AGR5201 ADVANCED STATISTICAL METHODS Semester 2019/2020 Lecture Experimental design: CRD vs RCBD What is an experiment ? • A planned inquiry to: – Obtain new facts – Confirm or deny the result of previous experiments • Results from experiment will aid in administrative decision: – Recommending a variety, procedure or pesticide Basic elements in experimental design • • • • Experimental unit (E.U) Treatment Replications Randomization Experimental unit (E.U) • Experimental unit: • Examples: – The unit of material which one application of treatment is applied – An animal (draw blood sample from an animal) – 10 chickens in a cage (fed on the same diet) – A row of corn (10 plants/row) – A pot of bean plants • It can be plant/pot or a few plants/pot Treatment • • Treatment: – A procedure whose effect is to be measured and compared with other treatments Examples: – Rate of nitrogen  kg/ha N, 100 kg/ha N, 150 kg/ha N – Spraying schedule – times a day, times a day, once a day – Different temperature  30oC, 35oC, 40oC, 60oC – The combination of different rate of N and P fertilizer Replication • • When a treatment appear more than once The functions: – To provide an estimate of experimental error – Improve precision  reduce std dev of a treatment mean Randomization • Function: to ensure we have a valid and unbiased estimate of: – Experimental error – Treatment means – Differences among treatment means Experimental error • • Characteristics of experimental material  variations! • Experimental error is the error variance or mean square error (MSE) in ANOVA • Experimental error  a measure of variation that exists among observation on experimental unit (E.U) treated alike Example: – Two pots of bean plants (two E.U) that have been treated in the same treatment produce different yield Variations in experimental units: • Two main sources: Inherent variability of material • The materials are not uniform Lack of uniformity in the physical conduct of experiment • • Different environmental conditions (light received, slope area) Different people measure differently Example • • An experiment was conducted to study the effect of different fertilizer on corn yield Three types of fertilizer were used as treatments and there were four replications in each treatment Corn plants were planted in a 6-meter row After four months of planting, the whole row will be harvested to determine the yield – Experimental unit?  The 6-meter row of corn – Treatment?  three types of fertilizer – # of replications?  four Randomization of treatment in CRD NOT RANDOM 0N • 120 N Replicated but biased RANDOM 100 N 0N 120 N 100 N 0N 120 N 100 N 0N • 100 N 120 N Replicated and randomized 120 N 0N 100 N 100 N 120 N 0N Advantages of a CRD • Flexibility – Any number of treatments and any number of replications – Don’t have to have the same number of replications per treatment (but more efficient if you do) • Simple statistical analysis • • Missing plots not complicate the analysis – Even if you have unequal replication Maximum error degrees of freedom Disadvantage of CRD • Low precision if the plots are not uniform Uses for the CRD • If the experimental site is relatively uniform: – lab – greenhouse Design construction • • • • No restriction on the assignment of treatments to the plots Each treatment is equally likely to be assigned to any plot Should use some sort of mechanical procedure to prevent personal bias Assignment of random numbers may be by: – – – lot (draw a number ) computer assignment (Excel) using a random number table Linear additive model for a CRD Yij = µ + τi + εij Where, Yij = the observation made on the j th experimental unit of the i th treatment µ = the overall population mean τi = effect of the i th treatment (µi - µ) εij = the unexplained portion of the observation made on the j th i treatment, the residual (Xij-µi) th experimental unit of the Hypothesis test (CRD) H0: µ1 = µ2 = = µi = = µt HA: not all µi’s are equal One-way ANOVA (CRD) Source of variation Treatment Degree of Sum of square freedom (df) (SS) t-1 (between groups) r ∑ (Y i − Y ) Mean square (MS) F value Pr < F SSTrt/ dfTrt MSTrt/MSE The Pr of F value i Error (N-1) – (t-1) SSE/dfE ∑ (Y (within group) ij − Y i ) i, j Total N-1 ( Y − Y ) ∑ ij i, j Y Yij Y i  Grand mean  Observation of i th treatment and j  Mean of treatment i th E.U The CRD Analysis We can:  Estimate the treatment means  Estimate the standard error of a treatment mean  Test the significance of differences among the treatment means Problem What can we when experimental units are highly heterogeneous and we can group them based on their heterogeneity?  Use a randomized complete block design (RCBD) Randomized complete block design (RCBD)  What is the difference between CRD and RCBD? CRD RCBD T2R4 T3R1 T2R2 T1R1 T3R2 T2R1 T1R3 T2R3 T1R2 T3R4 T3R2 T1R4 T3B1 T1B2 T1B3 T1B4 T2B1 T2B2 T3B3 T2B4 T1B1 T3B2 T2B3 T3B4 Block Block2 Block3 Block Linear additive model for RCBD Yij = µ + τi + βj +εij Where, Yij = the observation made on the j th experimental unit of the i th treatment µ = the overall population mean th τi = effect of the i treatment (µi - µ) th βj = effect of the j block (µj - µ) εij = the unexplained portion of the observation made on the j th i treatment, the residual (Xij-µi) th experimental unit of the Hypothesis test (RCBD) H0: µ1 = µ2 = = µi = = µt HA: not all µi’s are equal Characteristics of the RCBD When can we use the RCB design? • • • • • • When experimental units can be grouped into blocks And each block is large enough that it can contain all treatments Types of blocks Spatial - physical blocks, locations, laboratories, greenhouses Temporal - repetitions in time, on any scale Animals - size, age, sex, physiological state People - large # of visual ratings to be taken (plant breeding) Two-way ANOVA (RCBD) Source of Degree of freedom variation (df) Treatment Sum of square (SS) r ∑ (Y i − Y ) t-1 Mean square (MS) F value Pr < F SSTrt/ dfTrt MSTrt/MSE The Pr of F Trt SSBlk/dfBlk MSBlk/MSE The Pr of F Blk i Block r-1 t ∑ (Y j − Y ) j Error (N-1) – (t-1) – (r-1) SSE/dfE SST – SSTrt – SSBlk Total (t*r)-1 or N-1 ( Y − Y ) ∑ ij i, j Y Yij Y i Y i  Grand mean  Observation of i th treatment and j  Mean of treatment i  Mean of block j th E.U ... heterogeneity?  Use a randomized complete block design (RCBD) Randomized complete block design (RCBD)  What is the difference between CRD and RCBD? CRD RCBD T2R4 T3R1 T2R2 T1R1 T3R2 T2R1 T1R3 T2R3... the EXPERIMENTAL DESIGN = how you allocate the treatments to the experimental units Experimental design • • • Completely randomized design (CRD) Randomized complete block design (RCBD) Latin square... (SS) t-1 (between groups) r ∑ (Y i − Y ) Mean square (MS) F value Pr < F SSTrt/ dfTrt MSTrt/MSE The Pr of F value i Error (N-1) – (t-1) SSE/dfE ∑ (Y (within group) ij − Y i ) i, j Total N-1 (