Transcript A Bestiary of Experimental and Sampling Designs

A Bestiary of
Experimental and
Sampling Designs
REMINDERS
• The goal of experimental design is to minimize the
potential “sources of confusion” (Hurlbert 1984):
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6.
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.”
The design of an experiment
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The details of:
replication,
randomizations,
and
independence
We cannot draw blood from a stone
• Even the most sophisticated analysis
CANNOT rescue a poor design
Categorical variables
• They are classified into one or more
unique categories
– Sex (male, female)
– Trophic status (producer, herbivore,
carnivore)
– Habitat type (shade, sun)
Continuous variables
• They are measured on a continuous
numerical scale (real or integer
values)
– Size
– Species richness
– Habitat coverage
– Population density
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
Ordinate (vertical y-axis)
Ln (lambda)
2.000
Y=26.06X-2.99
r2=0.355
Fstat=25.84
1.000
0.000
-1.000
0
5
10
15
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Time since fire (years)
Abscissa (horizontal x-axis)
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Four classes of experimental design
Dependent
variable
Independent variable
Continuous
Categorical
Continuous
Regression
ANOVA
Categorical
Logistic regression
Tabular
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)
Four classes of experimental design
Dependent
variable
Independent variable
Continuous
Categorical
Continuous
Regression
ANOVA
Categorical
Logistic regression
Tabular
Regression designs
• Single-factor regression
• Multiple regression
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)
Plot #
Rodents/m2
Seeds
1
50
3.2
2
12
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n
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5.3
Variables
Plots
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
In regression (Model I)
• You are assuming that the value of the
independent variable is known exactly
and is not subject to measurement error
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
Multiple regression
• Two or more continuous predictor variables
are measured for each replicate, along with
the single response variable
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
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
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
Multiple regression
• It is a mistake to
think that a model
selection algorithm
can identify reliably
the correct set of
predictor variables
Four classes of experimental design
Dependent
variable
Independent variable
Continuous
Categorical
Continuous
Regression
ANOVA
Categorical
Logistic regression
Tabular
ANOVA designs
• Analysis of Variance
• Treatments: refers to the different
categories of the predictor variables
• Replicates: each of the observations made
ANOVA designs
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Single-factor designs
Randomized block designs
Nested designs
Multifactor designs
Split-plot designs
Repeated measurements designs
BACI designs
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)
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
Id #
Treatment
Replicate
Number of flowers
1
Watered
1
9
2
Not watered
1
4
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11
Watered
6
10
12
Not watered
6
2
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
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
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
Id # Treatment
Block Number of flowers
1
Watered
1
9
2
Not watered
1
4
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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
Randomized block designs
Valid blocking
Invalid blocking
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 layout
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 nonindependence 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
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
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
Id #
Treatment
Subsample
Replicate
Number of flowers
1
Watered
1
1
9
2
Watered
2
1
4
3
Watered
3
1
7
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19
Not watered
1
7
16
20
Not watered
2
7
10
21
Not watered
3
7
2
Advantages
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Subsampling increases the precision of
the estimate for each replicate in the
design
Allows to test two hypothesis:
1. First: Is there variation among treatments?
2. Second: Is there variation among replicates
within treatments?
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Can be extended to a hierarchical
sampling design
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
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
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.
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
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
Interactions
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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
Two-factor design
Substrate treatment
Granite
Predator
treatment
Unmanipulated
Cage Control
Predator
exclusion
Predator
intrusion
Slate
Cement
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
Disadvantages
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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
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
Split plot design
Substrate treatment
The subplot factor
Granite
Unmanipulated
Predator
treatment
The whole- Control
plot factor
Predator exclusion
Predator intrusion
Slate
Cement
Advantages
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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
Disadvantages
•
As with nested
designs, a very
common mistake is
for investigators to
analyze a split-plot
design as a two
factor ANOVA
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)
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
Advantages
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Efficiency
It allows each replicate to serve as its
own block or control
It allows us to test for interactions of time
with treatment
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
For repeated measures design it means that the variance
of the difference of observations between any pair of times
is the same
Disadvantages
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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
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
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
Four classes of experimental design
Dependent
variable
Independent variable
Continuous
Categorical
Continuous
Regression
ANOVA
Categorical
Logistic regression
Tabular
Tabular designs
• The measurements of these designs are
counts
• A contingency table analysis is used to
test hypotheses