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

1
Models for cost analysis in health care a
critical and selective review
Department of BioinformaticsLjubljana, 1st April
2005

• Dario Gregori
• Department 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
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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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.

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).

5
The CE Incremental Ratio

• Goal is to compare efficacy with costs
• T1, T2 treatment-groups of patients

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The Cost-Efficacy plane
?C
Upper Threshold
R1
R1c
Lower Threshold
R1B
R2A
?E
R1A
R2B
R2
R2c
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Dominance

• Laska Wakker work (late 80s)
• ?C lt 0, ?E gt 0 T1 is dominant
• ?C gt 0, ?E lt 0 T2 is dominant
• ?C gt 0, ?E gt 0 T1 more effective and more costly
• ?C lt 0, ?E lt 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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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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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 (OHagan 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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Proportionality on cost accumulation and censoring
Etzioni, 1999
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Accumulation under alternatives (without
covariates)
12
Censoring some conflicting definitions
Analysis Censoring definition Caveats
Administrative Cost till death (OHagan, 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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Cost distribution
zero-cost patients 2226
Min 1st Q Median Mean 3rd Q Max
99.42 1938 3913 7278 9014 89650
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Accumulation of costs over time
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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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Linear models

• Ordinary Least Square (OLS) model assumes the
  following form for the costs

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

• Problems
• normality is still assumed
• bias is
• thus, heteroscedasticity, if present, raises
  additional efficiency and inference problems on
  the transformed scale

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Log-normal models

• A particular case of transformation is the
  ln(Cij) N(?j, sj2) for two treatments j0,1
• In this case, E(Cij)exp(?j0.5 sj2) and a test
  of H0 ?1 ?20 is a test for the geometric
  means. This was argued to be less interesting for
  policy makers, but observing
• H0 exp(?10.5 s12) exp(?20.5 s22) implies
• H0 ?1 ?20 iff s12 s22
• 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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Box-Cox transform varying ?
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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

• 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

20
GLM models

• To avoid bias in transforming the costs directly,
  since

the idea is to model the transformation of the
expectation

• 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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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 approachs estimating
  equations

Given correct specification of Eyx µ(xß),
key issues relate to second-order or efficiency
effects This requires consideration of the
structure of v(yx)
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Variance determination

• Accommodates skewness related issues via
  variance weighting rather than transform/retransfo
  rm methods
• Assumes Varyx a E(yx)?
• a exp(xß)?
• For GLM, solutions are
• Adopt alternative "standard" parametric
  distributional assumptions,
• ? 0 (e.g. Gaussian NLLS)
• ? 1 (e.g. Poisson)
• ? 2 (e.g. Gamma)
• ? 3 (e.g. Wald or inverse Gaussian)
• Estimate ? 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 aµ?
• Given choice of ?, can form V(x) and conduct
  (more efficient) second-round estimation and
  inference

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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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Effect of skewness on the raw scale
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Effects of heavy tails on the log scale
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Effects of shape for Gamma
27
Effect of heteroschedasticity on the log scale
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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 gt 3, substantial gains from least
  squares or robust regression.
• Substantial gains in precision from estimator
  that matches data generating mechanism

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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(ck)
• Tobit and Adjusted Tobit models (Heckman type
  model)
• Two-part models

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The log(ck) 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 k1 arbitrarily
• Value of k matters, need grid search for optimum
• Poorly behaved (Duan 1983)
• Retransformation problem aggravated at low end

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 b0 xb e, but we only observe
• y 1, if y gt 0, and y 0 if y 0,

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The Tobit Model

• Can also have latent variable models that dont
  involve binary dependent variables
• Say y xb u, ux Normal(0,s2)
• But we only observe y max(0, y)
• The Tobit model uses MLE to estimate both b and s
  for this model
• Important to realize that b estimates the effect
  of x on y, the latent variable, not y

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Interpretation of the Tobit Model

• Unless the latent variable y is whats of
  interest, cant just interpret the coefficient
• E(yx) F(xb/s)xb sf(xb/s), so
• ?E(yx)/?xj bj F(xb/s)
• If normality or homoskedasticity fail to hold,
  the Tobit model may be meaningless

34
Tobit fit to diabetes data
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

36
Mixed models

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

• Thus, expectation is splitted in two parts,
• Pr(any use or expenditures)
• Full sample
• Use logit or probit regression
• 2. Level of use or expenditures
• Conditional on c gt 0 (subsample with c gt0)
• Use appropriate continuous model
• Estimates of mean costs are obtained using the
  Duans (1983) smearing estimator (mean of the
  exponentiated residuals)

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Diabetes two-part model
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Marginal effect in the two-part model
Continuous variable x
P(ygt0)0.54 E(YYgt0)7509.82 For year of
diabetes, this means ?logit 0.025 ?ols49.83 Ma
rginal effect is 208 per year of diabetes
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

where d is the censoring indicator, M(t) is the
cumulative cost up to time t and K() is the
Kaplan-Meier estimate
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)
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
  (lt25) 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
  gt30) with larger sample sizes (ngt200), the
  bootstrap confidence intervals based on the
  partitioned estimator has superior performance in
  terms of both efficiency and coverage accuracy

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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
42
Survival models

• The cost function is defined as

and the hazard of having an excess of costs is
modeled avoiding (Coxs model) or not (Weibull
model) the full specification of the baseline ?0
to avoid assumption of proportional accumulation
over time (Etzioni, 1999), an alternative model
can be the Aalen additive regression (Zigon, 2005)
where the hazard rate is a linear combination of
the variables x(c) and a(c) are functions
estimated from the data
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 ß(c)X ßX(c)
• Refer to other models (accelerated failure or
  additive hazards)

44
Diabetes Full cohort
45
Issues and models in cost-analysis
X satisfied, o partially satisfied
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

Both normality (Shapiro-Wilk test plt0.0001) and
proportionality in hazards (Grambsch-Therneau
test plt0.001) assumptions refused
47
Covariates effects
48
Estimates of the mean
49
Cost profiling
50
Effect of covariates (Aalen model) on ?(c)
51
One-year cost distribution
52
Cost distribution
53
Cost accumulation over time
54
Model coefficients
Significant coefficients in italic
55
Mean cost estimates
56
Patient profiling
57
Relative accuracy
Deviation () for the fitted model from the
observed data
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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)

59
Remarks - II

• Modeling costs is basically an exercise of
• fitting adequacy
• and
• 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