Supervisor : Prof:L.A. Leslie JayasekaraDepartment Of MathematicsUniversity Of Ruhuna

Name: W.J.JannidiSC/2010/7623

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CONTENT

• Dose-Response Data

• Probit Model

• Logit Model

• LC50 Value

• Application

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Dose-Response Data

• Dose - A quantity of a medicine or a drug

• Response- Any action or change of condition

1 death, condition well

Response

0 no death, not well

• Dose-Response Relationship The dose-response relationship describes the change in effect on an

organism caused by differing levels of doses.

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• Dose-Response CurveSimple X-Y graph

X- dose, log(dose)

Y- response, percentage response, proportion

• Information of CurvePotency - the amount of drug necessary to

produce a certain effect

Efficacy- the maximal response

Slope- effect of incremental increase in

dose

Variability- reproductively of data different for different organism

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Further…………

NOAEL :- No Observed Adverse Effect Level

LOAEL :- Low Observed Adverse Effect Level

Threshold :- No adverse effect below that dose

Probit Model

• IntrodutionProbit analyze is used to analysis many kinds of dose-response or binomial

response experiments in a variety of fields and commonly used in toxicology.

In probit model, the inverse standard normal distribution of the probability is modeled as a linear combination of the predictors.

i.e Pr(y=1|x)= Φ(xβ) where Ф indicates the C.D.F of standard normal distribution.

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)(......

2

1)( )()(

1

110

2

2

xx

e

nn

x

Xand

zzwheredzzX

• Likelihood Contribution

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For single observation

When yi=1,p.d.f is Ф 𝑥𝑖𝛽 and when yi=0 ,1 −Ф(𝑥𝑖𝛽)

Likelihood is [∅(𝑥𝑖𝛽)]𝑦𝑖 [1 − ∅(𝑥𝑖𝛽)]1−𝑦𝑖

For n observation

L β = [∅(xiβ)]y i [1 − ∅(xiβ)]1−y i

n

i=1

Log-likelihood function is

ln L β = yi∅(xiβ + 1 − yi ni=1 (1 − ∅(xiβ))]

∂lnL (β)

∂β=

y i−∅(x iβ)

∅(x iβ)(1−∅(x iβ) n

i=1 ∅(xiβ)xi′ And

𝜕2𝑙𝑛𝐿(𝛽)

𝜕𝛽𝜕𝛽′ = − ∅(𝑥𝑖𝛽)2

∅(xiβ)(1 − ∅(xiβ) 𝑥𝑖

′𝑥𝑖

𝑛

𝑖=1

• Marginal effects

Marginal Index Effects

partial effects of each explanatory variable on the probit

index function xiβ.

Marginal Probability Effects

partial effects of each independent variables on the probability that the

observed dependent variable yi = 1.

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if xi is a continuous variabl

MIE of xi =∂E y i x i

∂x i=

∂x iβ

∂x i= βi

if xi is a binary variable

𝑀𝐼𝐸 𝑜𝑓 𝑥𝑖 = 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥𝑖𝛽 𝑤ℎ𝑒𝑛 𝑥𝑖 = 1 𝑜𝑟 𝑥𝑖 = 0

• Relationship between MIE and MPE

MPE is proportional to the MIE of xi where the factor of

proportionality is the standard normal p.d.f. of Xβ.

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If xi is a continuous variable

MPE of xi =∂Pr(y=1)

∂x i=

∂∅(xiβ)

∂x i=

d∅ xiβ ∂(x iβ)

d(x iβ)∂x i= ∅(xiβ)

∂x iβ

∂x i

If xi is a binary variable

MPE of xi = ∅ x1β − ∅(x0β)

When xi is a continuous explanatory variable

MIE of xi =∂(xiβ)

∂xi and MPE xi = ∅(xiβ)

∂(xiβ)

∂xi

i.e MPE of xi = ∅ xiβ ∗ MIE of xi

• Goodness of fit test

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Judge by McFaddens pseudo R2

Measure for proximity of the model

lnL Mfull : Likelihood of model of interest

lnL Mintercept : Likelihood with all coefficients zero without intercept

Always holds that lnL Mfull ≥ lnL Mintercept

pseduo R2 = RMcF2 = 1 −

ln L M full

ln L M intercept ; 0 ≤ RMcF

2 ≤ 1

An increasing pseudo R2 may indicate a better fit of the model.

Logit Model

There are two type of logit models

Binary logit model : dependent variable is dichotomous

Multinomial logit model : dependent variable contains more than

two categories

Independent variables are either continuous or categorical in both

models.

