PP 17

What is blocking in experimental design?

Grouping experimental units into blocks (usually space or time) and applying each treatment level to one unit within each block.

What is a randomized complete block (RCB) design?

Treatment levels are randomly assigned to separate experimental units within each block. Randomization is restricted to units within blocks.

What is the main difference between RCB and repeated measures (RM) designs?

RCB applies treatments to different units within blocks; RM applies treatments successively to the same subjects.

In a repeated measures design, what are subjects?

The blocks that receive treatments successively or are measured repeatedly over time.

What are within-subjects effects?

Effects of the treatment or time measured repeatedly within the same subjects.

What are between-subjects effects?

Effects of differences among subjects (the blocks).

What is compound symmetry?

The assumption that variances and covariances among treatment responses are equal.

What does homogeneity of variances mean in repeated measures?

Variances among the different treatment or time responses are equal.

What does homogeneity of covariances mean?

Covariances between different treatment or time responses are equal.

What is sphericity?

The variances of differences between responses within subjects are equal for all pairs of treatments.

What is the difference between compound symmetry and sphericity?

Compound symmetry requires equal variances and covariances; sphericity only requires equal variances of pairwise differences.

How do you test for a linear trend over time using repeated measures?

Use weights −2, −1, 0, +1, +2; multiply each time mean by its weight and sum.

What does a linear trend test tell us?

Whether responses show an overall increase or decrease over time.

How do you test for a quadratic trend over time?

Use weights +2, −1, −2, −1, +2; multiply each time mean by its weight and sum.

What does a quadratic trend test tell us?

Whether responses show curvature, such as increasing then decreasing or decreasing then increasing.

What is required when using weighted trend tests for time?

Time intervals must be equally spaced.

How do you determine whether a linear or quadratic trend is significant?

Calculate weighted sums for each subject and test whether their mean differs significantly from zero.

What are the advantages of using mixed effects models for repeated measures?

Model subject variability, handle unbalanced data, allow flexible covariances/autocorrelation, and accommodate irregular measurement times and continuous time.

How does treating time as continuous differ from treating time as categorical?

Categorical time tests whether date-specific means differ; continuous time tests for linear or curved temporal trends.

What is a random intercepts model?

A model allowing subjects to have different starting response levels (vertical shifts).

When must we use a random intercepts model?

When treating Time as a categorical variable.

When can we use random intercepts OR random slopes and intercepts?

When treating Time as a continuous variable.

What variance-covariance structure does a random intercepts model implicitly assume?

Compound symmetry: equal variances and equal covariances among time points.

Why is sphericity not adequate when using a random intercepts model?

A random intercepts model specifically assumes compound symmetry, not just sphericity.

What is autocorrelation in repeated measures?

When nearby observations are more alike or correlated than observations farther apart.

What is autoregressive decay?

Observations closer together in time are more strongly correlated than observations farther apart.

How does autoregressive decay affect compound symmetry?

It violates compound symmetry because covariances decrease as the time between observations increases.

What happens when correlated errors violate compound symmetry?

Statistical power for tests of fixed effects becomes inflated.

How can mixed effects models account for autoregressive decay?

Use an AR(1) first-order autoregressive error structure.

What does AR(1) model?

Within-subject residuals are more strongly correlated for measurements taken closer together in time.

What is the difference between random intercepts and random slopes?

Random intercepts account for vertical shifts; random slopes account for different rates of change.

What does adding random slopes do to the compound symmetry assumption?

Relaxes the assumption of equal covariances among observations over time.

Do random slopes eliminate autoregressive decay?

No. AR(1) can still be used alongside random slopes and intercepts.

What is a Latin square design?

A design with two blocking factors (rows and columns), with one experimental unit per row-column combination.

What is the main restriction of a Latin square design?

Each treatment must occur exactly once in every row and every column.

What determines the replication level in a Latin square?

The number of treatments.

Why might a Latin square be replicated?

Small numbers of treatments limit degrees of freedom and statistical power.

What is a split plot design?

A blocked factorial design where one factor is applied between blocks and another factor is applied within blocks.

What are whole plot or between plot effects?

Treatment effects applied at the level of entire blocks or plots.

What are split plot or within plot effects?

Treatment effects applied to experimental units within each block or plot.

Why are split plot designs used?

Some treatments are more efficiently applied at larger scales, requiring a hierarchical blocking structure.

What is an example of a treatment applied at the whole plot level?

Irrigation, fertilizer, or pesticide applied to an entire plot.

In the caddisfly experiment, what was Factor A?

Hydroperiod: permanent versus ephemeral ponds (between plot factor).

In the caddisfly experiment, what was Factor B?

Pond identity nested within hydroperiod (random effect).

In the caddisfly experiment, what was Factor C?

Competition treatments randomly applied within each pond (within plot factor).

What is the general split plot linear model?

In the split plot model, what does μ represent?

The overall constant mean of the response variable.

In the split plot model, what does αᵢ represent?

