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Statistics

DIALLEL-SAS05

A Comprehensive Program for Griffing's and Gardner–Eberhart Analyses

^a Dep. of Agron., Iowa State Univ., Ames, IA 50011-1010
^b Dep. of Agron. and Environ. Manage., Louisiana State Univ. Agric. Cent., Baton Rouge, LA 70803-2110

^* Corresponding author (mkang{at}agcenter.lsu.edu)

Received for publication October 8, 2004.

	ABSTRACT

TOP ABSTRACT INTRODUCTION PROGRAM DESCRIPTION AN EXAMPLE DISCUSSION REFERENCES

 
Diallel mating designs provide to breeders useful genetic information, such as general combining ability (GCA) and specific combining ability (SCA), to help them devise appropriate breeding and selection strategies. Here we report a much-improved version of DIALLEL-SAS that was originally released in 1997. The new program, DIALLEL–SAS05, has a clear and user-friendly interface that was designed to meet users' needs for various diallel-cross design experiments. DIALLEL-SAS05 has major advantages over DIALLEL-SAS in that: (i) it analyzes not only all four Griffing's diallel methods (both fixed and random models), but it also computes Gardner–Eberhart's Analyses II and III; (ii) it provides desired results from diallel experiments with parent number from 4 to 12, (iii) it can analyze diallel data from any number of environments, and (iv) for a random-effects model, it provides estimates of GCA (²_g) and SCA (²_s) variances, which can be used to estimate additive (²_A) and dominance (²_D) variances, and ultimately narrow-sense heritability (h²). DIALLEL-SAS05 also provides information on GCA x ENV, SCA x ENV, reciprocal x ENV, maternal x ENV, and nonmaternal x ENV interactions, when applicable. DIALLEL-SAS05 should greatly improve researchers' efficiency in analyzing and interpreting diallel-cross data. The program code is available on a CD from the corresponding author.

Abbreviations: GCA, general combining ability • GEAN, Gardner–Eberhart analysis • SCA, specific combining ability

	INTRODUCTION

TOP ABSTRACT INTRODUCTION PROGRAM DESCRIPTION AN EXAMPLE DISCUSSION REFERENCES

 
DIALLEL MATING DESIGNS are an important tool in plant and animal breeding programs to obtain information on inheritance, such as GCA and SCA, of quantitative or complexly inherited traits (Hayman, 1954; Griffing, 1956; Fry, 2004). Griffing's (1956) diallel methods have been extensively used in crop plants (Evans et al., 1966; Stuthman et al., 1971; Borges, 1987; Pixley and Bjarnason, 1993; Kang et al., 1995, 1999; Zhang et al., 1996). A useful summary of important features of Griffing's four diallel methods is given in Zhang and Kang (1997)(Table 1).

Gardner and Eberhart (1966) proposed alternative analyses of data from diallel crosses produced from heterogeneous parents/populations ("varieties"). Gardner–Eberhart Analysis (GEAN) I is resource-intensive, requiring the evaluation of n parents, n(n – 1)/2 F₁ crosses, and inbred progeny of parents and crosses, but provides information on additive and dominance gene action, heterosis, and inbreeding depression (Gardner and Eberhart, 1966; Murray et al., 2003). The GEAN II is useful in evaluating n populations (varieties) and their n(n – 1)/2 F₁ crosses; variation among populations is partitioned into populations and midparent heterosis (Gardner and Eberhart, 1966; Hallauer and Miranda, 1988; Murray et al., 2003). Heterosis is further partitioned into average, variety, and specific heterosis, but additive and dominance parameters cannot be determined in Analysis II because they are confounded with the source of variation labeled "variety" (Murray et al., 2003). The GEAN III provides estimates of variety and GCA effects from an analysis that contains the following sources of variation: parents, parents vs. F₁ crosses, and F₁ crosses; GCA effects are estimated in a manner similar to Griffing's Method 4, Model 1 (Murray et al., 2003). Both GEAN II and GEAN III provide estimates of average heterosis and SCA (Gardner and Eberhart, 1966; Murray et al., 2003). Many researchers have fruitfully used GEAN for studying heterosis and estimating GCA and SCA in variety diallels (Crossa et al., 1987; Ali et al., 2001; Lee et al., 2003).

Murray et al. (2003) pointed out that formulas for the effects for GEAN II and GEAN III were nonintuitive both biologically and genetically and incorporated the number of parents as multipliers. They further opined that (i) the variety effects obtained in GEAN III were "unconstrained" estimates, whereas those from GEAN II were constrained estimates because of the assumption of no heterosis; and (ii) the results had implications for the use and interpretation of such effects. For further specific details about GEAN I, II, and III, the reader is referred to Murray et al. (2003).

