Field Crops Research 114 (2009) 361–373
Contents lists available at ScienceDirect
Field Crops Research
journal homepage: www.elsevier.com/locate/fcr
Grain yield increase in cereal variety mixtures: A meta-analysis of field trials
Lars P. Kiær a,b,*, Ib M. Skovgaard b, Hanne Østergård a
a
b
Biosystems Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark DTU, Frederiksborgvej 399, DK-4000 Roskilde, Denmark
Department of Basic Sciences and Environment, Faculty of Life Sciences, University of Copenhagen, Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark
A R T I C L E I N F O
A B S T R A C T
Article history:
Received 7 June 2009
Received in revised form 7 September 2009
Accepted 8 September 2009
Plant ecology theory predicts that growing seed mixtures of varieties (variety mixtures) may increase
grain yields compared to the average of component varieties in pure stands. Published results from field
trials of cereal variety mixtures demonstrate, however, both positive and negative effects on grain yield.
To investigate the prevalence and preconditions for positive mixing effects, reported grain yields of
variety mixtures and pure variety stands were obtained from previously published variety trials,
converted into relative mixing effects and combined using meta-analysis. Furthermore, available
information on varieties, mixtures and growing conditions was used as independent variables in a series
of meta-regressions. Twenty-six published studies, examining a total of 246 instances of variety
mixtures of wheat (Triticum aestivum L.) and barley (Hordeum vulgare L.), were identified as meeting the
criteria for inclusion in the meta-analysis; on the other hand, nearly 200 studies were discarded. The
accepted studies reported results on both winter and spring types of each crop species. Relative mixing
effects ranged from 30% to 100% with an overall meta-estimate of at least 2.7% (p < 0.001),
reconfirming the potential of overall grain yield increase when growing varieties in mixtures. The mixing
effect varied between crop types, with largest and significant effects for winter wheat and spring barley.
The meta-regression demonstrated that mixing effect increased significantly with (1) diversity in
reported grain yields, (2) diversity in disease resistance, and (3) diversity in weed suppressiveness, all
among component varieties. Relative mixing effect was also found to increase significantly with the
effective number of component varieties. The effects of the latter two differed significantly between crop
types. All analyzed models had large unexplained variation between mixing effects, indicating that the
variables retrievable from the published studies explained only a minority of the differences among
mixtures and trials.
ß 2009 Elsevier B.V. All rights reserved.
Keywords:
Barley
Crop diversity
Cultivar mixtures
Genotype–environment interactions
Heterogeneous environments
Meta-regression
Relative mixing effect
Wheat
1. Introduction
The mixed cultivation of different varieties of a crop species in
varietal seed mixtures represents a low-tech method to increase
and stabilize grain yields and to reduce the dependence on
pesticides (Smithson and Lenné, 1996). Cultivation of variety
mixtures of various crops is a characteristic trait of subsistence
agriculture (Harlan, 1975), and it has gained increasing importance
in industrialized countries also (e.g. Østergård and Jensen, 2005;
Finckh and Wolfe, 1997; Wolfe et al., 2008). In variety mixtures,
two or more component varieties are grown concurrently within
the same field, introducing diversity to the crop stand. The
hypothesis is that genetic, physiological, structural and phenological diversity among component varieties may drive beneficial
* Corresponding author at: Biosystems Division, Risø National Laboratory for
Sustainable Energy, Technical University of Denmark DTU, Frederiksborgvej 399,
DK-4000 Roskilde, Denmark. Tel.: +45 46774107.
E-mail address: [email protected] (L.P. Kiær).
0378-4290/$ – see front matter ß 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.fcr.2009.09.006
interactions between varieties and between varieties and environments. For example, the commercial cultivation of cereal variety
mixtures has been driven primarily by the aim of controlling foliar
diseases through the introduction of diversity in disease resistance
genes (Finckh and Wolfe, 1997). The possible benefits include
better overall utilization of resources and buffering against
variation in environmental factors, potentially resulting in higher
and more stable crop yields (Simmonds, 1962; Wolfe et al., 2008).
The former is the focus of the present study; for a review of the
latter, see Piepho (1998).
There is an ongoing effort to design favourable cereal variety
mixtures for various growing conditions and to learn about the
influence of various environmental and varietal characteristics
(e.g. Cowger and Weisz, 2008; Kaut et al., 2008; Newton and Guy,
2009). Convincing increases in grain yield have generally been
reported for cereal variety mixtures (e.g. see the papers in Further
reading), and positive overall differences, or mixing effects, were
found for reported grain yields in the review of Smithson and
Lenné (1996). However, in specific cases, many negative mixing
effects have also been reported, and most often both positive and
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L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
negative mixing effects are observed in the same trial (e.g. Finckh
and Mundt, 1992; Jedel et al., 1998). The mixing effect of a specific
variety mixture may be difficult to predict, partly due to the
complex processes underlying crop interactions and the
uncontrollable factors characteristic of many field trials. Consequently, results of individual trials may be limited in their
relevance to other mixtures and other growing conditions, and
information about underlying mechanisms may not be revealed.
One possible answer to this problem is to combine the mixing
effects found in different field trials, and to relate these to the
circumstantial differences between trials.
Meta-analysis, as a well-established statistical method, is
becoming the standard approach for evaluating experimental
results across studies, analysing the influence of various experimental factors, and assessing publication bias, that is, the tendency
of published results within a given research area to be more
significant than given by the meta-population of effects being
considered. Furthermore, variation between studies can be
modelled explicitly in the analysis. In this respect, a meta-analysis
differs fundamentally from other review types such as narrative
reviews and vote counting, being a quantitative synthesis of
reported results. Having been developed initially in educational
psychology (Glass, 1976) and medicine (Mann, 1990) it has been
increasing applied in ecology, starting with the work of Gurevitch
et al. (1992). Within agronomical sciences, meta-analysis is a
relatively novel method (e.g. Rosenberg et al., 2004; Tonitto et al.,
2006; Rotundo and Westgate, 2009).
In this study, we used established meta-analysis techniques to
investigate overall effects on grain yield when cereal varieties are
grown in mixtures, as compared to the average yield when
varieties are grown in pure stand. Furthermore, we analyzed to
which extent this relates to a number of mixture characteristics
and growing conditions. Previous reviews of cereal variety
mixtures have been narrative (e.g. Finckh et al., 2000; Mundt,
2002a) or semi-quantitative (Smithson and Lenné, 1996). To our
knowledge, this is the first quantitative review of cereal variety
mixtures. Wheat (Triticum aestivum L.), barley (Hordeum vulgare
L.) and oat (Avena sativa L.) were first chosen as focal species.
These are important cereal crops around the world, and results
from field trials of variety mixtures of each of these have been
published. An important issue in meta-analysis of field trials is
how to transform the reported measures of experimental
variation to mixing effect variances. Only a small number of
studies have previously used meta-analysis to combine data
from field trials (e.g. Miguez and Bollero, 2005; Leimu and
Koricheva, 2006; Paul et al., 2007). Very few of these have
addressed directly the issue of variation, which we will therefore
also do in this study.
