Estimation of Parameter Values

Nutrition Models Workshop

Luis Moraes

The Ohio State University

06/25/2017

1

Outline

• Nutrition models are VERY diverse• Combination of empirical, mechanistic, dynamic and static models

• Regression, linear and nonlinear mixed models, differential equations

• Today: Main approaches for estimating parameters in a variety of models• Some mathematical description

• Idea is for you to understand the reasoning and challenges of different approaches

• One exercise/demonstration in the end• Fit model with two approaches

2

Introduction

• Different types of models have been used for nutrition modeling• Compartmental, regression, meta-analysis, nonlinear mixed models, …..

• One feature is common to almost all these models• Parameters are needed to describe the system

• Quantify relationship between variables

3

Introduction

• Simple example: linear regression 0 1i i iY x

0 20 40 60 80 100

010

20

30

40

50

60

x

y

• Yi is the response variable for the ith observation• xi is the predictor variable in the ith observation• β0 is the intercept• β1 is the slope

• εi is the error, E[εi] = 0, Var[εi] = σ2 and εi are independent• i = 1, …, n

In matrix notation: y Xβ ε

4

1 1 1

0

1

1

1n n n

Y x

Y x

Introduction• In practice, parameter true values are unknown

• Estimators from a sample

• Parameters have to be optimal in some sense• Least square estimators minimize squared errors

• Maximum likelihood estimators maximize the likelihood function

5

Least Squares Estimators

6

Least Square Estimators

• The least squares estimators minimize the square errors:

• How do we get them???

• We can find points of minimum and maximum of a function using derivatives.

For example for f (x) = 160x – 16x2

7

2

0 1

1

n

i i

i

Q Y x

Set derivative to zero and “solve” for x:

160 – 32x = 0

x = 5

Second derivative test: −32

Least Squares Estimators

8

0 1

10

0 1

10

2

2

n

i i

i

n

i i i

i

QY x

Qx Y x

Setting these partial derivatives to zero, we construct the normal equations

0 1

1 1

2

0 1

1 1 1

n n

i i

i i

n n n

i i i i

i i i

Y nb b X

X Y b X b X

Least Squares Estimators

• The least square estimators are the solutions to the normal equations

• The concept extends to multiple regression

• General form of the least squares estimators:

9

2

1

1 1

0 1

/n n

i i i

i i

b x x Y Y x x

b Y b x

1

T T

b X X X Y

2

0 1 1 1 , 1

1

n

i i p i p

i

Q Y x x

Least Squares Estimators

• Estimates of the uncertainty associated with these parameters

• Estimator of the error’s variance

• Estimated variance-covariance matrix of the parameters

10

2

0 1 1 1 , 1

1

1 n

i i p i p

i

MSE Y b b x b xn p

1

TMSE

X X

Nonlinear Models

• So far, we can estimate parameters in linear models

• Many phenomena in biology are nonlinear• For example, reaction velocity vs. substrate concentration in an enzymatic reaction

• Before we start with nonlinear models, let’s clarify

is a nonlinear model

is a linear model

11

1

2 31 expi i

i

Yx

2

0 1 2i i i iY x x

Nonlinear Models

12

50

100

150

200

0.0 0.3 0.6 0.9

Substrate Concentration

Reaction R

ate

Michaelis-Menten Kinetics

max

M

V Sv

K S

Nonlinear Regression

13

,i i iY f x θ

f is the nonlinear function describing the relationship between Y and x

Michaelis-Menten example: max i

i i

M i

V xy

K x

• yi is the reaction rate for the ith observation

• xi is the associated substrate concentration

are the parameters to be estimated

• εi is the error, E[εi] = 0, Var[εi] = σ2 and independent

• i = 1, …, n

max, ii

M i

V xf x

K x

θ

T

max , MV Kθ

Least Squares Estimators

• For the simple linear regression model, least squares minimize

• For the nonlinear regression, the idea is the same: minimize

14

2

0 1

1

n

i i

i

Q Y x

2

1

n

i i

i

Q Y f x

Least Squares Estimators

• Solution to the normal equations are often difficult to obtain analytically

• Numerical Algorithms• For example, Gauss-Newton

• Require initial values to initialize numerical procedures

15

Gauss-Newton

• Default in PROC NLIN and nls()

