Upload
georgia-dodson
View
35
Download
2
Embed Size (px)
DESCRIPTION
Department of Bioinformatics Ljubljana, 1 st April 2005. Models for cost analysis in health care: a critical and selective review. Dario Gregori Department of Public Health and Microbiology, University of Torino Giulia Zigon, Department of Statistics, University of Firenze - PowerPoint PPT Presentation
Citation preview
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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.
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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).
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 5
The CE Incremental Ratio
Goal is to compare efficacy with costs
T1, T2 treatment-groups of patients
21
2112 EE
CC
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 6
The Cost-Efficacy plane
ΔC
ΔE
Upper Threshold
Lower ThresholdR1R1
cR1
B
R1
A R2
BR2c
R2
A
R2
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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)
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 10
Proportionality on cost accumulation and censoring
Etzioni, 1999
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 11
Accumulation under alternatives (without covariates)
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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)
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 24
Effect of skewness on the raw scale
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 25
Effects of heavy tails on the log scale
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 26
Effects of shape for Gamma
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 27
Effect of heteroschedasticity on the log scale
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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,
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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|
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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)
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 44
Diabetes Full cohort
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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)
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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)
Age=40 Years of Diabetes=10
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)
Age=70 Years of Diabetes=20
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)
Age=60 Years of Diabetes=15
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)
Age=65 Years of Diabetes=30
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)
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 51
One-year cost distribution
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 52
Cost distribution
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 53
Cost accumulation over time
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 54
Model coefficients
Significant coefficients in italic
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 55
Mean cost estimates
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 56
Patient profiling
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
Slide 57
Relative accuracy
Deviation (%) for the fitted model from the observed data
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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)
19/04/23
Department of Public Health and MicrobiologyUniversity of Torino
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