A Bestiary of Experimental and Sampling Designs

The design of an experiment 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) Abscissa (horizontal x-axis)

Four classes of experimental design Dependent variable Independent variable ContinuousCategorical Continuous RegressionANOVA Categorical Logistic regressionTabular

The analysis of covariance 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 ContinuousCategorical Continuous Regression ANOVA Categorical Logistic regressionTabular

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 variables) is responsible for rodent density (the response variable)

Plot #SeedsRodents/m 2 Vegetation cover n 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 In some cases, is useful to ensure that the distribution of predictor values is approximately uniform within the sample range, but…

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. This collinearity makes it difficult to estimate accurately regression parameters and to tease apart how much variation in the response variable is accurately 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 ContinuousCategorical Continuous Regression ANOVA Categorical Logistic regressionTabular

An alysis o f Va riance 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 ANOVA designs

Single factor designs It is one of the simplest, but most powerful, experimental designs, and it 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 #TreatmentReplicateNumber of flowers 1Watered19 2Not watered Wateredn10 nNot wateredn2

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

One effective way to incorporate environmental heterogeneity 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 #TreatmentBlockNumber of flowers 1Watered19 2Not watered14....WateredN10 nNot wateredN2

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 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 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 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 estimate the response of each replicate

Id #TreatmentSubsampleReplicateNumber of flowers 1Watered Not watered Not watered Not watered372

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 solution to inadequate replication

Randomized block designs Strictly speaking, the randomized block and the nested ANOVA are two-factor designs, but the second factor (blocks, or sub samples) is included only to control for sampling variation and is not the 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 more cost effective to run a single experiment than to run two separate experiments It allows to test for both main effects and for interaction effects

Interactions st Qtr2nd Qtr3rd Qtr4th Qtr West North

Orthogonal The key element of a proper factorial 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 single-factor design Substrate treatment GraniteSlateCement Predator treatment Unmanipulated 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 two 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 GraniteSlateCement 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 Repeated measurements designs

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

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 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. This design can be treated as a simple factor in a two way 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-way analysis of variance

Think outside the ANOVA Box Experimental regression

Four classes of experimental design Dependent variable Independent variable ContinuousCategorical Continuous RegressionANOVA Categorical Logistic regression Tabular

Tabular designs The measurements of these designs are counts A contingency table analysis is used to test hypothesis