models for cost analysis in health care: a critical and selective review

Models for cost analysis in health care: a critical and selective review

Dario GregoriDepartment of Public Health and Microbiology, University of Torino

Giulia Zigon, Department of Statistics, University of Firenze

Rosalba Rosato, Eva Pagano, Servizio di Epidemiologia dei Tumori, Università di Torino, CPO Piemonte

Simona Bo, Gianfranco Pagano, Dipartimento di Medicina Interna, Università di Torino

Alessandro Desideri, Service of Cardiology, Castelfranco Veneto Hospital

University of TorinoDepartment of Public Health and Microbiology

Department of BioinformaticsLjubljana, 1st April 2005

19/04/23

Department of Public Health and MicrobiologyUniversity of Torino

Slide 2

Outline

• Cost-effectiveness and cost-analisys• Problems in cost analisys of clinical data

– zero costs– skewness– censoring

• Models for cost data• Two case studies

– Diabetes costs in the Molinette cohort– COSTAMI trial

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Slide 3

The Molinette Diabetes Cohort

3892 subjects, including all type 2 diabetic patients, resident in region Piedmont, attending the Diabetic Clinic of the San Giovanni Battista Hospital of the city of Torino (region Piedmont, Italy) during 1995 and alive at 1st January 1996.

A mortality and hospitalization follow-up was carried over up to 30th June 2000.

A sub-cohort of 2550 patients having at least one hospitalization in the subsequent years was also identified.

Demographic data (age, sex) and clinical data relative to the year 1995 ( duration of disease or years of diabetes and number of other co-morbidities) were recorded.

Costs (in euros) for the daily and the ordinary hospitalizations have been calculated referring to the Italian DRG system.

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Slide 4

The COSTAMI study

• 487 patients with uncomplicated AMI were randomly assigned to three different strategies:– (132 patients) early (Day 3-5) use of pharmacological

stress echocardiography and discharge on days 7-9 in case of a negative test result ;

– (130 patients) pre-discharge exercise ECG, that is a maximum, symptom limited test on days 7-9, followed by discharge in case of a negative test result;

– (225 patients) clinical evaluation and hospital discharge in Day 7-9.

• The suggested strategy in case of a positive test for the strategy 1 and 2 was coronary angiography followed by ischaemia guided revascularisation (Desideri et. al, 2003).

A follow up of 1 year for medical costs was carried out. Cost of hospitalization was estimated referring to mean reimbursement for the diagnosis-related groups (DRG).

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Slide 5

The CE Incremental Ratio

Goal is to compare efficacy with costs

T1, T2 treatment-groups of patients

21

2112 EE

CC

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Slide 6

The Cost-Efficacy plane

ΔC

ΔE

Upper Threshold

Lower ThresholdR1R1

cR1

B

R1

A R2

BR2c

R2

A

R2

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Slide 7

Dominance

Laska & Wakker work (late 80’s)

ΔC < 0, ΔE > 0 T1 is dominant

ΔC > 0, ΔE < 0 T2 is dominant

ΔC > 0, ΔE > 0 T1 more effective and more costly

ΔC < 0, ΔE < 0 T1 less costly but less effective

If effects are equivalent or of no interest, then the approach is the analysis of costs alone

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Slide 8

Typical goals in cost-analysis

•To get an estimate of the mean costs of treating the disease

–In experimental settings: to test for differences among two or more groups–In observational settings: to identify patients/structure characteristics influencing costs

•To get an estimate of the expected costs, at a fixed time point, for specific types of patients (cost profiling)

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Slide 9

Typical problems in cost-analysis

• The possible large mass of observations with zero cost;

• The asymmetry of the distribution, given that there is a minority of individuals with high medical cost compared to the rest of the population

• Possible presence of censoring:– Right censoring due to loss at follow-up or administrative rule

(O’Hagan 2002)– Death censoring: dead patients are seen as lost at follow-up, to

compensate for higher/earlier mortality at lower costs (Dudley et al, 1993)

