a mixed model for the analysis of enzyme linked ... · nestlé research center a mixed model for...

Nestlé Research Center

A Mixed Model for the Analysis ofEnzyme Linked Immunosorbent Assay

Dominik Grathwohl

Nestlé Research Center

Lausanne

Switzerland

Nestlé Research Center16-NOV-2006 NRC/BAS - Dominik Grathwohl2

Enzyme Linked Immunosorbent Assay (ELISA)

Steps:(1) Plate is coated with a protein for capture.(2) Serum containing a mixture of antibodies is added. Antibodies specific for the

capture protein will be bound.(3) A secondary antibody linked to an enzyme is added. It binds specifically to

the first antibody.(4) Substrate is added, and is converted by enzyme to detectable form.

1) 2) 3) 4)


Cutoff analysis

0

1

2

3

500

1500

4500

1350

0

4050

0

1215

00

3645

00

1093

500

nega

tive

cont

rol

posit

iveco

ntro

l

Dilution of the serum

Op

tica

ld

en

sity


Cutoff analysisExplanation

Why does it work?

cjini

inidc

inidc

ds

sODf

sfOD

cj

cj

~

)(

)(

1

constant

1

1

Continuous inverse exists inmeasurement range


Cutoff analysisResume

Cutoff analysis is simple and robust

No calibration necessary

Producing ordered categorical data

Wasting information

Treatment effects in arbitrary units

Statistical analysis: Stage 1: Each subject has its own curve characteristic:

Value above the cutoff

Stage 2: Characteristics is analyzed by conventional statisticaltechniques (e.g. Wilcoxon-test).


Cutoff analysisRevisited

0.0000 0.0005 0.0010 0.0015 0.0020

0.0

0.1

0.2

0.3

0.4

Concentration

OD

B 1

Measurements of optical density (OD) over concentration of mouse B1,

concentration is presented in arbitrary units:25

1,

75

1,

225

1,

675

1,

2025

1,

6075

1,

18225

1

54675

1


Michaelis-Menten modelgeneralized Michaelis-Menten model

Leonor Michaelis 1875-1949Maud Menten 1879-1960

sK

sV

m

max

0max'

max

'max

0 , VVVsK

sVV

m

0.000 0.002 0.004 0.006 0.008 0.010

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Michaelis-Menten model

Concentration

OD

Vmax

Km

0.000 0.002 0.004 0.006 0.008 0.010

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

generalized Michaelis-Menten model

Concentration

OD

Vo


Nonlinear calibrationCalibrating step

0.0000 0.0005 0.0010 0.0015 0.0020

0.0

0.1

0.2

0.3

0.4

Concentration

OD

known

,,,:

:

'max0

01

01

01'max

0

m

inid

inidm

inid

KVVfit

sknown

sK

sVV

j

j

j


Nonlinear calibrationPredicting step

0.0000 0.0004 0.0008 0.0012

0.0

0.1

0.2

0.3

0.4

Concentration

OD

unknow n

0 e+00 4 e-04 8 e-04

0.0

0.1

0.2

0.3

0.4

Concentration

OD

unknow n

0 e+00 4 e-04

0.0

0.1

0.2

0.3

0.4

ConcentrationO

D

unknow n

0 e+00 2 e-04 4 e-04

0.0

0.1

0.2

0.3

0.4

Concentration

OD

unknow n

0

01

01'max

0'

max0 known,,, iniuini

inidm

inid

m sssK

sVVKVV

j

j


Nonlinear calibrationResume

Using all measurements Curve characteristics S0

ini is continuous Treatment effects in physical units Calibration sample necessary Needs nonlinear fitting software

Statistical analysis: Stage 1: Each subject has its curve characteristic: Stage 2: Characteristics is analyzed by conventional statistical techniques

(e.g. ANOVA).

The parameters: are method characteristics,they can be determined by the laws of thermodynamics.They can be considered as fixed effects!

Biological variability is described in , will be considered asrandom effect!

,,, 'max0 mKVV

uinis


Nonlinear mixed effect model

Mean model

Error model

Autocorrelation

Heteroscedasticity

Random effect

Probability model

22

ijijVar

2,0~ bi Nb

iiim

j

ii

i

j

trttrti

binitK

d

trtbij

inittrtidij

ijm

ij

ij

eeseewith

e

VV

sssK

sVV

1

log

1

'max

0

11

'max

0

11

0

10

,,

1

and

normalornormal log

)1(AR


Nestlé animal trial (Mouse)

n/group = 5, primary: IgE, secondary IgG1

B) Water (negative control)

E) Bacteria-DNA

F) Bacteria-DNAse

G) Calf-thymus-DNA (negative control)

Contrasts Expectation

E-B negative

F-B negative

G-B +/- zero

F-E ?