2. Materials and methods
2.1. Collection of studies
2.1.1. Database retrieval
The main criterion for inclusion of studies in the meta-analysis
was publication in a peer-reviewed journal included in The Science
Citation Index Expanded database (Web of Science, 2008),
spanning the period from 1900 to 19 January 2008. A wide
Boolean search was made on all possible combinations of typically
used wordings for variety mixtures and the common names for the
crop species of interest (Table 1). For consistency, we chose not to
include any unpublished results.
2.1.2. Filtering
As a first coarse filtering, only references contained in the
subject categories Agricultural Engineering, Agronomy, Biology,
Table 1
Boolean search expression for all possible combinations of a common variety
mixture term and a common term for the crops of interest. An asterisk denotes a
wildcard, representing any number of characters, including blanks.
Variety mixture terms
Crop terms
mix*
OR blend*
OR biblend*
OR multiblend*
OR multiline*
OR ‘‘inter* competition’’
OR ‘‘heterogeneous population*’’
OR ‘‘population diversity’’
OR ‘‘co* * genotypes’’
wheat
OR wheats
OR triticum
OR barley
OR barleys
OR ‘‘hordeum vulgare’’
OR oat
OR oats
OR ‘‘avena sativa’’
OR cereal
AND
Ecology, Multidisciplinary Agriculture, and Plant Sciences (in total
3526) were retained.
The titles and abstracts of the 3526 references were then
examined in order to identify potentially relevant studies.
References related to subjects such as livestock, intercropping,
toxicity and genetics were discarded, as were references with a
strictly phytopathological focus.
For a study to be accepted for the meta-analysis, a number of
criteria had to be met. A prerequisite was that the study provided a
relevant measure of experimental variation (see below). In the
optimal case, the study provided retrievable yields for mixtures as
well as component varieties in pure stand, either in absolute values
or relative to some standard yield. Acceptable exceptions were
studies reporting average yields in either of two cases: (1) across
component varieties for each mixture grown, as such values
corresponded with our method of calculating mixing effect (see
below) and (2) from field trials (see below) repeated over several
sites and/or years, but only when the number of these was
apparent and a measure of variation relating directly to the
estimates was retrievable. However, average yield results combined from different treatments could not be accepted. Finally, a
few studies were discarded due to the spatial designs of trials, and
a few studies were discarded due to selective reporting of results
(apparent or stated).
2.2. Data extraction
2.2.1. Field trial results
Most retrieved studies reported results from a single field trial,
whereas a few reported more than one. Yield data and experimental information were extracted from all obtained trials. Most
yield data were obtained directly from the presented tables,
whereas a few were obtained from digitized figures.
All included field trials were structured in either randomized
complete block designs or split-plot designs with replication.
Within each subplot of a trial plot was grown a genotypic entity,
which could be either a variety or a seed mixture of different
varieties. In most trials, more than one mixture was grown. Most
trials considered only one such replicated design within a given
locality within a given year (the combination hereafter termed
environment), whereas some trials included one such design in
two or more environments. For the split-plot designs, results
were always extracted separately for each level of the main-plot
factor, in effect making each such sub-experiment a randomized
block design. Considering b randomized and completely
replicated blocks nested within a number of environments,
possibly just one, a model for the yields in a variety trial is then
defined as
Y i jk ¼ mi þ A j þ B jk þ C i j þ ei jk ;
(1)
L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
where Yijk is the grain yield of the ith of v genotypic entities in the
kth of b blocks in the jth of e environments, mi is the expected mean
yield of genotypic entity i, while Aj, Bjk, Cij and eijk are independent
normally distributed random variables with variances s 2A , s 2B , s 2C
and s 2e , respectively. For all studies, estimates m̄1 ; . . . ; m̄v of the
mean yield for each genotypic entity were assumed to be
estimated from balanced variants of this model, using simple
averages and ordinary variance estimates. All reported yields were
transformed to the same scale (kg/ha) prior to further analysis.
The variance reported for each trial was used to estimate the
variance of the mixing effect(s) deriving from it. The applied
procedure, being described in detail in Appendix B, essentially
relies on the ability to compute the variance of a contrast between
two genotypic entities. The accepted studies reported, for each
trial, a table of variance components and/or an overall measure of
variation. Whenever available, estimates of variance of the mixing
effects were calculated directly from the variance components.
Overall measures of variation addressed either observations of
separate genotypic entities (coefficient of variation, CV, or mean
squared error, MSE), the mean of a group of observations (standard
error of mean, SE/SEM), or the pair-wise comparisons between
mean estimates (standard error of differences, SED, or least
significant difference, LSD). For all of these measures, a 0.05 level of
significance was assumed if nothing else was stated.
Among the field trials spanning more than one environment,
some were analyzed within each environment separately, whereas
others were analyzed in one overall analysis of all environments. In
those cases where an overall measure of variation was reported it
was not always clear whether the variance component for
environment, s 2A , had been included in the reported measure
together with s 2C and s 2e (cf. Appendix A). Despite being correct for
other purposes, such inclusion results in overestimation of the
contrast variances used to estimate the variance of the mixing
effects. We decided to include these trials anyway, as the result
was then at worst a downweighting of those mixing effects in the
meta-analysis (see below), thus doing no more harm than omitting
them.
2.2.2. Independent variables
The information retrieved and used in the meta-analyses as
independent variables was describing mixture properties (crop
type, indicators of diversity among component varieties, and
number of component varieties) and trial properties (seeding rate,
latitude, and altitude). Crop type was considered as a factor and
included in all analyses of the remaining independent variables.
Some variables were retrievable from all studies, whereas others
were retrievable from subsets of the studies.
Three indicators of diversity among the component varieties of
each mixture were defined, two being dummy variables that
described the criteria for mixing component varieties of a mixture,
namely diversity in disease resistance characteristics and diversity
in characteristics related to weed suppression (i.e. diversity in
plant height and/or earliness). For each of these criteria, mixtures
were designated the value of 1 when diversity was reported as a
mixing criterion in the study, or the value of 0 when nothing was
reported or similarity was reported as a mixing criterion.
The third indicator, component yield diversity (CYD), we
calculated as the coefficient of variation in the reported pure
stand grain yields of component varieties, weighting the yields of
components according to their proportion in the mixture, so that
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pk
2
SEweighted
i¼1 pi ðm̄i m̄ð p ÞÞ
CYD ¼
;
¼
Eweighted
m̄ð p Þ
P
where m̄ð p Þ ¼ ðm1 p1 þ . . . þ m̄k pk Þ is the expected mixture
yield, p1, . . ., pk denote the proportion of each of the k components
363
P
in the mixture, so that
pi ¼ 1, and m̄i are the corresponding
mean yields of varieties. Three studies involved multiline mixtures
of winter wheat (Gill et al., 1977, 1979; Malik et al., 1988). As these
reported only mean yields of parental pure stands, it was not
possible to calculate component yield diversity for mixtures in
these studies. To avoid a reduction in the number of observations
when modelling the influence of component yield diversity in
combination with covariates that was retrievable from all entries,
the mean component yield diversity value of the remaining winter
wheat studies was imputed for entries from these three studies.
