based on genetic models for doubled hap- Press WH, Flannery BP, Teukolsky SA, and Vetterling is more to plant mating systems than the
WT, 1986. Numerical recipes: the art of scientific com-
loid, backcross, recombinant inbred, var- puting. Cambridge, Massachusetts: Cambridge Uni- dichotomy of selfing versus outcrossing.
ious testcross, F2, and F3 progeny in which versity Press. With regard to genetic markers, the main
the phenotypes of flanking molecular Press WH, Flannery BP, Teukolsky SA, and Vetterling alternative has been to study the paternity
marker loci are used as independent vari- WT, 1988. Numerical recipes in C: the art of scientific of seeds with the aim of counting numbers
able values (Knapp et al. 1990). The sta- computing. Cambridge, Massachusetts: Cambridge
University Press. of fathers and determining multiple pa-
tistical models arising from these genetic
SAS, 1985. SAS IML user's guide, Version 5 ed. Cary,
ternity.5 Schoen and Clegg19 and Ritland13
models are nonlinear. The parameters of North Carolina: SAS. have developed a different approach to
these models are the means of quantitative SAS, 1987. SAS STAT guide for personal computers, paternity by developing parametric mat-
trait locus genotypes and recombination Version 6 ed. Cary, North Carolina: SAS. ing-system models that mathematically
frequencies between marker and quanti- describe a correlation of paternity.
tative trait loci. We are distributing SAS These approaches to describing the
(1987) PROC NLIN programs for estimat- mating system are based on the notion
ing the parameters of these models using that specific models and parameter values
maximum-likelihood methods. (PROC are largely sufficient to describe the dy-
NLIN is the SAS  procedure for es- A Series of FORTRAN namics of genotypic frequencies in pop-
timating the parameters of nonlinear Computer Programs for ulations. This quantitative framework al-
models.) The Gauss-Newton algorithm Estimating Plant Mating lows the actual recovery of information
(Gallant 1987) and ordinary or generalized about parameter values from the observed
least squares methods are used in our Systems
dynamics of genotypic frequencies at
programs. Maximum-likelihood estimates K. Ritland marker-gene loci. The extraction of infor-
of the parameters are output. In addition,
mation is most efficiently achieved by the
we are distributing PROC IML, PROC
The past decade has witnessed consid- statistical method of maximum likelihood,
REG, and PROC NLIN programs (SAS 1985,
erable development of models for describ- which was developed by the population
1987), which test hypotheses about gene
ing plant mating systems. The classical geneticist R. A. Fisher. This method also
effects and recombination frequencies us-
model of mixed mating, which assumes a allows the testing of hypotheses about the
ing Wald statistics or log-likelihood ratios
(Gallant 1987). The model and derivative mixture of selfing and random outcrossing fit of the model to the data and the setting
source code from our programs, which are and which was formulated for one locus, of confidence intervals for parameter val-
specific to the genetic and statistical was introduced by Fyfe and Bailey6 and ues. This note describes a series of avail-
models, can be used in C or FORTRAN has served as the starting point for more able programs that use parametric models,
implementations of Marquardt or Gauss- complex models. Along these lines, Brown maximum likelihood, and genetic marker
Newton nonlinear least-squares algo- and Allard2 and Clegg et al.4 introduced data to characterize the mating system in
rithms (Press et al. 1986, 1988). the use of progeny arrays for estimating plant populations.
maternal parentage along with the selfing The programs have several features in
Personal-computer, workstation, or rate and also popularized the use of co- common. All are written in FORTRAN and
mainframe versions of SAS (1987) can be dominant electrophoretic data. Ritland12 run on IBM-compatible personal com-
used to run these programs. They have and Schoen,17 by treating the progeny ar- puters. The source code, compiled code,
been tested on simulated and real data ray as the unit of observation, derived ex- and documentation will be written onto
bases with as many as 2,000 observations. plicit formulas for the joint estimation of 5lA-in DOS-formatted floppy diskettes,
Memory limitations have not been expe- parental and mating system parameters. which must be supplied by the requester.
rienced on personal computers having at In further elaborations of the mixed- Because programs are often dimensioned
least 640 K RAM. S.J.K. will distribute pro- mating model, Shaw et al.20 and Ritland to hold the largest possible data sets, gen-
gram copies, listings, and documentation. and Jain16 developed multilocus outcross- erally 640 K of memory is needed to run
To receive program copies, please send a ing models, and they emphasized their programs; an AT-compatible computer
DOS- or UNIX-formatted 3.5-in or 5.25-in ability to provide robust and precise es- with a math coprocessor is recommended
diskette. timates of selfing and their ability to at (information about Apple formats is avail-
least partially separate biparental in- able from the author).