• General requisite are– the censoring must be independent or non informative. This

condition is needed because the individuals still under observation must be representative of the population at risk in each group, otherwise the observed failure rate in each group will be biased

– the assumption of proportional hazards may be violated by the medical costs due to accumulation at different rates

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Slide 10

Proportionality on cost accumulation and censoring

Etzioni, 1999

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Slide 11

Accumulation under alternatives (without covariates)

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Slide 12

Censoring: some conflicting definitions Analysis Censoring

definitionCaveats

Administrative Cost till death (O’Hagan, 2003)

Only dead patients have complete follow-up history

Cost and survival are closely related

Loss at follow-up

Cost till death Only dead patients have complete follow-up history

Possible informative censoring

Death censoring Cost up to a pre-specified time (Harrell, 1993)

Only patients arrived alive at the end of follow-up are uncensored

Informative censoring

No-censoring (actual data)

Observed costs Downward bias in cost estimation

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Slide 13

Cost distribution

0 40000 80000 120000

010

0020

0030

00

Costs (€) full cohort

0 40000 80000 120000

050

010

0015

00

Costs (€) sub-cohort with one hospitalization (no-zero)

# zero-cost patients: 2226 Min 1st QMedian

Mean 3rd Q Max

99.42 1938 3913 7278 9014 89650

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Slide 14

Accumulation of costs over time

0 1 2 3 4

Follow-up

0

10000

20000

30000

40000

50000

Cum

ulat

ive

cost

up

to ti

me

of e

vent

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Slide 15

Studies with no-zero mass

• OLS on untransformed use or expenditures• OLS for log(y) to deal with skewness• Box-Cox generalization• Gamma regression model with log link• Generalized Linear Models (GLM)

• Robustness to skewness• Reduce influence of extreme cases• Good forecast performance• No systematic misfit over range of predictions• Efficiency of estimator

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Slide 16

Linear modelsOrdinary Least Square (OLS) model assumes the following form for the costs

ijji xc estimated via Gauss-Markov or ML, in this case requiring normality and constant variance on residualsTo reduce skewness in the residuals, the Box-Cox transform of ci can be used

0 if

0 if

ijji

ijji

xc

xc

)log(

1

Problems: – normality is still assumed– bias is

thus, heteroscedasticity, if present, raises additional efficiency and inference problems on the transformed scale

i

i

x

x

2

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Slide 17

Log-normal models

A particular case of transformation is the ln(Cij) ~ N(γj, σj2) for

two treatments j=0,1

In this case, E(Cij)=exp(γj+0.5 σj2) and a test of H0: γ1 – γ2=0 is a

test for the geometric means. This was argued to be less interesting for policy makers, but observing

H0: exp(γ1+0.5 σ12) = exp(γ2+0.5 σ2

2) implies

H0: γ1 – γ2=0 iff σ12= σ2

2

Making a test for the geometric means being equivalent to one on arithmetic means only in case of homogeneity of variances in the treatment groups

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Slide 18

Box-Cox transform varying λ

6 8 10 12

010

020

030

040

0

lambda=0 (log)

0 200 400 600

010

020

030

040

050

060

0

lambda=1/2

15 20 25 30 35

010

020

030

040

050

060

0

lambda=1/8

26 28 30 32 34 36

010

020

030

040

050

0

lambda=1/20

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Slide 19

The threshold-logit model

Utilized to model the probability of having costs in excess of a given threshold, usually chosen as the median q2 or the third quartile q3 in the cost distribution

2 3

1

1( )

1 ( )i h

j jj

p c qexp x

It does not requires normality, and can work also for very skewed cost-distributions.Problems:• it does not give an estimate of the mean costs, although it estimates the covariates’ effects

on costs• conclusions are sensitive to the threshold chosen, which, in addition is sample-based

