire
  population of interest
• Statistics allow us to estimate these values for
  the entire population based on multiple, random
  samples of the variable of interest
• The larger the number of samples, the closer the
  estimated measure is to the true population
  measure
• Statistics also allow us to efficiently compare
  populations to determine differences among them
• Statistics allow us to determine relationships
  between variables

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Statistical analysis of data
Heights of pine trees at 2 sites in Everglades
National Park

• Measures of central tendency
• Measures of dispersion and variability

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Measures of central tendency

• Where is the center of the distribution?
• mean (? or µ) arithmetic mean
• median the value in the middle of the ordered
  data set
• mode the most commonly occurring value
• Example data set 1, 2, 2, 2, 3, 5, 6, 7, 8, 9,
  10
• Mean (1 2 2 2 3 5 6 7 8 9
  10)/11 55/11 5
• Median 1, 2, 2, 2, 3, 5, 6, 7, 8, 9,10 5
• 1, 2, 2, 2, 3, 5, 6, 7, 8, 9,10,11
  (56)/2 5.5
• Mode 1, 2, 2, 2, 3, 5, 6, 7, 8, 9, 10 2

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Measures of dispersion and variability

• How widely is the data distributed?
• range largest value minus smallest value
• variance (s2 or s2) ..
• standard deviation (s or s)

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Measures of dispersion and variability
Example data set 0, 1, 3, 3, 5, 5, 5, 7, 7, 9,
10Variance 9.8Standard Deviation
3.29Range 10 Example data set 0, 10, 30,
30, 50, 50, 50, 70, 70, 90, 100Variance
980Standard Deviation 270.13Range 100
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Normal distribution of data

• A data set in which most values are around the
  mean, with fewer observations towards the
  extremes of the range of values
• The distribution is symmetrical about the mean

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Proportions of a Normal Distribution

• A normal population of 1000 body weights
• µ 70kg s 10kg
• 500 weights are gt 70kg
• 500 weights are lt 70 kg

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Proportions of a Normal Distribution

• How many bears have a weight gt 80kg
• µ 70kg s 10kg X 80kg
• We use an equation to tell us how many standard
  deviations from the mean the X value is located
  
• 
  
• We then use a special table to tell us what
  proportion of a normal distribution lies beyond
  this Z value
• This proportion is equal to the probability of
  drawing at random a measurement (X) greater than
  80kg

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Z table

• Look for Z value on table (1.0)
• Find associated P value (0.1587)
• P value states there is a 15.87 ((0.1587/1)x100)
  chance that a bear selected from the population
  of 1000 bears measured will have a weight greater
  than 80kg

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Probability distribution tables

• There are multiple probability tables for
  different types of statistical tests.
• e.g. Z-Table, t-Table, ?2-Table
• Each allows you to associate a critical value
  with a P value
• This P value is used to determine the
  significance of statistical results

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Using Excel

• Program used to organize data
• Produce tables
• Perform calculations
• Make graphs
• Perform statistical tests

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Organizing data in tables

• Allows you to arrange data in a format that is
  best for analysis
• The following are the steps you would use

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Performing calculations

• Allows you to perform several calculations
• Sum, Average, Variance, Standard deviation
• Basic subtraction, addition, multiplication
• More complex formulas

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Making graphs

• Bar Charts.
• Scatter Plots.

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Making graphs

• Bar Charts.
• Scatter Plots.

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Analyzing Data in Excel

• Statistical tests can be done to determine
• Whether or not there is a significant difference
  between two data sets (Students t-test)
• Whether or not there is a significant difference
  between more than two data sets (ANOVA)
• Whether or not there is a significant
  relationship between two variables (Regression
  analysis)

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Analyzing Data in Excel

• The following steps must be followed
• Choose an appropriate statistical test
• State H0 and HA
• Run test to produce Test Statistic
• Examine P-value
• Decide to accept or reject H0

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Analyzing Data in Excel

• Normally, you would have to calculate the
  critical value and look up the P value on a table
  
• All tests done in Excel provide the P value for
  you
• This P value is used to determine the
  significance of statistical results
• This P value must be compared to an a value
• a value is usually 0.05 or less (e.g. 0.01)
• Less than 5 chance that the null hypothesis is
  true
• The lower the a value the more certain we about
  rejecting the null Hypothesis
• First thing you must do is select which
  statistical test you want to perform
• This is how it is done..

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t-Tests

• Used to compare the means of two populations and
  answer the question
• Is there a significant difference between the
  two populations?
• Example Is there a significant difference
  between the average height of pine trees from 2
  sites in Everglades National Park?
• You cannot use this test to compare two different
  types of data (e.g. water depth data and soil
  depth data).
• It can only compare two sets of data based on the
  same data type (e.g. water depth data from two
  different sites)
• The two data sets that are being compared must be
  presented in the same units. (e.g. you can
  compare two sets of data if both are recorded in
  days. You cannot compare data recorded in units
  of days with data recorded in units of months)

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1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• Your Null Hypothesis is always
• There is no significant difference between the
  two compared populations (µ1 µ2)
• Your Alternative Hypothesis is always
• There is a difference between the two compared
  populations (µ1 ? µ2)

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1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0
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t-Tests
1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• When you run the test, look for the p-value
• If p gt 0.05 then fail to reject your Null
  Hypothesis and state that there is no
  significant difference between the two compared
  populations
• If p lt 0.05 then reject your Null Hypothesis and
  state that there is a significant difference
  between the two compared populations

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t-Tests
1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• When you run the test, look for the p-value
• Our results show P 0.09903
• Therefore P gt 0.05 (This means that there is
  greater than a 5 chance that our null hypothesis
  is true)
• So we must fail to reject the Null Hypothesis and
  state that there is no significant difference
  between the two compared populations

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ANOVA

• Used to compare the means of more than two
  populations and answer the question
• Is there a significant difference between the
  populations?
• Example Is there a significant difference
  between the average height of pine trees from 4
  sites in Everglades National Park?
• For comparing a particular feature of two or more
  populations, use a Single Factor ANOVA
• For comparing a particular feature of two or more
  populations, subdivided into two groups, use a
  Two Factor ANOVA

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1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• Your Null Hypothesis is always
• There is no significant difference between the
  compared populations (µ1 µ2 µ3 µ4 ..)
• Your Alternative Hypothesis is always
• There is a difference between the compared
  populations (µ1 ? µ2 ? µ3 ? µ4 ..)