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π x = E(y|x) =eβX

1 + eβX

A transformation of π(x) is

g x = ln π(x)

1−π(x) =𝛃𝐗

• Simple Logit Model

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π x =eβ0+β1x

1 + eβ0+β1x

Assume that β1 >0,

for negative values of x, eβ0+β1x → 0 as x → −∞

hence π x →0

1+0= 0

for very large value of x, eβ0+β1x → ∞ and hence π x →∞

1+∞= 1

when x = −β0

β1,β0 + β1x = 0 and hence π x =

1

1+1= 0.5

Thus β1 controls how fast π(x) rises from 0 to 1.

• Likelihood function

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Consider a sample of n independent observations of the pair (xi,yi) i=1,2….n

Pr y = 1|x = π x and Pr y = 0 x = 1 − π(x)

For the pair (xi,yi), likelihood function is π(xi)y i 1 − π(xi)

1−yi

Assume that observations are independent,

Likelihood function of n observation is L(β) = [π(xi)]y ini=1 [1 − π(xi)]1−yi

lnL β = yilnπ xi + (1 − yi)ln(1 − π(xi))

n

i=1

To find the value of β that maximizes the lnL(β), differentiate lnL(β) w.r.t β0

and β1 and set the resulting expressions equal to zero.

yi − π xi = 0 and xi yi − π xi = 0

• Significance of the CoefficientsUsually involves formulation and testing of a statistical

hypothesis to determine whether the independent variables in the

model are significantly related to the outcome variable.

1. Likelihood ratio test

2. Wald test

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D = −2ln likelihood of the fitted model

likelihood of the saturated moel = −2 ln likelihood ratio

D is called the deviance.

Let ,G = D(model without the variables) − D(model with the variables)

G = −2ln likelihood without the variable

likelihood with the variable ~xno of extra parameter

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Wj =βj

SEβ j

~x12

• Score testBased on the slope and expected curvature of the log-

likelihood function L(β) at the null value β0.

• Confidence interval

100(1-α)% C.I for the intercept and slope

• Multiple logistic model

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u β = ∂L(β)/ ∂β|β0 = u β0

−E[∂2L(β)/ ∂β2|β0] = τ β0

test st: u(β0)/[τ(β0)]1/2 ~N(0,1)

β 0 ± z1−α/2SE β0 and β 1 ± z1−α/2SE β1

𝑔 𝑥 = 𝑙𝑛𝜋(𝑥)

1 − 𝜋(𝑥)= 𝛽0 + 𝛽1𝑥1 + ⋯+ 𝛽𝑝𝑥𝑝

• Dichotomous independent variable

Independent variable has two categories and coded as 1

and 0.

• Polychotomous independent variablehas k>2 categories

Reference cell coding method

Ex: Risk of a disease

• Odds Ratio

Odds : For a probability π of success, odds are defined as

Ω=π/(1-π)

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Rate Risk(code) D1 D2

Less 0 0

Same 1 0

More 0 1

Independent variable X

Outcome variable(y) X=1 X=0

Y=1

Y=0

Total 1 1

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π 1 =eβ0+β1

1 + eβ0+β1

𝜋 0 =𝑒𝛽0

1 + 𝑒𝛽0

1 − 𝜋 1 =1

𝑒𝛽0+𝛽1 1 − 𝜋 0 =

1

1 + 𝑒𝛽0

OR =π(1)/[1 − π(1)]

π(0)/[1 − π(0)]= eβ1

95% CI of ln OR = ln(OR) ± 1.96SE[ln OR ]

95% CI of OR = eln(OR )±1.96SE [ln OR ]

• Relative riskRatio of the two outcome probabilities

RR=π(1)/π(0)

LC 50 Value

The concentration of the chemical that kills 50% of the test animals.

Use to compare different chemicals.

In general, the smaller the LC50 value, the more toxic the chemical. The opposite is also true: the larger the LC50 value, the lower the toxicity.

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• Method of Miller and Tainter

Ex:

The percentage dead for 0 and 100 are corrected before the

determination of probits using following formulas.

For 0%dead = 100(0.25/n)

For 100%dead =100(n-0.25/n)

Fitting linear regression model between log(dose) and probit

value, LC 50 is calculated.

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Dose Log(dose) % dead Corrected %

Probits

25 1.4 0 2.5 3.04

50 1.7 40 40 4.75

75 1.88 70 70 5.52

100 2 90 90 6.28

150 2.18 100 97.5 6.96

LC50= 57.54mg/kg

• Probit table

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Application

Laboratory experiment was carried out to evaluate the effect of

different botanicals such as Wara,Keppetiya and Maduruthala in

the control of root knot nematode (M. javanica) by

Prof:(Mrs)W.T.S.D.premachandra, Department Of Zoology.

Approximately 50 juveniles were dispensed into petridishes

containing different concentration extracts (100,80,60,40,20) of

the botanicals. After 48 hours, recorded number of deaths of each

petridishes.

