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PART 8. Two Stage & Joint Models. SEERMED DATA . Motivation:. End of Life Colorectal Cancer Costs. $500,000. $0. Expenditure. Professional Health-Care Services. HMO. Hospice. FFS. Medicare. Private Ins. Rejected Allowed Co-Pay Deductibles. Data.

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part 8

PART 8

Two Stage & Joint Models

BIO656--Multilevel Models

seermed data
SEERMED DATA

Motivation:

End of Life Colorectal Cancer Costs

$500,000

$0

Expenditure

BIO656--Multilevel Models

slide3

Professional

Health-Care Services

HMO

Hospice

FFS

Medicare

Private Ins.

Rejected

Allowed

Co-Pay

Deductibles

Data

Factors: Need-based Enabling Predisposing

Patient – Physician

Cancer Diagnosis

Claims

Terminal-Phase Costs

12

mos

Medicare Payments

BIO656--Multilevel Models

Death

slide4
Data

Patient – Physician

Cancer Diagnosis

Medicare Payments

Terminal-Phase Costs

12

mos

3

mos

BIO656--Multilevel Models

Death

seermed data5
SEERMED DATA

Motivation:

End of Life Colorectal Cancer Costs

$500,000

$0

Expenditure

BIO656--Multilevel Models

a normal distribution
A “Normal” Distribution

Density

Y

BIO656--Multilevel Models

a complex distribution
A Complex Distribution

Density

Y

BIO656--Multilevel Models

complex distributions mixtures of simple distributions
Complex Distributions  Mixtures of Simple Distributions

Mixtures-of-Experts Models (MEM)

Finite Mixture Models (FMM)

Density

Y

McLachlan, Peel. (2001), FMM

BIO656--Multilevel Models

Jacobs, Jordan. (1991), MEM, Neural Comp

a simple two part mixture
A simple, two-part mixture

$0

1. P(Y>0)

$+

2. E(Y|Y>0)

E(Y+)

BIO656--Multilevel Models

slide10

A Two-Part Model:(Intensity & Size)

IS – logit/lognormal

1. logit{ Pr(Yi>0) } = x

2. i.) log10(Yi+) = x + i

ii.) i ~ N(0,2)

0. “Tobit” model: Tobin (1958)

1. Selection (hurdle) models: (Amemiya 1984; Heckman 1976)

2. Zero-inflated models (Lambert 1992; Green 1994)

3. Two-part models (Manning 1981; Mullahy 1998)

BIO656--Multilevel Models

slide11

Another Two-Part Model:(Intensity & Size)

IS – Probit/log-Gamma

1. -1{ Pr(Yi>0) } = x

2. i.) log10{E( Yi+)} = x

ii.)Yi+~ (,)

BIO656--Multilevel Models

slide12

A Two-Part Model:The Intensity-Size GLM

IS – GLM

h1 binary data link function

h2 continuous data link function

f exponential family w/ dispersion 

BIO656--Multilevel Models

slide13

Multiple Levels 1

0

+

BIO656--Multilevel Models

monthly seermed data

Month 12

Monthly SEERMED Data

Month 11

12

10

Month 10

11

12

+

10

11

+

+

BIO656--Multilevel Models

hmrem1

Multiple Levels 2

0

0

+

+

Time

X

X

X

X

X

X

HMREM1

Month 12

f12

g1

g2

Month 11

f11

a

0

+

g1

g2

Month 10

f10

a

b

g1

g2

b

BIO656--Multilevel Models

a 2 part model
A 2-Part Model
  • Intensity:

logit( i) = x

  • Size:
    • i = x
    • Yi+ ~ f( i, )

BIO656--Multilevel Models

a longitudinal 2 part model

ui= ~ N,  =

ai0aa

bi0babb

A Longitudinal 2-Part Model
  • Intensity:

logit( ic) = x+ zai

  • Size:
    • ic = x + zbi
    • Yi+c~ f( ic, )

1. Olsen, Schafer, (2001)

2. Tooze, Grunwald, Jones, (2002)

3. Yau, Lee, Ng, (2002)

3. Random Effects:

BIO656--Multilevel Models

data analysis 3 general steps
Data Analysis: 3 General Steps
  • Exploration
  • Model Fitting and Estimation
  • Diagnostics

and the greatest of these is…

BIO656--Multilevel Models

slide19

Uncooked Spaghetti Plot

BIO656--Multilevel Models

monthly seermed data20

Month 12

Monthly SEERMED Data

Month 11

12

10

Month 10

11

12

+

10

11

+

+

BIO656--Multilevel Models

figure 5 seermed log10 month 1 2

Month 10 & Month 11 log10(Costs)

Bivariate Point Mass

Bivariate Continuous Distb.

Univariate Continuous Distbs.

