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What ? Why ? How?

What ? Why ? How?. EXPERIMENTAL DESIGN. Experimental Design.

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What ? Why ? How?

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  1. What ? Why ? How?

  2. EXPERIMENTAL DESIGN

  3. Experimental Design Experimental design is a set of rules used to choose samples from populations. The rules are defined by the researcher himself, and should be determined in advance. In controlled experiments, the experimental design describes how to assign treatments to experimental units, but within the frame of the design must be an element of randomness of treatment assignment. It is necessary to define

  4. Experimental Design..

  5. Basic Designs • Completely Randomized Design (CRD) • Randomized Block design (RBD) • Latin Square Design CRD is known as “One-way design”

  6. Designs commonly used in Animal Science • One-way design (no interaction effect) • Fixed effects • Random effects ii) Factorial design (interaction effect)

  7. Some important definitions Treatments : Whose effect is to be determined. For example i)you are to study difference in lactation milk yield in different breeds of cows. ….. Treatment is breed of cows. Breed 1, Breed 2… are levels ii) You intend to see the effect of 3 different diets on the performance of broilers. ….. Treatment is diet and diet1, diet2 and diet3 are levels (1,2,3)

  8. …..definitions Experimental units: Experimental material to which we apply the treatments and on which we make observations. In the previous two examples cow and broilers are the experimental materials and each individual is an experimental unit. Experimental error: The uncontrolled variations in the experiment is called experimental error. In each observation of example(i) there are some extraneous sources of variation (SV) other than breed of cow in milk yield. If there is no uncontrolled SV then all cows in a breed would give same amount of milk (!!!).

  9. …..definitions Replication: Repeated application of treatment under investigation is known as replication. In the example (i) no. of cows under each breed (treatment) constitutes replication. Randomization: Independence (unbiasedness) in drawing sample. Randomization, replication and error control are three principles of experimental design.

  10. Fixed Effects One-way ANOVA

  11. Fixed effects one-way ANOVA • Consider an experiment with 15 steers and 3 treatments (T1, T2, T3) • Following scheme describes a CRD NB: One treatment appeared 5 times. Equal no. of replication/treatment – not necessary in one-way ANOVA

  12. Fixed effects one-way ANOVA.. Data sorted by treatment for RANDOMIZATION T1 T2 T3

  13. Fixed effects one-way ANOVA..

  14. Fixed effects one-way ANOVA.. Model Where Yij = Observation of ith treatment in jth replication = Overall mean ti = the fixed effect of treatment i(denotes an unknown parameter) eij = random error with mean ‘0’ and variance ‘ ‘ The factor or treatment influences the value of observation

  15. Fixed effects one-way ANOVA.. Treatment 1 Treatment 2 Look the difference

  16. Fixed effects one-way ANOVA.. Problem 1: An expt. was conducted to investigate the effects of 3 different rations on post weaning daily gains (g) in 3 different groups of beef calf. The diets are denoted with T1, T2, and T3. Data, sums and means are presented in the following table.

  17. Fixed effects one-way ANOVA..

  18. One-way ANOVA: Hypothesis Null hypothesis Ho: There is no significant difference between the effect of rations on the daily gains in beef calves ie Effects of all treatments are same. Alternative hypothesis Ha: There is significant difference between the effect of rations on the daily gains in beef calves ie Effect of all treatments are not same.

  19. Commonly used level of significances p< 0.05, conf. interval = 95% ; p< 0.01, conf. interval = 99%

  20. Calculation of different Sum of Squares(SS) Total SS = Treatment SS = Error SS = Total SS – Treatment SS = T0-T= E say CF stands for correction Factor

  21. One-way ANOVA Table If the calculated value of F with (k-1) and (N-k) df is greater than the tabulated value of F with same df at 100α % level of significance, then the hypothesis may be rejectedie the effects of all the treatments are not same. Otherwise the hypothesis may be accepted. (N=Total no of observation, k=no of treatments)

  22. One-way ANOVA… • Grand Total (GT) = • CF = • Total Corrected SS = = 1268700 – 1261500 = 7200 4. Treatment SS = • Error SS = Total SS – Treatment SS = 7200-3640 = 3560

  23. ANOVA for Problem 1. The critical value of F for 2 and 12 df at α = 0.05 level of significance is F 0.05 (2,12 )= 3.89. Since the calculated F (6.13) > tabulated F or critical value of F(3.89), Ho isrejected. It means the experiments concludes that there is significant difference (p<0.05) between the effect different rations (at least in two) of calves causing daily gain. Now the question of difference between any two means will be solved by MULTIPLE COMPARISON TEST(S).

