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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t weig

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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t weig

ht (g

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y = y

Homogeneity of Error VarianceHomogeneity of Error Variance

05

101520253035

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Days after planting

Plan

t weig

ht (g

)

Homogeneity of Error VarianceHomogeneity of Error Variance

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Days after planting

Plan

t weig

ht (g

)

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

1

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

ae

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

22

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