Review of statistical methods for analysing healthcare resources and costs

“The four-part model in Duan (1983) (see Section 3.1.6) is also in essence a data components model.”

“Extended approaches based on the Poisson distribution by Cameron and colleagues (Cameron and Johansson, 1997; Cameron and Trivedi, 1986) (negative binomial and variance linear in the mean, Poisson polynomial model) and Grootendorst (1995) (zero-inflated Poisson and negative binomial) are used to…”

“mishandling of both non-users and significant users of health services) (Duan et al., 1983).”

“tion of the variance function could lead to inefficiencies, the mean function estimates are usually robust (Manning and Mullahy, 2001).”

“…reviewed included an informal comparison of different methods and approaches applied to the same data (Austin et al., 2003; Briggs and Gray, 1998; Buntin and Zaslavsky, 2004; Chen and Zhou, 2006; Cooper et al., 2007; Lipscomb et al., 1998; Manning and Mullahy, 2001; O’Hagan and Stevens, 2003;…”

“In this regard, Mullahy (1998) proposes two estimators using exponential conditional mean specifications.”

“Two common approaches to directly evaluate uncertainty in costs, effects and/or cost-effectiveness in randomised trials are the Central Limit Theorem and the bootstrap approach (Barber and Thompson, 2000; O’Hagan and Stevens, 2003; Thompson and Barber, 2000).”

“General methods for count data have been extensively described elsewhere (Cameron and Trivedi, 1998; Winkelmann, 2008).”

“, 1983; Leung and Yu, 1996), GLM or OLS models have been employed in the second part to evaluate mean costs, and truncated-at-zero Poisson, negative binomial (Grootendorst, 1995; Pohlmeier and Ulrich, 1995), or truncated Poisson-lognormal models (Winkelmann, 2004) to evaluate resource use (for a wider range of hurdle…”

“Log-linear (Duan et al., 1983; Leung and Yu, 1996), GLM or OLS models have been employed in the second part to evaluate mean costs, and truncated-at-zero Poisson, negative binomial (Grootendorst, 1995; Pohlmeier and Ulrich, 1995), or truncated Poisson-lognormal models (Winkelmann, 2004) to evaluate…”

“…generalized linear models (GLMs) a mean function (between the linear predictor and the mean) and a variance function (between the mean and variance on the original scale) are specified and the parameters are estimated given these structural assumptions (see general theory in Blough et al. (1999)).”

“Furthermore in the case of unknown separation (latent mixture models) and more than two components the identification of all components might be difficult because of the increasing overlap of distributions (Cameron and Trivedi, 2009).”

“The benefits of this proposition have not been observed in the case of the RAND HIE and are unlikely to hold generally in randomised studies when the interest is in the estimated mean (Duan et al., 1984; Maddala, 1985; Manning et al., 1987).”

“These include approaches based on mixtures of distributions, non-parametric density approximation, data components, model averaging and Markov chain methods.”

“…al., 1999), the estimation for the link and variance and the incorporation of the uncertainty from this estimation (including underlying heteroscedasticity) in the final parameter estimates is rarely done; for an exception see the Extended Estimating Equations approach in Basu and Rathouz (2005).”

“The benefits of this proposition have not been observed in the case of the RAND HIE and are unlikely to hold generally in randomised studies when the interest is in the estimated mean (Duan et al., 1984; Maddala, 1985; Manning et al., 1987).”

“An extended two-part model, which explicitly models and estimates the correlation between the logistic and lognormal part for repeated measures data, is suggested by Tooze (Tooze et al., 2002) and is shown to outperform the model with uncorrelated random effects.”