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Slide 20

GLM models

To avoid bias in transforming the costs directly, since

ii cgEcEg 11

the idea is to model the transformation of the expectation

jji xcEg

Where the distribution for the response is usually taken to be Gamma() and the link function– for additive effects as the identity function I()– for multiplicative models as the log()

allowing in this case back-transformation to avoid bias

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Slide 21

GLM and QL/GEE estimate

• Use data to find distributional family and link• Family “down weights” noisy high mean cases• Link can handle linearity• Note difference in roles from Box-Cox

– Box-Cox power addresses mostly symmetry in error.– GLM with power function addresses linearity of response on

scale to be chosen• GLM/GEE/GMM modeling approach’s estimating equations

Given correct specification of E[y|x] = µ(xβ), key issues relate to second-order or efficiency effects

This requires consideration of the structure of v(y|x)

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Slide 22

Variance determination

Accommodates skewness & related issues via variance weighting rather than transform/retransform methods

Assumes Var[y|x] = α × [E(y|x)]γ

= α × [exp(xβ)]γ

For GLM, solutions are• Adopt alternative "standard" parametric distributional assumptions,

– γ = 0 (e.g. Gaussian NLLS)– γ = 1 (e.g. Poisson)– γ = 2 (e.g. Gamma)– γ = 3 (e.g. Wald or inverse Gaussian)

• Estimate γ via:– linear regression of log((y- µ)2) on [1, log( µ)] (modified "Park test" by

least squares)– gamma regression of (y- µ)2 on [1, log( µ)] (modified "Park test"

estimated by GLM)– nonlinear regression of (y- µ)2 on αµγ

– Given choice of γ, can form V(x) and conduct (more efficient) second-round estimation and inference

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Slide 23

Monte Carlo Simulation (Mannings, 2000)• Data Generation

– Skewness in dependent measure• Log normal with variance 0.5, 1.0, 1.5, 2.0• Heavier tailed than normal on the log scale

– Mixture of log normals• Heteroscedastic responses• Std. dev. proportional to x• Variance proportional to x

– Alternative pdf shapes• monotonically declining or bell-shaped• Gamma with shapes 0.5, 1.0, 4.0

• Estimators considered– Log-OLS with

• homoscedastic retransformation• heteroscedastic retransformation

– Generalized Linear Models (GLM), log link– Nonlinear Least Squares (NLS)– Poisson– Gamma

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Slide 24

Effect of skewness on the raw scale

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Slide 25

Effects of heavy tails on the log scale

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Slide 26

Effects of shape for Gamma

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Slide 27

Effect of heteroschedasticity on the log scale

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Slide 28

Simulation summary

• All consistent, except Log-OLS with homoscedastic retransformation if the log-scale error is actually heteroscedastic

• GLM models suffer substantial precision losses in face of heavy-tailed (log) error term. If kurtosis > 3, substantial gains from least squares or robust regression.

• Substantial gains in precision from estimator that matches data generating mechanism

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Slide 29

The “zero” problem

• Problems with standard model– OLS may predict negative values– Zero mass may respond differently to covariates– These problems may be bigger when higher mass at 0

• Alternative estimators– Ignore the problem– ln(c+k) – Tobit and Adjusted Tobit models (Heckman type model)– Two-part models

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Slide 30

The log(c+k) solution

Solution: add positive constant k to costs

• Advantages– Easy– Log addresses skewness, constant deals with ln(0)

• Disadvantages– Zero mass may respond differently to covariates– Many set k=1 arbitrarily– Value of k matters, need grid search for optimum– Poorly behaved (Duan 1983)– Retransformation problem aggravated at low end

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Slide 31

Latent Variables

Sometimes binary dependent variable models are motivated through a latent variables model

The idea is that there is an underlying variable y*, that can be

modeled as y* = 0 +x + e, but we only observe

y = 1, if y* > 0, and y =0 if y* ≤ 0,

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Slide 32

The Tobit Model

Can also have latent variable models that don’t involve binary dependent variables

Say y* = x + u, u|x ~ Normal(0,2)

