April, 1947 Research Bulletin 347 Punched Card and Calculating Machine Methods for Analyzing Lattice Experiments Including Lattice Squares and the Cubic Lattice By PAUL G. HOMEYER, MARY A. CLEM AND WALTER T. FEDERER AGRICULTURAL EXPERIMENT STATION IOWA STATE COLLEGE OF AGRICULTURE AND MECHANIC ARTS STATISTICAL SECTION BUREAU OF AGRICULTURAL ECONOMICS UNITED STATES DEPARTMENT OF AGRICULTURE Cooperating AMES, IOWA ERRATA In table 6, page 48, 113.959 = E should read 113.959 = E b• In table 35, page 96, 63.219=E should read 63.219=Er. In table 56 (a), page 134, (c ') should read (ey'). In table 56(b), page 135, (c ') should read (cz'). Page 141, step 28, the reference in the first sentence to table 56(b) should read table 56(a). In table 57, page 145, E" should read Ee. Page 146, second sentence, the reference for the M + L' values to table 56 (b) should read table 56 (a) . Page 156, table 69, (h-3c) in the heading should read (h-30). Table 74, page 159, the heading for the second column (f-30) should read (f-3c). 0 in the first footnote should read c. Tabte 75, page 159, M in the heading should read M'. Page 167, second paragraph, Oe' should read ce'. CONTENTS Summary 30 Introduction 31 General comments about computations 33 Triple lattice 37 Procedure of computations using calculating machines 37 Calculations for the analysis of variance · . 45 Adjustments of variety total~ 48 Standard errors of the differences, between adjusted variety means . 50 Efficiency of the triple lattice design relative "to random- ized complete blocks 51 Coefficient of variation 51 Calculations using punched card machines 52 Coding 'data' and punching cards 52 Sorting and tabulating procedure 53 Calculations for the analysis of variance 56 Simple lattice 60 Procedure of computations using calculating machines. 60 Calculation~ for the analysis of variance . 65 Adjustments of variety totals 68 " St~ndard_ ~t:r0rs of the differences between adjusted var- Iety rneahs . 69 Efficiency of the simple lattice design relative to ran- domized c'omplete blocks . 70 Coefficient of variation 70 Calculations using punched card machines 71 Coding data and punching cards 71 Sorting and tabulating procedure 71 Calculations for the analysis of variance. 75 Balanced lattice . 76 Procedure of 'comp'utations using calculating machines. 77 Calculations for the analysis, of variance . 80 Adjustments of variety totals 81 Standard error of the differences between adjusted var- iety means 82 Efficiency of the balanced lattice design relative to ran- domized complete blocks . 82 28 Coefficient of variation 82 Calculations using punched card machines 82 Coding data and punching cards 82 Sorting and tabulating procedure 84 Calculations for the analysis of variance . 86 Lattice square with (k+1) /2 replicates. 87 Procedure of computations using calculating machines. 87 Calculations for the analysis of variance . 95 Adjustments of variety totals 97 Standard errors of the differences between adjusted val" iety means 98 Efficiency of the lattice square design relative to ran· domized complete blocks . 99 Coefficient of variation 99 Calculations using punched card machines 100 Coding data and punching cards 100 Sorting and tabulating procedure 100 Calculations for the analysis of variance. 106 Adjustments of variety totals 107 Lattice square with (k+l) replicates 109 Procedure of computations using calculating machines 109 Calculations for the analysis of variance . 116 Adjustments of variety totals 117 Standard error of the differences between adjusted var· iety means 119 Efficiency of the lattice square design relative to ran· domized complete blocks . 119 Coefficience of variation 119 Calculations using punched card machines 120 Coding data and punching cards 120 Sorting and tabulating procedure 120 Calculations for the analysis of variance. 123 Cubic lattice 126 Coding of varieties 126 Procedure of computations using calculating machines. 127 Calculations for the analysis of variance 142 Adjustments of variety totals 144 Standard errors of the differences between adjusted val" iety means 148 29 Efficiency of the cubic lattice . design relative to ran- domized complete blocks . 149 Coefficient of variation 149 Calculations using punched card machines 150 Coding data and punching cards 150 Sorting and tabulating procedure 150 Calculations for the analysis of variance 159 Adjustments of variety totals 161 Additional lattice designs . 166 Quadruple lattice, quintuple lattice and higher order lattice designs 167 Duplications of the quadruple lattice, quintuple lattice and higher order lattice designs 168 Other lattice designs 169 References cited. 