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Paper Summary

Monotone Smoothing with Application to

Dose-Response Curves and

the Assessment of Synergism

Colleen Kelly and John Rice, Biometrics,

Vol. 46, No. 4, (Dec., 1990), pp. 1071-1085


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Outline

Introduction

Monotone Smoothing

Assessing SynergismBootstrap Method

Procedure

ApplicationResult


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Monotone Smoothing

0 1 1

1

To monotone smoothing is based on spline

function. Given a partition

... of an interval [ , ]

a spline function is a polynomial on each

interval [ , ).

We shall refer to the as "kno

p

j j

i

a b a b

ts".


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Monotone Smoothing

Given data points ( , ) ( 1,..., ), a smoothing

spline is the function that minimizes

i i x y i n

1

2 2

1

( ) ( ( )) [ ( )]n

n x

i ix

i

Q Y x x dx

The minimizing is a cubic spline with

a knot at every data point.


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Monotone Smoothing

as 0, tends to an interpolant of the data.

The smaller is, the quicker the running time,

especially when 0.

p

as , tends to the LS line fit to the data.


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Monotone Smoothing

Knot placement should depend on the density

of data points or on local behavior of curve.

Thus we control smoothing only with .

1. Chosen the amount of smoothing to apply .

2. Leave-one-out cross-validation.

3. Doing a preliminary QR decomposition.


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Assessing Synergism

Synergistic if its effect is more than

would be expected form what is known

about the effect of the individuak

constituent.

Antagonistic if the effect is less.

Asynergistic if effect is the same.


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Assessing Synergism

Definition of synergism:

Let the repsone to dosages and of two

drug be denote by ( , ).

Assume 0 0, and ( ,0) (0, ) are

monotone.

x y

f x y

x y f x f y


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Assessing Synergism

For given response value let and be

those dosages such that ( ,0) (0, ) .

if ( , ) , 1.

Synergistic if left side is less than 1.

Antagonistic if it is greater than 1.

z z

z z

z z

z x y

f x f y z

x y f x y z x y


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Bootstrap Method

Parametric bootstrap samples

Model could be binomial or poisson.

2

*( )

i

*(b) *( )

, ~(0, ), independent.bootstrap residuals

: 1... : 1...

bootstrap sample for Y is Y .

b

b

Y X

i n b B

X


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Procedure

Monotone spline

Parametric bootstrap sample

Check synergism


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Application I

Data form Le Pelley and Sullivan

(1936).

Alcoholic solutions of rotenone,

pyrethrins, and a mixture of the two.

The mixture series was 1:5 ratio and

1:15 ratio.


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Application I

2

2

Calculations

for the 1:5 ratio, (3) 81.19 was synergism.

for the 1:15 ratio, (3) 11.72 which is

not significant at 0.05.


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Application I

Bliss (1939);

Finney, using probit analysis (1952);

Chou and Talalay with median effect

analysis (1983)

All investigators concluded there was

synergism in the 1:5 ratio and slight

synergism in the 1:15 ratio.


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Application II

Data form Dr Stephen Howell of the UCSD

Cancer Center.

Ovarian cancer may be treated by VP-16 and

DPM.

There are 6 concentration levels for each

individual dose-reponse curve, with 18

observations.There are 4 combination doses ratio 1:150 of

VP-16/DPM.


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Application II

Parametric bootstrap based on Poisson

model for the random variability under

asynergy.

Responses are to increasing

concentrations of a conmbination of both

drugs in a 1:150 ratio.


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Reference

Howell, S. B., Sanga, R., Vick, J., and Abramson, I. (1989a).

Comparison of the Synergistic Potentiation of Etoposide,

Doxorubicin, and Vinblastine Cytotoxicity by Dipyridamole.

Cancer Research 49, 4147-4153.

, , lecture5, lecture6, lecture7.


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Thank you for your listening.

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