The number and proportions of component varieties were
available for each mixture. A measure of the effective number of
components (ECN) in a mixture was defined as
ECN ¼ ð1 gð pi ÞÞ k;
where g(pi) is the Gini coefficient of inequality, equal to the average
of the absolute values of the proportion differences between all
pairs of component varieties (Gini, 1955). The Gini coefficient
ranges between 0 and 1, with 0 indicating perfect equality.
The variables describing trial properties, that is, site latitude,
site altitude, and seeding rate, were used as descriptors of the
growing conditions of the crop plants. These variables were
retrievable from 26, 13, and 15 of the studies, respectively. Some
information on latitude was obtained from the publications
directly, while some was derived by investigating further the
described field locations. All trials but one were conducted in the
northern hemisphere; and prior to analysis latitudes were
transformed to distance to the equator. For trials where results
were combined across multiple sites the average latitude was used,
since the sites were in all cases relatively closely located. Altitude
was used directly in terms of meters above sea level. Seeding rate
was standardized to number of seeds per m2.
2.2.3. Measure of relative mixing effect
To describe the grain yield difference of growing varieties in
mixture compared to pure stands, a measure of relative mixing
effect was calculated for each mixture as
erel ¼
m̄mix Pk
Pk
i¼1
i¼1
pi m̄i
pi m̄i
;
(2)
P
where m̄mix is the observed mean yield of the mixture, ki¼1 pi m̄i is
the weighted mean yields of the component pure stands, and pi, m̄i ,
and k are defined as above. In the rest of the paper, use of the
wording mixing effect refers to this relative measure of mixing
effect. The variance of this measure is approximately (see
Appendix B)
P
2
1 þ ki¼1 p2i v̂
Varðerel Þ ;
2
Pk
i¼1 pi m̄i
(3)
2
where v̂ is given in Appendix B. Notice that average component
performance is estimated more precisely than mixture performance, being averaged over k times as many observations as the
mixture. For this reason, the estimated variance is expected to
decrease with increasing number of components, and mixtures
with higher numbers of components are expected to contribute
more to the final meta-estimate.
2.3. Meta-analysis
2.3.1. Random effects model
An overall meta-estimate of mixing effect was derived by
considering the mixing effects (erel) as i in 1, . . ., m independent,
asymptotically normally distributed, and approximately unbiased
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L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
effect size estimates:
Y i ¼ u þ ui þ ei ;
(4)
where ei Nð0; s2i Þ, and s2i are the assumed known variances of Yi,
var(Yi) = var(erel), calculated according to Eq. (3). Then, u denotes
the central meta-effect, ui Nð0; t 2 Þ, where t2 denotes the
between-effects random variation, or residual heterogeneity. To
estimate u, the original effect size estimates were first weighted
with the inverse of their respective variances, divided by the sum
of all inverse variances (Hedges and Olkin, 1985).
2.3.2. Mixed effects model
The significance of the residual heterogeneity, t2, was tested in
each meta-analysis (see below). Furthermore, this between-effects
random variation was sought explained with the independent
variables described above by including them as explanatory factors
and covariates in the model, leading to a mixed effects metaanalysis model of the form
ûi ¼ b0 þ b1 X 1 þ . . . þ bq X q þ ui þ ei ;
(5)
where b0, b1, . . ., bq are regression coefficients, X1, . . ., Xq are vectors
of observations for each of q covariates, or dummy variables coding
the effects of the factors, and ui and ei are defined above. The
significance of the relationship between each covariate and the
effect sizes in such a meta-regression model was evaluated by
testing the hypothesis that the corresponding fitted parameter was
zero. The amount of residual heterogeneity was then evaluated as
described below.
2.3.3. Estimation and test of residual heterogeneity
Residual heterogeneity was estimated using restricted maximum-likelihood (REML) estimation (Viechtbauer, 2005). The
absence of heterogeneity is usually tested using Q (Cochran,
1954), which under a fixed effects H0 (t2 = 0) is given as the
weighted sum of squared differences between individual mixing
2
P
2
effects and the meta-effect, Q ¼ m
i¼1 wi ðû i u Þ , where wi ¼ n̂i .
2
Under the hypothesis, Q follows a x distribution with m 1
degrees of freedom (df), with m being the number of effect sizes
included in the meta-analysis.
For comparability reasons, Q may be better reported as the
percentage of variation across effect sizes that is due to
heterogeneity rather than chance (Higgins and Thompson, 2002;
Higgins et al., 2003), having I2 = 100% (1 df/Q). In contrast to Q,
I2 can be directly compared between meta-analyses with different
numbers of studies and different combinations of covariates, and it
was thus used to quantify the importance of introducing a
covariate or a factor to a meta-regression model.
2.3.4. Modelling approach
First, we tested whether there was an overall effect of mixing,
using a model similar to Eq. (4). Using a meta-regression model
with crop type as factor variable, we then tested the hypothesis
that crop types had similar overall mixing effects, with the
alternative hypothesis that they were different.
Crop type specific relationships between the remaining
independent variables (covariates) and mixing effect were then
tested in parallel. Thus, for each covariate, a meta-regression
model was constructed following the form ûi ¼ u þ b1;cðiÞ þ
b2;cðiÞ X 2 þ ui þ ei , where c(i) denotes the crop type of the ith
mixing effect, b1,c(i) denotes the crop type specific intercept, b2,c(i)
denotes the crop type specific regression coefficient in relation to
the covariate, and the rest are defined as above. This model was
used to test the hypothesis that crop types had the same
relationships with the variable (identical regression coefficients).
If this hypothesis was acceptable the meta-regression model was
reduced to the form û i ¼ u þ b1;cðiÞ þ b2 X 2 þ ui þ ei , where b2 is the
common regression coefficient in relation to the covariate. This
model was then used to test whether mixing effect did vary with
the variable at all. The marginal explanatory ability of each
covariate was evaluated by the reduction in I2 accomplished when
including the covariate in a model, as compared to the model
without it.
Following these regressions, all covariates with crop type
specific or general relationships with mixing effect were combined
in a (multiple) meta-regression model (q > 2 in Eq. (5)), using no
interactions between covariates other than those with the crop
type factor. Insignificant terms in this model were then removed
step-wise (one at a time) until a model with only significant terms
was obtained.
All meta-analyses were carried out running the metafor (W.
Viechtbauer) package in R 2.9.1 (R Development Core Team, 2009),
in which all other computations were also run. As the algorithms in
metafor are not able to handle factors, crop type was coded as three
dummy variables. The test of the hypothesis of similar mixing
effects of crop types and the tests of the hypotheses of crop type
specific relationships with each covariate were done with Wald
tests at the 5% level, using the covariance matrix of the estimates
for calculation of the test statistic.