From the Department of Crop Science, Oregon State breeding from uniparental inbreeding. All programs use maximum likelihood
University, Corvallis (Knapp) and the Department of Taking advantage of the increased degrees to find estimates and the chi-square sta-
Experimental Statistics, Clemson University, South
Carolina (Bridges). This work was partially funded by of freedom offered by multilocus analyses, tistic to test the fit of data to the model.
a grant from the Medical Research Foundation of Or- Ritland and El-Kassaby14 introduced and In some programs, likelihood ratio tests
egon and is Oregon Agricultural Experiment Station studied the use of multilocus models for are possible. In most programs, the data
Technical Paper No. 8985. Address reprint requests to
Dr. Knapp, Department of Crop Science, Oregon State the estimation of outcrossing and pollen is family structured: several progeny of a
University, Corvallis, OR 97331. gene pools for individual plants. Recent single plant are sampled to form a progeny
modifications of the single-locus mixed- array, and several plants are sampled from
mating model included incorporation of the population. At the marker loci, parent
the effects of population structure11 and genotype may be either known or un-
References incorporation of different modes of inher- known. Because of this family structure,
Gallant RA, 1987. Nonlinear statistical models. New itance (tetraploidy1) and different types of
York: John Wiley and Sons. the parental parameters (gene frequen-
selfing (intergametophytic selfing vs. in- cies, inbreeding coefficients) are also es-
Knapp SJ, Bridges WC, and Birkes D, 1990. Mapping
quantitative trait loci using molecular marker linkage tragametophytic selfing73). timated in addition to the mating system
maps. Theor Appl Genet (in press). Workers have also emphasized that there parameters.
Brief Communications 2 3 5
Table 1. Some differences between the computer programs, and total sizes of source code, compiled dividual-level estimates are discussed in
code, and documentation (in bytes) Ritland and El-Kassaby14 and Ritland and
No. alleles Separate p's Hedrick (in prep.).
Program Total sample size" allowed estimated* Iterative method Size (K) 2. ML2T. This program estimates out-
MLT 1,000 3 Yese
NR" or EM' 160 crossing rates for two groups within a pop-
ML2T 1,000 3 Yes' NR (/), EM (p) 120 ulation. All other parameters are assumed
ESR Unlimited 3 No NR or EM 160 identical among the two groups, so that
DOMT Unlimited 2 No NR or EM 90
CSP 1,000 3 No EM 140 except for the two lvalues this program is
TETRAT 1,000 2 No NR (0, EM O ) 120 much like MLT. ML2T can be used for
FERNT 2,000 6 No NR or EM 100 gynodioecious populations, distylous
populations, and experimental manipula-
" Number of individuals among all arrays; all programs allow up to 100 arrays (except FERNT, which does not
use arrays). tion of factors determining outcrossing rate
'Means that pollen and ovule gene frequencies are separately estimated. involving two treatments.
They can be constrained to equal each other. 3. ESR. This program estimates param-
NR = Newton-Raphson method. eters of the "effective selfing" model. This
* EM = Expectation-maximization method. model describes the joint effects of uni-
parental and biparental inbreeding in
terms of separate, single-locus selfing rates
To maximize the likelihood function, the is encoded by negative integers. A brief for inbred vs. outbred parents.11 The es-
programs use one of two iteration options: example, with four loci and with two small timation procedure is described in Rit-
(1) the Newton-Raphson method, which families where parentage is unknown in land.12 If more than one locus is used, a
is fast and gives statistically valid esti- the first family and partially known in the minimum variance average of single-locus
mates of outcrossing greater than 1.0; and second, is estimates is found. This program does not
(2) the expectation-maximization method, directly estimate the rate of biparental in-
which is slow to converge and constrains pgi pgm idh est breeding. No current method gives un-
estimates within the bound of zero to 1.0, (a4,4i2) biased estimates of biparental inbreeding;
but is less likely to suffer numerical prob- faml 6 1 3 2 tm-ts gives a lower bound for selfing due
lems.12 faml 5 1 3 0 to biparental inbreeding (see MLT).