The effect of Factor A, the between plot factor.

In the split plot model, what does βⱼ₍ᵢ₎ represent?

The nested plot effect (Factor B), which is almost always random.

In the split plot model, what does γₖ represent?

The effect of Factor C, the within plot factor.

In the split plot model, what does αγᵢₖ represent?

The interaction between the between plot factor A and within plot factor C.

In the split plot model, what does βγⱼ₍ᵢ₎ₖ represent?

The interaction between within plot factor C and nested plot factor B.

What does the residual term ε represent in the split plot model?

Random or unexplained error associated with an individual observation.

In a classic split plot ANOVA, which factors are assumed fixed and random in this lecture?

Factors A and C are fixed; the nested effect of B is random.

What is the F-ratio for testing the between plot effect A?

What is the F-ratio for testing the nested plot effect B(A)?

What is the F-ratio for testing the within plot effect C?

What is the F-ratio for testing the interaction A × C?

What is the F-ratio for testing B(A) × C?

What does the between plot effect A represent in the caddisfly experiment?

Effects of hydroperiod (permanent vs. ephemeral), averaging across competition treatments.

What does the nested effect B(A) represent?

Differences among individual ponds nested within each hydroperiod, averaging across competition treatments.

What does the within plot effect C represent?

Effects of the six competition treatments within ponds, averaging across hydroperiods.

What does the interaction A × C represent in a split plot design?

Whether the effect of the within plot treatment C changes depending on the level of between plot factor A.

What would an A × C interaction look like in a figure?

Non-parallel lines showing that treatment effects change depending on the between plot treatment.

What does the B(A) × C interaction represent?

Whether the effects of treatment C depend on the identity of the individual plot nested within A.

When can the B(A) × C interaction be tested?

Only when there are replicate observations of each C treatment within plots; otherwise, it is ignored.

What is a repeated measures design with between-subjects effects?

Subjects are randomly allocated to treatment groups, and responses are recorded repeatedly over time within subjects.

How is a repeated measures design with between-subjects effects analyzed?

Using the classic split plot framework.

In repeated measures with between-subjects effects, what is Factor A?

The between-subjects treatment factor.

In repeated measures with between-subjects effects, what is Factor B?

Subjects nested within levels of A; almost always random.

In repeated measures with between-subjects effects, what is Factor C?

Time, the within-subjects factor.

What does A × C represent in repeated measures with between-subjects effects?

The interaction between the between-subjects treatment and time.

What does B(A) × C represent in repeated measures with between-subjects effects?

The interaction between time and individual subjects nested within treatment groups.

How do the ANOVA table and F-ratios for repeated measures with between-subjects effects compare to classic split plot designs?

They are identical.

In the tree respiration experiment, what was Factor A?

Tree species (between-subjects factor).

In the tree respiration experiment, what was Factor B?

Individual trees nested within species.

In the tree respiration experiment, what was Factor C?

Temperature measurements (10, 20, 30, and 40°C) within subjects.

What does the between-subjects effect A represent in the tree experiment?

Differences among species, averaging across all temperatures.

What does B(A) represent in the tree experiment?

Differences among individual trees within each species.

What does the within-subjects effect C represent in the tree experiment?

Effects of temperature, averaging across all species.

What does the interaction A × C represent in repeated measures?

Whether the effect of time or temperature changes depending on the between-subjects treatment.

What does an A × C interaction look like when there is NO interaction?

Parallel response lines across the time or temperature levels.

What does an A × C interaction look like when there IS an interaction?

Non-parallel response lines, indicating that responses change differently among groups.

What is a crossover interaction?

When the effects of the between-subjects groups reverse over time.

What happens to species effects in a crossover interaction?

The relative effects of species reverse over time.

Why can interactions make other factor effects difficult to interpret?

Interactions can obscure main effects, making p-values of between-subjects effects questionable.

When can the B(A) × C interaction be tested in repeated measures?

Only when there are replicate observations of each C treatment per date.

What assumptions apply to between plot/subjects effects?

Homogeneity of variances and normality of error terms for tests of Factor A.

What is the independence assumption for treatments within blocks or subjects?

Treatments within blocks must be independent and must not affect each other.

What are spillover and carryover effects?

Spillover occurs in split plots; carryover occurs in repeated measures. Both mean one treatment affects another.

What assumption applies to tests involving within plot/subjects factors?

Sphericity or compound symmetry of the variance-covariance structure.

Which effects require the sphericity or compound symmetry assumption?

Tests of C and A × C.

What corrections can be used when sphericity is violated?

Greenhouse-Geisser or Huynh-Feldt corrections using epsilon.

What does an epsilon estimate far from 1 indicate?

A greater deviation from sphericity.

If given a blocked experimental design, what should you be able to identify?

Which effects are between plot/subjects effects and which are within plot/subjects effects.

If a two-way factorial design is converted into a split plot design, what changes?

One factor becomes a between plot effect and the other becomes a within plot effect.