2.4. Sensitivity and bias analysis
In addition to the described tests of residual heterogeneity, the
robustness of the results was investigated with an approach
similar to jackknifing; the relative changes in parameter estimates
and/or significance levels of covariates were assessed by leaving
out trials one at a time and re-doing each meta-analysis. Using this
leave-one-out procedure, the sensitivity of the covariate parameter estimates (see above) to the effects studied was evaluated as
the proportional change following exclusion. Likewise, trials of
particular influence on parameter estimates could be identified
and scrutinized. Based on the reduced data set without these
outliers, meta-regressions of each and all covariates were then
made in parallel to the approach described above.
The extent of publication bias was assessed by use of standard
regression and rank-based procedures. Both techniques are based
on the expectation that effect sizes will be symmetrically
distributed around the overall meta-estimate, with more precise
effect size estimates lying closer to the meta-estimate. Absence of
the most positive or negative effect sizes due to publication bias
would then lead to asymmetry in this funnel-shaped relationship
between effect sizes and corresponding standard errors. This was
first assessed by using the standard errors of effect sizes as a
regression covariate (Egger et al., 1997). Furthermore, the number
of effect sizes ‘missing’ in order to obtain funnel plot symmetry
was estimated using the simple, nonparametric trim-and-fill
method (Duval and Tweedie, 2000), and the overall meta-estimate
obtained when including these was estimated.
3. Results
Database retrieval followed by examination of titles and
abstracts resulted in 213 potentially relevant reference studies
on field trials that involved variety mixtures of wheat, barley or oat
(list available from the authors). A total of 28 references fulfilled
the criteria for being included in the meta-analysis. Among these
were only two studies on oat variety mixtures, which were
considered an insufficient number for oat to be included. The
resulting 26 studies of wheat and barley (see Further reading)
provided 575 combinations of 246 different mixtures and 114
different trials (on average 5 mixtures reported from each trial),
and the trials were carried out in 12 different countries (Table 2).
L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
365
Table 2
Summary of the retrieved mixture data grouped per crop type.
Crop type
No. studies
Winter wheat (WW)
Spring wheat (SW)
Winter barley (WB)
Spring barley (SB)
12
5
4
5
Total
26
No. trials
No. mixtures
No. mixture results
Trial countries
54
28
16
16
118
22
34
72
238
87
118
132
Australia, Hungary, India, Nepal, Pakistan, USA
Canada, England, USA
Germany, USA
Canada, Germany, Northern Ireland, Wales
114
246
575
The studies were published from 1969 to 2005, and all but five
were reported in English (four in German and one in Hungarian).
3.1. Meta-regression
Effect sizes (relative yield increase in mixtures compared to
pure stands) ranged from 30% to +100%, with the overall metaestimate of mixing effect being significantly different from 0 (3.5%
(s.e. = 0.4); p < 0.001). Wheat and barley had overall mixing effects
of 3.9% and 2.6%, respectively (p < 0.001 in both cases). A metaregression of mixing effect against crop species, growing period
(winter or spring type), and the interaction of these identified that
the latter was significant (p < 0.001). Mixing effects differed
significantly between the four crop types, being significant within
the winter wheat and spring barley types (5.7% (s.e. = 0.6) and 5.1%
(s.e. = 0.8), respectively; in both cases p < 0.001), and insignificant
within the spring wheat and winter barley types (1.0% and 0.4%,
respectively). Accordingly, overall mixing effects were similar
within each pair of groups (not shown). For comparison, the simple
average across all effects was 3.7% (10.7), whereas simple averages
within winter wheat, spring wheat, winter barley, and spring
barley were 6.1% (11.4), 2.1% (11.8), 0.4% (7.1), and 6.0% (9.1),
respectively (standard deviations in parentheses). Crop type was
therefore included as a four-level factor in all subsequent
regressions involving the remaining covariates, the results of
which are presented in Table 3 and Fig. 1.
Disease resistance diversity was reported as a mixing criterion
in 60% (68) of all trials. Weed suppressiveness diversity was used
as a mixing criterion in 12% (14) of the trials. Based on the retrieved
set of trials, neither of these dummy variables were found to have a
significant overall effect on the mixing effect (Table 3).
Both of the remaining mixture characteristics, namely component yield diversity and effective number of components, on the
other hand, had a significant positive overall relationship with
mixing effect (Table 3). The grain yield increased on average 43%
with each unit change in component yield diversity and 1% per
additional effective mixture component. The relationship with the
effective number of components differed significantly between
crop types with estimates of 0.13 and 0.01 within spring wheat and
winter wheat, respectively (each significantly different from zero),
being significantly different between spring wheat and each of the
other three types (not shown).
Latitude turned out to be the only covariate related to trial
conditions that was retrievable from all studies. Mixing effects
changed significantly with distance to the equator within spring
wheat and winter wheat, the oppositely directed regression
estimates (0.069 103 and 0.023 103, respectively;
Table 3) being significantly different (p < 0.001). No changes were
found within spring barley and winter barley (Table 3; Fig. 1). The
reported grain yields of genotypic entities (mixtures as well as
component varieties) increased clearly (not shown) with distance
to the equator (ranging from 2933 to 6260 km).
The relationship between mixing effect and each of the
covariates site altitude and seeding rate was similar among crop
types and not significantly different from zero (Table 3; Fig. 1).
Reported grain yields decreased clearly (not shown) with altitude.
Through reduction of the multiple meta-regression model
combined from all significant terms, a model was obtained that
Table 3
Covariates investigated and number of studies, number of trials from which they were obtained, number of mixing effects with information on the covariate, test probabilities
(see foot note), estimates of change in relative mixing effect per unit change of the covariate (unit in brackets), and 95% confidence limits (CL). Where significant differences
between crop type specific regression coefficients were identified (p(H1)), estimates and CL are provided for each crop type. Probabilities given in bold indicate significance at
5% level.
Covariate (abbreviation)
Number of included
Hypothesis tests
p(H1)
a
Change in relative mixing effect
b
p(H2)
Estimate
95% CL
Studies
Trials
Effects
Disease resistance diversity used
as a mixing criterion, DRD
Weed suppression diversity used
as a mixing criterion, WSD
26
114
575
0.28
0.19
0.011
0.0056; 0.028
26
114
575
0.081
0.98
0.0002
0.018; 0.018
Effective no. of components, ECN
Winter wheat
Spring wheat
Winter barley
Spring barley
26
114
575
0.0017
0.0010
0.0099
0.010
0.13
0.0041
0.0075
0.0040;
0.0042;
0.065;
0.029;
0.040;
Component yield diversity, CYD
Seeding rate (m2)
23
15
109
67
514
341
0.35
0.95
Latitude (103 km)
Winter wheat
Spring wheat
Winter barley
Spring barley
21
109
500
<0.001
0.087
Altitude (m)
13
80
358
0.71
0.98
a
b
<0.001
0.58
0.43
0.018 103
0.0096
0.023
0.069
0.015
0.000
0.00 103
Test probability of the hypothesis H1 that the crop type specific regression coefficients of mixing effect on the covariate are identical.
Test probability of the hypothesis H2 that the common regression coefficient is significantly different from 0.
