Making a curved line straight Data Transformation & Regression.
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Transcript of Making a curved line straight Data Transformation & Regression.
Making a curved line straightMaking a curved line straight
Data Transformation Data Transformation & Regression& Regression
Last ClassLast Class
Predicting the dependant variable Predicting the dependant variable and standard errors of predicted and standard errors of predicted values.values.
Outliers.Outliers.Need to visually inspect data in Need to visually inspect data in
graphic form.graphic form.Making a curved line straight.Making a curved line straight.
Transformation.Transformation.
Early Growth Pattern of PlantsEarly Growth Pattern of Plants
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Days after planting
Plan
t weig
ht (g
)
Early Growth Pattern of PlantsEarly Growth Pattern of Plants
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ht (g
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00.511.522.533.5
y = Ln(y)
Early Growth Pattern of PlantsEarly Growth Pattern of Plants
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ht (g
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y = y
Homogeneity of Error VarianceHomogeneity of Error Variance
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101520253035
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Days after planting
Plan
t weig
ht (g
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Homogeneity of Error VarianceHomogeneity of Error Variance
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Days after planting
Plan
t weig
ht (g
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y =Ln(y)
Growth CurveGrowth Curve
Y = eY = exx
Growth CurveGrowth Curve
Y = Log(x)Y = Log(x)
Sigmoid Growth CurveSigmoid Growth Curve
Sigmoid Growth CurveSigmoid Growth Curve
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Accululative Accululative Normal Normal
DistributionDistribution
Sigmoid Growth CurveSigmoid Growth Curve
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Accululative Accululative Normal Normal
DistributionDistribution
TT
-- ƒƒ((dddd
TT
Sigmoid Growth CurveSigmoid Growth Curve
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Accululative Accululative Normal Normal
DistributionDistribution
TT
-- ƒƒ((dddd
TT
Probit AnalysisProbit Analysis
• Group of plants/insects exposed to Group of plants/insects exposed to different concentrations of a specific different concentrations of a specific stimulant (i.e. insecticide).stimulant (i.e. insecticide).
• Data are counts (or proportions), say Data are counts (or proportions), say number killed.number killed.
• Usually concerned or interested in Usually concerned or interested in concentration which causes specific concentration which causes specific event (i.e. LD 50%).event (i.e. LD 50%).
Probit Analysis ~ ExampleProbit Analysis ~ Example
Insecticide concentration (%)
0.37 0.75 1.5 3 6 12 24
Number larvaekilled
0 1 8 11 16 18 20
Proportion killed 0 0.05 0.40 0.55 0.80 0.90 1.00
Level 0 1 2 3 4 5 6
0
0.2
0.4
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0.8
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1.2
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Insecticide concentration (%)
Pro
por
tion
larv
ae k
ille
dProbit Analysis ~ ExampleProbit Analysis ~ Example
Estimating the MeanEstimating the Mean
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Level
Pro
po
rtio
n l
arv
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kil
led yy= 50% Killed= 50% Killed
x x ~ 2.8~ 2.8
Estimating the Standard DeviationEstimating the Standard Deviation
2.82.8
2.82.8
22
Estimating the Standard DeviationEstimating the Standard Deviation
2.82.8
22
95% 95% valuesvalues
Estimating the Standard DeviationEstimating the Standard Deviation
2.82.8
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95% 95% valuesvalues
= 1.2= 1.2
Estimating the Standard DeviationEstimating the Standard Deviation
Probit AnalysisProbit Analysis
Conc. NumberKilled
Prop.Killed
(x) (z)(x-2.8)/1.2
Probit()
0.375 0 0.00 0 -2.33 0.00990.750 1 0.05 1 -1.50 0.06681.500 8 0.40 2 -0.67 0.25253.000 11 0.55 3 0.17 0.56626.000 16 0.80 4 1.00 0.841312.000 18 0.90 5 1.83 0.966624.000 20 1.00 6 2.67 0.9962
Probit AnalysisProbit Analysis
-0.5 -0.3 0 0.25 0.5 0.75 1 1.25 1.5
Log10 (concentration)
Pro
bit
(p)
Probit AnalysisProbit Analysis
Probit (Probit () = ) = + + . Log . Log1010(concentration)(concentration) = -1.022 = -1.022 ++ 0.202 0.202
= 2.415 = 2.415 ++ 0.331 0.331
LogLog1010 (conc) to kill 50% (LD-50) is probit 0.5 = 0 (conc) to kill 50% (LD-50) is probit 0.5 = 0
0 = -1.022 + 2.415 0 = -1.022 + 2.415 xx LD-50 LD-50
LD-50 = 0.423LD-50 = 0.423
10100.4230.423 = 2.65% = 2.65%
ProblemsProblems
Obtaining “good estimates” of the Obtaining “good estimates” of the mean and standard deviation of the mean and standard deviation of the data.data.