170 SUMMARY Punched card and calculating machine methods are given for analyzing data from lattice experiments. Numerical examples are worked out for the triple lattice, simple lattice, balanced lattice, lattice square with (k+ 1) / 2 replicates, lattice square with (k+1 ) replicates and cubic lattice designs. In additi011, brief computational directions are given for the quadruple lat­ tice, quintuple lattice and higher order lattices and for dupli­ cations of these designs. The use of the punched card machine method materially de­ creases the time and the chance for errors in making the com­ putations, particularly when the total number of plots is more than about 150 or when more than one ' character is to be analyzed. The formulas given are appropriate for any k and for an:­ number of repetitions of the basic plan for the simple lattice, triple lattice, quadruple lattice and higher order lattice designs and for the cubic lattice design and for any k for the lattice square with (k + 1)/2 replicates, lattice square with (k + 1) replicates and the balanced lattice designs. Punched Card and Calculating Machine M ethods for Analyzing Lattice Exper­ iments Including Latti~e Squares ancl the Cubic Lattice1 By PAUL G. HOMEYER', MARY A. CLEM' AND WALTER T. FEDERER' During the past few years there has been a considerable in­ crease in the use of lattice designs, introduced by Yates in 1936 and subsequently, particularly to test large numbers of varieties of various crops. Since the method of analysis is complicated and the number of varieties tested is usually large, computation of the data tends to become burdensome. For a number of years the Iowa State College Statistical Laboratory has been developing punched card5 and calculating machine methods to decrease the time and effort and to lessen the chance for error in analyzing data from lattice6 experiments. While' all of the computations can be made with any of the standard calculating machines, the use of punched card machines for certain of the computations materially lessens the time and the chance for errors, particularly when the total number of plots is about 150 or more. The object of this bulletin is to give the procedures for these machine methods. Any of the standard calculating machines and card-punching, sorting and tabulating machines may be used. Instructions for operating 1 Project 890 of the Statistical Section of the Iowa Agricultural Experi­ ment Station in cooperation with the Bureau of Agricultural Economics, United States Department of Agriculture. The authors are especially in­ debted to R. J. Monroe for his assistance in the early stages of developing punched card machine methods for analyzing lattice experiments. Also, the authors acknowledge the assistance, suggestions and criticisms of George W. Snedecor, W. G. Cochran, Campbell C. Mosi~r, Gertrude M. Cox a nd R. L. Anderson. 2 Research Assistant Professor. Statistical Section, Iowa Agricultural Ex­ periment Station; also Agent, Burea u of Agricultural Economics, United States Department of Agriculture. 3 Technician, Statistical Section, Iowa Agricul tural Experiment Station ; also Agent, Bureau of Agricultural Economics, United States . Department of Agriculture. 'Agricultural Statistician, Bureau of Agricultural Economics, United States Department of Agriculture. • Also known as the Hollerith, tabulating machine or electric accounting machine method. • The term lattice is used to include all experimental designs with the word lattice in their titles. For definitions of the various lattice designs, see the general references given in the introduction. 32 these machines may be obtained from the manufacturers and are beyond the scope of this bulletin. The following references may be helpful in using punched card machines: Bachne (1), Brandt (2), Dunlap (6) and Hartkemeier (9). To illustrate the numerical calculations, examples are worked out for the triple lattice, simple lattice, balanced lattice, lattice square with (k+l) / 2 replicates, lattice square with (k+1) replicates and cubic lattice designs. 7 They are given in detail so that computers will be able to follow the directions with a minimum of supervision. In addition, computational directions are given briefly for the quadruple lattice, quintuple lattice and higher order lattices al~d for duplications of these designs. Ref­ erences are given showing the method of analysis of other lattice designs. The punched card machine methods of analysis given in this bulletin can be adapted rather easily to these other de- signs. The method of analysis used is that worked out by Yates (16, 18 and 19) for the recovery of inter-block information. Ref­ erences are made to the literature giving the mathematical de­ velopment, the field plans and the method of analysis of the designs discussed. Details of the designs are given only when they are considered necessary to clarify the directions for compu­ tations. General references that will be helpful in laying out and analyzing lattice designs are Yates (13, 14, 15, 16, 17, 18 and 19), Yates and Cochran (20), Cochran and Cox (4;) and Goulden (8 ). Cert[!,in steps given in the computations for both the punched card and calculating machine methods may be modified or com­ bined and others eliminated if desired. The types of machines and ability and training of computers available should guide such decisions. The tabulating machine sheets are shown for each tabulation given in this bulletin; however, only the first and last few list­ ings are given when the tabulation is long.