obtaining regionalized maps of the spatial distribution of an insect plague across time and on the whole crop growth stage,Define the “convenient” distance between (pheromone) traps,Software: SAS, S+, GS+, GIS software

1. The role of a biometrician in an
International Agricultural Center:
service and research in an
interdisciplinary frame
Some ideas and examples
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2. Interdisciplinary collaboration
• No man (or woman) is an island
• Few problems are simple problems
• Problems are better solved using teams of specialists
working together
• The biometrician needs scientist’s data to propose and
test methodologies and the scientists need the analytical
tools to address their problems
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3. Should be possible for the Biometrician to
participate in all or some of
• Planning of surveys and experiments
• Data processing, analysis and interpretation
• Design of novel tools/methodologies for analysis
• Writing and/or editing results
• Seminars and courses on new methodologies
and software (the free software issue)
• Capacity building (internal and external)
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4. Trying to
• Allow researchers to work in depth within
their disciplines and problems, not on
methods for data analysis, computer
routines, software, etc.
• Supply new points of view and new
methodological tools on the research work
• Improve the quality of inferences
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5. Four levels of participation
1. Routine analysis and known problems: short
and fast responses
2. New problems, known methodologies: we
need think together
3. New problems, existing methodology: we need
to understand, study and propose solutions
4. New problems, unknown methodology: we
need do some methodological research
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6. Some examples of
interdisciplinary collaboration
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7. Experimental Design
(continuous traits, Mixed Model)
• Spatial analysis of an Experimental Design
• Variety trial designed as a row-column design
with 64 varieties planted in two contiguous
replicates laid out in 8 rows and 16 columns
Nguyen and Williams, 1993
(Austral. J. Stat. 35: 363-370)
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8. Layout (r=2, t=64)
-lattice (0,1,2)
r r
e o column
p w
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1 5 19 55 23 27 38 64 44 14 13 59 25 45 49 11 43
1 2 2 36 62 29 34 40 52 54 28 26 58 42 32 6 39 53
1 3 37 3 17 24 15 50 47 56 16 30 31 60 7 18 8 21
1 4 20 35 57 63 46 12 41 10 22 1 51 61 33 48 4 9
2 5 58 44 15 25 18 53 5 22 55 36 33 29 12 47 8 1
2 6 41 17 63 37 38 19 26 21 28 4 6 45 56 46 59 42
2 7 14 9 62 48 34 31 60 13 20 7 64 3 39 40 10 23
2 8 24 52 11 54 51 57 35 2 50 27 30 61 43 49 32 16
5,3 =0 5,2 =1 5,44 =2
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9. Statistical models
yij i j ij
RCBD
yij i j k ( j) l( j) ijkl
Row-Col
Local control of field heterogeneity
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10. • Assumptions on the “error” RV: V( ) = 2R
Usual GLM: dij: distance in time and/or space
Independence
Repeated measures
Spatial analysis
1 0 0 0 ... 0
0 1 0 0 ... 0
0 0 1 0 ... 0
1 f (d12 ) ... f (d1n )
2
R
0 0 0 1 ... 0 2
f (d 21 ) 1 ... f (d 2 n )
R
... ... ... ... ... ... ... ... ... ...
0 0 0 0 ... 1 f (d n1 ) f (d n 2 ) ... 1
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11. three distance functions f(d ij )
1.00
f(d)
0.50
0.00
0 10 20 30 40 50
distance
spherical exponential gaussian
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12. Spatial statistical model
entry
yijkl i ijkl
ε ~ N (0, R)
Where:
i = 1,2,…, t (entries)
j = 1,2,…, J (replications)
k = 1,2,.., K (rows)
l = 1,2,…, L (columns)
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13. Results
Precision of estimates and test of hypothesis
Average Standard Error of ls-means
rcbd row-col spat-sph
Average 0.5076 0.4041 0.4526
Standard Errors of some ls-means differences
Label rcbd row-col spat-sph
5 vs 3 0 0.7179 0.5256 0.4737
5 vs 2 1 0.7179 0.4833 0.4286
5 vs 14 2 0.7179 0.5232 0.4906
5 vs 44 2 0.7179 0.5159 0.4938
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14. conclusion
When possible it is convenient to use
layouts that allow the spatial analysis with
the objective of reaching lower standard
errors for the contrasts
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15. Spatial distribution of two insect plagues
(apple and peach trees) on a region
• Grafolita (Cydia
molesta)
• Carpocapsa
pomonella
• Pheromone traps
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16. Objectives
• To obtain regionalized maps of the spatial
distribution of an insect plague across time and
on the whole crop growth stage
• Define the “convenient” distance between
(pheromone) traps
• Software: SAS, S+, GS+, GIS software, R?