Make a calculated guess, use Make a calculated guess, use iteration to get “better fit” to iteration to get “better fit” to observed data.observed data.
Where Straight Lines MeetWhere Straight Lines Meet
Optimal AssentOptimal Assent
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Optimal AssentOptimal Assent
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Y1=a1+b1x
Optimal AssentOptimal Assent
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Y1=a1+b1x
Y2=a2+b2x
Optimal AssentOptimal Assent
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Y1=a1+b1x
Y2=a2+b2x
tt =[b =[b11-b-b22]/se(b)]/se(b)
= ns= ns
Optimal AssentOptimal Assent
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Y1=a1+b1x
Y3=a3+b3x
Optimal AssentOptimal Assent
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Y1=a1+b1x
Y3=a3+b3x
tt =[b =[b11-b-b33]/se(b)]/se(b)
= ***= ***
Optimal AssentOptimal Assent
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Y1=a1+b1x
Y3=a3+b3x
Optimal AssentOptimal Assent
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Y1=a1+b1x
Y3=a3+b3x
tt =[b =[b11-b-bnn]/se(b)]/se(b)
= ***= ***Yn=an+bnx
Optimal AssentOptimal Assent
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Y1=a1+b1x
Y3=a3+b3x
Y3=a3+b3x
Yield and NitrogenYield and NitrogenN applied (lb/acre)
Seed Yield (lb/acre)
50 921 60 997 70 1086 80 1214 90 1299 100 1341 110 1370 120 1402 130 1409
What application of What application of nitrogen will result in nitrogen will result in
the the optimumoptimum yield yield response?response?
Intersecting LinesIntersecting Lines
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N application
See
d Y
ield
(lb
/acr
e)
Intersecting LinesIntersecting Lines
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N application
See
d Y
ield
(lb
/acr
e) Y = 2.81x + 1055.10
Y = 9.01x + 466.60
Intersecting LinesIntersecting Lines
tt = [b = [b1111 - b - b2121]/average se(b)]/average se(b)
6.2/0.593 = 10.45 6.2/0.593 = 10.45 ** , With 3 df , With 3 dfIntersect = same value of yIntersect = same value of y
bb1010 + b + b1111x = y = bx = y = b2020 + b + b2121xx
x = [bx = [b2020 - b - b1010]/[b]/[b1111 - b - b2121]]
= 94.92 lb N/acre= 94.92 lb N/acre
with 1321.83 lb/acre seed yieldwith 1321.83 lb/acre seed yield
Intersecting LinesIntersecting Lines
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N application
See
d Y
ield
(lb
/acr
e) Y = 2.81x + 1055.10
Y = 9.01x + 466.60
94.92 lb N/acre94.92 lb N/acre
1321.83 1321.83 lb/acrelb/acre
LinearLinear
Y = bY = b00 + b + b11xxQuadraticQuadratic
Y = bY = b00 + b + b11x + bx + b2 2 xx22
CubicCubic
Y = bY = b00 + b + b11x + bx + b2 2 xx2 2 + b+ b3 3 xx33
Bi-variate DistributionBi-variate DistributionCorrelationCorrelation