LATIN SQUARE DESIGN (LS) -With the Latin Square design you are able to control variation in two directions.

Transcription

1 Facts about the LS Design LATIN SQUARE DESIGN (LS) -With the Latin Squae design you ae able to contol vaiation in two diections. -Teatments ae aanged in ows and columns -Each ow contains evey teatment. -Each column contains evey teatment. -The most common sizes of LS ae 5x5 to 8x8 Advantages of the LS Design 1. You can contol vaiation in two diections.. Hopefully you incease efficiency as compaed to the RCBD. Disadvantages of the LS Design 1. The numbe of teatments must equal the numbe of eplicates.. The expeimental eo is likely to incease with the size of the squae. 3. Small squaes have vey few degees of feedom fo expeimental eo. 4. You can t evaluate inteactions between: a. Rows and columns b. Rows and teatments c. Columns and teatments. Effect of the Size of the Squae on Eo Degees of Feedom SOV Df x 3x3 4x4 5x5 8x8 Rows Columns Teatments Eo (-1)(-) Total Whee numbe of ows, columns, and teatments. -One way to incease the Eo df fo small squaes is to use moe than one squae in the expeiment (i.e. epeated squaes). 1

2 Example Two 4x4 Latin squaes. SOV Df Squaes sq 1 1 * Row(squae) sq(-1) 6 * Column(squae) sq(-1) 6 Teatment -1 3 Squae x Teatment (sq-1)(-1) 3 * Eo sq(-1)(-) 1 Total sq 1 31 *Additive acoss squaes. Whee sq numbe of squaes. Examples of Uses of the Latin Squae Design 1. Field tials in which the expeimental eo has two fetility gadients unning pependicula each othe o has a unidiectional fetility gadient but also has esidual effects fom pevious tials. Gadient. Animal science feed tials. Gadient 1 3. Insecticide field tial whee the insect migation has a pedictable diection that is pependicula to the dominant fetility gadient of the expeimental field. 4. Geenhouse tials in which the expeimental pots ae aanged in a staight line pependicula to the glass walls, such that the diffeence among ows of pots and distace fom the glass wall ae expected to be the majo souces of vaiability. A D C B B C A D D A B C C B D A

3 Randomization Pocedue -Depends on the type of Latin Squae you use. 3x3 Latin Squae -Stat with the standad squae and andomize all columns and all but the fist ow A B C B C A 3 C A B Standad squae Randomize columns 3 1 C A B A B C B C A Randomize all but the fist ow C A B B C A A B C 4x4 Latin Squae -Randomly choose a standad squae. -Randomize all columns and all but the fist ow. 5x5 Latin Squae -Randomly choose a standad squae. -Randomize all columns and ows. 3

4 Analysis of a Single Latin Squae Example Gain yield of thee maize hybids (A, B, and D) and a check (C). Row Column 1 Column Column 3 Column 4 Row ( R ) (B) 1.10 (D) 1.45 (C) (A) (C) (A) (D) 1.90 (B) (A) (C) (B) (D) (D) 1.90 (B) (A) (C) Column total ( C ) Step 1. Calculate teatment totals. Teatment Total A B C 4.70 D Step. Compute the Coection Facto (CF). CF Y Step 3. Calculate the Total SS TotalSS Y ij ( )

5 Step 4. Calculate the Row SS Row RowSS ( ) Step 5. Calculate the Column SS. Col Col. SS ( ) Step 6. Calculate the Teatment SS Y TtSS i. ( ) Step 7. Calculate the Eo SS Eo SS Total SS Row SS Column SS Tt SS

6 Step 8. Complete the ANOVA table SOV Df SS MS F Row Column Tt Tt MS/Eo MS 6.60 * Eo (-1)(-) Total Step 9. Calculate the LSD. LSD t α EoMS.447 (.015) Linea Model Yij( t) µ + β i + κ j + τ t + ε ij( t) whee: µ the expeiment mean. β i the ow effect, κ the column effect, j τ t the teatment effect, and ε the andom eo. ij(t) 6