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17. traps location
(25 x 26 km)
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18. Semivariogram (geostatistics)
Sill point 2600
2400
2200
2000
semivar
1800
1600
1400
Nugget (intercept)
1200
1000
0 5000 10000 15000
distance (m)
Range
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19. Spherical :
3
d ij d ij
f (d ij ) 1 1.5 0.5 I (d ij )
Exponential :
d ij
f (d ij ) e
Gaussian : (range): distance from
2
which observations are
independent
d ij
f (d ij ) e 2
f (d ii ) 1
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20. A linear mixed model is used to
selecting the best fit model, and
estimating the range
2
yij V( ) R F
ij
F { f (d ij )}
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21. • Null model: the independence model
R0= 2I, in which ij = 0 for all ij
• Alternative model: the spatially related model:
R = 2F,
F={f(dij)} matrix
• Using a Likelihood Ratio Test (LRT)
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22. Results
Model ˆ (m) -2 Log(L) (1) -2 log( ) P( 2> 2
c)
Carpocapsa
Independence 1169
Spherical 4818 1156 12.7 0.002 **
Gaussian 2256 1157 11.9 0.003 **
Exponential 1343 1157 11.8 0.003 **
Grafolita
independence 1785
Spherical 1290 1782 3.5 0.174 ns
Gaussian 604 1781 4.0 0.135 ns
Exponential 527 1780 4.6 0.100 ns
(1) smaller -2Log(L) is better JFD

23. Prediction (interpolation)
Prediction of non observed values using the
observed values plus the knowledge of a good
spatial model: KRIGING procedures
Krige, Danie G. (1951). "A statistical approach to some basic mine
valuation problems on the Witwatersrand". J. of the Chem., Metal.
and Mining Soc. of South Africa 52 (6): 119–139.
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24. Carpocapsa
Mean Standard Range Number of traps
deviation
60 49 6-259
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25. Forecasting
(a more complex model)
• Forest inventory and growth models
– Non linear (but linearized models)
– structured equations models
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26. Growth forestry model
• Eucalyptus (Bicostata)
• In a region
• Using the inventory data sets (yearly data)
• n = 2461
- from 506 plots (300 m2, circular)
- Measured each year (from 3 to 15)
• “difference” models
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27. First data set
(coefficient estimation and model fit)
nplot Y1 Y2 H1 H2 BA1 BA2 N1 N2 V1 V2 S
100 3 4 12.5 16.9 15.7 22.2 1367 1367 53.2 108.0 22.3
100 4 5 16.9 17.7 22.2 26.1 1367 1367 108.0 135.9 24.2
100 5 6 17.7 22.9 26.1 30.3 1367 1367 135.9 183.9 21.7
100 6 7 22.9 23.3 30.3 33.2 1367 1333 183.9 219.3 25.0
100 7 8 23.3 23.3 33.2 36.6 1333 1333 219.3 244.9 23.3
100 8 9 23.3 23.9 36.6 38.2 1333 1333 244.9 264.6 21.7
100 9 10 23.9 24.1 38.2 39.7 1333 1333 264.6 275.7 21.1
100 10 11 24.1 25.7 39.7 41.7 1333 1333 275.7 304.9 20.3
100 11 12 25.7 25.0 41.7 46.4 1333 1333 304.9 327.0 20.9
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28. An external equation
(dominant height and site index)
2
1 exp( 1 A2 )
h2 h1
1 exp( 1A )
1
ˆ2
1 exp( ˆ1 7)
S h1
1 exp( ˆ1 A1 )
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29. And 5 structured non-linear
(but linearized) difference equations
density
1. ln2~ln1+I(lA2-lA1)+ 1
basal area-1
2. lg1~I(1/A1)+ln1+lh1+I(lh1/A1)+I(S*A1/100) + 2
basal area-2
3. lg2~lg1+I(1/A2-1/A1)+I(ln2-ln1)+I(lh2-lh1)+I(lh2/A2-
lh1/A1)+I(S*(A2/100-A1/100)) + 3
Volume-1
4. lv1~ln1+lh1+lg1+I(S*A1/100) + 4
Volume-2
5. lv2~lv1+I(ln2-ln1)+I(lh2-lh1)+I(lg2-lg1)+I(S*(A2/100-
A1/100)) + 5
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30. characteristics
