A Bestiary of Experimental and Sampling Designs

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A Bestiary of Experimental and Sampling Designs
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REMINDERS

• The goal of experimental design is to minimize
  the potential sources of confusion (Hurlbert
  1984)
• Temporal (and spatial) variability
• Procedure effects
• Experimenter bias
• Experimenter-generated variability (random
  error)
• Inherent variability among experimental units
• Non-demonic intrusion
• it is the elementary principles of experimental
  design, not advanced or esoteric ones, which are
  most frequently and severely violated by
  ecologists.

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The design of an experiment

• The details of
• replication,
• randomizations,
• and independence

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We cannot draw blood from a stone

• Even the most sophisticated analysis CANNOT
  rescue a poor design

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Categorical variables

• They are classified into one or more unique
  categories
• Sex (male, female)
• Trophic status (producer, herbivore, carnivore)
• Habitat type (shade, sun)

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Continuous variables

• They are measured on a continuous numerical scale
  (real or integer values)
• Size
• Species richness
• Habitat coverage
• Population density

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Dependent and independent variables

• The assignment of dependent and independent
  variables implies an hypothesis of cause and
  effect that you are trying to test.
• The dependent variable is the response variable
• The independent variable is the predictor variable

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Ordinate (vertical y-axis)
Abscissa (horizontal x-axis)
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Four classes of experimental design
Dependent variable Independent variable Independent variable
Continuous Categorical
Continuous Regression ANOVA
Categorical Logistic regression Tabular
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The Analysis of Covariance (ANCOVA)

• It is used when there are two independent
  variables, one of which is categorical and one of
  which is continuous (the covariate)

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Four classes of experimental design
Dependent variable Independent variable Independent variable
Continuous Categorical
Continuous Regression ANOVA
Categorical Logistic regression Tabular
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Regression designs

• Single-factor regression
• Multiple regression

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Single-factor regression

• Collect data on a set of independent replicates
• For each replicate, measure both the predictor
  and the response variables.
• Hypothesis seed density (the predictor variable)
  is responsible for rodent density (the response
  variable)

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Plot Seeds Rodents/m2
1 50 3.2
2 12 11.7
. . .
n 300 5.3
Plots
Variables
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Single-factor regression

• You assume that the predictor variable is a
  causal variable changes in the value of the
  predictor would cause a change in the value of
  the response
• This is very different from a study in which you
  would examine the correlation (statistical
  covariation) between two variables

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In regression (Model I)

• You are assuming that the value of the
  independent variable is known exactly and is not
  subject to measurement error

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Assumptions and caveats

• Adequate replication
• Independence of the data
• Ensure that the range of values sampled for the
  predictor variable is large enough to capture the
  full range of responses by the response variable
• Ensure that the distribution of predictor values
  is approximately uniform within the sample range

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Multiple regression

• Two or more continuous predictor variables are
  measured for each replicate, along with the
  single response variable

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Assumptions and caveats

• Adequate replication
• Independence of the data
• Ensure that the range of values sampled for the
  predictor variables is large enough to capture
  the full range of responses by the response
  variable
• Ensure that the distribution of predictor values
  is approximately uniform within the sample range

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Multiple regression

• Ideally, the different predictor variables
  should be independent of one another in reality,
  many predictor variables are correlated (e.g.,
  height and weight)
• This collinearity makes it difficult to estimate
  accurately regression parameters and to tease
  apart how much variation in the response variable
  is associated with each of the predictor
  variables

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Multiple regression

• As always, replication becomes important as we
  add more predictor variables to the analysis.
• In many cases it is easier to measure additional
  predictor variables than is to obtain additional
  independent variables
• Avoid the temptation to measure everything that
  you can just because it is possible

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Multiple regression

• It is a mistake to think that a model selection
  algorithm can identify reliably the correct set
  of predictor variables

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Four classes of experimental design
Dependent variable Independent variable Independent variable
Continuous Categorical
Continuous Regression ANOVA
Categorical Logistic regression Tabular
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ANOVA designs

• Analysis of Variance
• Treatments refers to the different categories of
  the predictor variables
• Replicates each of the observations made