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1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0
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ANOVA
1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• When you run the test, look for the p-value
• If p gt 0.05 then fail to reject your Null
  Hypothesis and state that there is no
  significant difference between the compared
  populations
• If p lt 0.05 then reject your Null Hypothesis and
  state that there is a significant difference
  between at least two of the compared populations

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ANOVA
1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• When you run the test, look for the p-value
• Our results show P 0.002197
• Therefore P lt 0.05 (This means that there is less
  than a 5 chance that our null hypothesis is
  true)
• So we must reject your Null Hypothesis and state
  that there is a significant difference between
  at least two of the compared populations

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ANOVA

• Remember
• The ANOVA result will only tell you that
• None of the data sets are significantly different
  from each other
• OR
• At least two of the data sets among the data sets
  being compared are significantly different
• If there is a significant difference between at
  least two data sets, it will not tell you which
  two.

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Two-way ANOVA

• Used to compare the means of more than two
  populations that are subdivided into two or more
  groups and answer the question
• Is there a significant difference between the
  populations?
• Example Is there a significant difference
  between the average height of pine trees from 4
  sites in Everglades National Park, during the wet
  and dry season?

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Two-way ANOVA
1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• When you run the test, look for the interaction
  p-value
• If p gt 0.05 then fail to reject your Null
  Hypothesis and state that there is no
  significant difference between the compared
  populations
• If p lt 0.05 then reject your Null Hypothesis and
  state that there is a significant difference
  between at least two of the compared populations

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Two-way ANOVA
1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• Our results show P 0.2888
• Therefore P gt 0.05 (This means that there is a
  greater than a 5 chance that our null hypothesis
  is true)
• So we must fail to reject the Null Hypothesis and
  state that there is no significant difference
  between the compared populations

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Regression analysis

• Used to determine whether or not there is a
  linear relationship between two variables and
  answer the question
• Is there a significant linear relationship
  between two variables?
• Example Is there a significant relationship
  between the average height of pine trees and soil
  depth in Everglades National Park?
• It basically creates an equation (or line) that
  best predicts Y values based on X values.
• You cannot use this test to compare populations.
  It only compares variables.
• You are looking at two different variables (e.g.
  water depth (cm) and plant abundance (no. of
  individuals), so the data sets do not have to be
  presented in the same units

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1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• Your Null Hypothesis is always
• There is no significant linear relationship
  between the two variables
• Your Alternative Hypothesis is always
• There is a significant linear relationship
  between the two variables

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• R squared how well y can be predicted by x,
  i.e. how strong the linear relationship is
  between the two variables.
• The closer R square is to 0, the less well it
  fits the data.
• The closer R square is to 1, more it fits the
  data.

• Example R square value of 0.04
• The regression line does not fit the data well
• Many of the points lie far from the line, so
  there is not a defined linear relationship
  between the two variables
• x cannot be used to predict y
• Example R square value of 0.94
• The regression line fits the data well
• The points all lie fairly close to the line, so
  there is a defined linear relationship between
  the two variables
• x can be used to predict y

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1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0
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Regression analysis
1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• When you run the test, look for the Significance
  F or Sample p-value
• If p gt 0.05 then fail to reject your Null
  Hypothesis and state that There is no
  significant linear relationship between the two
  variables
• If p lt 0.05 then reject your Null Hypothesis and
  state that There is a significant linear
  relationship between the two variables

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Regression analysis
1. Choose an appropriate statistical test2.
State H0 and HA 3. Run test to produce Test
Statistic4. Examine P-value5. Decide to accept
or reject H0

• When you run the test, look for the p-value
• Our results show Significance F or Sample p-value
  1.65E08 0.0000000165
• Therefore P lt 0.05 (This means that there is less
  than a 5 chance that our null hypothesis is
  true)
• So we must reject your Null Hypothesis and state
  that There is a significant linear relationship
  between the two variables
• Next look at the R squared value
• Our results show R squared 0.975
• Therefore the line fits the data well
• x can be used to predict y

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Ecological study

• What is the aim of the study?
• What is the main question being asked?
• What are your hypotheses?
• Collect data
• Summarize data in tables
• Present data graphically
• Statistically test your hypotheses
• Analyze the statistical results
• Present a conclusion to the proposed question

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• Aim To determine whether or not there are
  changes in heights of Pine trees with distance
  from the edge of a forest trail in Everglades
  National Park.
• Hypotheses
• HO There is no significant relationship between
  distance from the edge of the trail and Pine tree
  height
• HA There is a significant relationship between
  distance from the edge of the trail and Pine tree
  height
• Results

Average tree height of pine trees along transect
from forest trail to interior forest at ENP

• P 1.65E-08 Since P lt 0.05, reject Ho
• Therefore, there is a significant relationship
  between distance from the edge of the trail and
  Pine tree height
• R Square 0.97, so there is a strong positive
  linear relationship between distance from the
  trail and plant height

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Assignment Worksheet 1

• Three questions
• T-test
• Single factor ANOVA and Two-way ANOVA
• Regression analysis