Figure 5: Seermed log10 month 1 & 2

Density

0

0

Expenditure 11

Expenditure 10

BIO656--Multilevel Models

5

5

slide22

bb

aa

ba

PRISM plot: Month 10 & 11 SEERMED Costs

Paired

Response

Intensity

Size

Mixture

plot

BIO656--Multilevel Models

slide23

PRISM Matrix: Months 10-12

BIO656--Multilevel Models

seermed mrem

Intensity: Probit, Logistic

Size: Lognormal, Gamma

ui= ~ N,  =

ai0a

bi0bab

2

2

SEERMED MREM
  • Intensity:

h1( ic) = 0+1Obs+2Male+3Obs*Male+ ai

  • Size:
    • h2( ic) = 0 + 1Obs + 2Male + bi
    • Yi+c~ f( ic, )
  • Random Effects:

BIO656--Multilevel Models

estimation

Likelihood:

Li()

Estimation

Whoa.

But:

Non-Linear Mixed Model (NLMM)

  • PQL, MCEM, MCMC, …
  • Adaptive Quadrature – Newton-Raphson

Zeger, Karim (1991); Davidian, Giltinan, (1993); Pinheiro, Bates (1995);

Mcculloch (1997); Booth et al. (2001); Rabe-Hesketh, et al. (2004)

BIO656--Multilevel Models

estimation sas
Estimation: SAS

procnlmixed data=SEERMED;

parms / data=parms_start;

*- 1) logistic: logit{Pr( Y>0 | a )} = Xalpha + a = “eta0” -*;

eta0 = alpha0_c + alpha1_c*obs + alpha2_c*male + alpha3_c*obsmale + a;

pi_c = exp(eta0) / (1+exp(eta0));

*- 2) log-normal: E( log(Y) | Y>0, b ) = XB + b = “eta1” -*;

eta1 = beta0_c + beta1_c*obs + beta2_c*male + b;

*- log-likelihood -*;

pi=CONSTANT('PI');

if y=0 then ll1 = 0;

else ll1=-.5*log(2*pi*sigma**2)-.5*((log10y-eta1)/sigma)**2;

ll = (1-Gpos)*log(1-pi_c) + Gpos*log(pi_c) + Gpos*(ll1);

model y ~ GENERAL(ll);

RANDOM a b ~ NORMAL([0,0],[tau_aa, tau_ba, tau_bb]) SUBJECT=id;

run;

BIO656--Multilevel Models

estimation sas better
Estimation: SAS (better)

procnlmixed data=sanfran qpoints=10;

parms / data=parms_start;

*-logit-*;

eta0 = alpha0_c + alpha1_c*obs + alpha2_c*male + alpha3_c*obsmale + a;

expeta = exp(eta0);

pi_c = expeta / (1+expeta);

tau_aa = exp(logtau_a)**2;

*-lognormal-*;

eta1 = beta0_c + beta1_c*obs + beta2_c*male + b;

phi = 10**(log10phi); *std dev of log10(Y+1)|b;

tau_bb = (10**(log10tau_b))**2;

*- RE Var -*;

rho_ba = (exp(2*zrho_ba) - 1) / (exp(2*zrho_ba) + 1);

tau_ba = rho_ba*(tau_aa*tau_bb)**.5;

*- log-likelihood -*;

pi=CONSTANT('PI');

if y=0 then ll1 = 0; else ll1=-.5*log(2*pi*phi**2)-.5*((log10y-eta1)/phi)**2;

ll = (1-Gpos)*log(1-pi_c) + Gpos*log(pi_c) + Gpos*(ll1);

model y ~ GENERAL(ll);

RANDOM a b ~ NORMAL([0,0],[tau_aa, tau_ba, tau_bb]) SUBJECT=id;

odsoutput ParameterEstimates = parms_new;

run;

BIO656--Multilevel Models

seermed mrem results 1
SEERMED MREM Results 1

BIO656--Multilevel Models

profile ll alpha3

c

MREM Profile Likelihood Plots for 3

Profile ll (alpha3)

Probit*-

Lognormal

Probit*-

Gamma

Logit-

Lognormal

Scaled Profile Likelihood

Logit-

Gamma

LR  6

BIO656--Multilevel Models

c

Intensity model Obs*Male interaction term (3)

seermed mrem results 2
SEERMED MREM Results 2

BIO656--Multilevel Models

slide31

bb

aa

ba

PRISM plot: Month 10 & 11 SEERMED Costs

Paired

Response

Intensity

Size

Mixture

plot

BIO656--Multilevel Models

seermed mrem results 232
SEERMED MREM Results 2

But do these models fit?…

BIO656--Multilevel Models

slide33

Data vs. MREM Models

Obs: ,Y

BIO656--Multilevel Models

Exp: P, L,G

slide34

Diagnostic PRISM Matrix: lognormal IS-GLMM Residuals

Expected

Observed

BIO656--Multilevel Models

slide35

Diagnostic PRISM Matrix: lognormal IS-GLMM Residuals

Expected

Observed

BIO656--Multilevel Models

slide36

Review & Related Work

MEM

MREM

HMREM

HMMMM

Ideas

  • Simple Combinations of Simple Models

+

0

2. Complex (Multi-Level) Data:

BIO656--Multilevel Models

Many Models & Many Pictures

12

slide37

Data vs. HMREM Models

Data vs. HMMMM Models

BIO656--Multilevel Models

slide38

Review & Related Work

  • These ideas are not just for Zero-Inflated Data
  • Latent Variables are useful for “connecting” things