• results from an equation are independent variables on another
• “residuals” ( 1, 2,…, 5) are correlated and, possibly, do not have
homogeneous variances
• it is necessary a simultaneous estimation process
• 22 regression coefficients
• Possible methods: OLS, SUR, 2SLS, 3SLS
• Software: Systemfit in “R” (freeware)
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31. model
evaluation
External:
Internal:
adjust of estimated model
Model fit on a randomly
on other “independent”
selected sub-dataset
sub-dataset
Year by year Long term
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32. Second data set: forecasting
(prediction and external fit)
nplot Y1 Y2 HD1 BA1 N1 V1 S
100 3 4 12.5 15.74 1366.7 53.2 22.3
100 3 5 12.5 15.74 1366.7 53.2 22.3
100 3 6 12.5 15.74 1366.7 53.2 22.3
100 3 7 12.5 15.74 1366.7 53.2 22.3
100 3 8 12.5 15.74 1366.7 53.2 22.3
100 3 9 12.5 15.74 1366.7 53.2 22.3
100 3 10 12.5 15.74 1366.7 53.2 22.3
100 3 11 12.5 15.74 1366.7 53.2 22.3
100 3 12 12.5 15.74 1366.7 53.2 22.3
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33. Observed correlations between
residuals (3SLS-method)
density basal-1 basal-2 vol-1 vol-2
density 1.0000 0.0993 0.0362 -0.0861 -0.0234
basal-1 1.0000 -0.4837 -0.7354 0.1089
basal-2 1.0000 0.2935 -0.6356
vol-1 1.0000 -0.4777
vol-2 1.0000
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34. Predicted growth curves
(4 plots)
obs pred Lower Upper
500
400
volume (m3)
300
200
100
0
4 6 8 10 12 4 6 8 10 5 7 9 11 5 7 9 11
Age
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35. Competitiveness
Comparison of the socio-economic
competitiveness of three regional
blocks: Mercosur, NAFTA, EU
(UNDP)
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36. Competitiveness drivers
Driver Nr. variables
1. Knowledge 11
2. Innovation platform 12
3. Connectivity 12
4. Infrastructure 8
5. Macroeconomic variables 9
6. Social cohesion 13
Total 65
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37. Objectives and Methods
• Generating a Competitiveness Index (CI)
– STAGE 1: PCA per driver
– STAGE 2: PCA using the first PC from STAGE 1
(CI = scores on the first PC from STAGE 2)
• Improvement comparison: advances by driver
– Relative improvement = (2000 vs. 1990)
– Weighted average per driver
(Weight = participation of each driver into the CI)
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38. Driver Nr. vars. Contribution Explained
to CI Variability(1)
Knowledge 11 20.1 74.5
I Platform 12 18.4 66.1
Connectivity 12 21.6 82.5
Infrastructure 8 13.4 78.0
Macro Econ -vars 9 5.6 47.8
S-cohesion 13 20.9 62.4
Sum 65 100
(1) First PC within driver
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39. Driver-year CI
2.0
1.5 KNOW
1.0
IPLAT
0.5
CONEC
score
0.0
INFR
-0.5
-1.0 MACV
-1.5 COHE
-2.0
MS90 MS95 MS00 NF90 NF95 NF00 EU90 EU95 EU00
block-year
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40. One dimension CI
3.50
0.00
-3.50 -2.50 -1.50 -0.50 0.50 1.50 2.50 3.50
MS NF UE
-3.50
Competitiveness Index
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41. RELATIVE IMPROVEMENT 2000/1990
(weighted means)
KNOWLEDGE
0.8
0.6
S-COHESION 0.4 I-PLATFORM
0.2
0.0
MACRO-VARS CONECTIVITY
INFRASTRUCTURE
MS NF UE
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42. Association mapping
• The mixed model on association mapping
• Example:
– 46 wheat genotypes
– 374 markers (DArT)
– 5 traits: w1000, leaf rust, steam rust, maturity,
yield
– environments: 15, 17, 5, 15, 10
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43. Components
Mixed
Model
Relations
Markers Phenotypic Underlying
between
