AGRI 2400 Lecture 25-26 - RBC ANOVA

Randomized Block Designs

• blocks (blocking variables) can be delineated around any suspected or known source of variation in time or space

• within each block, experimental units (replicates) are randomly selected and randomly assigned experimental treatments

Why Include a Block Factor

• to account for known or expected variation that would otherwise make it more difficult to detect patterns of association between your variables of interest

  • e.g. a study of pathogen risk for canola based on precipitation may block by soil type

• sometimes blocking is required due to operational constraints

RBC Design Simplest Case

• where every block includes one replicate of each treatment level of your indep. variable of interest

RBC Example Experiment

• A, B, C, D correspond to 4 different fertilizers types (indep. variable)

• field = block

• dependent variable = canola yield

  • field 1 = A, B, C, D

  • field 2 = A, D, C, B

  • field 3 = C, A, B, D

RBC ANOVA Steps

• calculate among groups (treatment) SS and df

• calculate among groups (block) SS and df

• calculate within groups (residual) SS and df

• (optional) calculate total SS and df

• calculate MS (three this time)

• construct F-ratios (two this time)

• determine p-values (two this time)

Constructing the F-Ratios

• in a RBC design there are 2 sets of hypotheses that can be tested, and thus 2 F-ratios

• you may not care about the p-value resulting from the block, however

RBC ANOVA Assumptions

• each sample is randomly selected and independent

• ratio or interval scale measurement of dependent variable

• residuals are nomally distributed

• equal (homogeneous) variances among treatment groups

• no outliers

• additivity between blocks and treatments

Did Randomization Occur?

• random allocation of experimental units to treatments within blocks

• randomized sequence of while measuring each experimental unit

• measurements made for one experimental unit must not have an influence on the measurements taken on another

Beware of Pseudo Replication

• extremely common to mistaken use observation unit instead of experimental unit whenever subsampling occurs

• observational units (subsamples) are generally not independent

Additivity Between Treatments and Blocks

• this assumption states that the difference in the mean response between any 2 treatments is the same in all blocks

• the overall mean of the responses from each treatment may vary among blocks, but the differences must be constant

• the assumption of additivity means there should be no interaction between treatments and blocks

What is an Interaction?

• when the effect of one indep. variable on the dep. variable depends on the state (value) of a second indep. variable (i.e. state dependence)

• if an interaction is present, and there is only one replicate per treatment per block, you would not proceed with the analysis since it will result in misleading or incorrect results

• access by producing an interaction plot

The ‘Cost’ of Blocks

• including a blocking term reduces the df available for the SSwithin groups and therefore it reduces the power of the test, unless the block effect is meaningful

• blocks should be only used when there is a known or likely source of variation that the need to account for

  • e.g. field plots