Plant Science 547

Biometrics for Plant Scientists

Association Between Association Between CharactersCharacters

Effect of One Treatment on AnotherEffect of One Treatment on Another

Test hypothetical models for Test hypothetical models for biological systems. biological systems. To explain To explain relationships (i.e. linear, quadratic, relationships (i.e. linear, quadratic, etc., orthogonal contrasts).etc., orthogonal contrasts).

To predict To predict the values of one the values of one variable according to set values of variable according to set values of another.another.

Possible Relationships of InterestPossible Relationships of Interest

PredictPredict optimal nitrogen application to optimal nitrogen application to maximize seed yield.maximize seed yield.

Determine deficiencies in national supply Determine deficiencies in national supply of specific agricultural products by relating of specific agricultural products by relating yield to weather related characters (rainfall, yield to weather related characters (rainfall, sunshine, etc).sunshine, etc).

Explain Explain relationship between plant biomass relationship between plant biomass and time after seeding to select for more and time after seeding to select for more insect tolerant cultivars.insect tolerant cultivars.

Charles DarwinCharles DarwinBorn in England, Born in England,

educated in Scotland.educated in Scotland.The father of evolutionThe father of evolutionMost famous for his Most famous for his

travels on the travels on the Beagle Beagle to to the Galapagos Islands.the Galapagos Islands.

““Survival of the fittest”.Survival of the fittest”.Wrote Wrote The Origin of The Origin of

Species.Species.

HistoryHistory19th Century - Charles Darwin.19th Century - Charles Darwin.Francis Galton: In the “ law of universal Francis Galton: In the “ law of universal

regressionregression” “ each peculiarity in a man is ” “ each peculiarity in a man is shared by his kinsman, but shared by his kinsman, but on averageon average to a to a lesser degree”.lesser degree”.

Karl Peterson & Andrew Lee (statisticians) Karl Peterson & Andrew Lee (statisticians) survey 1000 fathers and sons height.survey 1000 fathers and sons height.

Using this data set Galton, Peterson and Lee Using this data set Galton, Peterson and Lee formulated formulated regression analyses.regression analyses.

Regression ModelsRegression Models

Dependant variableDependant variable (of interest). (of interest). Usually donated by yUsually donated by y

One, or more, One, or more, independent independent variablevariable on which the dependant on which the dependant variable is related in a specific variable is related in a specific manner. Usually donated by x, xmanner. Usually donated by x, x11, ,

xx22, etc. , etc.

Common Types of RegressionCommon Types of Regression

Simple linear regression: Simple linear regression: y=by=b00+b+b11xx

Non-linear regression: Non-linear regression: y=by=b00+b+b11x+bx+b22xx22; y=e; y=exx; y=ln(x); y=ln(x)

Multiple regression: Multiple regression: y=by=b00+b+b11xx11+b+b22xx22

Nitrogen application Nitrogen application vv Seed yield Seed yield

020406080

100120140

10 20 30 40 50 60 70 80 90 100

Nitrogen application

Seed

Yiel

d

020406080

100120140

0 20 40 60 80 100 120

Nitrogen application

Seed

Yiel

d

Y = bY = boo + b + b11xx

bo

bb11

Nitrogen application Nitrogen application vv Seed yield Seed yield

Simple Linear RegressionSimple Linear Regression

b1 = [SP(x,y)/SS(x)]

Y = bY = boo + b + b11xx

SP(x,y) = (xi-x)(yi-y)

SP(x,y) = (xy) - [(x) (y)]/n

SS(x) = (xi-x)2

SS(x) = (x2) - [(x)]2/n

Simple Linear RegressionSimple Linear Regression

Y = bY = boo + b + b11xx

bo = mean(y) = b1 x mean(x)

Linear regression exampleLinear regression example

Sex-linked mutations in Sex-linked mutations in Drosophila.Drosophila.

x-variable is dosage of radiation x-variable is dosage of radiation (1000’s rads).(1000’s rads).

y-variable is the percentage of y-variable is the percentage of mutation observed in mutation observed in DrosphilaDrosphila populations.populations.

Mutation Frequency in Mutation Frequency in DrosophilaDrosophila

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Radiation dose

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Linear Regression ExampleLinear Regression Example

Dosage in 1000 rads

(x)

% Mutation

(y) x2 y2 xy

0.5 1.1 0.25 1.21 0.55 1.0 1.9 1.00 3.61 1.90 2.0 5.0 4.00 25.00 10.00 3.0 7.3 9.00 53.29 21.90 5.0 11.0 25.00 112.00 55.00

11.5 26.3 39.25 204.11 89.35

Linear Regression ExampleLinear Regression Example

SS(x) = (xi2)-[(xi)]2/n

39.25 - (11.5)2/5

12.800

Mean (y) = (yi)/n

11.5/5

2.30

Linear Regression ExampleLinear Regression Example

SS(x,y) = (xiyi)-[(xi) (yi)]/n

89.35 - (11.5 x 26.3)/5

28.860

b1 = SP(x,y)/SS(x)

28.860/12.800 = 2.255

b0 = y - b1x

5.26 - 2.255 x 2.30 = 0.735

Mutation Frequency in Mutation Frequency in DrosophilaDrosophila

0.0

2.0

4.0

6.0

8.0

10.0

12.0

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Radiation dose

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Y = 0.0735 + 2.255 xY = 0.0735 + 2.255 x

Analysis of Variance ~ RegressionAnalysis of Variance ~ Regression

Total variation of the dependant Total variation of the dependant variable (the one of interest).variable (the one of interest).