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Response variable

1 when death is occur

y

0 no death

Independent variables

Concentration

Plant type

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Plant type D1 D2

Maduruthala 0 0

Keppetiya 1 0

Wara 0 1

• The Logistic Model

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call:

glm(formula=data$ dead ˜data$Concentration + data$plant.f, family = binomial(link

= ”logit”))

Deviance Residuals:

Min 1Q Median 3Q Max

-2.4779 -0.5636 -0.1515 0.5664 2.5410

Coefficients:

Estimate Std. Error z value Pr(> |z|)

(Intercept) -5.735211 0.131744 -43.53 <2e-16 ***

Concentration 0.063686 0.001481 42.99 <2e-16 ***

Keppetiya 2.388994 0.086474 27.63 <2e-16 ***

Wara 2.702167 0.089134 30.32 <2e-16 ***

Null deviance: 10374.8 on 7499 degrees of freedom

Residual deviance:6350.3 on 7496 degrees of freedom

AIC: 6358.3

Pseduo Rsq= 0.3879

Fitted values

All independent variables are significant.

For every one unit change in concentration, the log odds of death

(versus no death) increases by 0.0636.

Having a death with Keppetiya plant, versus a death with a

Maduruthala plant, changes the log odds of death by 2.3889. And also

Having a death with Wara plant, versus a death with a

Maduruthala plant, changes the log odds of death by 2.7021.

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π (x) =e−5.735+0.0636Con +2.3889Keppetiya +2.7021Wara

1 + e−5.735+0.0636Con +2.3889Keppetiya +2.7021Wara

Estimated logit,

g x = ln(ODDS) = −5.735 + 0.0636Con + 2.3889Keppetiya + 2.7021Wara

We can test for an overall effect of plant using the Wald test.

Wald test:

test st:= 1036.2 df = 2 P(>X2) = 0.00

The overall effect of plant is statistically significant.

Odds Ratios and their 95%CI:OR 2.5% 97.5%

Intercept 0.003230201 0.002486319 0.00416747

Concentration 1.065757750 1.062704680 1.06889510

Keppetiya 10.902516340 9.215626290 12.93485861

Wara 14.912011210 12.541600465 17.78769387

An increase of one unit in Concentration is associated with 1.0658 increase in

the odds of having a death. Keppetiya increases the odds of having a death

than Maduruthala by 10.903.

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• The Probit Model

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call:

glm(formula=data$ dead ˜data$Concentration + data$plant.f, family =

binomial(link = ”probit”))

Deviance Residuals:

Min 1Q Median 3Q Max

-2.5347 -0.5739 -0.1043 0.5857 2.5829

Coefficients:

Estimate Std. Error z value Pr(> |z|)

Intercept -3.2911468 0.0683004 -48.19 <2e-16 ***

Concentration 0.0371691 0.0007816 47.56 <2e-16 ***

Keppetiya 1.3218435 0.0475294 27.81 <2e-16 ***

Wara 1.5192155 0.0486710 31.21 <2e-16 ***

Null deviance:10374.8 on 7499 degrees of freedom

Residual deviance:6346.2 on 7496 degrees of freedom

AIC: 6354.2

The predicted probability of death is

Pr(y=1|x)=π(x)=Φ(−3.2911 + 0.0372Con + 1.3218Keppetiya +

1.5192Wara)

All independent variables are significance and has positive effect

from each variables.

For every one unit change of Concentration, the c.d.f of standard

normal distribution is increase by 0.0372.

Having a death with Keppetiya plant, versus a death with a

Maduruthala plant, changes c.d.f of death by 1.3218.

Having a death with Wara plant, versus a death with a

Maduruthala plant, changes c.d.f of death by 1.5192.

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• Marginal effects

Probability of having a death changes by 0.88% for

every one unit change of Concentration.

Having a death in Keppetiya is 31.3% more likely than

in Maduruthala.

And also

Having a death in Wara is 35.98% more likely than

in Maduruthala.

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Concentration Keppetiya Wara

0.008802969 0.313060054 0.359804831

• Comparison of LC50 values

Lowest LC50 value means that highest effect on death.

Wara plants extract has the lowest LC50 value.

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Plant LC50 using Logit

model

LC50 using probit

model

Maduruthala 90.1729 88.4704

Keppetiya 52.6116 52.9382

Wara 47.6871 47.6317

The maximal response has been obtained

by Wara plant extract.

That is,

it has highest efficacy than others. Potency of

Wara is also highest value but no more

differ from Keppetiya.

Maduruthala plant extract has shown

lower potency and lower efficacy.

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• Conclusions

According to the LC50 values and other toxic

measures , Wara is recommended as the effective

botanical than other botanicals.

Also,

It is enough, add 47.63mg/ml of Wara plant extract to

kill 50% of the Nematode population.

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