0.016
0.016
0.20
0.021
0.025
0.29; 0.57
0.08 103; 0.044 103
0.021;
0.036;
0.032;
0.016;
0.055;
0.0014
0.010
0.11
0.045
0.055
0.03 103; 0.03 103
366
L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
Fig. 1. Meta-regression plots of the relations between relative mixing effect and (a) reported use of disease resistance diversity as a mixing criterion, (b) reported use of
weed suppression diversity as a mixing criterion, (c) component yield diversity, (d) effective number of component varieties, (e) seeding rate, (f) latitude and (g)
altitude. Data points are shown for winter wheat (open triangles), spring wheat (crosses), winter barley (full circles), and spring barley (full squares). Regression lines
are shown for winter wheat (dashed), spring wheat (solid), winter barley (dotted), and spring barley (dot-dashed) in the range of the underlying covariate values.
Regression lines are only shown in cases where regression coefficients are significantly different from zero and/or significantly different between crop types (cf.
Table 3).
L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
described the combined relationship of mixing effect with crop
type (p < 0.001) and component yield diversity (CYD; p < 0.001),
as well as the crop type specific influence of the effective number of
components (ECN; p = 0.0024). This can be written as
ûi ¼ u þ b1;cðiÞ þ b2 CYDi þ b3;cðiÞ ECNi þ ui þ ei ;
where all are defined above.
3.2. Sensitivity and bias analysis
3.2.1. Sensitivity
The high level of variability in mixing effects is apparent from
the regression plots (Fig. 1). Still, the estimate of the relationship
between component yield diversity and mixing effect was
generally robust, as seen from the leave-one-out procedure. The
regression coefficient estimate changed 5% to 8% relative to the
estimate based on all trials, except for two trials that caused
changes of 28% and 23%, respectively, when excluded. The
estimate was consistently positive and remained highly significant
after all exclusions, with a level of variation that was generally low
(CV in the range 0.13–0.21).
The estimate of the influence of effective component number on
mixing effect was less robust. Relative changes in the regression
coefficient estimate were in the range 4% to 5%, except for six
trials. The exclusion of each of these resulted in relative changes in
the regression coefficient of 44%, 17%, 14%, 23%, 30%, and 48%,
respectively. Again, the estimate was consistently positive and
remained significant after all but one exclusion, namely the one
responsible for the largest reduction in the estimate (p = 0.077).
The level of estimate variation was moderate (CV in the range
0.22–0.57).
The trials causing extraordinary changes in the estimates of
component yield diversity were both reported by Kovacs and
Abranyi (1985; see Further reading). The extraordinary changes in
the estimates of effective number of components were caused by 2
out of 3 trials in Malik et al. (1988), the trial in Gallandt et al.
(2001), 1 out of 2 trials in Kovacs (1985), the trial in Gill et al.
(1977), and the trial in Gill et al. (1979), respectively (all listed in
Further reading). Combined, these ‘outlier’ trials contributed with
83 mixing effect sizes, all on winter wheat.
Based on the reduced data set of 492 mixing effects, a metaregression approach similar to that described above showed a
reduction in the overall mixing effect to 2.7% (p < 0.001) and a
reduction of the mixing effect in winter wheat to 4.6% (p < 0.001).
Mixing effects were still found to differ significantly between crop
types (p < 0.001). In general, the significance levels of the various
parameter estimates within the winter wheat group of trials did
not change as a consequence of trial exclusion (not shown), except
for weed suppressiveness diversity becoming significant
(p < 0.001). Furthermore, significant differences between crop
types were found with respect to both the effective number of
components and latitude, and effective number of components and
component yield diversity were each still found to have a
significant overall influence on mixing effect. Additionally, a
number of previously unidentified relationships became apparent
when analysing the reduced data set. Firstly, the general relationship between disease resistance diversity and mixing effect
became significant, with a positive parameter estimate of 0.021
(CL: 0.007; 0.035; p = 0.0034). No difference among crop types was
found. Secondly, the general relationship between weed suppression diversity and mixing effect became significantly positive
(p = 0.010), and a significantly different relationship was found
between crop types (p = 0.0034), the relationship being significant
within winter wheat (p < 0.001) and almost so within spring
wheat (p = 0.083), whereas not within the barley types.
367
Table 4
Parameter estimates of the reduced combined model based on the data set with
particularly influential trials excluded (see text).
Covariate
Overall
[WW]
[SW]
[WB]
[SB]
CYDa
DRDb
Latitude (103 km)
Change in relative mixing effect
Estimate
95% CL
0.17
0.27
0.12
0.05
0.27; 0.08
0.37; 0.18
0.2; 0.034
0.14; 0.045
0.22
0.036
0.018
ECNc
[WW]
[SW]
[WB]
[SB]
0.022
0.09
0.007
0.017
WSDd
[WW]
[SW]
[WB]
[SB]
0.053
0.031
0.058
0.009
a
b
c
d
0.083; 0.37
0.02; 0.052
0.005; 0.031
0.012; 0.031
0.04; 0.14
0.024; 0.01
0.039; 0.006
0.03; 0.075
0.0003; 0.063
0.003; 0.11
0.016; 0.033
Diversity in component variety yields.
Diversity in component variety disease resistance.
Effective number of components in mixtures.
Diversity in component variety traits related to weed suppression.
Reduction of the multiple meta-regression model combined
from these terms, still based on the reduced data set, resulted in a
model describing the concurrent relationship of mixing effect with
crop type, component yield diversity (CYD; overall), disease
resistance diversity (DRD; overall), latitude (LAT; overall), effective
number of components (ECN; crop type specific), and weed
suppression diversity (WSD; crop type specific):
ûi ¼ u þ b1;cðiÞ þ b2 CYDi þ b3 DRDi þ b4 LAT i þ b5;cðiÞ ECNi
þ b6;cðiÞ WSDi þ ui þ ei ;
(6)
where all are defined as above. The regression coefficient estimates
from a model equivalent to Eq. (6), leaving out the common
intercept, are provided in Table 4.