Partition into variation Partition into variation accountable by the regression accountable by the regression model (linear or other): model (linear or other): Sum of Sum of squares for regression.squares for regression.

Other, non-explaiable variation: Other, non-explaiable variation: Residual sum of squares.Residual sum of squares.

Linear Regression ExampleLinear Regression Example

Total SS = SS(y) = (yi2)-[(yi)]2/n

204.11 - (26.3)2/5 = 65.772

Regression SS = [SP(x,y)]Regression SS = [SP(x,y)]22/SS(x)/SS(x)

[28.860][28.860]22/12.800 = 65.070/12.800 = 65.070

Residual SS [Residual SS [22ResRes]] = Total SS - Regression SS = Total SS - Regression SS

65.772 - 65.070 = 0.70265.772 - 65.070 = 0.702

Analysis of Variance RegressionAnalysis of Variance Regression

Source d.f. S.Sq M.Sq F

Regression 1 65.070 65.070 371.8 ***

Residual 3 0.702 0.234

Total 4 65.772

Residual can be tested if observations are replicated

t-test and regressiont-test and regression

t-tests and Regressiont-tests and RegressionIs the regression slope significantly

greater than zero?

t = b-0/se(b)b/se(b)

se(b) = {[SS(y) - [b1 SP(x,y)]]/[(n-2)SS(x)]

{[65.772 - [2.255 x 28.860]]/[(3 x 12.800]

= 0.134

t-tests and Regressiont-tests and Regression

Things to NoteThings to Note

se(b) = se(b) = {[SS(y) - [b{[SS(y) - [b11 SP(x,y)]]/[(n-2)SS(x)] SP(x,y)]]/[(n-2)SS(x)]

se(b) = se(b) = [Residual MSq/SS(x)][Residual MSq/SS(x)]

= = [[22ResRes/SS(x)]/SS(x)]

[SS(y) - [b[SS(y) - [b11 SP(x,y)]] = Residual SS SP(x,y)]] = Residual SS

[SS(y) - [b[SS(y) - [b11 SP(x,y)]]/(n-2) = Residual MSq SP(x,y)]]/(n-2) = Residual MSq

t-tests and Regressiont-tests and Regression

Is the intercept significantly Is the intercept significantly

different from different from aa??

t = bt = b00--aa/se(b/se(b00))

se(bse(b00) = ) = {{22ResRes . [1/n + [mean(x). [1/n + [mean(x)22/SS(x)]]}/SS(x)]]}

{0.234 {0.234 xx [1/5+[(2.3 [1/5+[(2.322/12.800]]}/12.800]]}

= 0.378= 0.378

Predicting the dependant variable!Predicting the dependant variable!

Predicting the Dependant VariablePredicting the Dependant Variable

Y = 0.0735 + 2.255 xY = 0.0735 + 2.255 x

At x = 2.5 - 1000 RadsAt x = 2.5 - 1000 Rads

y = 0.073 + 2.255 y = 0.073 + 2.255 x x 2.5 = 5.71052.5 = 5.7105

se(yse(ypp) = ) = {{22 [1+1/n+(x [1+1/n+(xpp-x)-x)22/SS(x)]}/SS(x)]}

How accurate is this estimation?How accurate is this estimation?

-

Linear Regression ExampleLinear Regression Example

Predicting at x = 2.5Predicting at x = 2.5

se(yse(ypp) = ) = {0.234 [1+1/5+(2.5-2.3){0.234 [1+1/5+(2.5-2.3)22/12.800]}/12.800]}

se(yse(ypp) = 0.531) = 0.531

yyp p = 5.7105 = 5.7105 ++ 0.531 0.531

Linear Regression ExampleLinear Regression Example

Predicting at x = 4.5Predicting at x = 4.5

se(yse(ypp) = ) = {0.234 [1+1/5+(4.5-2.3){0.234 [1+1/5+(4.5-2.3)22/12.800]}/12.800]}

se(yse(ypp) = 0.663) = 0.663

yyp p = 10.2205 = 10.2205 ++ 0.663 0.663

Mutation Frequency in Mutation Frequency in DrosophilaDrosophila

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Radiation dose

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