Nonlinear mixed effect modelIgG1: Model selection, parameter estimates

Description Parameter Lower Estimate Upper

background velocity V0 0.056 0.057 0.058

maximum velocity Vmax 0.269 0.426 0.674

power trans. substrate 0.685 0.749 0.818

treatment effects 0B 56.0 181.7 589.0

1E 0.198 0.554 1.551

1F 0.191 0.536 1.500

1G 0.064 0.181 0.514

random effect standard deviation b 0.555 0.815 1.198

autocorrelation parameter 0.137 0.357 0.544

power of the variance function -1.881 -0.996 -0.110

residual standard deviation 0.042 0.101 0.245

IgG1 analyzed by nonlinear mixed effect model,parameter estimates and 95% confidence intervals

AIC(normal) = -1323.99AIC(log-normal) = -1324.23

Normal vs. log-normal


Nonlinear mixed effect modelIgG1: Quality of the fit

Fitted values

Sta

nd

ard

ize

dre

sid

ua

ls

-3

-2

-1

0

1

2

-2.5 -2.0 -1.5

Standardized residuals

Qu

an

tile

so

fsta

nd

ard

no

rma

l

-2

-1

0

1

2

-3 -2 -1 0 1 2


Nonlinear mixed effect modelIgG1: Prediction

Concentration

log

(OD

)

-3.0

-2.5

-2.0

-1.5

0.0000 0.0005 0.0010 0.0015 0.0020

:id { 11 } :id { 12 }

0.0000 0.0005 0.0010 0.0015 0.0020

:id { 13 } :id { 14 }

0.0000 0.0005 0.0010 0.0015 0.0020

:id { 15 }

:id { 31 } :id { 32 } :id { 33 } :id { 34 }

-3.0

-2.5

-2.0

-1.5

:id { 35 }-3.0

-2.5

-2.0

-1.5

:id { 36 } :id { 37 } :id { 38 } :id { 39 } :id { 40 }

:id { 41 }

0.0000 0.0005 0.0010 0.0015 0.0020

:id { 42 } :id { 43 }

0.0000 0.0005 0.0010 0.0015 0.0020

:id { 44 }

-3.0

-2.5

-2.0

-1.5

:id { 45 }


Nonlinear mixed effect model, cutoffIgG1: Treatment differences

Mixed

95 % two-sided confidence intervals

-4 -3 -2 -1 0 1 2

F-E

G-B

F-B

E-B

( )

( )

( )

( )

-4 -3 -2 -1 0 1 2

Cutoff

95% two-sided confidence intervals

F-E

G-B

F-B

E-B


Nonlinear mixed effect modelIgE: Model selection, parameter estimates

Description Parameter Lower Estimate Upper

background velocity V0 0.041 0.043 0.044

maximum velocity Vmax 0.326 0.762 1.777

power trans. substrate 0.579 0.646 0.721

treatment effects 0B 1.012 9.773 94.417

1E 0.064 0.504 4.002

1F 0.121 0.960 7.615

1G 0.023 0.189 1.533

random effect standard deviation b 1.137 1.628 2.333

autocorrelation parameter 0.623 0.775 0.871

power of the variance function -2.020 -1.653 -1.286

residual standard deviation 0.317 0.440 0.612

IgE analyzed by nonlinear mixed effect model,parameter estimates and 95% confidence intervals

AIC(normal) = -1048.54AIC(log-normal) = -1073.40

Normal vs. log-normal


Nonlinear mixed effect modelIgE: Quality of the fit

Fitted values

Sta

nd

ard

ize

dre

sid

ua

ls

-1

0

1

2

3

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5


Qu

an

tile

so

fsta

nd

ard

no

rma

l

-2

-1

0

1

2

-1 0 1 2 3


Nonlinear mixed effect modelIgE: Prediction

Concentration

log

(OD

)

