A Bestiary of Experimental and Sampling Designs

REMINDERS

The goal of experimental design is to minimize the potential “sources of confusion” (Hurlbert 1984):

1. Temporal (and spatial) variability

2. Procedural effects

3. Experimenter bias

4. Experimenter-generated variability (“random error”)

5. Inherent variability among experimental units

6. 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 experimentThe details of:

•Replication •Randomization•Independence

… are these always obvious in biological research? Are they system-dependent?

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)– Species

Continuous variables

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

NOTE: Discrete random variables such as counts are still considered continuous variables because they represent a numerical scale and not a category…

Dependent and independent variables

• The assignment of dependent and independent variables implies a 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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0 5 10 15 20 25 30

Time since fire (years)

Ln

(la

mb

da

)

Y=26.06X-2.99r2=0.355Fstat=25.84

Ordinate (vertical y-axis)

Abscissa (horizontal x-axis)

By convention independent variables are plotted in the x-axis and dependent variables in the y-axis… in this example we are implying that lambda (population growth) depends or is affected directly by time since fire…

Four classes of experimental design

Dependent (response) variable

Independent (predictor) 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.

• e.g. Hypothesis: seed density (the predictor variable) is responsible for rodent density (the response variable).

Plot # Seeds Rodents/m2

1 50 3.2

2 12 11.7

. . .

n 300 5.3

Variables

Plots

• 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.

Single-factor regression

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.

A

B

What is different between these two designs?

Would the conclusions be different?

A

B

What is different between these two designs?

Would the conclusions be different?

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.

These are the same assumptions as for the single-factor regression BUT additionally…

Multiple regression• Ideally, the different predictor variables

should be independent of one another; however 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 collect additional predictor variables on the same replicates than to obtain additional independent replicates.

• Avoid the temptation to measure everything that you can just because it is possible.

• Think about measuring variables that are meaningful for you study system!

Multiple regression

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

Four classes of experimental design

Dependent variable

Independent variable

Continuous Categorical

Continuous Regression ANOVACategorical Logistic regression Tabular

• Analysis of Variance

• Treatments: refers to the different categories of the predictor variables.

• Replicates: each of the observations made.

ANOVA designs

• Single-factor designs

• Randomized block designs

• Nested designs

• Multifactor designs

• Split-plot designs

• Repeated measurements designs

• BACI designs (before-after-control-impact)

ANOVA 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).

• 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

Single-factor designs

Id # Treatment Replicate Number of flowers

1 Watered 1 9

2 Not watered 1 4

. . . .

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 background noise is much stronger than the signal of the treatments, the experiment may have low power, and therefore the analysis may not reveal treatment differences unless there are many replicates.

• 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

Randomized block designs

Valid blocking Invalid blocking

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

. . .

11 Watered 6 10

12 Not watered 6 2

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.

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.

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.

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 (if we analyze them assuming independence is it an example of pseudoreplication)

• 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

. . . . .

19 Not watered 1 7 16

20 Not watered 2 7 10

21 Not watered 3 7 2

Advantages

• 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?

• 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

• 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.

Multifactor designs

• the main effects are the additive effects of each level of one treatment averaged 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.

Interactions

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Which of these graphs are showing interactions between direction (west or north) and quarter (1st to 4th)?

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 and so on…

• If some of the treatment combinations are missing we end with a confounded design.

Two-factor design

Substrate treatment

Granite Slate Cement

Predator treatment

Unmanipulated

Cage Control

Predator exclusion

Predator intrusion

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• 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 designSubstrate treatment

The subplot factor

Granite Slate Cement

Predator treatment

The whole-plot factor

Unmanipulated

Control

Predator exclusion

Predator intrusion

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.

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).

• 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… why do you think this is?

Repeated measurements designs

Advantages

• Efficiency.

• It allows each replicate to serve as its own block or control.

• It allows us to test for interactions between treatments and time.

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 always the same… i.e. there is the same variance between t1 and t2, as between t2 and t3, etc..

For repeated measures design it means that the variance of the difference of observations between

any pair of times is the same

This assumption is unlikely to be met in biological systems because of their temporal memory!

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.

Alternatives

1. To set enough replicates so that a different set is sampled 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 i.e. instead of height at age 0 and height at age 1 use growth…

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.

… we will cover this later on