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ANOVA designs

• Single-factor designs
• Randomized block designs
• Nested designs
• Multifactor designs
• Split-plot designs
• Repeated measurements designs
• BACI designs

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Single-factor designs

• It is one of the simplest, but most powerful,
  experimental designs
• Can readily accommodate studies in which the
  number of replicates per treatment is not
  identical (unequal sample size)

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Single-factor designs

• In a single-factor design, each of the treatments
  represent variation in a single predictor
  variable or factor
• Each value of the factor that represents a
  particular treatment is called a treatment level

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Id Treatment Replicate Number of flowers
1 Watered 1 9
2 Not watered 1 4
. . . .
11 Watered 6 10
12 Not watered 6 2
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Good news, bad news

• This design does not explicitly accommodate
  environmental heterogeneity, so we need to sample
  the entire array of background conditions
• This means the results can potentially be
  generalized across all environments, but
• If the noise is much stronger than the signal of
  the treatments, the experiment may have low
  power, and the analysis may not reveal treatment
  differences unless there are many replicates

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Randomized block designs

• An effective way to incorporate environmental
  heterogeneity into a design
• A block is a delineated area or time period
  within which environmental conditions are
  relatively homogeneous
• Blocks can be placed randomly or systematically
  in the study area, but should be arranged so that
  the environmental conditions are more similar
  within blocks than between them

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Randomized block designs

• Once blocks are established, replicates will
  still be assigned randomly to treatments, but a
  single replicate from each of the treatments is
  assigned to each block

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Id Treatment Block Number of flowers
1 Watered 1 9
2 Not watered 1 4
. . .
11 Watered 6 10
12 Not watered 6 2
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Caveats

• Blocks should have enough room to accommodate a
  single replicate of each of the treatments, and
  enough spacing between replicates to ensure their
  independence
• The blocks themselves also have to be far enough
  apart from each other to ensure independence of
  replicates among blocks

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Randomized block designs
Valid blocking
Invalid blocking
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Advantages

• It can be used to control for environmental
  gradients and patchy habitats
• It is useful when your replication is constrained
  by space or time
• Can be adapted for a matched pair lay-out

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Disadvantages

• If the sample size is small and the block effect
  weak, the randomized block design is less
  powerful than the simple one-way layout
• If blocks are too small, you may introduce
  non-independence by physically crowding the
  treatments together (e.g., nectar-removal and
  control plots on p. 152 of Gotelli Ellison)
• If any of the replicates are lost, the data from
  the block cannot be used unless the missing
  values can be estimated indirectly

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Disadvantages

• It assumes that there is no interaction between
  the blocks and the treatments
• BUT, replication within blocks will indeed tease
  apart main effects, block effects, and the
  interaction between blocks and treatments. It
  will also address the problem of missing data
  from within a block

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Nested designs

• It is any design in which there is subsampling
  within each of the replicates
• In this design the subsamples are not independent
  of one another
• The rational of this design is to increase the
  precision with which we estimate the response of
  each replicate

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Id Treatment Subsample Replicate Number of flowers
1 Watered 1 1 9
2 Watered 2 1 4
3 Watered 3 1 7
. . . . .
19 Not watered 1 7 16
20 Not watered 2 7 10
21 Not watered 3 7 2
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Advantages

• Subsampling increases the precision of the
  estimate for each replicate in the design
• Allows to test two hypothesis
• First Is there variation among treatments?
• Second Is there variation among replicates
  within treatments?
• Can be extended to a hierarchical sampling design

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Disadvantages

• They are often analyzed incorrectly
• It is difficult or even impossible to analyze
  properly if the sample sizes are not equal
• It often represents a case of misplaced sampling
  effort
• Subsampling is not a solution to inadequate
  replication

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Randomized block designs

• Strictly speaking, the randomized block and the
  nested ANOVA are two-factor designs, but the
  second factor (i.e., the blocks or subsamples) is
  included only to control for sampling variation
  and is not of primary interest

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Multifactor designs

• the main effects are the additive effects of each
  level of one treatment average over all levels of
  the other treatment
• the interaction effects represent unique
  responses to particular treatment combinations
  that cannot be predicted simply from knowing the
  main effects.

