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SEERMED DATA

Motivation:

End of Life Colorectal Cancer Costs

$500,000

$0

Expenditure

BIO656--Multilevel Models

Health-Care Services

HMO

Hospice

FFS

Medicare

Private Ins.

Rejected

Allowed

Co-Pay

Deductibles

DataFactors: Need-based Enabling Predisposing

Patient – Physician

Cancer Diagnosis

Claims

Terminal-Phase Costs

12

mos

Medicare Payments

BIO656--Multilevel Models

Death

Data

Patient – Physician

Cancer Diagnosis

Medicare Payments

Terminal-Phase Costs

12

mos

3

mos

BIO656--Multilevel Models

Death

SEERMED DATA

Motivation:

End of Life Colorectal Cancer Costs

$500,000

$0

Expenditure

BIO656--Multilevel Models

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 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

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

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

ai0aa

bi0babb

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

- Exploration
- Model Fitting and Estimation
- Diagnostics

and the greatest of these is…

BIO656--Multilevel Models

BIO656--Multilevel Models

Month 10 & Month 11 log10(Costs)

Bivariate Point Mass

Bivariate Continuous Distb.

Univariate Continuous Distbs.

Figure 5: Seermed log10 month 1 & 2Density

0

0

Expenditure 11

Expenditure 10

BIO656--Multilevel Models

5

5

aa

ba

PRISM plot: Month 10 & 11 SEERMED Costs

Paired

Response

Intensity

Size

Mixture

plot

BIO656--Multilevel Models

BIO656--Multilevel Models

Size: Lognormal, Gamma

ui= ~ N, =

ai0a

bi0bab

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

Li()

EstimationWhoa.

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

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)

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

BIO656--Multilevel Models

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

BIO656--Multilevel Models

aa

ba

PRISM plot: Month 10 & 11 SEERMED Costs

Paired

Response

Intensity

Size

Mixture

plot

BIO656--Multilevel Models

MEM

MREM

HMREM

HMMMM

Ideas

- Simple Combinations of Simple Models

+

0

2. Complex (Multi-Level) Data:

BIO656--Multilevel Models

Many Models & Many Pictures

12

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

BIO656--Multilevel Models

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

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

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

BIO656--Multilevel Models

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

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

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