3.2.2. Publication bias
Assessments of publication bias were made for effects within
each crop type separately, given that these were found to have
different mean meta-estimates, and that the retrieved original
papers each reported on only one of the crop types. Funnel plots for
each of these groups are shown in Fig. 2. The regression test for the
spring barley group indicated funnel plot asymmetry (p < 0.001)
due to missing negative effect sizes (positive slope). The trim-andfill adjustment found that 24 negative effects of low-to-intermediate
precision were missing from this subset of effects (Fig. 2). Adding
these effects to the data set resulted in a much lower but still highly
significant overall meta-estimate of mixing effect in spring barley
variety mixtures (3.0%; p 0.001). For spring wheat, the regression
test indicated funnel plot asymmetry (p = 0.012) due to missing
positive effect sizes (negative slope). However, no effects were found
to be missing by the trim-and-fill method. The regression test for the
winter barley group gave no indication of funnel plot asymmetry
(not shown), and the trim-and-fill adjustment found a single effect of
low precision to be missing from this subset. Variation among
estimates of mixing effect within the winter barley group was as
large among the less precisely estimated effects as among the more
precisely estimated effects, leading to a ‘column’ rather than funnel
shape (Fig. 2). For winter wheat, regression test gave no indication of
funnel plot asymmetry (not shown), and the trim-and-fill adjust-
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L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
Fig. 2. Funnel plots for each of the crop types (a) winter wheat, (b) spring wheat, (c) winter barley, and (d) spring barley, as well as the combined groups (e) winter wheat plus
spring barley and (f) spring wheat plus winter barley, plotting relative mixing effects against their value of precision (inverse standard error, SE1). All original effects are
shown as filled circles, except the winter wheat effects identified by sensitivity analysis, which are shown as open squares. Effects suggested added by the trim-and-fill
method (see text) are shown as open circles.
ment found no effects to be missing from this subset. It appears from
the funnel plot (Fig. 2a) that the subset of effects from the
particularly influential winter wheat trials spanned the range of
precision in this group, and that these made up for the largest
deviations from overall funnel shape.
growing conditions, our analyses show that this covers significant
differences between crop types and relationships with a number of
covariates. These will be discussed in the following.
3.3. Unexplained variation between effect sizes
Despite large residual heterogeneity in all meta-regression
models, the conclusions drawn were fairly robust. A small number
of trials each had extraordinary influence on the parameter
estimates from meta-regression models. Yet, the estimates
remained consistently positive, and highly significant in all cases
except one. The result from simple meta-analysis based on all but
these trials supported those based on the full data set, and the
reduced multiple regression model included more significant
parameter estimates. Accordingly, the fit of the multiple regression
model was substantially better. It can be argued that by
disregarding a minor group of dominating data, additional
relationships between mixing effect and covariates were disclosed
and already identified relationships were strengthened.
Our results confirm the findings in the review of Smithson and
Lenné (1996) that relative mixing effects tend to be higher in
wheat as compared to barley. A significant overall meta-effect of
mixing was found within winter wheat and spring barley but not
within spring wheat nor winter barley. These inverse patterns in
the two crop species may derive from differences in cultivation
practice; in most countries included in this analysis, wheat has
usually been cultivated as a winter crop, whereas barley has
usually been cultivated as a spring crop (e.g. Fischbeck, 1989;
Webster and Williams, 1989). This tendency is likely to affect the
range and general availability of regionally adapted varieties.
Additionally, the more extensive cultivation of these crop types
could ultimately result in higher disease pressures, which again
could lead to larger mixing effects. Finally, cultivation practices are
likely to affect research experience with the four crop types; the
The amount of variation not explained by the model, the
residual heterogeneity, was generally large and highly significant
in all meta-regressions (p 0.001). Based on the full data set, each
covariate reduced residual heterogeneity by no more than 0.7%,
when compared to the model without covariates, except component yield diversity, which reduced residual heterogeneity by 3.9%.
The covariates in the reduced, combined model reduced heterogeneity by a total of 4.3%.
Despite removal of outlier trials, a significant level of unexplained
variation was still found in all models, although it was generally
reduced to approximately half of that in models including the seven
trials. This reduction was attributable principally to the removal of
the trials reported by Kovacs (1985). The covariates included in the
reduced, combined model jointly reduced this lower level of
heterogeneity by 24.2%. The crop type specific effective component
number, crop type specific weed suppression diversity, and
component yield diversity contributed with approximately equal
weight to this reduction, whereas disease resistance diversity and
latitude contributed with minor reductions.
4. Discussion
The results obtained through meta-analysis confirm the
potential of cereal variety mixtures as a means of obtaining
higher grain yields, on average, compared to growing the crop in
pure stand. Further, being based on a range of genotypes and
4.1. Meta-regression
L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
number of peer-reviewed scientific studies on winter wheat is
approximately double the number of studies on spring wheat, as
indicated by a search on Web of Science (2009), whereas the
inverse is true for the barley crop types. Our findings imply that
effects of variety mixing should always be approached at crop type
level rather than at species level.
A number of covariates related to component diversity were
found to have large influence on mixing effect. Hence, diversity in
component vigour in the specific growing environments (the
component yield diversity) was found to be more important for the
successful mixing of varieties than any other variable included in
the meta-analysis. This was found to be equally important within
all crop types, based on the unreduced as well as the reduced data
set, and component yield diversity was the variable being most
robust to removal of trials in the leave-one-out procedure. Based
on the unreduced data set, component yield diversity was in
principle the only source of explanation for variation in mixing
effect. The purpose of selecting and combining specific varieties
was not always addressed explicitly in the reported studies,
however, in none of them was large component yield diversity
reported as being deliberate. Clay and Allard (1969) used the
difference in yield between two varieties as a measure of
component diversity in a number of binary mixtures, but found
no consistent relation with mixing effect.
Furthermore, based on the reduced data set, mixing effect was
found to be generally higher in mixtures reported to be
diversifying disease resistance levels than in those that were
not. The general expectations from previous reviews of cereal
variety mixtures were thus confirmed (e.g. Smithson and Lenné,
1996; Finckh et al., 2000). Additionally, mixtures with reported
diverse weed suppressiveness were found to provide higher
overall yields than did mixtures for which this was not reported.
This latter relationship differed between crop types, being
significant for mixtures of winter wheat only and almost so for
mixtures of spring wheat. Many studies on the subject have shown
that wheat is generally more vulnerable to weed competition than
barley (e.g. Satorre and Snaydon, 1992 and references therein;
Dhima and Eleftherohorinos, 2001). Regarding the trials studied in
the present review, such general difference between the species
could explain larger beneficial interactions when combining
diverse competition-related traits in wheat mixtures.
The importance of component diversity in traits such as disease
resistance, weed suppressiveness, or any other type of environmental response, is conditioned by many factors in the individual
experiments, such as the level of occurrence of the relevant biotic
or abiotic stresses. Information on the actual presence of diseases
and weeds was reported in only few of the retrieved studies and
was therefore not useful for analysis. The realized advantage of
mixtures with diverse component responses could therefore be
anything from high to not present. With this in mind, it seems
likely that even stronger relationships would have been found had
this information been consistently available and accountable for.
Effective number of component varieties (ECN) in mixtures had
significant positive influence on mixing effect overall, but this
relationship was significant only for wheat mixtures. Crop type
specific ECN was part of the final model based on both the full and
the unreduced data set, respectively. Both the number (e.g. Huhn,
1985; Newton et al., 1997) and proportions (e.g. Sarandon and
Sarandon, 1995; Juskiw et al., 2001) of components in variety
mixtures have previously been shown to influence mixing effect.
The ECN embedded both elements, the expectation being that
component varieties with more uneven proportions will tend to
have a lower level of interaction and hence a smaller mixing effect.