  24. Multiple Comparison among Group Means (Mean separation)

  25. Multiple comparison: Least Significant Difference(LSD) test LSD compares treatment means to see whether the difference of the observed means of treatment pairs exceeds the LSD numerically. LSD is calculated by where is the value of Student’s t with error df at 100 % level of significance, s2 is the MS of error and r is the no. of replication of the treatment. For unequal replications, r1 and r2 LSD=

  26. Duncan’s Multiple Range Test(DMRT) Duncan (1995) made , the level of significance a variable from test to test. The Least Significant Range (LSR) is defined by The value of significant studentized range (SSR) is given in Duncan (1955). In case, a pair of means differs by more than its LSR, they are declared to be significantly different.

  27. Random Effects One-way ANOVA: Difference between fixed and random effect

  28. One-way ANOVA, random effect For unbalanced cases n is replaced with

  29. Advantages of One-way analysis(CRD)

  30. Two-way ANOVA Suppose you intend to study the effectiveness of 3 different types of feed in 4 different strains of hybrid broilers. You need to distribute your treatments (3, feed) in a way so that birds of each of the strains (4, blocks) receive each type of feed. Randomization of the samples are to be ensured in an efficient way. Total no. of records = No. of treatments x No. of Blocks x No. of replication (2 in this case) per treatment (3x4x2=24)

  31. Why doing this kind of expt. ? You want to know

  32. Two-way ANOVA

  33. Two-way ANOVA Observations can be shown sorted by treatments and blocks yijkindicates experimental unit ‘k’ in treatment’ i ‘and block’ j ‘

  34. Statistical model in two-way ANOVA i = 1,…,a; j = 1,…,b; k = 1,….,n Where yijk= observation k in treatment i and block j μ= overall mean ti= effect of treatment i βj = effect of block j tβij= the interaction effect of treatment I and block j eijk= random error with mean 0 and variance Ϭ2 a = no. of treatments; b= no. of blocks; n= no. of obs in each treatment x block combination.

  35. Sum of Squares, Degrees of Freedom and Mean Squares in ANOVA

  36. Example: Two-way design Recall that the objective of the experiment previously described was to determine the effect of 3 treatments (T1, T2, T3) on average daily gain of steers, and 4 blocks were defined. However, in this example 6 animals (3x2) are assigned to each block. Therefore, a total of 4x3x2 = 24 steers were used. Treatments were assigned randomly to steers within block.

  37. Example: Two-way design The data are as follows

  38. Two-way: Computations 1. Grand Total = 2. Correction term for the mean = 3. Total SS= 4. Treatment SS=

  39. Two-way: Computations… 5. Block SS = 6. Interaction SS 7. Residual SS =

  40. ANOVA TABLE F value for treatment : F = 4012.79/175.83 = 22.82 F value for interaction: F = 1347.90/175.83 = 7.67

  41. Conclusion The critical value for testing the interaction is F0.05,6,12 = 3.00, and for testing treatments is F0.05,2,12 = 3.89. So at p = 0.05 level of significance, H0 is rejected for both treatments and interaction. Inference: There is an effect of treatments and the treatment effects are different in different blocks.

  42. A practical example of one-way ANOVA Problem: Adjusted weaning weight (kg) of lambs from 3 different breeds of sheep are furnished below. Carry out analysis for i) descriptive Statistics ii) breed difference. Suffolk: 12.10, 10.50, 11.20, 12.00, 13.20, 10.90,10.00 Dorset: 11.50, 12.80, 13.00, 11.20, 12.70 Rambuillet: 14.20, 13.90, 12.60, 13.60, 15.10, 14.70, 13.90, 14.50

  43. Analysis by using SPSS 14Descriptive Statistics

  44. ANOVA (F test) a) ANOVA

  45. Mean Separation Post hoc tests Homogenous subsets Wean Duncan

  46. Interpretation of results

  47. You are going to be an Animal Scientist!!!! Do you know Statistics????? Booo----

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