“20: 897–916 (2011) DOI: 10.1002/hec (Austin et al., 2003; Basu, 2005; Briggs et al., 2005; Briggs and Gray, 1998; Chen and Zhou, 2006; Deb and Burgess, 2003; Dinh and Zhou, 2006; Duan et al., 1983; Dudley et al., 1993; Gilleskie and Mroz, 2004; O’Hagan and Stevens, 2003; Thompson and Barber, 2000;…”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“…employed in the second part to evaluate mean costs, and truncated-at-zero Poisson, negative binomial (Grootendorst, 1995; Pohlmeier and Ulrich, 1995), or truncated Poisson-lognormal models (Winkelmann, 2004) to evaluate resource use (for a wider range of hurdle models please refer to Hilbe (2007)).”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“A Weibull survival model was suggested for the hazard of initiating mental health service use as duration on treatment (time from initiating treatment to end of year) was indicative of cost (Keeler et al., 1988).”

“Two and three-parameter lognormal and log-logistic distributions are used by Nixon and Thompson (Nixon and Thompson, 2004; Thompson and Nixon, 2005) to model costs, and lognormal and Weibull models are used by Marazzi et al. (1998) to model length of stay in hospital.”

“An implementation of the approach in evaluating mean length of stay shows better in-sample fit of a phase-type distribution compared with lognormal and gamma models (Faddy et al., 2009).”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“…include: a bivariate Poisson distribution (Cameron and Johansson, 1998), negative binomial marginal distribution and copula functions (Cameron et al., 2004), independent Poisson distributions with conditional mean functions that depend on correlated latent effects (Chib and…”

“Recently, Basu and Manning (2009) concluded, ‘No current method is optimal or dominant for all cost applications.”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“Thus, the approaches for back transformation are dominated by Duan’s non-parametric smearing when the error term is homoskedastic (Duan, 1983) and its variants when the error term is heteroskedastic across the compared groups (Ai and Norton, 2000; Duan et al., 1983; Manning, 1998).”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“such as the semiparametric Cox proportional hazards model (Austin et al., 2003; Basu et al., 2004; Dudley et al., 1993; Lipscomb et al., 1998) and the parametric Weibull proportional hazards model (Dudley et al.”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“Santos-Silva and Windmeijer suggest more appropriate modelling of spells of care and number of visits per spell through a Poisson model for the first part (number of spells) and a logarithmic model for the second part (number of visits per spell) (Santos-Silva and Windmeijer, 2001).”

“…al., 2005), more complex specification of the mean–variance relationships (Basu, 2005; Basu et al., 2006; Blough et al., 1999) and GLM estimators that are more robust to outliers (and potentially more suitable for very heavy-tailed data) (Cantoni and Ronchetti, 2001; Cantoni and Ronchetti, 2006).”

“A hurdle model with a generalised logistic model in the first part is proposed by Gurmu (1998) and a modified second part in the two-part model is suggested by Mullahy (1998).”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“In circumstances where few of these issues exist then a sample size in the hundreds may be large enough, but where these issues are all present then a sample size of thousands may nevertheless be too small.”

“…to account for skewness and have been shown to provide more efficient estimates but these modifications are subject to a degree of subjectivity in the choice of transformation and do not generally allow for adjustment for covariates (Chen and Zhou, 2006; Dinh and Zhou, 2006; Zhou and Dinh, 2005).”

“…2001), a multivariate overdispersed Poisson mixture model (Gurmu and Elder, 2000), bivariate Poisson-lognormal mixture and bivariate negative binomial regression models (Munkin and Trivedi, 1999), and a bivariate zero-inflated binomial regression for count data with excess zeros (Wang, 2003).”

