Exercise.13 Formation of ANOVA table for Latin square design (LSD) and comparison of means using critical difference values

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1 xercise.13 Formation of NOV table for Latin square design (LS) and comparison of means using critical difference values Latin Square esign When the experimental material is divided into rows and columns and the treatments are allocated such that each treatment occurs only once in each row and each column, the design is known as L S. In LS the treatments are usually denoted by etc. For a x LS the arrangements may be Square 1 Square 2 Square 3 nalysis The NOV model for LS is Y ijk µ + r i + c j + t k + e ijk r i is the ith row effect c j is the jth col effect t k is the kth treatment effect 1

2 The analysis of variance table for LS is as follows: Sources of Variation egrees of Freedom Sum of Squares Mean Squares F-ratio Rows t-1 RSS RMS RMS/MS olumns t-1 SS MS MS/MS Treatments t-1 TrSS TrMS TrMS/MS rror (t-1)(t-2) SS MS Total t 2-1 TSS F table value F [t-1),(t-1)(t-2)] degrees of freedom at % or 1% level of significance Steps to calculate the above Sum of Squares are as follows: orrection Factor (F) Total Sum of Squares (TSS) Row sum of squares (RSS) olumn sum of squares (SS) Treatment sum of squares (TrSS) rror Sum of Squares TSS-RSS-SS-TrSS These results can be summarized in the form of analysis of variance table. alculation of S, S(d) and values 2

3 S where r is the number of rows S(d) x S t x S(d) where t table value of t for a specified level of significance and error degrees of freedom Using value the bar chart can be drawn and the conclusion may be written. Problem elow are given the plan and yield in kgs/plot of a x Latin square experiment on the wheat crop carried out for testing the effects of five, manorial treatments,,,, and. denotes control R R R R R , 2 72, 3 82, 4 77, 80 ; GT 393 nalyze the data and state your conclusions. nalysis 1. orrection factor where GT is the grand total r is number of rows, and c is number of columns 3

4 2. Total SS F SS due to rows (SSR) 4. SS due to columns (SS) To get SS due to treatments, first find the totals for each treatment using the given data as follows: Treatment () T 1 4 T 2 8 T 3 88 T T SS due to treatments SS due to error TSS SSR SS SST

5 7. Table for analysis of variance Source of variation f SS MS Variance ratio F Rows olumns Treatments rror Total ** Highly significant F value at % level & 1% level ** The observed highly significant value of the variance ratio indicates that there are significant differences between the treatment means. S.. of the difference between the treatment means (S) r where MS indicates the error mean square and indicates the number of replications. i.e. Sd ritical difference Sd x t % at df x Summary of results 1.33 totals. Treatment means will be calculated from the original table on treatment Treatments % Mean yield in kgs / plot onclusion represented symbolically The treatment have been compared by setting them in the descending order of their yields. Treatments :

6 Mean yields In kgs/plot The treatment is the best of all. The treatments and do not differ significantly each other. The yield obtained by applying every one of the manorial treatment is significantly higher that obtained without applying any manure. Learning xercise 1. n oil company tested four different blends of gasoline for fuel efficiency according to a Latin square design in order to control for the variability of four different drivers and four different models of cars. Fuel efficiency was measured in miles per gallon (mpg) after driving cars over a standard course. Fuel fficiencies (mpg) For 4 lends of Gasoline (Latin Square esign: lends Indicated by Letters -) ar Model river I II III IV These data are from Ott: Statistical Methods and ata nalysis, 4th ed., uxbury, 1993, page 8. (Similar data are given in the th edition by Ott/Longnecker, in problem 1.10, page 889.) nalyse the data and draw yor conclusion. 2. The numbers of wireworms counted in the plots of Latin square following soil fumigations (L,M,N,O,P) in the previous year were olumns P(4) O(2) N() L(1) M(3) M() L(1) O() N() P(3) O(4) M(8) L(1) P() N(4) N(12) P(7) M(7) O(10) L() L() N(4) P(3) M() O(9) nalyse the data and draw your conclusions. Rows 3. The following layout presents the observations made on treatments,,, and in an experiment of paddy crop by adopting LS. The figures indicate the grain yield of paddy in kg/plot. nalyse the data and draw your conclusion. Rows olumns

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