Completely Randomised Designs
- randomization and replication
- treatments allocated at random to experimental units
- each treatment repeated equal number of times
- allocation
o by lot
- Analysis
o provides one-way classified data according to levels of single factor (treatment)
o Anova
§ how much variation is due to differences between treatments
§ how much variation is due to differences within each set of observations for same treatment
o linear statistical model
§ y = u + α + ε
· y -- response
· u -- general mean
· α -- effect of treatment
· ε -- error
o error components assumed to be independently and normally distributed.
· total variation partitioned
o variation among treatments (treatment)
o variation among units within treatment units
- Advantages
o flexible design
o simple statistical analysis if some missing data / loss of info small in comparison to other designs
o design provides maximum degrees of freedom for estimate of error variance à which increases precision for small experiments.
o maximum use of experimental units à all experimental material can be used.
- Disadvantages
o if experimental material is not uniform, precision is low.
§ whole variation among experimental units is included in residual variance
- Applications
o physical science laboratory experiments
o greenhouse studies, where experimental material is uniform.
o recommended for situations where large fraction of units likely to be destroyed or fail to respond.
o where total number of units is limited.
Randomised Block Designs
- experimental units sorted into subgroups à where it is believed subgroups will respond differently to treatment
- Analysis
o two way ANOVA
§ blocks , treatment , error
o response = general mean effect (overall mean) + treatment effect + block effect + error
| Source of variation | Degrees of freedom (df) | Sum of Squares | Mean Square | F Statsitic |
| Treatment | p-1 | SST | MST = SST / (p-1) | F = MST / MSE |
| Block | r – 1 | SSB | MSB = SSB / (r-1) |
|
| Error | (p-1) ( r-1) | SSE | MSE = SSE / (p-1))r-1) |
|
| Total | rp - 1 | SStotal |
|
|
p treatments
r blocks
- Advantages
o design more efficient than CRD à improves precision by removing one source of variation from experimental error
o no restrictions on number of treatments or replicates
o any number of blocks can be used so long as each treatment replicated same number of times in each block.
o stat analysis simple and rapid
- Disadvantages
o not suitable for large number of treatments
o efficiency decreases as number of treatments and hence block size increases
o in analysis à missing data can cause some difficulty.
- applications
o provides unbiased estimates of means of blocking categories à additional information obtained
o remove one source of variation
Latin Square Design
- control 2 sources of variation
- rows and columns
- number of treatments = number of replications [?]
- p treatments require p * p experimental units
- [is fig 2.4 on page 14 correct?]
- Analysis
o three way anova
o response = general mean effect (overall mean) + row effect + column effect + treatment effect + error.
| Source of variation | Df | SS | MS | F Statistic |
| Rows | p-1 | SSR | MSR = SSR / (p-1) | Fr = MSR / MSE |
| Columns | p-1 | SSC | MSC = SSC / (r-1) | Fc = MSC / MSE |
| Treatment | p-1 | SST | MST = SST / (p-1) | Ft = MST / MSE |
| Error | (p-1) (p-2) | SSE | MSE = SSE / (p-1)(p-1) |
|
| Total | p^2 -1 | SSTotal |
|
|
- Advantages
o control two sources of variation
o more than one factor can be investigated simultaneously and with fewer trials
o even with missing data, analysis relatively simple.
- disadvantages
o number of treatments restricted to number of replications [?]
o fundamental assumption à there is no interaction between different factors à may not be true
o missing plots – when several units are missing à complex analysis
- Applications
o agriculture / food manufacturing / engineering / dairy farm
Factorial Designs
- CRD / RBD / LSD à primarily concerned with comparison and estimation of effects of a single set of treatments
- factorial à treatments are all the combinations of different factors
o effects of each factor
o interaction effects à variation in one factor as result of different levels of other factors
Nested Designs
- nested or hierarchical design
- advantages
o study effects of sources of variability that manifest themselves over time.
Repeated Measures
- all subjects participate under all levels of the independent variable
- controls for subject heterogeneity
o problem
§ practice effects
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