Naturally because of cumbersome calculations, researchers have felt the need to develop computer programs for analyzing diallel data. Program codes have been developed for use in crop breeding (Burrow and Coors, 1994; Kang, 1994; Magari and Kang, 1994; Zhang and Kang, 1997) and tree breeding (Wu and Matheson, 2000, 2001; Xiang and Li, 2001). Murray et al. (2003) reported a SAS computer program for GEAN II and GEAN III; the program used the general linear model with cell means to estimate various genetic effects.

The DIALLEL-SAS program of Zhang and Kang (1997) has been extensively used—for example, to analyze diallel-cross data in maize (Zea mays L.) (Kang et al., 1999, 2005; Naidoo et al., 2002; Lee et al., 2003), rapeseed (Brassica napus L.) (Goffman and Becker, 2001), wheat (Triticum aestivum L.) (Le Gouis et al., 2002; Hakizimana et al., 2004), durum wheat (Triticum turgidum L. var. durum) (Solomon and Labuschagne, 2004), alfalfa (Medicago sativa L.) (Guines et al., 2002; Riday and Brummer, 2002a, 2002b), and white lupin (Lupinus albus L.) (Lagunes-Espinoza et al., 2003). The usefulness of DIALLEL-SAS can be enhanced, however. The limitations of DIALLEL-SAS (Zhang and Kang, 1997) are (i) while it can be easily adapted for a particular diallel set (Griffing's Methods 1 to 4) for two environments, the user must modify the code to analyze data from more than two environments, which can be cumbersome; (ii) it provided results only for a fixed-effects model; (iii) it required extensive and laborious program modifications for parent number other than n = 5 used in the example programs in Zhang and Kang (1997); and (iv) it did not include Gardner and Eberhart (1966) Analyses II and III.

Currently, there is no known computer program that handles analyses of both Griffing's four diallel methods and Gardner–Eberhart Analyses II and III. Thus, a need exists for a comprehensive program that computes not only Griffing's diallel analyses—both fixed- and random-effects models—but also Gardner–Eberhart Analyses II and III. We report here the development of such a program called DIALLEL-SAS05.

	PROGRAM DESCRIPTION

TOP ABSTRACT INTRODUCTION PROGRAM DESCRIPTION AN EXAMPLE DISCUSSION REFERENCES

 
The general linear models used for analyzing Griffing's diallel methods are the same as those reported in Zhang and Kang (1997). Because Zhang and Kang (1997) did not discuss GEAN, we deemed it appropriate to include the following brief discussion about them.

In the current program, DIALLEL-SAS05, statistical models used for GEAN2 and GEAN3 are as follows (Gardner and Eberhart, 1966; Murray et al., 2003):

and

where Y_jj' = mean of a parent when j = j' and mean of a cross when j j'; µ_v = mean of all parents; v_j and v_j' are effects for variety j and j', respectively, when parents are included in the analysis (restriction: v_j = 0); = average heterosis; = 0 when j = j' and = 1 when j j'; h_j and h_j' = variety heterosis for variety j and j', respectively; s_jj' = specific heterosis in GEAN2 and SCA in GEAN3 (restriction s_jj' = s_jj = 0); x_jj' = the cross effect for the mating between varieties j and j'; and g_j and g_j' are GCA effects for variety j and j', respectively.

Program Components
The DIALLEL-SAS05 program consists of two major parts. The first part is a user-interface program called "Diallel.sas." This is where the user inputs data and invokes necessary macro functions. The second part is called "DialAll.sas," which includes seven major macro functions and other utility macros. A flow chart for functional relationship among different macros is given in Fig. 1 .

View larger version (38K): [in this window] [in a new window]   Fig. 1. Flow chart for function calls among different macros.

	AN EXAMPLE

TOP ABSTRACT INTRODUCTION PROGRAM DESCRIPTION AN EXAMPLE DISCUSSION REFERENCES

 
The best way to familiarize the user with the operation of DIALLEL-SAS05 is through an example. We use a factitious diallel-cross dataset for Griffing's Method 1 (abbreviated data shown below after the CARDS statement) for five parents and two environments.

OPTIONS PS=56 LS=100 mLOGIC PAGENO=1;

%INCLUDE 'C:\DIALLEL\DIALALL.SAS';

DATA DT1;

INPUT I J REP ENTRY YIELD ENV;

IF I=J THEN AVGH=0;ELSE AVGH=1;

CARDS;

1 1 1 1 10.5 1

1 1 2 1 10.7 1

1 2 1 2 11.9 1

1 2 2 2 12.0 1

1 3 1 3 14.5 1

1 3 2 3 14.2 1

1 4 1 4 9.0 1

1 4 2 4 8.5 1

1 5 1 5 13.5 1

1 5 2 5 14.2 1

.

more data records

.

