Experimental Studies. Statistical techniques for Experimental Data. Experimental Designs can be grouped. Experimental Designs can be grouped

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1 Experimental Studies Statistical techniques for Experimental Data Require appropriate manipulations and controls Many different designs Consider an overview of the designs Examples of some of the analyses will come later Important to recognize Categorical versus Continuous variables Experimental Designs can be grouped Dependent versus Independent variables Response versus predictor Y versus X axis Experimental Designs can be grouped Doesn t t include: ANCOVA (analysis of covariance) Both a continuous and an categorical independent variable MANOVA (multivariate ANOVA) Multiple response variables Experimental Designs can be grouped Regression Logistic Regression ANOVA Tabular 1

2 Regression Designs Independent variable measured on a continuous scale If dependent variable is measured on a continuous scale = linear (or nonlinear) regression If dependent variable is measured on an ordinal scale (0/1) then logistic regression Collect data on a set of independent replicates (one dependent [Y] and one independent [X] variable) Observational study: neither variable is manipulated and range of X and Y dictated by natural responses Is density of rodents controlled by the availability of seeds? Sample 20 independent plots with each plot representing a different abundance of seeds In an experimental study the levels of the predictor (X) variable are manipulated You would manipulate seed density and use regression to examine the effects on rodent density Y = a + bx 1 Need to capture the full range of variation in X (predictor) Ensure an even distribution of samples within the natural range of the predictor variable 2

3 Multiple Regression Two or more continuous predictor variables are measured for each replicate In addition to seeds you believe that rodent density is also influenced by vegetation structure Measure rodent density, seed density, vegetation structure Multiple Regression Y = a + bx 1 + cx 2 Need to avoid multicolinearity (need independence in the predictor variables) ANOVA = Analysis of Variance Categorical predictor and continuous response variable Treatments: Different categories of the predictor variable (Species: Moose, Elk, Sheep) Represent different Manipulations (Fertilizer: K, P, none) # of categories = # of s ANOVA Replicate Multiple observations are made for each In most designs, replicates need to be independent of the other replicates within the ANOVA Single-factor Design Each represents variation in a single predictor or factor Each value of the factor that represents a particular is called a level Response of planted seedlings to four levels of fertilizer ANOVA Multi-factor Design Treatments cover two (or more) different factors Application of four levels of Nitrogen and four levels of Phosphorus 4x4 = 16 levels with each applied in various combinations to all replicates of a 3

4 ANOVA Multi-factor Design Effects in ANOVA Main effects The additive effects of each level of one averaged over all of the levels of other s Effect of N averaged over the responses to P levels Effects in ANOVA Interaction effects The unique response of particular combinations Interaction between browsing repellent and a fertilizer Need to examine often leads to use of Factorial Designs Single Factor ANOVA One of simplest, but most powerful designs One-way layout Compare means among two or more groups One-way Layout Place out replicates of 3 types of tile, randomly in an inter-tidal tidal zone Return after fixed time to examine barnacle recruitment One-way Layout Can handle unequal number of replicates ( unbalanced ) Tests for differences among s 4

5 One-way Layout Does not accommodate environmental heterogeneity ( noise or in this case other factors present in the tide pool) Noise: Y ij = μ + A i + ε ij One-way Layout Randomized Block Design Delineate (or block ) the s along a temporal or environmental gradient Each block contains exactly one replicate of the Randomized Block Design Each block relative small to ensure uniform environmental effect within the block Efficient way to control for environmental variability Randomized Block Design Randomized Block Design Disadvantages Statistical cost less powerful than simple one-way design Small blocks may affect independence If replicates in a block are lost, remaining data in that block can t t be used Assumes there is not interaction between blocks and s (can t t test for) 5

6 Nested Designs Sub-sampling within each replicate In this case three sub- samples taken from each replicate In other cases may need to use because experimental units are not true replicates (willows growing in a clearcut as opposed to planting out random willows with the ) Nested Designs Advantages Sub-samples increase precision of estimates Test for variation among and within s Nested Designs Disadvantages Incorrectly analyzed as one-way designs (not independent) Where is most appropriate use of sampling effort (among or within s)? Multifactor designs: two-way way layouts Extend one-way principles to two or more factors Assign two or more factors to a in place of only one Factorial Design Factorial Design Two or more factors tested simultaneously in one experiment Fully crossed If every level of one (substrate) occurs with every level of the other (predator access) 6

7 Factorial Design Factorial Design Same design tests for both main effects and interactions of the main effects Substrate * Predator Disadvantage Number of combinations 12 combinations with 10 replicates = 120 total replicates Split-Plot Design Single Plot is split into sub-plots (from agriculture) Each receiving a different Unlike randomized block design, a second is also applied Split-Plot Design Efficient use blocks for the application of two s... Does not allow you to look at interaction effects Repeated Measures Design Repeated Measures Design Multiple observations on an individual are not independent of each other so look at a within-subjects factor Multiple observations on the same replicate are collected at different times 7

8 Repeated Measures Design Intensively studying the behaviour of few individuals over an extended period of time Environmental Impacts over Time Before and After Control Impact (BACI) Form of repeated measures design with several measurements taken before and after the Environmental Impacts over Time If properly used very powerful Help separate individual (site) effects from effects Often not used properly Single site (not randomly chosen) Random versus Fixed Effects What do your s units represent? Random Effects your inference is to the broader population they were taken from (e.g., a biogeoclimatic subzone) Fixed Effects inference is only to those units that were sampled (e.g., specifically chosen licks) Affects the calculation of the F ratio... Tabular Designs Both predictor an response variables are categorical Contingency table analyses 8