data (DArT) data structure
genotypes
Binary Groups Coancestry
traits
values (co-variables) (Parentage)
(Y1,…,Ye)
{0, 1} Q matrix K matrix
STRUCTURE K=2pij or similarity
no linked markers from markers
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44. Data set (subset)
(one environment, 7groups, 4 markers)
geno y2 gr1 gr2 gr3 gr4 gr5 gr6 gr7 m45 m46 m47 m48
1 8.6 0.001 0.994 0.001 0.001 0.001 0.001 0.001 0 1 1 1
2 12.9 0.996 0.001 0.001 0.001 0.000 0.000 0.001 1 1 1 1
3 11.8 0.001 0.001 0.001 0.001 0.001 0.993 0.002 1 0 1 1
4 12.3 0.281 0.003 0.001 0.002 0.462 0.199 0.052 1 1 0 0
5 10.0 0.002 0.007 0.003 0.239 0.136 0.610 0.005 1 0 1 1
. . . . . . . . . . . . .
45 10.8 0.001 0.001 0.991 0.002 0.001 0.002 0.002 0 1 1 1
46 7.0 0.001 0.002 0.948 0.042 0.004 0.001 0.002 0 1 1 1
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45. Proposition
• The marker is associated to the phenotype if the
trait average for genotypes owning a “0” is “very”
different to the trait average for genotypes owning
a “1”
• “very”= statistically different in a test of hypothesis
• 3 possible models: one way anova, Q model, Q+K
model
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46. Q+K model (one site)
fixed random
g 1
yij i l 1
(l )
ˆ(
qijl ) j (i ) ij
Xβ Zu ε
i = 1,2 (two states of marker)
i = 1 if the ith marker is present in jth genotype
ˆ(
qijl )= membership probability of the jth genotype to the lth group
j(i) = genotype nested in the marker state (i=1 or i=0) JFD

47. Variance-covariance matrices
V (ε) 2
I
2
V (u ) A a
• A is known: coefficient of parentage or some
similarity using non linked markers
• Variances should be estimated
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48. matrix K (= A)
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49. Bonferroni bound
(when tests are not independent)
• Example: m=46 tests, =0.05,
P[reject at least one Ho/Ho true] =
1-(1- )m = = 1-(0.95)46 = 0.906
• if you want 0.05 “on all tests” type I error
Reject Ho when
1
ˆ 1 (1 0.05) 46
0.001114
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50. Results (yield)
(10 environments, 374 DArT, Q+K model)
0.10 0.05
Site DART F p p<=0.00028 p<=0.000138
348 wPt-0086 14.9740 1.26E-04 1 1
275 wPt-1859 14.3168 1.77E-04 1 0
279 wPt-3992 15.1771 1.13E-04 1 1
194 wPt-4115 15.3556 1.04E-04 1 1
163 wPt-4487 24.1353 1.27E-06 1 1
234 wPt-9075 19.5032 1.30E-05 1 1
424 wPt-9930 14.7447 1.42E-04 1 0
total 7 5
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51. Methodological Research
1. The Ward-MLM three-way strategy for
classifying genetic resources in multiple
environments
2. Sampling strategies for conserving
diversity when forming core subsets
using genetic markers
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52. Interdisciplinary group
• Jose Crossa CIMMYT-Biometrics
• Suketoshi Taba CIMMYT-Maize Genebank
• Marilyn Warburton CIMMYT-USDA-Molecular
genetics
• Sarah Hearne IITA- Molecular genetics
• Steve A. Eberhart USDA-Genebank
• Jose Villaseñor C. P.-mathematics
• Jorge Franco UDELAR-CIMMYT-Biometrics
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53. The Ward-MLM three-way strategy
for classifying genetic resources in
multiple environments: evaluate GxE
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Franco et al (2003) Crop Sci 43; 1249-1258

54. Properties of the Ward-MLM strategy
1. It assigns the observations to an optimal number of
clusters based on membership probability (is a
statistical method)
2. It uses discrete an continuous variables simultaneously
3. It follows the optimization of two objective functions
(minimum variance within group, and maximum Log-