But we only observe y = max(0, y*)

The Tobit model uses MLE to estimate both and for this model

Important to realize that estimates the effect of x on y*, the latent variable, not y

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Slide 33

Interpretation of the Tobit Model

Unless the latent variable y* is what’s of interest, can’t just interpret the coefficient

E(y|x) = (x/)x + x/, so

∂E(y|x)/∂xj = j (x/)

If normality or homoskedasticity fail to hold, the Tobit model may be meaningless

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Slide 34

Tobit fit to diabetes data

Value Std. Error z p(Intercept) 5510.8 1474.4643 3.737 0.000186Age 16.94 22.2739 0.761 0.446917Sex -62.85 424.3257 -0.148 0.88225Years.Diabetes 50.48 25.0192 2.018 0.043613Pat.1 2134.09 605.1603 3.526 0.000421Log(scale) 9.09 0.0167 544.008 0

8.2 8.4 8.6 8.8

Linear predictor

-4-2

02

Dev

ianc

e re

sidu

als

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Slide 35

Tobit – some notes

• Only works well if dependent variable is censored Normal

• Places many restrictions on parameters, error term

• Hypersensitive to minor departures from normality

• (Almost) never recommended for health economics

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Slide 36

Mixed models

On the basis of the basic rule of expectation one can partition

0|0| iiii ccEcPxcE

Thus, expectation is splitted in two parts,1. Pr(any use or expenditures)

Full sampleUse logit or probit regression

2. Level of use or expendituresConditional on c > 0 (subsample with c >0)Use appropriate continuous model

Estimates of mean costs are obtained using the Duan’s (1983) smearing estimator (mean of the exponentiated residuals)

xcn

xxxcE ii )ln(exp1

exp|

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Slide 37

Diabetes two-part model

Logit modelValue Std. Error t value

(Intercept) -2.17186743 0.229258 -9.473445Age 0.02991614 0.003507 8.531373Sex 0.10780381 0.067253 1.602964Years.Diabetes 0.02408149 0.004125 5.837866Pat.1 0.6860717 0.1064 6.448012OLS model

Value Std. Error t value(Intercept) 5125.61 1428.88 3.59Age 28.02 21.33 1.31Sex 483.89 413.26 1.17Years.Diabetes 49.83 24.24 2.06Pat.1 2596.41 566.67 4.58

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Slide 38

Marginal effect in the two-part model

Continuous variable x

P(y>0)=0.54

E(Y|Y>0)=7509.82

For year of diabetes, this means

Βlogit = 0.025

Βols=49.83

Marginal effect is 208€ per year of diabetes

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Slide 39

Weighted-regression models

To adjust for censoring, the basic idea is to weight the costs for the inverse of the probability of being alive, mimicking the basic Horvitz-Thompson estimator.Thus, the Bang-Tsiatis (2000) basic estimator is

n

i

K

jjij

jijiji

i TK

tMtM

ncE

1 1

11

Bang-Tsiatis (2000) proposed an improved version accounting for cost-history lost due to censoring, allowing the cost function M() and the Kaplan-Meier to be estimated in each of the K intervals, defined optimally according to Lin (1993)

n

i i

iii TK

M

ncE

1 )(

1)(

where δ is the censoring indicator, M(t) is the cumulative cost up to time t and K() is the Kaplan-Meier estimate

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Slide 40

Improving estimation (Jiang, 2004)

Bootstrap confidence interval had much better coverage accuracy than the normal approximation one when medical costs had a skewed distribution.

When there is light censoring on medical costs (<25%) the bootstrap confidence interval based on the simple weighted estimator is preferred due to its simplicity and good coverage accuracy.