7 Latin Squae - Combined Analysis Acoss Squaes -The squaes can be at the same location, o thee diffeent locations, o thee diffeent yeas, etc. Example Thee 3x3 Latin squaes Squae 1 R 41 (B) 5 (C) 15 (A) 81 SS Row (A) 3 (B) 4 (C) 76 SS Column (C) 1 (A) 1 (B) 55 SS Teatment C SS Eo Squae R 7 (C) 8 (B) 3 (A) 58 SS Row (A) 17 (C) 9 (B) 30 SS Column 110. (B) 4 (A) 17 (C) 43 SS Teatment 534. C SS Eo Squae 3 R 43 (B) 7 (C) 17 (A) 87 SS Row (A) 34 (B) 6 (C) 8 SS Column (C) 14 (A) 3 (B) 61 SS Teatment C SS Eo Step 1. Test the homogeneity of the Eo MS fom each squae using Batlett s Chisquae test. Step 1.1 Calculate the Eo SS fo each squae. Step 1. Calculate the Eo MS fo each squae. 7

8 Step 1.3 Calculate the Log of each Eo MS Squae Eo SS Eo df Eo MS Log Eo MS s log s i Step 1.4 Calculate the Pooled Eo MS (s p ) s p si # sq Step 1.5 Calculate Batlett s χ χ [( sqlog s ) log s ].306( Eodf ) p ( sq + 1) 1+ 3* sq * Eodf Whee Eo df df fo one squae. i χ [( 3log9.67).9374] ( 3 1).306() *3* Step 1.6 Look up the Table χ -value at the 99.5% level of confidence and df #sq-1. χ 0.005;df 10.6 Step 1.7 Make conclusions Since χ calc < χ table we fail to eject H o : σ 1 σ σ 3 at the 99.5% level of confidence; thus, we can do the combined analysis acoss squaes 8

9 Step. Calculate Teatment Totals fo each squae. Teatment Squae 1 Squae Squae 3 TRT A B C Squae Step 3. Calculate the Coection Facto (CF). CF Y... sq * 573 3*3 1, Step 4. Calculate the Total SS ( ) TotalSS,60.67 Step 5. Calculate the Squae SS SquaeSS Sq ( ) Step 6. Calculate the Row(Squae) SS (Additive acoss squaes) Row(Squae) SS Row 1 SS + Row SS + Row 3 SS

10 Step 7. Calculate the Column(Squae) SS (Additive acoss squaes) Column(Squae) SS Column 1 SS + Column SS + Column 3 SS 89.3 Step 8. Calculate the Teatment SS TtSS TRTi sq * ( ) 3*3 1,174. Step 9. Calculate the Squae X Teatment SS. Sq * TtSS ( SqXTt) SquaeSS TtSS ( ) SquaeSS TtSS Step 10. Calculate Eo SS (Additive acoss squaes) Eo SS Eo 1 SS + Eo SS + Eo 3 SS Eo SS Step 11. Complete the ANOVA Table. SOV Df SS MS F (Squaes and Tt ae Fixed effects) Squae Sq Non-valid F-test Row(Sq) Sq(-1) Non-valid F-test Column(Sq) Sq(-1) Non-valid F-test Tt Tt MS/Eo MS ** Sq X Tt (sq-1)(-1) Sq X Tt MS/Eo MS.49 ns Eo Sq(-1)(-) Total Sq

11 Conclusions: 1. The non-significant Squae X Teatment inteaction indicates that teatments esponded similaly in all squaes. Table 1. Mean fo the squae x teatment inteaction. Teatment Squae A B C LSD(0.05) ns The significant F-test fo Teatment indicates that aveaged acoss all squaes, thee wee diffeences between teatments. 3. Table. Mean fo the teatment main effect aveaged Acoss squaes. Teatment Mean A 1.3 B 8.1 C 3. LSD(0.05) 3.6 Step 1. Calculate LSD s Squae X Tt: Nomally, you would not calculate this LSD because the F-test fo the inteaction was non-significant. Howeve, if it would have been significant, you would have calculated the LSD using the following method: LSD SqXTt t a / ; eodf EoMS (9.67) 3 This LSD would be used fo compaisons only in Table 1. 11

12 Teatment: LSD Tt t a / ; eodf EoMS sq * (9.67) 3*3 This LSD would only be used fo compaisons in Table. 1