Basic Concepts of Experimental Design & Standard Design ( Statistics )

This gives the basic description of Design and Analysis of Experiment . This is one of the most important topic in Statistics and also for Mathematics and for Researchers-Scientists

1. Basic Concepts of Experimental Design & Standard Design Presented by HASNAT ISRAQ ISLAMIC UNIVERSITY, BANGLADESH

2. Contents Introduction of experimental design Basic principles of experimental design Analysis of Variance Linear statistical model Fixed effect model Random effect model Mixed effect model Standard design Completely Randomized design(CRD) Randomized Block Design(RBD) Latin Square Design(LSD) COMPARISION BETWEEN CRD , RBD & LSD

3. Introduction of Experimental Design In experimental design experiment is the planned research conducted for certain period to obtain new facts or to confirm or refute the results of previous study. And here the term of the experimental design refers to a plan for assigning experimental units to treatment conditions in a systematic way so that the results can give meaningful output. Purposes of experimental design : Obtain maximum information from fewest experiments & minimize time spent in gathering data. Quantify effects from different factors using analysis. Determine if a factor’s effect is significant.

4. Introduction.. Requirement for a good experiments Absence Of Bias or free from systematic error Absence Of Bias or free from systematic error Precision In experimental design there are two types of data: One way classified data Two way classified data Choosing a set of treatment Selection of experimental unit Specification of the number of experimental units Allocating the treatments to the experimental units Specification of the measurements Specification of the grouping of experimental unit Experimental design consists of these steps :

5. Basic principle of experimental design According to Prof. R.A. Fisher , there are three types of basic principle of experimental design. They are given below : Replication : Replication means repetition of basic treatments under investigation. For example , A farmer wants to know whether new type of fertilizer will give him better yields. He will frame his investigation in terms of some suitable hypothesis . Randomization : Randomization is the process of distributing the treatments to the experimental unit purely by chance mechanism. Local control : Local control is the procedure of reducing and controlling error variation by arranging the experimental units in blocks.

6. Analysis of Variance Analysis of variance is the separation of variance ascribable to one group of causes from the variance ascribable to other group. It is also denoted by ANOVA . Purpose of ANOVA : It identifies the causes of variation and sort out corresponding components of variation with associated degrees of freedom . It provides for test of significance based on F distribution . Assumption of ANOVA : Observations and errors are independently distributed. Observations and errors conform to normal distributions with equal variances. Treatment and environmental effects are additive ..

7. Source of variation DF SS MS F Between classes k - 1 Σ ᵢ r ᵢ ( ȳᵢ - ȳ)² = Within classes n - k Σ ᵢ Σ j r ᵢ ( y ij - ȳ)² = Source of variation DF SS MS F Between classes k - 1 Σ ᵢ r ᵢ ( ȳᵢ - ȳ)² Within classes n - k Σ ᵢ Σ j r ᵢ ( y ij - ȳ)² Anova Table for one-way classification Let, ȳᵢ = The mean value of ith class ȳ = The grand mean r ᵢ = Number of observation n = Total number of observation Degrees of freedom = DF Sum square = SS Mean Square = MS ANOVA TABLE FOR ONE-WAY CLASSIFICATION

8. Source of variation DF SS MS F Between levels of A r - 1 SS(A) = Between levels of B k - 1 SS(B) = Error (r-1)(k-1) SS(E) Source of variation DF SS MS F Between levels of A r - 1 SS(A) Between levels of B k - 1 SS(B) Error (r-1)(k-1) SS(E) Anova Table for one-way classification Let, r = number of observation Degrees of freedom = DF Sum square = SS Mean Square = MS ANOVA TABLE FOR TWO-WAY CLASSIFICATION  

9. LINEAR STATISTICAL MODEL A linear statistical model is a linear function parameters and also involves one or more error terms. Linear statistical model is to classify as , Fixed effect model (FEM) Random effect model (REM) Mixed effect model (MEM) In many situations in the experiment we use these linear statistical model individually .Now we describe these individual model in the next slide.

10. Fixed Effect Model A model in which all the assignable factors have fixed effects and only the error effect is random is called a fixed effect model. yᵢ = β₀ + β₁α ᵢ + Ԑ ᵢ ; where α ᵢ have fixed effect. and Ԑ ᵢ is random. Example of fixed effect model : If we want male or female in a experiment it will be FEM. Also Insecticide sprayed or not. Upland or lowland etc.

