Plant Science 547

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Plant Science 547 Biometrics for Plant Scientists
Association Between Characters
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Effect of One Treatment on Another

• Test hypothetical models for biological systems.
  To explain relationships (i.e. linear, quadratic,
  etc., orthogonal contrasts).
• To predict the values of one variable according
  to set values of another.

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Possible Relationships of Interest

• Predict optimal nitrogen application to maximize
  seed yield.
• Determine deficiencies in national supply of
  specific agricultural products by relating yield
  to weather related characters (rainfall,
  sunshine, etc).
• Explain relationship between plant biomass and
  time after seeding to select for more insect
  tolerant cultivars.

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• Charles Darwin
• Born in England, educated in Scotland.
• The father of evolution
• Most famous for his travels on the Beagle to the
  Galapagos Islands.
• Survival of the fittest.
• Wrote The Origin of Species.

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History

• 19th Century - Charles Darwin.
• Francis Galton In the law of universal
  regression each peculiarity in a man is
  shared by his kinsman, but on average to a lesser
  degree.
• Karl Peterson Andrew Lee (statisticians) survey
  1000 fathers and sons height.
• Using this data set Galton, Peterson and Lee
  formulated regression analyses.

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Regression Models

• Dependant variable (of interest). Usually donated
  by y
• One, or more, independent variable on which the
  dependant variable is related in a specific
  manner. Usually donated by x, x1, x2, etc.

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Common Types of Regression

• Simple linear regression
  yb0b1x
• Non-linear regression yb0b1xb2x2 yex
  yln(x)
• Multiple regression yb0b1x1b2x2

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Nitrogen application v Seed yield
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Nitrogen application v Seed yield
b1
bo
Y bo b1x
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Simple Linear Regression
Y bo b1x
b1 SP(x,y)/SS(x)
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
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Simple Linear Regression
Y bo b1x
bo mean(y) b1 x mean(x)
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Linear regression example

• Sex-linked mutations in Drosophila.
• x-variable is dosage of radiation (1000s rads).
• y-variable is the percentage of mutation observed
  in Drosphila populations.

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Mutation Frequency in Drosophila
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Linear Regression Example
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Linear 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
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Linear 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
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Mutation Frequency in Drosophila
Y 0.0735 2.255 x
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Analysis of Variance Regression

• Total variation of the dependant variable (the
  one of interest).
• Partition into variation accountable by the
  regression model (linear or other) Sum of
  squares for regression.
• Other, non-explaiable variation
  Residual sum of squares.

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Linear Regression Example
Total SS SS(y) ?(yi2)-?(yi)2/n 204.11 -
(26.3)2/5 65.772
Regression SS SP(x,y)2/SS(x) 28.8602/12.800
65.070
Residual SS ?2Res Total SS - Regression
SS 65.772 - 65.070 0.702
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Analysis of Variance Regression
Residual can be tested if observations are
replicated
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t-test and regression
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t-tests and Regression
Is 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
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t-tests and Regression
Things to Note
se(b) ?SS(y) - b1 SP(x,y)/(n-2)SS(x)
SS(y) - b1 SP(x,y) Residual SS
SS(y) - b1 SP(x,y)/(n-2) Residual MSq
se(b) ?Residual MSq/SS(x) ??2Res/SS(x)
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t-tests and Regression
Is the intercept significantly different from a?
t b0-a/se(b0)
se(b0) ??2Res . 1/n mean(x)2/SS(x)
?0.234 x 1/5(2.32/12.800 0.378
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Predicting the dependant variable!
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Predicting the Dependant Variable
Y 0.0735 2.255 x
At x 2.5 - 1000 Rads y 0.073 2.255 x 2.5
5.7105
How accurate is this estimation?
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se(yp) ??2 11/n(xp-x)2/SS(x)
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Linear Regression Example
Predicting at x 2.5 se(yp) ?0.234
11/5(2.5-2.3)2/12.800 se(yp) 0.531 yp
5.7105 0.531
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Linear Regression Example
Predicting at x 4.5 se(yp) ?0.234
11/5(4.5-2.3)2/12.800 se(yp) 0.663 yp
10.2205 0.663
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Mutation Frequency in Drosophila
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