“, 2004), independent Poisson distributions with conditional mean functions that depend on correlated latent effects (Chib and Winkelmann, 2001), a multivariate overdispersed Poisson mixture model (Gurmu and Elder, 2000), bivariate Poisson-lognormal mixture and bivariate negative binomial regression models (Munkin and…”

“20: 897–916 (2011) DOI: 10.1002/hec (Austin et al., 2003; Basu, 2005; Briggs et al., 2005; Briggs and Gray, 1998; Chen and Zhou, 2006; Deb and Burgess, 2003; Dinh and Zhou, 2006; Duan et al., 1983; Dudley et al., 1993; Gilleskie and Mroz, 2004; O’Hagan and Stevens, 2003; Thompson and Barber, 2000;…”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“…included an informal comparison of different methods and approaches applied to the same data (Austin et al., 2003; Briggs and Gray, 1998; Buntin and Zaslavsky, 2004; Chen and Zhou, 2006; Cooper et al., 2007; Lipscomb et al., 1998; Manning and Mullahy, 2001; O’Hagan and Stevens, 2003; Zhou, 2002).”

“Two and three-parameter lognormal and log-logistic distributions are used by Nixon and Thompson (Nixon and Thompson, 2004; Thompson and Nixon, 2005) to model costs, and lognormal and Weibull models are used by Marazzi et al. (1998) to model length of stay in hospital.”

“…included an informal comparison of different methods and approaches applied to the same data (Austin et al., 2003; Briggs and Gray, 1998; Buntin and Zaslavsky, 2004; Chen and Zhou, 2006; Cooper et al., 2007; Lipscomb et al., 1998; Manning and Mullahy, 2001; O’Hagan and Stevens, 2003; Zhou, 2002).”

“More robust estimation procedures via alternatives to maximum likelihood have been recently suggested (Lu et al., 2003).”

“…and kurtosis in data; (2) incorporate adjustment for covariates and extend to cost-effectiveness analysis and (3) be applicable to data sets of small to moderate sample sizes usually available in randomised studies, where these issues were not explicitly considered by the authors themselves.”

“For these methods the individual healthcare resource use or cost data is transformed in order to overcome problems of skewness and comparison of means on the transformed scale (Briggs and Gray, 1998) or OLS type approaches (as above) to model the transformed dependent variable are employed (Ai and Norton, 2000; Duan e…”

“A Bayesian implementation of a two-part model in WinBUGS is illustrated by Cooper et al. (2003).”

“An approach based on a finite Markov chain is suggested to estimate resource use over different phases of health care (Coxian phase-type distribution) and evaluate total cost by attaching unit costs to these phases (Marshall et al., 2007).”

“The four-part model in Duan (1983) (see Section 3.”

“General methods for count data have been extensively described elsewhere (Cameron and Trivedi, 1998; Winkelmann, 2008).”

“Alternative approaches of direct numerical integration over fully parameterized error distributions are proposed by Abrevaya (2002) and Ai and Norton (2000), while Ai and Norton (2008) propose a more general semi-parametric approach.”

“Lambert et al., also illustrate the possibility to increase the flexibility using a two-part model for cost components with excess zeros (Lambert et al., 2008).”

“Regression of the quantiles of costs (Wang and Zhou, 2010) has been shown to have competitive and more robust performance compared to the smooth quantile ratio estimator (Dominici et al., 2005), maximum likelihood estimator, generalised linear models based on quasilikelihood (Blough and Ramsey, 2000), and the internal…”

“Conigliani and Tancredi (2006; 2009) showed that the performance of Bayesian model averaging depends on what models are averaged over and whether there was a model included that fits the data well; otherwise an approach based on mixtures of distributions seemed more appropriate in terms of coverage.”

“Regression of the quantiles of costs (Wang and Zhou, 2010) has been shown to have competitive and more robust performance compared to the smooth quantile ratio estimator (Dominici et al., 2005), maximum likelihood estimator, generalised linear models based on quasilikelihood (Blough and Ramsey,…”

“The applications published to date refer to components of costs or resource use modelled as bivariate or multivariate normally or lognormally distributed data (Hahn and Whitehead, 2003; Lambert et al., 2008).”

“A second is models which allow for potentially heavy upper tails, while allowing flexible (or even non-parametric) distributions for the bulk of the data (Conigliani and Tancredi, 2005).”

