 Basic Research
– Researcher makes hypothesis and conducts a single experiment to test it
– The hypothesis is modified and another experiment is conducted
– Combined analysis of experiments is seldom required
– Experiments may be repeated to
• Provide greater precision (increased replication)
• Validate results from initial experiment
 Applied Research
– Recommendations to producers must be based on multiple locations and seasons that represent target environments (soil types, weather patterns)

 Often called MET = multi-environment trials
 How do treatment effects change in response to differences in soil and weather throughout a region?
–
What is the range of responses that can be expected?
 Detect and quantify interactions of treatments and locations and interactions of treatments and seasons in the recommendation domain
 Combined estimates are valid only if locations are randomly chosen within target area
– Experiments often carried out on experiment stations
– Generally use sites that are most accessible or convenient
–
Can still analyze the data, but consider possible bias due to restricted site selection when making interpretations

 Complete ANOVA for each experiment
– Do we have good data from each site?
– Examine residual plots for validity of ANOVA assumptions, outliers
 Examine experimental errors from different locations for heterogeneity
– Perform F Max test or Levene’s test for homogeneity of variance
– If homogeneous, perform a combined analysis across sites

 Differences in means across sites are often greater than treatment effects
 Does not prevent a combined analysis, but may contribute to error heterogeneity if there are associations between means and variances
 If heterogeneous:
– Break sites into homogeneous groups and analyze separately
– Use a transformation
– Use a generalized linear mixed model that accounts for error structure

Y ijk
=

+
 i
+
 j(i)
+
 k
+ (

) ik
+
 ijk

= mean effect
 i
 j(i)
 k
 ik
 ijk
= i th location effect
= j th block effect within the i th location
= k th treatment effect
= interaction of the k th
= pooled error treatment in the i th location
 Environments = Locations = Sites
 Blocks are nested in locations
– SS for blocks is pooled across locations

 Obtain a preliminary estimate of interaction of treatment with environment or season
 Will we be able to make general recommendations about the treatments or should they be specific for each region or site?
– Error degrees of freedom are pooled across sites, so it is relatively easy to detect interactions
– Consider the relative magnitude of variation due to the treatments compared to the interaction MS
– Are there rank changes in treatments across environments (crossover interactions)?

Source df SS MS
Location
Blocks in Loc.
l-1 l(r-1)
SSL M1
SSB(L) M2
Treatment t-1 SST M3
Loc. X Treatment (l-1)(t-1) SSLT M4
Pooled Error l(r-1)(t-1) SSE M5
Expected MS
 2 e
 r
 2
TL
 t
 2
R ( L )
 rt
 2
L
 2 e
 2 e
 2 e
 2 e
 t
 2
R ( L )
 r
 2
TL
 r
 2
TL
 rl
 2
T
F for Locations = (M1+M5)/(M2+M4)
Satterthwaite’s approximate df
N1’ = (M1+M5) 2 /[(M1 2 /(l-1))+(M5 2 /((l)(r-1)(t-1)))]
N2’ = (M2+M4) 2 /[(M2 2 /(l-1))+(M4 2 /((l)(r-1)(t-1)))]
F for Treatments = M3/M4
F for Loc. x Treatments = M4/M5

Fixed Locations
• constitute the entire population of environments
OR
• represent specific environmental conditions (rainfall, elevation, etc.)
Source
Location
Blocks in Loc.
Treatment
Loc. X Treatment df l-1 l(r-1) t-1
(l-1)(t-1)
SS
SSL
SSB(L)
SST
SSLT
MS Expected MS
M1 t 2
R(L) rt 2
L
M2
M3
M4 e e t 2
R(L) r rl 2
T
2
LT
Pooled Error l(r-1)(t-1) SSE M5
 2 e e
F for Locations = M1/M2
F for Treatments = M3/M5
F for Loc. x Treatments = M4/M5

Treatments are fixed, Locations are random
Source df SS MS
Location
Blocks in Loc.
l-1 l(r-1)
SSL M1
SSB(L) M2
Treatment t-1 SST M3
Loc. X Treatment (l-1)(t-1) SSLT M4
Pooled Error l(r-1)(t-1) SSE M5
Expected MS
 2 e
 t
 2
R ( L )
 rt
 2
L
 2 e
 t
 2
R ( L )
 2 e
 2 e
 2 e
 r
 2
TL
 r
 2
TL
 rl
 2
T
F for Locations = M1/M2
F for Treatments = M3/M4
F for Loc. x Treatments = M4/M5
 SAS uses slightly different rules for determining Expected MS
 No direct test for Locations for this model