• Approximate the nonlinear model with linear terms

• Taylor series expansion and least squares as for linear regression

• Denote the least squares estimates g and the initial values

• Approximation around starting values:

16

0 (0) (0) (0)

0 1 1, , , pg g g g

0

10 0

0 =

,, ,

pi

i i k k

k k

f xf x f x g

θ g

θθ g

Gauss-Newton

• Model approximation

• It is a linear model!

• Estimate parameters by least squares:

• Update:

17

0

100

0

0 0 0

,,

pi

i i k ikk k

f xY f x g

θ g

θg

Y D β ε

1

0 0 T 0 0 T 0

b D D D Y

1 0 0 g g b

Gauss-Newton

• Evaluation criteria:

• Start the process again with as the initial values

• Repeat procedure until is negligible

• Estimate of error’s variance:

• Other methods available, e.g. Nelder-Mead and Marquardt

18

2

0 0

1

n

i i

i

SSE Y f

1g

1s sSSE SSE

2

1

, /n

i i

i

MSE Y f x n p

g

Compartmental Models

• Traditionally used in nutritional modeling• Roots on pharmacokinetics and differential calculus

19

GutCompartment

Drug in ka Blood Compartment

ke

Compartmental Models

• Functional forms described in terms of differential equations• Instead of the “integrated form”

• Strategy for parameter estimation• Expected mean represented by a compartmental model f

• If f cannot be obtained analytically, it has to be solved numerically • Euler, Runge-Kutta4, lsoda

• Can use nonlinear least squares but have to numerically solve f at iteration

• Modern software estimate using maximum likelihood

20

Maximum Likelihood Estimation

• Another strategy for parameter estimation

• For regression models with independent , estimators coincide with least squares estimators

• Estimators maximize the likelihood function• Parameter values that are in best agreement with the data

21

2~ 0,i N

Maximum Likelihood Estimation

22

• p (y | μ σ2) is the density function: How likely y is at each value

• The likelihood function is: – “How likely the whole data is with that set of parameters values”

– MLE: “maximize the likelihood of getting the observed data”

2 2 2

1 1, | ,..., | , | ,n nL y y p y p y

2

2

22

1| , exp

22

yp y

Maximum Likelihood Estimation

• Linear Regression Example:

• It is easier to work with the log-likelihood

• To find parameters that maximize the likelihood we…• Take the derivative with respect to each parameter and set to zero. Also need

second derivative test

23

0 1i i iY x

0 1 0

2

1 22 1

2 1 1, | , , exp

22,

n

i

in i xL y y y

22 2

0 1 1 0 12

1log , , | , , log 2 log

2 2 2n i i

i

n nL y y y x

Mixed Models

• Modern mixed modeling relies heavily on likelihood methods

• Extension of (non)linear models with both fixed and random effects

• Probably the “type” of model you need to analyze your data or construct your nutrition model

• There’s a whole workshop on mixed models in this meeting

24

Nonlinear Mixed Models

• Nonlinear functional forms• Michaelis-Menten, logistic, exponential, Gompertz, …

• Random effects that “enter the model” nonlinearly

• Allow you to model nonlinear clustered, logitudinal data • Records from the same animal, treatment means from the same study

25

Nonlinear Mixed Models

• yij is the jth record on the ith “subject” or cluster

• xij is the associated predictor variable

• θi is the vector of subject specific parameters

θi = β + bi bi ~ N(0, Ψ)

• εij is the random error ~ N(0, σ2)26

,ij ij i ijy f x θ

“Fixed” “Random”

Nonlinear MM Maximum Likelihood

• There is more than one source of variability• Between subjects and within subjects

• To represent the generative process of the data we need to take both into account• Joint density of the response and the random effects