“Two and three-parameter lognormal and log-logistic distributions are used by Nixon and Thompson (Nixon and Thompson, 2004; Thompson and Nixon, 2005) to model costs, and lognormal and Weibull models are used by Marazzi et al. (1998) to model length of stay in hospital.”

“Another survival type approach suggested in the literature is the Aalen regression model with additive hazard function (Pagano et al., 2008).”

“This model shows good performance when the proportional hazards assumption holds; a test for proportional hazards is suggested by Basu and Manning (2006).”

“This model controls for skewness and heteroscedasticity and approximates the distribution of the data based on the modified Park test [53, 54].”

“Although cost data are often right skewed because a few patients use very high amounts of resource, producing a distribution that may violate the assumptions of standard statistical tests, it is not clear that the many alternative approaches suggested, such as bootstrapping, provide more reliable test results.”

“The general consensus is that estimating the mean cost using a logarithmic regression model leads to biased estimation [2,4-6].”

“To account for this, we followed a recommendation to apply standard regression using a bootstrap estimate of standard error, and to repeat the analysis with and without the extreme values [49].”

“In addition, most analyses fail to appropriately account for the shared correlation among cost variables (11).”

“We also extracted whether any measures of dispersion, such as standard deviation (SD), around the mean cost values were given [22].”

“Cost data are often substantially positively skewed (as in this study) and have traditionally been handled using nonparametric methods such as rank-order statistics.(45) We used the nonparametric Mann–Whitney U-test to evaluate differences in median costs between the two groups.”

“In line with literature recommendations, we can assume a near normality of the sample means, as the sample size is sufficiently large and the proportion of zero costs relatively small (,2%) (30).”

“Generalized linear model (GLM), which could manage skewness and heteroscedasticity, was employed to explore relationships between potential covariates and EQ5D utility scores as well as VAS scores [39, 40].”

“As both length of stay and hospital reimbursement are non-negative and right-skewed, generalised linear models with a gamma distribution and a log-link were specified.(16) For continuous variables, non-linearity was incorporated using restricted cubic splines.”

“A confirmatory analysis of the family and link functions was conducted using the modified Park test [21, 22].”

“Costs were trimmed back to the 99th percentile to remove extreme outliers (27,28).”

“This pooled dataset was then used to run multivariable generalized estimating equation (GEE) models with a gamma family and log link [28] to estimate the per-infection attributable cost as measured by an adjusted risk difference between infection patients and their uninfected controls.”

“In our two-stage model, the first part P (cost > 0) was modelled using a logit model; then, conditional on the healthcare spending being positive, the value of the spending was modelled using gamma regressions with log link [28, 29].”

“compared between groups using generalized linear modelling (gamma family, identity link) as recommended to account for the highly skewed nature of cost data (Mihaylova et al., 2011).”

“[15] and the literature already mentioned.”

“Although more research is needed to provide actionable guidance for applied researchers (Mihaylova et al., 2011; Neelon et al., 2016), understanding the nature of the zeros in the data, using theory to distinguish between a single or two-part data-generating process, and reviewing fit diagnostics…”

“In a review study, Mihaylova and colleagues recommended that the two-part model be used for modeling expenditure data with excessive zeros [47].”

“Different statistical techniques have been used for presenting the cost results in mean costs including bootstrapping and using the gamma distribution [29,54].”

“For the base case analysis, generalized linear models (GLMs) were developed using the gamma distribution with a log-link function [26].”

“” [69] and an identity link [12,15] may be analytically sufficient seems not to be the case with our data, although the LMM and log-OLS models were relatively easy to fit and interpret.”

“(Authors’ note: Health care cost data have long been observed to exhibit substantial skewness with long right-sided tails, commonly resulting in large SDs often approaching or exceeding the value of the mean.(26))”

“Likewise, for CEAs that use end points from CRTs as parameters in a decision model, any probabilistic sensitivity analysis should carefully justify the distributional assumptions of these input parameters.(26,54) To summarize, statistical methods in CEAs that use CRT have to recognize clustering and correlation, as we…”

“Cost data are substantially positively skewed and have traditionally been handled using nonparametric methods [42].”