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Multifactor designs

• In a multifactor design, the treatments cover two
  (or more) different factors, and each factor is
  applied in combination in different treatments.
• In a multifactor design, there are different
  levels of the treatment for each factor

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Multifactor designs

• Why not just run two separate experiments?
• Efficiency. It is often more cost effective to
  run a single experiment than to run two separate
  experiments
• A multifactor design allows you to test for both
  main effects and for interaction effects

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Interactions
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50
40
West
30
North
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10
0
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
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Orthogonal

• The key element of a proper multifactorial design
  is that the treatments are fully crossed or
  orthogonal every treatment level of the first
  factor must be represented with every treatment
  level of the second factor
• If some of the treatment combinations are missing
  we end with a confounded design

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Two-factor design
Substrate treatment Substrate treatment Substrate treatment Substrate treatment
Granite Slate Cement
Predator treatment Unmanipulated
Predator treatment Cage Control
Predator treatment Predator exclusion
Predator treatment Predator intrusion
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Advantages

• The key advantage is the ability to tease apart
  main effects and interactions between factors.
  The interaction measures the extent to which
  different treatment combinations act additively,
  synergistically, or antagonistically

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Disadvantages

• The number of treatment combinations can quickly
  become too large for adequate replication
• It does not account for spatial heterogeneity.
  This can be handled by a simple randomized block
  design, in which each block contains exactly one
  of the treatment combinations
• It may not be possible to establish all
  orthogonal treatment combinations

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Split-plot designs

• It is an extension of the randomized block design
  to two treatments
• What distinguishes a split plot design from a
  randomized block design is that a second
  treatment factor is also applied, this time at
  the level of the entire plot

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Split plot design
Substrate treatment The subplot factor Substrate treatment The subplot factor Substrate treatment The subplot factor Substrate treatment The subplot factor
Granite Slate Cement
Predator treatment The whole-plot factor Unmanipulated
Predator treatment The whole-plot factor Control
Predator treatment The whole-plot factor Predator exclusion
Predator treatment The whole-plot factor Predator intrusion
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Advantages

• The chief advantage is the efficient use of
  blocks for the application of two treatments
• This is a simple layout that controls for
  environmental heterogeneity

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Disadvantages

• As with nested designs, a very common mistake is
  for investigators to analyze a split-plot design
  as a two factor ANOVA

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Repeated measurements designs

• It is used whenever multiple observations on the
  same replicate are collected at different times
  (It can be thought of as a split-plot in which a
  single replicate serves as a block, and the
  subplot factor is time)

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Repeated measurements designs

• The between-subjects factor corresponds to the
  whole-plot factor
• The within-subjects factor corresponds to the
  different times
• The multiple observations on a single individual
  are not independent of one another

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Advantages

• Efficiency
• It allows each replicate to serve as its own
  block or control
• It allows us to test for interactions of time
  with treatment

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Circularity

• Both the randomized block and the repeated
  measures designs make a special assumption of
  circularity for the within-subjects factor.
• It means that the variance of the difference
  between any two treatment levels in the subplots
  is the same

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For repeated measures design it means that the
variance of the difference of observations
between any pair of times is the same
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Disadvantages

• In many cases the assumption of circularity is
  unlikely to be met for repeated measures
• The best way to meet the circularity assumption
  is to use evenly spaced sampling times along with
  knowledge of the natural history of your
  organisms to select the appropriate sampling
  interval

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Alternatives

1. To set enough replicates so that a different set
   is censused at each time period. With this
   design, time can be treated as a simple factor in
   a two-factor analysis of variance
2. Use the repeated measures layout but collapse the
   correlated repeated measures into a single
   response variable for each individual, and then
   use a simple one-factor analysis of variance

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Think outside the ANOVA Box

• Many ecological experiments test a continuous
  predictor at only a few values so they can be
  shoehorned into an ANOVA design
• One Alternative Experimental regression design

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Four classes of experimental design
Dependent variable Independent variable Independent variable
Continuous Categorical
Continuous Regression ANOVA
Categorical Logistic regression Tabular
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Tabular designs

• The measurements of these designs are counts
• A contingency table analysis is used to test
  hypotheses

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