1. Statistics in Science  Statistical Analysis & Design in Research Structure in the Experimental Material PGRM 10

2. Statistics in Science  Blocking – the idea Detecting differences between treatments depends on the background noise (BN) BN is: –caused by inherent differences between the experimental units –measured by the residual (error) mean square RMS (alternatively! MSE) Comparing treatments on similar units would reduce background noise With blocks of units of differing contributing characteristics we measures the variation due to blocks and reduce residual variation

3. Statistics in Science  Blocking – the benefit Reducing background noise: Gives more precise estimates Allows a reduction in replication, without loss of power (the probability of detecting an effect of a specified size) Reduces cost!

4. Statistics in Science  Blocking and experimental material Examples 1.A field: with fertility increasing from top to bottom With 3 treatments group plots into BLOCKS of 3, starting at top and continuing to bottom. Randomise treatments within each block

5. Statistics in Science  Block Design How many replicates per treatment? What is the experimental unit? What is the block?

6. Statistics in Science  Example 2 drugs (A, B) to control blood pressure 100 subjects – randomly assign 50 each to A and B Valid - but is it efficient? If subjects are heterogenous - likely to be a large variation ( 2 ) in the responses within each group. Design may not be very efficient.

7. Statistics in Science  Factors affecting BP variation

8. Statistics in Science  Blocking and experimental material 1.100 subjects are selected to compare new drug to control BP with a Control Block into pairs by age & weight (believed to affect BP) In each pair one is selected at random to receive the new drug, the other receives Control Alternatively – see next slide

9. Statistics in Science  Groups (Blocks)

10. Statistics in Science  Groups (Blocks)

11. Statistics in Science  Blocking and experimental material Examples 1.A field: with fertility increasing from top to bottom With 3 treatments group plots into BLOCKS of 3, starting at top and continuing to bottom. Randomise treatments within each block 2.100 subjects are selected to compare new drug to control BP with a Control Block into pairs by age & weight (believed to affect BP) In each pair one is selected at random to receive the new drug, the other receives Control 3.3 products to be compared in 15 supermarkets: All 3 compared in each supermarket, regarded as BLOCKS

12. Statistics in Science  Blocking and experimental material Examples (contd) 4.A crop experiment will take 5 days to harvest. The material is blocked into 5 sets of plots, and treatments assigned at random within each set A BLOCK of plots is harvested each day Here: day effects, such as rain etc will be allowed for in the ANOVA table, not clouding the estimation of treatment effects, and reducing residual variation.

13. Statistics in Science  Blocking factors in your work area?

14. Statistics in Science  Reasons to BLOCK 1.Reduce BN (as above) 2.Material is naturally blocked (eg identical twins) so using this a part of the design may reduce BN 3.To protect against factors that may influence the experimental outcomes, and so cloud comparison of treatments 4.To assess block variation itself eg day to day variation large may indicate a process that is not well controlled.

15. Statistics in Science  Typical Randomised Block Design (RBD) Layout Block 1T3T1T2T4 2T2T3T1T4 3T1T2T3T4 4T2T4T1T3 5T4T2T3T1 6T3T1T4T2 4 treatments T1 – T4  BLOCKS of size 4 Example of random allocation within blocks :

16. Statistics in Science  ANOVA table SourceDFSSMSFPr > F Treatmentst – 1TSSTMSTMS/RMSSmall? Blocksb – 1BSSBMSBMS/RMSSmall? Residual(t-1)(b-1)RSSRMS Totaltb - 1 each treatment occurs once in each block t treatments b blocks tb experimental units MS = SS/DF

17. Statistics in Science  Example PGRM pg 10-2 Compare effect of washing solution used in retarding bacterial growth in food processing containers. Only 3 trials can be run each day, and temperature is not controlled so day to day variability is expected. BLOCKS: day Treatments: 2%, 4%, 6% of active ingredient Randomisation: 3 containers randomly allocated to 3 treatments on each of 4 days. Response: bacterial count on each container each day (low score = cleaner)

18. Statistics in Science  Example (contd) DaySolution(%)Count 1213 1410 165 2218 2420 266 3218 3417 367 4230 4431 4610 Day,Solution(%),Count 1,2,13 1,4,10 1,6,5 2,2,18 2,4,20... Note:  Response values in a single column  Extra column to identify BLOCK (day) TREATMENT (solution) csv ExcelExcel

19. Statistics in Science  SAS GLM code proc glm data = randb; class solution day; model score = solution day; lsmeans solution; lsmeans day; estimate ‘2-6’ solution 1 0 -1; estimate ‘linear ok?’ solution 1 -2 1; quit;

20. Statistics in Science  GLM OUTPUT: ANOVA 425.17 + 322.92 = 748.09 So the Model SS has been partitioned into TREATMENT (solution) and BLOCK (Day)

21. Statistics in Science  GLM OUTPUT: means

22. Statistics in Science  ANOVA table

23. Statistics in Science  More Blocking – Latin square designs

24. Statistics in Science  Latin Square design – blocking by 2 Sources of variation Variation in milk yield among cows is large (CV% = 25) Variation in Yield across lactation is large Use different treatments in sequence on each cow Need to allow for a standardisation period (1- 2) weeks between treatments

25. Statistics in Science  Data Columns for period,cow and treatment codes

26. Statistics in Science  SAS GLM code proc glm data = latinsq; class period cow treat; model yield = period cow treat; lsmeans treat; lsmeans period; lsmeans cow; estimate ‘1v2’ treat 1 -1 0 0 ; Run;

27. Statistics in Science  Results Means Cow and Period removed much variation

28. Statistics in Science  Conclusions on Latin square design CV greatly reduced to 6% - When the effect of period is allowed for, repeated measurements within a cow are not very variable. Periods and cows are nuisance variables. Sometimes the row and column variables are of interest in themselves and so design is very efficient – information on 3 factors. (e.g. treatments, machines, operators). Useful for screening but questionable whether short term results would apply for the long term.