“It is well recognized that statistical analysis of healthcare cost data poses a number of di culties [16].”

“0%) had zeromedical expenditures, the age-, race-, sex-, and BMI-specific annual health care expenditures for both patients with diabetes and those without diabetes were estimated using a twopart model (29,30).”

“The skewness and kurtosis of the sample data met the normal distribution criteria [50,51].”

“1) suitable for nonparametric dependent variables [19, 20].”

“where Y represents the cost and X represents the vector of covariates [18].”

“For each subject, continuous confounders (Z1; Z2; Z3; Z4) were generated from bivariate normal distributions with the following means, standard deviations and correlation: Z1 Z2 N 2 4 ; 1 0:2 0:2 1 Z3 Z4 N 2 4 ; 1 0:2 0:2 1 : The treatment indicator t was randomly generated from a Bernoulli distribution with parameter…”

“As is typical for healthcare cost data, the distribution of total cost was highly skewed, resulting in heteroskedasticity of the residuals in the model (10).”

“The Heckman [27] model does not address skewness specifically; however, with a large-enough sample, the central limit theorem should provide confidence that sample means will be close to normal [29].”

“For a comprehensive review of statistical methods to analyse HCE data, see also Dodd et al. (2006) and Mihaylova et al. (2011).”

“, high positive skewness and heteroscedasticity), we used generalized linear models with a log link function and gamma family distribution for expenditure outcomes [27].”

“Cost outcomes were modeled using two-part models due to excessive zero cost (16, 17).”

“’s [29] literature review confirms that 2PM models perform better.”

“Because of the right-skewedness of expenditures data (ie, most expenditures are seen in only a small proportion of the population), twopart models were utilized to study expenditures.(27) Two-part models are often used to model health care expenditures, and are the product of: (1) the probability that any given indivi…”

“Because of the right skewedness of expenditures data (ie, most expenditures are seen in only a small proportion of the population), 2-part models were used to study expenditures.(21) Two-part models are often used to model healthcare expenditures and are the product of: (1) the probability that any given individual ha…”

“We therefore use a fixed effects Poisson model (FEP) in order to deal with these distributions and also avoid retransformation problems of taking natural logs of the outcome variables (Manning and Mullahy 2001, Buntin and Zaslavsky 2004, Mihaylova et al. 2011, Sparrow et al. 2014).”

“and there is a growing number of papers on regression model analysis and cost-benefit comparative analysis based on statistics [36,37].”

“The total medical healthcare expenditure is a sum of office-based medical provider expenditure, hospital outpatient expenditure, emergency room expenditure, inpatient hospital (including zero night stays) expenditure, prescription medicine expenditure, dental expenditure, home health care expenditure and other medical…”

“To estimate the per capita age-specific quality of life losses and healthcare expenditures for the competing diseases, two-part regression models were fitted, because the distributions of both outcomes were skewed.(24,25) The first part of the model predicted the probability of the outcome variable being 1 (for the qu…”

“FEP models are well suited to dealing with such outcomes (Mihaylova et al., 2011; Buntin and Zaslavsky, 2004; Manning and Mullahy, 2001).”

“Because the out-of-pocket spending data is skewed to the right and was not normally distributed, a generalized linear model (GLM) with a log-link function using a gamma distribution was applied, which is the most suitable option for cost data analysis analyzing both mean and variance functions and adjusting the right-…”

“Generalized estimating equations specifying a poisson distribution and identity link were used to estimate mean costs over the year.[40, 41] Separate models were performed to estimate cost rates for medication use (one week) and lost productivity (one month).”

“The cost-effectiveness analysis used seemingly unrelated regression (SUR), for which we rely on the near normality of differences in sample means and differences in sample variances in large samples.(31) 32 We implemented SUR using the—sureg—command in”