For heavily censored cost data (censoring rate >30%) with larger sample sizes (n>200), the bootstrap confidence intervals based on the partitioned estimator has superior performance in terms of both efficiency and coverage accuracy

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Slide 41

Censored estimation (diabetes cohort)

Mean estimate SE

Lin estimate (administrative censoring)

5856 249

Cox estimate (death censoring at 4 years)

33896 1249

No-censoring estimate

4488.18 129.44

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Slide 42

Survival models

The cost function is defined as

ccPcS ii ( ) ( ) (1 ( ))c f c F c

and the hazard of having an “excess” of costs is modeled avoiding (Cox’s model) or not (Weibull model) the full specification of the baseline λ0

01

( ) ( ) ( )h

i h j jj

c x c exp x

to avoid assumption of proportional accumulation over time (Etzioni, 1999), an alternative model can be the Aalen additive regression (Zigon, 2005)

01

( ) ( ) ( )h

i h j jj

c x c x c

where the hazard rate is a linear combination of the variables x(c) and α(c) are functions estimated from the data

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Slide 43

Survival approach – some notes

Coefficients are interpretable as the “risk” of having costs greater than actual ones

If proportionality does not hold, then

• Baseline cost-hazard with strata• Partition of the costs axis• Model non-proportionality by cost-dependent covariates β(c)X =

βX(c)• Refer to other models (accelerated failure or additive hazards)

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Slide 44

Diabetes Full cohort

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Slide 45

Issues and models in cost-analysis

Skweness Zero-cost Censoring Mean estimation

)|( xcE i

Original scale models

OLS (ci) X

Tobit/adjusted tobit X X

GLM (gamma, log-gamma)

X X X

Transformed response

OLS log(ci+k) O X O

Threshold logit models X O

Survival models

Parametric (Weibull) X X X

Semiparametric (Cox Proportional hazard)

X X O

Mixed models X X X

Weighted regression

Robis-Rotnizky 1995 Chao-Tsiatis 1997 Bang-Tsiatis 2000

X X X

X= satisfied, o = partially satisfied

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Slide 46

Estimates on the Molinette Cohort

We compared performances of the survival models with two “benchmarks” widely (and often inappropriately) used in the literature, OLS and Threshold-logit model, using the non-zero costs cohort

N Median 1st q, 3rd q Sex Female 1270 3617 1872, 8424

Male 1280 4290 2047, 9700 Co-morbidities 1 No 2187 3704 1850, 8386 Yes 363 5943 2765, 12950 Years of Diabetes [0, 4) 480 3552 1641, 8452

[4, 10) 594 3728 1922, 8009 [10, 18) 691 4007 1886, 9363 [18, 48] 785 4307 2142, 9671

Age [22.1, 59.2) 638 2891 1425, 7261 [59.2, 66.2) 638 3684 1872, 8121 [66.2, 72.6) 637 4844 2395, 10940 [72.6, 90.8] 637 4517 2333, 9411

Overall 2550 3913 1938, 9014

Both normality (Shapiro-Wilk test p<0.0001) and proportionality in hazards (Grambsch-Therneau test p<0.001) assumptions refused

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Slide 47

Covariates effects

Models 1 2 3 4 5

Intercept Age Sex (M vs F)

Years of diabetes

N. co-morbidities

OLS 2155.15 53.70 829.80 59.02 2946.98 (SE=1220.02) (SE=18.50) (SE=360.65) (SE=21.36) (SE=474.93) Logistic 2nd q -2.102 0.026 0.346 0.006 0.539

(SE=0.283) (SE=0.004) (SE=0.081) (SE=0.004) (SE=0.110) Logistic 3rd q -2.565 0.017 0.233 0.005 0.682

(SE=0.330) (SE=0.004) (SE=0.093) (SE=0.005) (SE=0.111) Weibull 3.0683 0.0577 0.2032 0.0439 1.3073

(SE=0.348) (SE=0.005) (SE=0.107) (SE=0.006) (SE=0.160) Cox – -0.0196 -0.0938 -0.0149 -0.4829

– (SE=0.001) (SE=0.03) (SE=0.001) (SE=0.051) Aalen 4.611 0.023 0.873 -0.078 -1.504