likelihood)
4. It allows the estimation of the quality of the resulting
clustering (average of the assignment probabilities)
5. It can be used on 3-way data sets (genotype ×
environment × trait). JFD

55. Example
256 Caribbean maize genotypes
Discrete variable :
• Agronomic Scale (1=poor, 2=regular, 3=good)
Continuous variables:
• Days to anthesis (DA)
• Plant height (PH, cm)
• Days to senescence of the ear leave (DS)
• Ear length (EL, cm)
Three environments (Mexico)
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56. Information to the 3-way analysis
(first 5 observations)
NOBS DA1 PH1 DS1 EL1 DA2 PH2 DS2 EL2 DA3 PH3 DS3 EL3 AS1 AS2 AS3
1 73.5 222.4 45.5 14.7 91.6 210.0 48.1 17.3 59.8 262.8 45.8 21.0 1 2 3
2 71.4 225.7 47.5 14.9 90.1 209.8 40.0 13.1 61.7 262.3 43.9 15.5 2 2 2
3 64.5 165.8 51.1 13.5 75.6 171.3 56.5 16.4 54.3 224.1 47.4 17.5 2 3 2
4 73.5 180.3 47.5 12.2 91.9 176.2 49.7 16.3 61.1 234.8 55.0 17.5 2 2 3
5 74.5 200.5 50.4 15.3 93.0 186.6 56.8 16.2 61.2 249.1 51.4 19.0 3 3 3
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57. Canonical representation (87%)
6
4 G1
G2
2
Can 2 (EL+,PH+, 8%)
G3
0
-10 -8 -6 -4 -2 0 2 4 6 8 10 G4
-2 G5
G6
-4
G7
-6
G8
-8
Can 1 (DS+, DA-, EL+,79% )
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58. Agronomic performance of genotypes
Group good Regular Poor Non id.
(GGG,GGR) (RPP,PPP) (GRP)
1 (46) 46
2 (25) 18 5 2
3 (27) 19 8
4 (53) 53
5 (24) 24
6 (22) 13 9
7 (32) 7 19 6
8 (27) 3 24
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59. Average values groups 5 and 8
DA PH DS EL AgS
G E1 E2 E3 E1 E2 E3 E1 E2 E3 E1 E2 E3 E1 E2 E3 n
5 75 94 63 213 218 266 45 52 49 15 16 19 3.0 3.0 2.9 24
8 81 102 70 218 219 257 38 43 39 13 15 16 1.3 1.4 1.0 27
m 77 97 66 214 218 266 42 48 45 14 16 18 1.9 2.0 2.4 256
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60. group X environment interaction
(no COI)
110 280
100
DA 270
PH
260
90 250
env 1 240 env 1
80 env 2 env 2
env 3 230 env 3
70 220
210
60
200
50 190
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
group group
55 20
DS 19
EL
50 18
17
env 1 env 1
45 env 2 16 env 2
env 3 env 3
15
40 14
13
35 12
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
group group
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61. Agronomic scale (COI**)
3 .5
3 .0
2 .5
env 1
2 .0 env 2
env 3
1.5
1.0
0 .5
0 1 2 3 4 5 6 7 8 9
g r o up
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62. Variable relations vs Environment
(between groups)
0.8
PH3
0.6 PH2
Q1 PH1
0.4 0.29
EL1
0.53
0.2 -0.17
EL3
Dimension 2
Env 1
EL2 env 2
0.0
env 3
DS1 Q
-0.94
-0.2 DS3 -0.87 DA1
DA3
DS2 -0.88
-0.4 DA2
Q2
-0.6
Q3
-0.8
-1.0 -0.5 0.0 0.5 1.0 1.5
Dimension 1
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63. Sampling strategies for conserving diversity
when forming core subsets using genetic
markers
• DEF: A core collection (or core subset) is a
sample from a large germplasm collection that
contains, with a minimum of repetitiveness, the
maximum possible genetic diversity of the
species in question (Frankel and Brown, 1984)
• Forming core subsets requires sampling
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64. STEP 1. Numerical classification
• The most used methods are Ward (minimum
variance within cluster), and UPGMA (average of
distances)
• They require an initial matrix of distances between
genotypes
• With SSR we can use genetic distances (Modified
Rogers, Cavalli-Sforza & Edwards). Both are Euclidian
metrics
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65. STEP 2. drawing accessions from clusters
(how many accessions from each cluster?)