The statistical variances of estimates are faml 6 1 2 1 4. DOMT. This program estimates t, p,
found with the bootstrap method. This fam2 - 2 0 - 1 0 and F for marker loci that have dominant
method involves redrawing progeny ar- fam2 2 1 1 3 alleles. The procedure, which uses prog-
rays, recomputing estimates, and exam- eny arrays, is described in Ritland and
ining the distribution of bootstrap esti- . . . etc. Ganders.15 There is an option to estimate
mates, which is assumed to approximate population substructure effects (see ESR).
the statistical distribution of the original The following provides short descrip- 5. CSP. CSP denotes "correlated selfing
estimate (thus many arrays are required). tions of the seven available programs; their and paternity"; the model and estimation
This method involves considerable com- documentation files supply more details, procedure are described in Ritland.13 The
puter time but avoids the complexity and and Table 1 gives additional information unit of observation is a pair of sibs. The
assumptions of the alternative method in- as well. Space requirements are given so program estimates (1) the correlation of
volving computation of the Fisher infor- that the number of floppy disks needed selfing and (2) the correlation of out-
mation matrix. The programs can also out- may be calculated (either 360 K or 1.2 MB crossed-paternity between these sibs. The
put each bootstrap estimate to a diskfile disks may be sent). latter parameter allows inferences about
for further statistical analyses. 1. MLT. MLT denotes "multilocus t," or, the extent of multiple paternity of fruits or
The data consist of individual genotypes alternatively, "maximum-likelihood t." This capsules. Either correlation and/or the
at one or more marker loci. For three al- program is based on the multilocus model selfing rate can be constrained to zero,
leles denoted A,, A2, and A3, most pro- of Ritland and Jain.16 It is the oldest and thus simplifying the model. An additional
grams require genotypes to be coded as largest program; most programs below are grouping variable is attached to the family
A,A, = 1, A,A2 = 2, A^ = 3, A,A3 = 4, modifications of the MLT code. This pro- identification to allow computation of
AjA3 = 5, /4y43 = 6, and missing = 0. The gram simultaneously estimates: (1) tm, the within-fruit correlations. Although the
tetraploid program requires data to be multilocus outcrossing rate; (2) ts, the sin- program can infer parentage, the assay of
coded as (for two alleles A and a) AAAA gle-locus outcrossing rate (averaged over parent genotype is desirable for this anal-
= 1, AAAa = 2, AAaa = 3, Aaaa = 4, aaaa loci); (3) F, the inbreeding coefficient of ysis.
= 5, and missing = 0, and the fern program maternal parents; and (4) the p's, the pol- 6. TETRAT. This program estimates sin-
has codes for individual alleles. All data len and ovule gene frequencies, either sep- gle-locus t, multilocus /, gene frequency p,
files consist of a line for locus names, a arately or as an average. The bootstrap and parental inbreeding coefficients for
line with a FORTRAN format for the data, method allows a confidence interval to be autotetraploids, assuming no double re-
and subsequent lines for data, with one placed on tm-t^ a positive difference in- duction. The single-locus progeny proba-
individual genotype per line. A family dicates biparental inbreeding. The pro- bilities given in Barrett and Shore1 were
identifier precedes each genotype, and in- gram can also compute estimates of t and/ generalized to the multilocus case. Be-
dividuals are grouped by family. If par- or p for each family and compute the prob- cause tetraploids have four alleles at a lo-
entage is known, the first line of the family ability that each progeny is selfed. The cus, there are four inbreeding coefficients
is the parent genotype, and this genotype statistical properties and uses of such in- inferred in parents, involving combina-
2 3 6 The Journal of Heredity 1990:81(3)
tions of two, three, and four alleles. Indi- proaches to mating system analysis (such application of a statistical method for evaluating its
vidual-plant estimates may also be per- as the generalized, nonreductionist, but importance. Am J Bot 1987; 74:1173-1183.
formed. aparametric mating-preference models of 9. Lynch M. Estimation of relatedness by DNA finger-
print. Mol Biol Evol 1988; 5.584-599.