11. Random Effect Model (REM) A model in which all the factors have random effects is known as a random effect model .Here the investigator is concerned with the population of levels of each factor of which only a random sample of levels of each factor is present in the experiment. yᵢ = β₀ + β₁α ᵢ + Ԑ ᵢ ; here the factors effect α ᵢ are random variables. Example of Random Effect Model : Individuals with repeated measures Block within a field Split plot within a plot

12. Mixed Effect Model A model in which some factors have fixed effects while others have random effects is known as mixed effects model. yᵢ = β₀ + β₁α ᵢ + β₂ϒ ᵢ + Ԑ ᵢ ; here α ᵢ have fixed effects and ϒ ᵢ have mixed effects. Thus an experiment in which it seems reasonable to regard one effect as fixed and other effect as random is presented by a mixed effect model.

13. Standard Designs Various types of experimental design are employed in experimental studies. There are some basic experimental design such as, Completely Randomized design(CRD) Randomized Block Design (RBD) Latin Square Design (LSD)

14. Completely Randomized Design In completely randomized design, all treatments are randomly allocated among all experimental subjects. This allows every experimental unit, i.e., plot,animal , soil sample, etc., to have an equal probability of receiving a treatment. This is suitable for only the experiment material is homogenous.( ex:laboratory experiments,green house studies etc ) = + τ ᵢ + ; = observation , τ ᵢ = Effect of the ith treatment.  

15. Source of variation DF SS MS F Treatments k - 1 Σ ᵢ r ᵢ ( ȳᵢ - ȳ)² = Error n - k Σ ᵢ Σ j r ᵢ ( y ij - ȳ)² = Source of variation DF SS MS F Treatments k - 1 Σ ᵢ r ᵢ ( ȳᵢ - ȳ)² Error n - k Σ ᵢ Σ j r ᵢ ( y ij - ȳ)² Anova Table for data of CRD Let, ȳᵢ = The mean value of ith class ȳ = The grand mean r ᵢ = Number of observation n = Total number of observation Total n - 1 ANOVA TABLE FOR CRD

16. Randomized Block Design (RBD) Experimental units are grouped into blocks according to know or suspected variation which is isolated by the blocks. Subject are sorted into homogeneous groups, called blocks and the treatments are then assigned at random within the blocks. Most widely used experimental designs in agricultural research. The design also extensively used in the fields of biology, medical, social sciences and also business research. = + β ᵢ + τ j + ; = observation , β ᵢ = Effect of the ith treatment, τ ᵢ = Effect of the jth treatment.  

17. Source of variation DF SS MS F Blocks r - 1 k Σ ᵢ r ( ȳᵢ. - ȳ)² = = Treatment k - 1 r ( - ȳ) 2 = = Error (r-1)(k-1) - - + ȳ) = = Total rk-1 Source of variation DF SS MS F Blocks r - 1 Treatment k - 1 Error (r-1)(k-1) Total rk-1 Anova Table for data of RBD Let, ȳᵢ. = Mean of ith block ȳ .j = Mean of jth treatment ANOVA TABLE FOR RBD

18. Latin Square Design (LSD) Two Directional heterogeneous unit. Blocks are in rows and column. Rows and columns are equal. LSD is treated as a logical extension of RBD. It is widely used in agriculture , industry , medical and psychological experiments. In general A B C B C A C A B This is a 3 X 3 Latin Square.

19. Source of variation DF SS MS F Rows r - 1 R Columns r-1 C Treatment r-1 T Error (r-1)(r-2) E Total -1 Source of variation DF SS MS F Rows r - 1 R Columns r-1 C Treatment r-1 T Error (r-1)(r-2) E Total Anova Table for data of LSD Let , R = sum square of rows T = Sum square of Treatment C = Sum square of columns E = Sum square of error ANOVA TABLE FOR LSD

20. COMPARISION OF CRD , RBD & LSD CRD RBD LSD CRD is the homogenous experimental unit and have only one block. RBD is the one directional heterogenous experimental unit. LSD is the two directional heterogenous experimental unit. row Block 1 Block 1 Block 2 Block 1 Block 2 Block 3 Block 4 column Here row and column are equal.

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