Allocation methods : are methods for determining the
number of observations to be randomly drawn from each
stratum (cluster)
Optimal (Neyman): proportional to the size and variability
of the cluster
P: proportional to the cluster size
L: proportional to the log of the cluster size
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66. D-method: proportional to any measure of the
cluster diversity
di
ni n t = 1,2,…,number of clusters
d
t t
ni: number of accessions to be drawn from ith cluster
di: average of distances between accessions within
the ith cluster
n: size of the core subset
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Franco et al (2006) Crop Sci 46; 854-864

67. Diversity
Distances among
Diversity indexes
Individuals
(allele richness)
or groups
Expected Non informative Informative
heterozygosis Markers Markers
(He) p {0,1} p [0,1]
Number of
Simple Matching Modified Rogers
effective alleles
(Ne)
Cavalli-Sforza and
Shannon Index Jaccard
Edwards
Nei and Li
Euclidian
(Dice)
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68. Example: three maize data sets
(SSR markers)
Obs. Alleles Markers Values Missing
Bulks 275 186 24 [0, 1] 1.5 %
Landraces 521 209 26 {0,.5,1} no
Populations 25 209 26 [0, 1] no
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69. 24 stratified strategies + MSTRAT
(20 independent core subsets)
Code Cluster Distance Allocation methods
1 Modified P, L,
to Ward Rogers D: MR,SH,HE,NE
12 Cavalli-S & P, L,
Edwards D: CE,SH,HE,NE
13 Modified P, L,
To Average Rogers D: MR,SH,HE,NE
24 (UPGMA)
Cavalli-S & P, L,
Edwards D: CE,SH,HE,NE
25 MSTRAT (non stratified method)
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70. Means of 20 reps (different core subsets)
Bulk data set
MRs CEs SHs HEs NEs
Best strategy (24)† 0.503a 0.578a 4.411b 0.626b 2.980b
M-Strategy 0.467b 0.559b 4.478a 0.644a 3.166a
Entire collection 0.440 0.521 4.399 0.620 2.937
δ'(%) 1.37 1.20 0.31 0.77 1.60
Landraces data set
MRs CEs SHs HEs NEs
Best strategy (24) 0.653a 0.719a 4.525a 0.619a 2.963a
M-Strategy 0.637b 0.704b 4.504b 0.599b 2.808b
Entire collection 0.635 0.700 4.467 0.591 2.742
δ'(%) 0.48 0.37 0.26 0.82 1.23
† (24) UPGMA – Cavalli-sforza & Edwards – Number of effective alleles
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71. Bulks
3
2
1
0
Dim-2
-4 -3 -2 -1 0 1 2 3 4
-1
-2
-3
-4
Dim-1
Data U-CE-NE (24) MSTRAT
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72. Landraces
3
2
1
Dim-2
0
-3 -2 -2 -1 -1 0 1 1 2 2 3
-1
-2
-3
Dim-1
Data U-CE-NE (24) MSTRAT
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73. Conclusions
• When constructing core subsets of individuals
(landraces/accessions) the D allocation method used
with a stratified sampling strategy was better than the M
strategy
• For bulks and populations the M-strategy was better for
diversity indexes.
• Some stratified sampling strategies (21-24) were always
better showing the higher average distance (MR, and
CE) between accessions
• All 25 strategies selected non-informative alleles but the
M strategy selected less than the others.
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74. Uses of D allocation method
• D method was used define reference
collections for inbred lines and populations
of maize from Mexico with Marilyn
Warburton from CIMMYT
• D method was used in a collaboration with
Sarah Hearne of IITA to define the
reference germplasm collection of cowpea
accessions
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75. Current research
A method for classifying genotypes
using phenotypic and genotypic
information simultaneously
Phenotypic: continuous and categorical
Genotypic: SSR, DART, SNP
We have a draft
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