7. FERNT. In vascular cryptograms, ga- Miiller-Starck and Gregorius10), require
10. Muller-Starck G, and Gregorius H-R. Analysis of
metophytes may self (intragametophytic entirely different models and computer mating systems in forest trees. In: Proceedings of the
selfing) and sporophytes may also self (in- programs. The programs offered in this Second International Conference on Quantitative Ge-
tergametophytic selfing, the normal type note are meant to encourage further work netics (Weir BS, Eisen EJ, Goodman MM, and Namkoong
G, eds). Held at North Carolina State University, Ra-
of selfing). This program jointly estimates on plant mating systems, both theoretical leigh, NC, May 31-June 5, 1987. Sunderland, Massa-
the rates of intragametophytic selfing vs. and empirical. chusetts: Sinauer; 1988:573-595.
intergametophytic selfing, based on two- 11. Ritland K. Correlated matings in the partial selfer,
From the Department of Botany, University of Toronto,
locus genotypic frequencies. Thus, it re- Canada. This research was supported by grants from Mimulus guttatus. Evolution 1989; 43:848-859.
quires at least two polymorphic loci, and NSERC of Canada. Address reprint requests to Kermit 12. Ritland K. Joint maximum likelihood estimation of
it also assumes inbreeding equilibrium. Ritland, Department of Botany, University of Toronto, genetic and mating structure using open-pollinated
Toronto, Ontario M5S 3B2, Canada. progenies. Biometrics 1986; 42:25-43.
This is the only program that does not
require progeny arrays. The model is de- 13. Ritland K. The effective proportion of self-fertiliza-
tion with consanguineous matings in inbred popula-
scribed in Ritland, D. Soltis, and P. Soltis.17 tions. Genetics 1984; 106:139-152.
Hedrick7 and Holsinger8 discuss other es- 14. Ritland K, and El-Kassaby YA. The nature of in-
1. Barrett SCH, and Shore JS. Variation and evolution
timation procedures that are based on sin- of breeding systems in the Tumera ulmifolia L. com- breeding in a seed orchard of Douglas-fir as shown by
gle-locus models. plex (Turneraceae). Evolution 1987; 41:340-354. an efficient multilocus model. Theor Appl Gen 1985;
The use of these programs involves 2. Brown AHD, and Allard RW. Estimation of the mating
system in open-pollinated maize populations using 15. Ritland K, and Ganders FR. Covariation of selfing
models and assumptions that are dis- isozyme polymorphisms. Genetics 1970; 66:133-145. rates with parental gene fixation indices within pop-
cussed further in the references and doc- ulations of Mimulus guttatus. Evolution 1987; 41:760-
3. Charlesworth D. A method for estimating outcross- 771.
umentation for these programs. These ing rates in natural populations of plants. Heredity
1988;61:469-471. 16. Ritland K, and Jain SK. A model for the estimation
programs are occasionally updated, and of outcrossing rate and gene frequencies using n in-
current users should write for the newest 4. Clegg MT, Kahler AL, and Allard RW. Estimation of dependent loci. Heredity 1981; 47:35-52.
versions as well as for information about life cycle components of selection in an experimental
plant population. Genetics 1978; 89:765-792. 17. Ritland K, Soltis D, and Soltis P. A two-locus model
other programs written since this note was for the joint estimation of intergametophytic and in-
5. Ellstrand NC. Multiple paternity within the fruits of tragametophytic selfing rates. J Hered 1990 (in press).
published. These programs are not meant the wild radish Raphanus sativus. Am Nat 1984; 123:
to be the be-all and end-all of mating sys- 810-828. 18. Schoen DJ. Mating system estimation via the one
pollen parent model with the progeny array as the unit
tem analysis. Other classes of data, such 6. Fyfe JL, and Bailey NTJ. Plant breeding studies in of observation. Heredity 1988; 60:439-444.
as DNA fingerprints (which are difficult to leguminous forage crops. 1. Natural crossbreeding in
19. Schoen DJ, and Clegg MT. Estimation of mating
interpret genetically9) and fitness in nat- winter beans. J Agric Sci 1951; 41:371-378.
system parameters when outcrossing events are cor-
ural populations (e.g., using the viabilities 7. Hedrick PW. Population genetics of intragameto- related. Proc Nat Acad Sci USA 1984; 81:5258-5262.
phytic selfing. Evolution 1987; 41:137-144.
of selfed vs. open-pollinated progenies to 20. Shaw DV, Kahler AL, and Allard RW. A multilocus
8. Holsinger KE. Gametophytic self-fertilization in estimator of mating system parameters in plant pop-
estimate outcrossing3), as well as other ap- homosporous plants: development, evaluation, and ulations. Proc Nat Acad Sci USA 1980; 78:1298-1302.
Brief Communications 2 3 7