5 1 1 21 22.1 2

5 1 2 21 21.0 2

5 2 1 22 21.9 2

5 2 2 22 20.2 2

5 3 1 23 20.2 2

5 3 2 23 20.8 2

5 4 1 24 21.2 2

5 4 2 24 20.6 2

5 5 1 25 20.2 2

5 5 2 25 21.0 2

;

RUN;

/* FOR GRIFFING'S METHOD 2, DO THE FOLLOWING TO DELETE RECIPROCAL CROSSES */

DATA DT2;SET DT1;

IF I>J THEN DELETE;

RUN;

/* FOR GRIFFING'S METHOD 3, DELETE INBRED LINES AS FOLLOWS */

DATA DT3;SET DT1;

IF I=J THEN DELETE;

RUN;

/* FOR GRIFFING'S METHOD 4, DELETE INBRED LINES AND RECIPROCAL CROSSES AS FOLLOWS */

DATA DT4;SET DT1;

IF I>=J THEN DELETE;

RUN;

/* TO MAKE DATASET FOR GARDNER-EBERHART ANALYSIS II AND III WHEN PLOT MEAN DATA ARE USED, DO THE FOLLOWING (NOTE: YOU WILL NEED TO REPLACE YIELD IN ‘AVG(YIELD)’ WITH APPROPRIATE VARIABLE BEING ANALYZED): */

PROC SQL;

CREATE TABLE DT5 AS

SELECT I,J,ENTRY,AVG(YIELD) AS AVG_Y,AVGH

FROM DT1

GROUP BY I,J,ENTRY;

/* GENERATING DESIGN MATRICES ONLY */

%MGENERATOR(NUM_P=5,DSN=DT1, METHOD=1);

%GARDNERANAL2(NUM_P=5, DSN=DT2);

%GARDNERANAL3(NUM_P=5, DSN=DT2);

/* FOR GRIFFINGS FIXED-EFFECTS MODEL */

%DIALANALFIXMODEL(NUM_P=5,METHOD=

1,YVAR=YIELD,ENV=2,REP=2,DSN=DT1);

%DIALANALFIXMODEL(NUM_P=5,METHOD=

2,YVAR=YIELD,ENV=2,REP=2,DSN=DT2);

%DIALANALFIXMODEL(NUM_P= 5,METHOD=

3,YVAR=YIELD,ENV=2,REP=2,DSN=DT3);

%DIALANALFIXMODEL(NUM_P= 5,METHOD=

4,YVAR=YIELD,ENV=2, REP=2, DSN=DT4);

/* FOR GRIFFINGS RANDOM-EFFECTS MODEL */

%DIALANALRANMODEL(NUM_P=5, METHOD=

1,YVAR=YIELD, ENV=2, REP=2, DSN=DT1);

%DIALANALRANMODEL(NUM_P=5, METHOD=

2,YVAR=YIELD, ENV=2, REP=2, DSN=DT2);

%DIALANALRANMODEL(NUM_P= 5, METHOD=

3,YVAR=YIELD, ENV=2, REP=2, DSN=DT3);

%DIALANALRANMODEL(NUM_P=5, METHOD=

4,YVAR=YIELD, ENV=2, REP=2, DSN=DT4);

/* GARDNER AND EBERHART ANALYSES */

/* FOR GARDNER-EBERHART ANALYSIS II AND ANALYSIS III, RECIPROCAL CROSSES ARE DELETED(SEE DATA DT2 ABOVE) */

TITLE "GARDNER AND EBERHART ANALYSIS II";

%DIALANALYSISII(NUM_P=5, YVAR=YIELD,

ENV=2,REP=2, DSN=DT2, MSE=–1,

DFE=0);

TITLE "ANALYSIS II, MSE=1.2";

%DIALANALYSISII(NUM_P=5, YVAR=AVG_Y,

ENV=1,REP=1, DSN=DT5, MSE=1.2,

DFE=28);

TITLE "GARDNER AND EBERHART ANALYSIS III";

%DIALANALYSISIII(NUM_P=5, YVAR=YIELD,

ENV=2,REP=2,DSN=DT2, MSE=–1,

DFE=0);

TITLE "ANALYSIS III, MSE=1.2";

%DIALANALYSISIII(NUM_P=5, YVAR=AVG_Y,

ENV=1,REP=1,DSN=DT5, MSE=1.2,

DFE=28);

RUN;

QUIT;

In the above SAS statements relative to the Gardner–Eberhart analyses, if MSE 0, the program recognizes that the data have multiple replications. If MSE is >0 (actual value to be provided by user), the program recognizes that data represent means across replications.

For brevity, only sample outputs for Griffing's Method 1 for a fixed-effects model or Model 1 (Fig. 2) and a random-effects model or Model 2 (Fig. 3) are shown. The results for GEAN II and III are given in Fig. 4 and 5 , respectively. That part of the output that is identical for GEAN II and GEAN III has been omitted from Fig. 5. Only essential information of various outputs has been shown in Fig. 2 through 5.