Varieties fixed, Locations random
PROC GLM ;
Class Location Rep Variety;
Model Yield = Location Rep(Location) Variety Location*Variety;
Random Location Rep(Location) Location*Variety/ Test ;
Source Type III Expected Mean Square
Location Var(Error) + 3 Var(Location*Variety) +
7 Var(Rep(Location)) + 21 Var(Location)
Dependent Variable: Yield
Source DF Type III SS Mean Square F Value Pr > F
Location 1 0.505125 0.505125 0.20 0.6745
Error 5.8098
15.027788 2.586644
Error: MS(Rep(Location)) + MS(Location*Variety) - MS(Error)

Source df SS MS
Years l-1
Blocks in Years l(r-1)
SSY M1
SSB(Y) M2
Treatment t-1 SST M3
Years X Treatment (l-1)(t-1) SSYT M4
Pooled Error l(r-1)(t-1) SSE M5
Expected MS
 2 e
 t
 2
R ( Y )
 rt
 2
Y
 2 e
 t
 2
R ( Y )
 2 e
 2 e
 2 e
 r
 2
TY
 r
 2
TY
 ry
 2
T
F for Years = M1/M2
F for Treatments = M3/M4
F for Years x Treatments = M4/M5

 Can analyze as a factorial
Source
Years df y-1
Locations
Years x Locations
Block(Years x Locations) l-1
(y-1)(l-1) yl(r-1)
 Can determine the magnitude of the interactions between treatments and environments
– TxY, TxL, TxYxL
 For a simpler interpretation, consider all year and location combinations as “sites” and use one of the models presented for multilocational trials

Combined Lab or Greenhouse Study (CRD)


Assume Treatments are fixed, Trials are random
A “trial” is a repetition of a replicated experiment
Source
Trial
Treatment df l-1 t-1
SS
SSL
SST
MS
M1
M2
Trial x Treatment (l-1)(t-1) SSLT M3
Pooled Error lt(r-1) SSE M4
Expected MS e e r r rt
2
LT
2
L
2
LT rl e
 2 e
2
T
F for Trials = M1/M4 ( SAS would say M1/M3 )
F for Treatments = M2/M3
F for Trials x Treatments = M3/M4
 If there are no interactions, consider pooling SSLT and SSE
– Use a conservative P value to pool (e.g. >0.25 or >0.5)

 Assumptions for this example:
– locations and blocks are random
–
Treatments are fixed
Source
Total
Location
Blocks in Loc.
df lrt-1 l-1 l(r-1)
SS
SSTot
SSL
SSB(L)
Treatment t-1 SST
Loc. X Treatment (l-1)(t-1) SSLT
Pooled Error l(r-1)(t-1) SSE
MS F
M1 M1/M2
M2
M3 M3/M4
M4 M4/M5
M5
 If Loc. x Treatment interactions are significant, must be cautious in interpreting main effects combined across all locations

PROC MIXED or PROC GLIMMIX

Genotype by Environment Interactions (GEI)
 When the relative performance of varieties differs from one location or year to another…
– how do you make selections?
– how do you make recommendations to farmers?

Genotype x Environment Interactions (GEI)
 How much does GEI contribute to variation among varieties or breeding lines?
P = G + E + GE
P is phenotype of an individual
G is genotype
E is environment
GE is the interaction
DeLacey et al ., 1990 – summary of results from many crops and locations
70-20-10 rule
E: GE: G
20% of the observed variation among genotypes is due to interaction of genotype and environment

 Many approaches for examining GEI have been suggested since the 1960’s
 Characterization of GEI is closely related to the concept of stability. “Stability” has been interpreted in different ways.
– Static – performance of a genotype does not change under different environmental conditions
(relevant for disease resistance, quality factors)
– Dynamic – genotype performance is affected by the environment, but its relative performance is consistent across environments. It responds to environmental factors in a predictable way.

 CV of individual genotypes across locations
 Regression of genotypes on an environmental index
– Eberhart and Russell, 1966
 Ecovalence
– Wricke, 1962
 Superiority measure of cultivars
– Lin and Binns, 1988
 Many others…

 Rank sum index (nonparametric approach)
 Cluster analysis
 Factor analysis
 Principal component analysis
 AMMI
 Pattern analysis
 Analysis of crossovers
 Partial Least Squares Regression
 Factorial Regression