Plant interactions such as competition for light, nutrients and
water are well documented to be larger among closely spaced
plants, yet, no relationship between seeding rate and mixing effect
369
was found. This could be due to lack of experimental information
on this topic in the majority of the retrieved trials. Another possible
explanation is that the effect of seeding rate on the level of crop
plant interaction, and hence mixing effect, may be non-linear,
increasing at lower rates and decreasing above a certain level. The
latter was examined further by fitting a meta-regression model
with a polynomial relationship between seeding rates and mixing
effects, however, even though the obtainable seeding rates covered
the spectrum well above and below ordinary rates, no significant
fit was obtained (not shown).
Analyses of the reduced and unreduced data sets both identified
significant differences among crop types in the relationship
between latitude and mixing effect. However, only after the
exclusion of outlier trials were these relationships strong enough
to matter in the final model. There, two significant but oppositely
directed crop type specific regression estimates were found, being
negative within winter wheat and positive within spring wheat. No
apparent explanation for this pattern was found.
4.2. Sensitivity and bias analysis
All of the outlier trials were on winter wheat, which was also
the crop type with the largest number of trials and mixing effects
contributing to the meta-analysis. It is therefore worth mentioning
that the significance levels of the various parameter estimates
within the winter wheat group of trials did not change as a
consequence of excluding these trials (not shown), except for weed
suppressiveness diversity becoming significant (p < 0.001).
The outlier trials were particular for differing reasons, each
highlighting important general aspects of the sensitivity of metaanalyses based on field trial data. The influential trials reported in
Kovacs and Abranyi (1985) were regular, medium-sized variety
trials, having large influence on the level of residual heterogeneity
in the data set. The reason for this appears to be a combination of a
few varieties providing extraordinarily low yields in pure stand,
leading to very large mixing effects, and a relatively small
experimental variation. The trial reported in Gallandt et al.
(2001) spanned 33 environments, resulting in the very smallest
variance of the derived mixing effects (app. six times as small as
the average; see the ‘tip’ of the funnel in Fig. 2) and consequently
the very highest weight in the analysis (cf. Appendix A). In
addition, the trial contributed with 15 effect sizes (the fourth
highest number from any trial). The remaining of the influential
trials (Gill et al., 1977, 1979; Malik et al., 1988) were all mediumsized multiline trials contributing with average sized mixing
effects and normal levels of experimental variation. The extraordinary impact of each of these trials on the influence of the
effective number of components on mixing effect was most likely
due to the high number of component varieties (6 or 7 in each). The
reason for this would be that (1) the approximated variance of the
relative mixing effect (Appendix B) was defined so that mixtures
with higher component number contributed more to the final
meta-estimate, and (2) overall 86% of mixing effects were based on
mixtures with 2 or 3 components, making the slope of the
relationship highly dependent on the mixing effect of trials with
such high component numbers (see Fig. 1d).
Meta-analyses within many research fields have indicated that
the results published on a given topic comprise a non-representative proportion of all studies made (e.g. Duval and Tweedie, 2000;
Jennions and Møller, 2002; Stanley, 2005). Typically, negative
effects of low-to-intermediate precision are missing, the interpretation being that results from smaller studies that contradict
the conventional notion are less likely to reach publication. In the
present study, the phenomenon was indicated for the retrieved
studies on spring barley variety mixtures. Assuming that these
were representative, this may reflect a deliberate effort to present
370
L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
the potential benefits of growing varieties in mixtures; however, it
is unresolved why no similar indications were found for the other
crop types. Perhaps a more important observation is that imputing
of the retrieved effects with those ‘missing’ did still result in a
significantly positive meta-estimate of mixing effect.
4.3. Unexplained variation between effect sizes
The available covariates covered the range of potential sources
of variation to a small extent. For example, plant characteristics of
direct importance for interspecific competition, such as earliness,
height or root growth, were reported for no more than 8%, 17% and
0% of the varieties, respectively, being of no use in meta-analysis.
Likewise, the studies gave limited information on environmental
factors of direct importance for plant growth, such as weeds,
diseases and nutrient levels. Accordingly, the meta-analysis
revealed a significant amount of unexplained variability in all of
the tested models, both in the reduced and unreduced data sets,
highlighting the problem of making general conclusions from
single field trials of variety mixtures.
Substantial between-study variance was also found in previous
meta-analyses of field trial data (e.g. Paul et al., 2006). Certainly,
studies of ecological phenomena are generally influenced by many
uncontrollable environmental factors and often vary significantly
in their estimates of effect. On the other hand, Shah and Dillard
(2006), in a meta-regression of yield losses due to common rust in
sweet corn, found that the factor variety explained 70% of the
identified heterogeneity. In that case, information on varietal
characteristics would be of greater importance than further
information on environmental factors. The sparse varietal and
experimental information in the retrieved studies, combined with
the large number of different mixtures and varieties among the
retrieved trials, impeded a similar analysis here.
4.4. Retrieval of information
Considering the number of published field trial studies of
cereal variety mixtures, relatively few met the criteria of
suitability for meta-analysis. In addition to the problem of
sparsely reported covariate information and condensed yield
reports, a surprisingly large proportion (more than half) of
otherwise suited studies did not report a measure of experimental variation. Poorly reported studies and lack of reported
variances have previously been mentioned as the biggest
challenge to meta-analysis in ecology (Gurevitch and Hedges,
1999). Given the solid statistical foundation of field trial designs,
one would expect this barrier to be less extensive for agricultural
data; however, the same has been repeatedly reported in
previous meta-analyses of crop performance (e.g. Vila et al.,
2004; Tonitto et al., 2006).
Weighting on the basis of variances is the most common
method of combining effect sizes in ecological meta-analyses, and
has also been used in meta-analyses of agronomic effects identified
from reported field trials (e.g. Ainsworth et al., 2002). However,
alternatives are frequently used, including equal weighting (e.g.
Varðm̄1 m̄2 Þ
¼ Var
¼ Var
¼ Var
Tonitto et al., 2006; Rotundo and Westgate, 2009) and weighting
by sample size (e.g. Hedges and Olkin, 1985; Gurevitch and Hedges,
1999; Leimu and Koricheva, 2006). The latter is often set to the
number of replications (e.g. Olkin and Shaw, 1995; Adams et al.,
1997; Paul et al., 2005). Using alternative weighting could thus
have increased the number of suited studies, and hence the
number of retrieved effect sizes. On the other hand, it would also
have yielded meta-results of poorer precision than an appropriately weighted average (consider for example the standard
errors of the crude means).
5. Conclusion
By means of meta-analysis, we were able to consolidate the
potentials of seed mixtures of wheat and barley to provide increased
grain yields. Additionally, our results support that this is crop type
specific. We were also able to demonstrate significant relationships
with the effective number of component varieties in mixtures, as
well as several types of diversity between components, thus
supporting the hypothesis that positive mixing effects may derive
from beneficial interactions between the component varieties.
Our analyses thus demonstrate the potential of meta-analysis
as a method of combining published results from field trials to test
agronomically relevant hypotheses. On the other hand, the
explanatory power of the retrievable variables was relatively
low, implying that the obtained results may be of low predictive
utility. This was probably due to the fact that only few variables
were available, and that none of these described directly the
growing environment of the crops.
