Experimental designs

Three components of experimental design:

block structure: organisation etc. of experimental units
treatment structure: reflect questions expt set up to answer
experimental design: allocation of treatments to EU(experimental unit)s

Many possible experimental designs, with same two goals:

treatment comparisons as precise as possible
estimates free of bias and inferences valid

Completely randomised design

Blocks: single group Treatments: $t$ treatments, any structure possible Allocation: randomly assign treatment $i$ to $r_i$ EU’s

Randomised complete block design

Block structure: $b$ groups Treatments: $t$ treatments, any structure possible Allocation: randomly assign each treatment to one EU in each block

Model: $y_ij = \mu + t_i + b_j + e_ij$ , $e_ij ~ NID(0, \sigma^2)$ .

Advantages: cost, block means cancel out when comparing treatments Assumptions: no interaction between treatment and block, error distribution same for all treatments and blocks

Test for interaction

Possible to test for interaction using Tukey’s test which detects curvilinear relationship between $y - \hat{y}$ and $\hat{y}$ (transformable non-additivity).

Calculate residuals
Calculate square predicted values $q_ij = (y_{i.} + y_{.j} - y_{..})^2$
Calculate residuals for $q$ ’s
Plot residuals vs. q-residuals. Interaction will show up as linear trend
Calculate $P = \sum(q-resid)(resid)$ , $Q = \sum(q-resid)^2$ , slope = $P / Q$ , $SS_slope = P^2 / Q$
Partition error SS in anova into non-add (slope) and diff, perform F-test

Randomised complete block design with replication

Model: $y_ij = \mu + t_i + b_j + (tb)_ij + e_ij$ , $e_ij ~ NID(0, \sigma^2)$ .

Can relax assumption of no block-treatment interaction (other possibilities: live with it, you less divergent blocks). Have $b$ blocks of $r \times t$ units, each recieving one randomly assigned treatment. Can now separate true replication from interaction, and have more sensitivity for testing significance of differences. However, will be more expensive.

In industrial expt, usually minimise expt’s and maximise factors, assume low variability because well-controlled processes: RCBD with replication not appropriate. In social and life sciences, usually have fewer factors and more error: RCBD with replication useful and appropriate

Repeated measures design

Repeated measures taken on same EU. Has two-way structure similar to RCBD, but subject is not group of EU’s with randomly applied treatments, nevertheless can still be appropriate to use same analysis.

Balanced imcomplete block design

In RCBD designs number of EU’s in a block constrained to be multiple of number of treatments (all treatments replicated in each block). What do we do if block size is smaller than number of treatments? BIBD.

Model: $y_ij = \mu + t_i + b_j + e_ij$ , $e_ij ~ NID(0, \sigma^2)$ . Same model as RCBD, but not all combinations included so need different analysis: use regression model to correct for effects of correlation.

Hidden power: if A & B are paired $n$ times, estimate of their difference is as precise if they were measure $2n$ times! Relies on assumptions of block-treatment independence and careful balancing.

$b$ blocks of $k$ EU’s, $t$ treatments occuring $r$ times, $bk = tr$ . Each pair of treatments occurs in $\lambda = r(k-1) / (t-1)$ blocks Relative efficiency: $e = t\lambda / rk$

When we can model blocks as random effects, block totals can be analysed to provide another set of estimated treatment effects, an inter-block analysis. Indepedent of above analysis (intra-block analysis). Weighted average of the two can be more precise than either set separately.

Latin square designs

Three factors (usually 2 blocking, 1 treatment) with same number of levels. Want to compare treatments removing variability from blocks. RCBD would need 64 runs for factors with four levels, can we make do with less? Latin square only needs 16, one treatment tested for each combination of blocks in such a way that effects can be separated and eliminated.

Model: $y_ij = \mu + t_i + r_j + c_k + e_ijk$ , $e_ij ~ NID(0, \sigma^2)$ Key assumption: no interaction between rows, columns and treatments.

Error term in model ($e_ijk$ ) represents variability if same conditions repeated independently – not true error variance, but can estimate it assuming no interaction. However, because is type of factorial design if we decide one factor is unimportant we can collapse the design and analyse without (sim. to stepwise regression, controversial if overdone). Analyse using standard ANOVA: means can be calculated directly because all other effects will cancel out.