View larger version (129K): [in this window] [in a new window]   Fig. 2. Output from DIALLEL-SAS05 program for Griffing's Diallel Method 1, Model 1 (fixed effects) for five parents.

View larger version (40K): [in this window] [in a new window]   Fig. 3. Output from DIALLEL-SAS05 program for Griffing's Diallel Method 1, Model 2 (random effects) for five parents.

View larger version (34K): [in this window] [in a new window]   Fig. 4. Output from DIALLEL-SAS05 program for Gardner–Eberhart Analysis II for five parents.

View larger version (43K): [in this window] [in a new window]   Fig. 5. Output from DIALLEL-SAS05 program for Gardner–Eberhart Analysis III for five parents.

	DISCUSSION

TOP ABSTRACT INTRODUCTION PROGRAM DESCRIPTION AN EXAMPLE DISCUSSION REFERENCES

DIALLEL-SAS05 is more user-friendly, comprehensive, and less cumbersome to modify than the DIALLEL-SAS program of Zhang and Kang (1997). The major features of the new program are

1. The user can analyze, without much modification to the program, 4- to 12-parent diallel-cross data for any number of environments for any of Griffing's four methods and Gardner and Eberhart Analysis II and III. The only modifications users need to make are in the macro functions with respect to number of replications, environments, and parents; dataset name; and variable to be analyzed.
   
2. Seven macro functions are provided that serve as "user interfaces" for different purposes and for ease.
   
3. The user can generate a linear matrix for any number of parents by extending the program code to diallels with more than 12 parents.
   
4. The variances and LSDs for various comparisons described by Griffing (1956) have been included.
   
5. The adjusted F values and P values have been calculated and tabulated for the random model (Model 2).
   
6. Estimates of ²_g, ²_s, ²_r, and their variances are computed for the random-effects model to estimate ²_A, ²_D, and h². To compute ²_A and ²_D (e.g., from the information given in Fig. 3), one would do the following:
   

When inbreeding coefficient (F) of parents = 0 (no inbreeding)

When inbreeding coefficient (F) of parents = 1 (100% inbreeding)

Software Installation and Support
The DIALLEL-SAS05 program was developed and tested on SAS 9.0 on workstation with Windows XP (SAS Inst., 2002). It can run on any SAS version 8.0 or above, however. The program is expected to run on other SAS versions without any significant problems/modifications, but we did not test it on older SAS versions.

The software is contained in two files: Diallel.sas and DialAll.sas. The DialAll.sas should be saved in a directory in the user's local PC or a server that the user can access. For example, the user may save it in C:\Documents\DialAll.sas. The other file, Diallel.sas, may be saved in the same folder or in a different folder in user's personal computer. Once the two files have been saved, the user would open Diallel.sas with the SAS software. Upon opening Diallel.sas, the user will need to change the %include file path to the user's own specific path (%include ‘C:\Documents\DialAll.sas’;). Now, the user is ready to input data and run any of the seven macro functions. To analyze data for more than one trait, the user would need to have as many statements as the number of traits. For example, to analyze five-parent diallel data on yield and testwt for Griffing's Method 1, Model 1, user must include the following statements:

%DIALANALFIXMODEL(NUM_P=5,METHOD=

1,YVAR=YIELD,ENV=2,REP=2,DSN=DT1);

%DIALANALFIXMODEL(NUM_P=5,METHOD=

1,YVAR=TESTWT,ENV=2,REP=2,DSN=DT1);

In summary, DIALLEL-SAS05 has the following major functions: (i) it deals with both fixed- and random-effects models relative to all four Griffing's diallel methods, (ii) it computes Gardner–Eberhart's Analyses II and III, (iii) it is effective for analyzing diallel experiments with number of parents ranging from 4 to 12, (iv) it can analyze diallel data for any number of environments, and (v) it provides information on GCA x ENV, SCA x ENV, reciprocal x ENV, maternal x ENV, and nonmaternal x ENV interactions, when applicable. The user can generate a linear matrix for any number of parents in a diallel cross by easily extending the program code to diallel designs with more than 12 parents. DIALLEL-SAS05 is expected to greatly enhance researchers' efficiency in analyzing and interpreting diallel-cross data.

The DIALLEL-SAS05 software will be provided free of charge to those affiliated with academic institutes. The user may request a copy of the software on a CD, along with documentation/instructions regarding DIALLEL-SAS05, by sending US$10 to the corresponding author to defray the cost of a CD and shipping and handling. Outputs for the five-parent example for all four methods of Griffing (both fixed and random effects) and for GEAN II and GEAN III will be provided along with other documentation for the software.

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TOP ABSTRACT INTRODUCTION PROGRAM DESCRIPTION AN EXAMPLE DISCUSSION REFERENCES

 

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