Issues To Consider when Estimating Health Care Costs with Generalized Linear Models (GLMs): To Gamma/Log Or Not To Gamma/Log? That Is The New Question

Transcription

1 Issues To Consder when Estmatng Health Care Costs wth Generalzed Lnear Models (GLMs): To Gamma/Log Or Not To Gamma/Log? That Is The New Queston ISPOR 20th Annual Internatonal Meetng May 19, 2015 Jalpa Dosh, Henry Glck, and Andrew Brggs Health Economcs Evaluatons: Polcy Relevant Parameter Adopton and dffuson of new medcal treatments depend ncreasngly on analyss of costs and costeffectveness. Polcy relevant parameter: dfferences n the arthmetc, or sample, mean costs Cost-effectveness ratos ( C / E) and NMB ([WTP E] - C) requre an estmate of C and E, the dfferences n arthmetc means From a budgetary perspectve, decson makers can use the arthmetc mean to determne how much they wll spend on a program Patent-level Health Economcs Analyses Analyses often rely on patent-level data from randomzed clncal trals or observatonal studes (.e. prospectve regstres and secondary data). Multvarate analyses of costs are recommended for RCTs and are advantageous for several reasons despte randomzaton Multvarate analyses of costs are necessary for observatonal studes but may not be suffcent However, patent-level health care cost data rarely meet assumptons of the ordnary least squares (OLS) regresson. Assume constant varance Assume E(y/x)=Σβ X 1

2 Typcal Dstrbutons Of Cost Data (I) Typcal Dstrbutons of Cost Data (II) Common feature of cost data s rght-skewness (.e., long, heavy, rght tals) Data tend to be skewed because: Can not have negatve costs Most severe cases may requre substantally more servces than less severe cases Certan events, whch can be very expensve, occur n a relatvely small number of patents A mnorty of patents are responsble for a hgh proporton of health care costs Multvarable Technques Used for the Analyss of Cost Old Favorte Ordnary least squares regresson predctng the log transformaton of costs after randomzaton (log OLS) New Favorte Generalzed Lnear Models (GLM) 2

3 Potental Problems wth the Old Favorte: Log OLS For economc analyss, the outcome of nterest s the dfference n untransformed costs (e.g., Congress does not approprate log dollars. Frst Bank wll not cash a check for log dollars ) Thus, the results on the transformed scale must be retransformed to the orgnal scale There s a very real danger that the log scale results may provde a very msleadng, ncomplete, and based estmate...on the untransformed scale, whch s usually the scale of ultmate nterest (Mannng, 1998) Ths ssue of retransformaton...s not unque to the case of a logged dependent varable. Any power transformaton of y wll rase ths ssue Generalzed Lnear Models (GLM) GLM models have the advantages of the log models, but Don t requre normalty or homoscedastcty, Evaluate the log of the mean, not the mean of the logs, and thus Don t rase problems related to retransformaton from the scale of estmaton to the raw scale To buld them, we must dentfy a "lnk functon" and a "famly (based on the data at hand) GLM Relaxes OLS Assumptons Ablty to choose among dfferent famles relaxes assumpton of constant varance Gauss: constant varance Posson: varance proportonal to mean Gamma: varance proportonal to square of mean Inverse gauss: varance proportonal to cube of mean Ablty to choose among dfferent lnks relaxes assumpton that y/x= Σβ X (OLS) or E(ln(y)/x)=Σβ X (Log OLS) 3

4 The Lnk Functon Lnk functon drectly characterzes how the lnear combnaton of the predctors s related to the predcton on the orgnal scale e.g., predctons from the dentty lnk -- whch s used n OLS -- equal: Y = β X ˆ The Lnk Functon Stata s power lnk provdes a flexble lnk functon It allows generaton of a wde varety of named and unnamed lnks, e.g., power 2: uˆ = (β X ) 0.5 power 1 = Identty lnk; uˆ = β X power.5 = Square root lnk; uˆ = (β X ) 2 power.25: uˆ = (β X ) 4 power 0 = log lnk; uˆ = exp(β X ) power -1 = recprocal lnk; uˆ = (β X ) -1 power -2 = nverse quadratc; u = (β X ) -0.5 ˆ The Log Lnk Log lnk s most commonly used n lterature When we adopt the log lnk, we are assumng: ln(e(y/x))=xβ GLM wth a log lnk dffers from log OLS n part because n log OLS, we are assumng: E(ln(y)/x)=Xβ ln(e(y/x) E(ln(y)/x).e. log of the mean mean of the log costs 4

5 ln(e(y/x) E(ln(y)/x) Varable Group 1 Group 2 Observatons Arthmetc mean Log, arth mean cost * Natural log Arth mean, log cost * Dfference = 0; Dfference = Comparson of Results of GLM Gamma/Log and log OLS Regresson Varable Coeffcent SE z/t p value GLM, gamma famly, log lnk Group Constant Log OLS Group Constant Real New Favorte: GLM (gamma/log) Gamma/log lnk GLM s beng commonly used n the lterature Most often wthout any dagnostc testng of ts ft for the data at hand So s that the correct famly/lnk combnaton for all datasets and studes? 5