-3

-2

-1

0

0.00 0.01 0.02 0.03 0.04

:id { 11 } :id { 12 }

0.00 0.01 0.02 0.03 0.04

:id { 13 } :id { 14 }

:id { 15 } :id { 31 } :id { 32 }

-3

-2

-1

0

:id { 33 }

-3

-2

-1

0

:id { 34 } :id { 35 } :id { 36 } :id { 37 }

:id { 38 } :id { 39 } :id { 40 }

-3

-2

-1

0

:id { 41 }

-3

-2

-1

0

:id { 42 }

0.00 0.01 0.02 0.03 0.04

:id { 43 } :id { 44 }

0.00 0.01 0.02 0.03 0.04

:id { 45 }


Nonlinear mixed effect model, cutoffIgE: Treatment differences

-6 -4 -2 0 2

Cutoff

95% two-sided confidence intervals

F-E

G-B

F-B

E-B

Mixed

95 % two-sided confidence intervals

-6 -4 -2 0 2

F-E

G-B

F-B

E-B

( )

( )

( )

( )


Nonlinear model,IgE: All parameters free

Concentration

log

(OD

)

-3

-2

-1

0

0.00 0.01 0.02 0.03 0.04

:ID { 11 } :ID { 12 }

0.00 0.01 0.02 0.03 0.04

:ID { 13 } :ID { 14 }

:ID { 15 } :ID { 31 } :ID { 32 }

-3

-2

-1

0

:ID { 33 }

-3

-2

-1

0

:ID { 34 } :ID { 35 } :ID { 36 } :ID { 37 }

:ID { 38 } :ID { 39 } :ID { 40 }

-3

-2

-1

0

:ID { 41 }

-3

-2

-1

0

:ID { 42 }

0.00 0.01 0.02 0.03 0.04

:ID { 43 } :ID { 44 }

0.00 0.01 0.02 0.03 0.04

:ID { 45 }


Parameter estimates

ID V0 Vmax eta beta sigma11 -3.17 -0.92 -0.33 5.80 0.02112 -3.22 4.75 -0.44 -5.01 0.02313 -3.25 -0.52 -0.67 0.85 0.02814 -3.21 2.35 -0.24 0.48 0.01615 -3.19 -1.74 -0.53 2.30 0.02431 -3.10 -1.22 -0.33 3.86 0.02232 -3.22 -1.57 -0.53 3.83 0.02033 -3.08 -0.90 -0.33 2.50 0.05334 -3.17 -0.67 -0.39 2.99 0.04435 -3.13 11.03 -0.57 -19.85 0.01736 -3.11 5.43 -0.58 -8.72 0.04237 -3.09 -0.71 -0.31 3.34 0.01738 -3.15 -0.41 -0.42 3.66 0.00839 -3.13 0.19 -0.38 2.25 0.05440 -3.09 5.90 -0.42 -9.07 0.01041 -3.12 -1.35 0.62 -6.86 0.03842 -3.18 2.52 -0.16 0.95 0.01443 -3.21 -1.44 -0.46 3.72 0.01244 -3.25 9.08 -0.68 -18.40 0.01345 -3.22 -1.22 -0.49 2.89 0.025


Identifying stable subpopulation

35

44

39

11

33 34

37 13

38

41

31

45 15

32

43

14

42

12

36

40

02

46

81

01

2

He

igh

t

-2 0 2 4 6 8 10

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Vmaxe

ta


Nonlinear mixed effect modelOn “stable” subpopulation (quality)

Fitted values

Sta

nd

ard

ize

dre

sid

ua

ls

-2

-1

0

1

2

-3.0 -2.5 -2.0 -1.5 -1.0


Qu

an

tile

so

fsta

nd

ard

no

rma

l-2

-1

0

1

2

-2 -1 0 1 2


Nonlinear mixed effect modelOn “stable” subpopulation (prediction)

Concentration

log

(OD

)

-3.0

-2.5

-2.0

-1.5

-1.0

0.00 0.01 0.02 0.03 0.04

:id { 11 } :id { 13 }

0.00 0.01 0.02 0.03 0.04

:id { 15 }

:id { 31 } :id { 32 }

-3.0

-2.5

-2.0

-1.5

-1.0

:id { 33 }

-3.0

-2.5

-2.0

-1.5

-1.0

:id { 34 } :id { 37 } :id { 38 }

:id { 39 } :id { 41 }

-3.0

-2.5

-2.0

-1.5

-1.0

:id { 43 }

-3.0

-2.5

-2.0

-1.5

-1.0

:id { 45 }


Nonlinear mixed effect modelResume

Mechanistic model with clear interpretation ofparameters

Enables to identify cross reactions

Enhance power of analysis by using all measurements

No calibration necessary

Treatment effects in arbitrary units

Needs nonlinear fitting software

Statistical analysis: Stage 1 and Stage 2 are merged together.

The calibration step is done “en passant”.

a mixed model for the analysis of enzyme linked ... · nestlé research center a mixed model for...

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