In order to understand and predict more optimal variety
mixtures of wheat and barley we find that: (1) further work should
try to separate the effects of the potential mechanisms and
interactions acting in variety mixtures; (2) more information on
the growing conditions of varieties and mixtures should be collected
and reported from original field trials; and (3) retrievable measures
of trial variation should be reported to a larger extent in order to
facilitate more substantial overall (meta-)analyses of mixing effects.
Acknowledgements
The authors would like to thank colleagues in SUSVAR (COST
action 860) for fruitful early discussions on the potential for metaanalysis of variety mixtures, and the anonymous reviewers for
suggestions that improved the manuscript. The work was partially
funded by a grant from the Research School for Organic Agriculture
and Food Systems.
Appendix A
Consider any variety trial design that fits into the framework of
Eq. (1), for example a complete balanced block design, nested within
several environments. The variance of a linear contrast between the
estimated means for two genotypic entities from such a trial, m̄1 and
m̄2 , say, is then
P P
j
P P
j
P
k Y 1 jk
Y 2 jk
be
ð
k m1 þ A j þ B jk þ C 1 j þ e1 jk Þ ðm2 þ A j þ B jk þ C 2 j þ e2 jk Þ
jC1 j
e
C2 j
PP
þ
j
e
k ð 1 jk
¼2
be
2
be
e2 jk Þ
s2
þ e ¼ 2v2
e
be
sC
;
L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
where v2 is defined by the last equality. This calculation uses the
convenient design feature that genotypes share environment and
block and hence share the A and B terms in the model.
2
Having provided the mean square components MSe ¼ ŝ e and
2
2
2
MSC ¼ b ŝ C þ ŝ e , for example, in an ANOVA table, v can be estimated
by solely extracting the mean square of the interaction:
v̂2 ¼
ðMSC MSe Þ=b MSe MSC
þ
¼
e
be
be
In single-site trials there is only one environment behind each
estimated effect measure, hence MSC is zero, and
MS
v̂2 ¼ e ;
be
where we keep the denominator for compatibility, although e = 1
in this case.
Some studies do not provide the MS quantities directly, but report
instead some other measure of precision, yielding another expression
2
of v̂ . These measures and expressions are given below.
SE of the mean (for a genotypic entity), sometimes denoted SEM, is
the standard error of a sample mean of a genotypic entity, which in a
single-environment trial is
ŝ e
SEM ¼ pffiffiffiffiffiffi ;
be
so that
v̂2 ¼ SEM2 :
Note that, for a multi-environment trial, SE of the mean involves
the variance components for environments and blocks, and the
shown relationship would overestimate v2, resulting in downweighting of effects in the meta-analysis.
CV is the coefficient of variation, defined as
CV ¼
ŝ e
:
m̄i
It is occasionally reported as a measure of variation, giving
v̂2 ¼
m̄2i CV 2
be
;
again with e = 1 for single-environment trials, while it provides
insufficient information in a multi-environment trial. In this
expression the mean yield for the entire trial was used.
MSE, or MSe as denoted above, may be obtained as the residual
mean square from an ANOVA table, estimating s 2e so that
v̂2 ¼
MSE
:
be
SED, the standard error of differences between any two varietal
estimates, is given as
qffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
SEDðm̄1 m̄2 Þ ¼ Varðm̄1 m̄2 Þ ¼ 2v̂ ;
yielding
v̂2 ¼
SED2
:
2
LSD, the least significant difference, being defined as
qffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
LSDðm̄1 ; m̄2 Þ ¼ t a=2;d f Varðm̄1 m̄2 Þ ¼ t a=2;d f 2v̂ ;
gives
v̂2 ¼
LSD2
;
2ta2 =2;d f
371
where df denotes the degrees of freedom within groups. LSDa was
occasionally termed GDa (German studies), CDa (Indian studies),
and SzD100Sa (Hungarian study), respectively.
Thus, for multi-environment studies SED, LSD or the variance
2
components were used to compute v̂ , while any of the measures
listed above could be used for single-environment trials. In rare
cases, a trial may be based on more than one environment and yet
report an estimate of precision that reflects only the residual
variation. If such cases report that genotype–environment interactions were tested for and not found significant it would still be
possible to extract the correct variance estimate. Alternatively,
variation would be somewhat underestimated, resulting in a
relatively higher weighting of effect sizes in the meta-analysis. On
the other hand, estimates from single-environment studies also, by
design, do not include a genotype–environment interaction in the
analysis, and hence these variances similarly, although by necessity,
ignores the contribution to the effect variance from such an
interaction and places this part of the variation in the betweenstudy variance. Thus, in both cases the interaction between
genotypes and environment becomes part of the heterogeneity
variance component in the meta-analysis. Refinements of the metaanalysis regarding the variance model of the heterogeneity,
distinguishing how many years and places are behind each effect
estimate, have not been attempted in the present paper, as it requires
substantial theoretical and logical development of the theory of
meta-analyses.
Appendix B
The variances of relative mixing effects were derived from the
2
estimates of the quantity v̂ , as given in Appendix A. The variance
estimate of the difference between the yields of a variety mixture, m̄,
P
and the weighted average yield of its k components, c̄ ¼ ki¼1 pi m̄i , is
thus given as
!
k
X
2
2
Varðm̄ c̄Þ ¼ 1 þ
pi v̂ ;
i¼1
where p1, . . ., pk denote the proportion of each of the components in
P
the mixture, so that
pi ¼ 1. The variance of the relative mixing
effect (see Eq. (2))
m̄ c̄
erel ¼
c̄
cannot be calculated exactly, since it is a non-linear function of the
data. A standard variance approximation for the ratio, using a firstorder Taylor approximation, gives
m̄ c̄
Varðerel Þ ¼ Var
c̄
!
Eðm̄ c̄Þ 2 Varðm̄ c̄Þ Varðc̄Þ 2 Covðm̄ c̄; c̄Þ
þ
:
Eðc̄Þ
Eðm̄ c̄ÞEðc̄Þ
Eðm̄ c̄Þ2
Eðc̄Þ2
This expression is not generally computable from the information
given in the papers, but by far the largest contribution to this
expression is from the first of the three terms in the last parenthesis,
which is computable. Take, for example, a mixture with mean relative
yield advantage up to 5 percent and a realistic 10% coefficient of
variation. Then, a mixture with three components in equal proportions grown in four complete blocks gives a standard error of erel equal
to 0.0577 according to the approximation using only the first of the
372
L.P. Kiær et al. / Field Crops Research 114 (2009) 361–373
three terms. In comparison, the full expression is between 0.0577 and
0.0585. Therefore, the approximation
P
2
1 þ ki¼1 p2i v̂
Varðm̄ c̄Þ
Varðerel Þ ¼
P
2 ;
k
Eðc̄Þ2
i¼1 pi m̄i
was used in the meta-analysis. The error thereby introduced is
negligible compared to estimation error of the variance in each
study, being ignored by the meta-analysis, and errors due to
assumptions of variance homogeneity made in the individual
studies and inherited by the meta-analysis.
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