27

Maximum Likelihood

28

• For the linear regression

• The likelihood function is:

• For the nonlinear mixed model, we need to compute the marginal density of the responses:

2 2 2

1 1, | ,..., | , | ,n nL y y p y p y

2

2

22

1| , exp

22

yp y

2 2| , , | , , |p p p d y β Ψ y b β b Ψ b

Maximum Likelihood

• Bad news from a estimation perspective

• The likelihood function to estimate the parameters requires integrating the joint density with respect to the random effects

• The integral often does not have a closed form expression

• Approximation of the likelihood function

29

Exercise

• Let’s go back to the compartmental model

30

11

21 2

a

a e

dyk y

dt

dyk y k y

dx

• y1 is the amount of drug in the gut compartment• y2 is the amount of drug in the blood compartment• ka is the absorption rate (1/hr)• ke is the elimination rate (1/hr)

Exercise

• Data from Davidian and Giltinan (1995)

• 12 subjects received a single oral dose of theophylline• Anti-asthmatic drug

• Single oral dose at time zero

• Measurements of blood concentrations of drug at 11 time points over a 25 hour period

31

Exercise

32

Time (hr)

Concentr

ation (

mg /

L)

0

2

4

6

8

10

0 5 10 15 20 25

Exercise

• We only observe data from the second compartment

• The differential equation for the second compartment can actually be solved analytically

• We will estimate the parameters in two different ways• Analytical solutions: nonlinear mixed model

• Numerical solutions for differential equations in a mixed model framework

33

Exercise

• The analytical solution to the second differential equation is

• As a nonlinear mixed model

34

( ) exp expe a

e a

a e

DoseC t t

k kk k

Cl k kt

,

,

,

exp expi e a i

ij e ij a i ij ij

i a i e

Dose k kC k t k t

Cl k k

2

2

2

,

00~ 0, and ~ ,

00a

i Cl

ij

a i k

ClN N

k

Exercise

• All three parameters must be positive

• Reparameterize model with parameters on a log scale

where

35

exp( ) exp exp exp exp

exp exp

e a

e a

a e

lk lk lCllk lk

lk l

DoseC t t t

k

log , log and loge ae alk lk lC Clk k l

Exercise

• First method• Fit nonlinear mixed model in R

36

Exercise

• Second method• Nonlinear mixed model but solve differential equations numerically: lsoda

solver

• Observation Equation:

37

11

21 2

1

2

0

0 0

a

a e

dyk y

dt

dyk y k y

dx

y Dose

y

2

e

y tC t k

Cl

Bayesian Inference

• Combines prior information with new data: update of knowledge

• All parameters are treated as random variables• Prior distributions for parameters

• Inference is based on the posterior distribution

• Bayes theorem

38

| |p g Lθ y θ θ y

Posterior DataPrior

From: https://www.r-bloggers.com/the-beta-prior-likelihood-and-posterior/

Bayesian Inference

• Inference based on posterior• Combines prior information with the observed data

• Particularly suited for models built with many parameters that good biological knowledge is available

• Many freely software available

• Including with differential equations “solvers”

• We don’t have to have a known or tractable posterior: Markov Chain Monte Carlo (MCMC)

39

Example of a Multivariate Nonlinear Model

• From Strathe et al. (2012). J. Agri. Sci. 150:764-774

40

F∙(ME-MEM)

kf1-kpkp

MEM

(1-F)∙(ME-MEM)

ME

MEPD MELD

HPPD LD

1-kf

Courtesy of A. B. Strathe and adapted after van Milgen and Noblet (1999)

Example for a Multivariate Nonlinear Model

41

explog

/

PDmaxmax

PDmax

b

f p

PBWPD

D BWBW

LD ME BW PD

BW

k a k

1

~ . , . and ~ . , .p fk N k N2 20 60 0 10 0 80 0 10

From Strathe et al. (2012). J. Agri. Sci. 150:764-774

Multivariate Model:

Priors from literature

Credible Intervals for the Posterior