(SE=5.744) (SE=0.067) (SE=1.503) (SE=0.118) (SE=0.576)

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Slide 48

Estimates of the mean

Models Estimated expectation 95% C.I. OLS 7278 7222.88, 7333.12

Logistic 2nd q 0.500 0.480, 0.519 Logistic 3rd q 0.2502 0.2334, 0.2670

Weibull 8269 8154.698, 8383.302 Cox 8717.984 7881.01, 9554.95

Aalen 8077.735 7493.737, 8661.733

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Slide 49

Cost profilingCrude OLS Weibull Cox Aalen costs (95% C.I.) (95% C.I.) (95% C.I.) (95% C.I.)

Age=40 Years of Diabetes=2

Sex=F Co-morbidities=0

3388 4421 236.40 1517.058 3936.722 (4365.88

4476.12) (102.3689 370.4311)

(1229.894 1804.222)

(3272.815 4600.629)


Sex=F Co-morbidities =1

7894 7840 1242 4594.555 5108.192 (7784.88

7895.12) (1107.969 1376.031)

(3521.434 5667.676)

(4043.500 6172.884)


Sex=M Co-morbidities =1

8077.704 10870 13347 16401.33 7637.626 (10814.88

10925.12) (13212.97 13481.03)

(13488.79 19313.88)

(6401.272 8873.980)


Sex=F Co-morbidities=1

5724.294 9209 4909 9806.214 6411.435 (9153.88

9264.12) (4774.969 5043.031)

(8006.951 11605.477)

(5374.243 7448.626)


Sex=M Co-morbidities=0

5527.482 8246 4199 8835.986 5377.089 (8190.88

8301.12) (4064.969 4333.031)

(7363.122 10308.850)

(4574.917 6179.260)

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Slide 50

Effect of covariates (Aalen model) on Λ(c)

Time

Cu

mu

lativ

e r

eg

ress

ion

fun

ctio

n

0 10000 20000 30000 40000 50000

05

10

15

Constant

Time

Cu

mu

lativ

e r

eg

ress

ion

fun

ctio

n

0 10000 20000 30000 40000 50000

-0.1

5-0

.05

0.0

5

Age

Time

Cu

mu

lativ

e r

eg

ress

ion

fun

ctio

n

0 10000 20000 30000 40000 50000

-1.5

-0.5

0.5

Sex

Time

Cu

mu

lativ

e r

eg

ress

ion

fun

ctio

n

0 10000 20000 30000 40000 50000

-0.1

00

.0

Years.Diabetes

Time

Cu

mu

lativ

e r

eg

ress

ion

fun

ctio

n

0 10000 20000 30000 40000 50000

-2.5

-1.0

0.0

Pat.1

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Slide 51

One-year cost distribution

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Slide 52

Cost distribution

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Slide 53

Cost accumulation over time

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Slide 54

Model coefficients

Significant coefficients in italic

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Slide 55

Mean cost estimates

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Slide 56

Patient profiling

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Slide 57

Relative accuracy

Deviation (%) for the fitted model from the observed data

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Slide 58

Remarks - I

First papers appeared in late ’80 in medical literature, and a decade before in the econometrical literature

Censored costs estimators appeared in Lin, 1997 and still growing research (Bang, 2002, 2003)

Still high interest is in the statistical aspects of no-censoring fitting approaches (Basu, HE, 2004, Etzioni, HE, 2005)

Need for a comprehensive simulation study under complex situations (censoring and non proportional accumulation in particular)

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Slide 59

Remarks - II

Modeling costs is basically an exercise of

fitting adequacyand

bias reduction

however, it does also have strong impact on public health aspects, like economic planning and resource allocation, based on optimal prediction of future costs (patient profiling).

Nevertheless, caution has to be used in choosing the model and interpreting results, which can be a finding due to an artifactual representation of real cost process, as a consequence of inappropriate assumptions made on data

models for cost analysis in health care: a critical and selective review

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