BIO656--Multilevel Models

opportunistic infection idu
Opportunistic Infection & IDU

Always Users

Interview: Reported Drug Use

Intermittent Users

Never Users

Interview: Reported No Drug Use

Opportunistic Infection

Each Line Represents 1 subject’s time in the study

BIO656--Multilevel Models

Day in Study

6 months prior to 1st interview

jointly analyze survival ois
Jointly Analyze Survival & OIs

1) logistic model:

logit{ Pr(OIij | ai) } = 0 + 1SUij + 2SUij*HCuseij + 3AUij+ 4Periodj + ai

2) Survival Model:

log{ (t) } = 0 + 1SUij + 2AUij + ai

3) Latent Effects:

ai ~ N(0,)

Guo & Carlin (2004)

BIO656--Multilevel Models

slide42

Warning!

  • But “Buyer Beware”
    • -- Model Assumptions
    • -- Identifiability
    • -- Model Fit
    • -- Marginalize & Check whenever possible
  • MLMs require even more due-diligence than usual

BIO656--Multilevel Models

references
References
  • Mixture Models:
    • McLachlan, G. J. and Peel, D. (2001), Finite mixture models, John Wiley & Sons.
    • Jacobs, R. A. and Jordan, M. I. (1991), “Adaptive mixtures of local experts. Neural Computation,” Neural Computation, 3, 79–87.
  • Two-Part Models:
    • Tobin, J. (1958), “Estimation of Relationships for Limited Dependent Variables,” Econometrica, 25, 24–36.
    • Amemiya, T. (1984), “Tobit models: A survey,” Journal of Econometrics, 24, 3–61.
    • Heckman, J. (1976), “The common structure of statistical models of truncation, sample selection, and limited dependent variables, and a sample estimator for such models,” The Annals of Economic Development and Social Measurement, 5, 475–592.
    • Lambert, D. (1992), “Zero-inflated Poisson regression, with an application to defects in manufacturing,” Technometrics, 34, 1–14.
    • Green, W. (1994), “Accounting for excess zeros and sample selection in Poisson and negative binomial regression models,” Working Paper EC-94-10, Department of Economics, New York University
    • Manning, W., Newhouse, J., Orr, L., Duan, N., Keeler, E., Leibowitz, A., Marquis, M., and Phelps, C. (1981), “A two-part model of the demand for medical care: Preliminary results from the health insurance experiment,” in Health, Economics, and Health Economics, eds. van der Gaag, J. and Perlman, M., pp. 103–104.
    • Mullahy, J. (1998), “Much ado about two: reconsidering retransformation and the two part model in health economics,” Journal of Health Economics, 17, 247–281.

BIO656--Multilevel Models

references44
References
  • Longitudinal 2-part models
    • Olsen, M. K. and Schafer, J. L. (2001), “A two-part random-effects model for semicontinuous longitudinal data,” Journal of the American Statistical Association, 96, 730–745.
    • Tooze, J. A., Gunward, G. K., and Jones, R. H. (2002), “Analysis of repeated measures data with clumping at zero,” Statistical Methods in Medical Research, 11, 341–355.
    • Yau, K. K. W., Lee, A. H., and Ng, A. S. K. (2002), “A zero-augmented gamma mixed model for longitudinal data with many zeros,” The Australian and New Zealand Journal of Statistics 44, 177–183.
  • Estimation:
    • Zeger, S. L. and Karim, M. R. (1991), “Generalized linear models with random effects: A Gibbs sampling approach,” Journal of the American Statistical Association, 86, 79–86.
    • Davidian, M. and Giltinan, D. M. (1993), “Some general estimation methods for nonlinear mixed-effects models,” Journal of Biopharmaceutical Statistics, 3, 23–55.
    • Pinheiro, J. C. and Bates, D. M. (1995), “Approximations to the log-likelihood function in the nonlinear mixed-effects model,” Journal of Computational and Graphical Statistics,4, 12–35.
    • McCulloch, C. E. (1997), “Maximum likelihood algorithms for generalized linear mixed models,” Journal of the American Statistical Association, 92, 162–170.
    • Booth, J. G., Hobert, J. P., and Jank, W. (2001), “A survey of Monte Carlo algorithms for maximizing the likelihood of a two-stage hierarchical model,” Statistical Modelling: An International Journal, 1, 333–349.
    • Rabe-Hesketh, S., Skrondal, A., and Pickles, A. (2004), “Maximum likelihood estimation of limited and discrete variable models with nested random effects,” Journal of Econometrics, in press.
  • Other:
    • Guo, X. and Carlin, B.P. (2004), ``Separate and Joint Modeling of Longitudinal and Event Time Data Using Standard Computer Packages," The American Statistician, 58 16--24.

BIO656--Multilevel Models