How to Model Health Care Costs

2.2.1. Box-Cox Procedure for Transformation 

Initially, the transformations of cost values have been preferred in solving non-linear and asymmetrical relationships. The Box-Cox transformations of variable y are used in the normalization of error distribution, stabilize error variance, and make the relationship between y and x more linear. In order for transformation to be used, the values of the dependent variable y must be positive.The only parameter that is estimated is here is λ. When commanding PROC TRANSREG, the only variable that is transformed is Y. SAS on the other hand implement Box Cox transformation for regression. During this practice, the optimal λ is selected to allow data transformation and fitting of the regression model. 

SAS Example: Box-Cox Transformation 

2.2.2. Log Transformation of Health Care Costs 

Log transformation is commonly used for addressing skewness and nonlinearity in healthcare costs. It transforms non-linear multiplicative relationships into linear equations as well as enhances precision and diminishes the effects of outliers. Log transformations utilize positive data; therefore, the addition of a constant value to zero eliminates cases of patients with zero costs of being left-out of the model. 

SAS Example: Estimation with Log Transformed Cost 

Step 1: Cost variable transformation: 

1 is constant and is added to each observation. 

Step 2: running the standard OLS model with log transformed dependent variable: 

Ref: see table 2.9: Regression Coefficients of Log Transformed Cost Models 

PROCREG can similarly be used for white heteroskedasticity. 

2.2.3. Retransformation Issues 

There are several issues of retransformation. First, policy decisions are always based on actual dollar values requiring retransformation of log-scaled predictions. Secondly, when errors achieve linearity, normality and homoscedasticity assumptions, retransformation presents no problem. In geometric the results from the log transformed model are considered as means instead of arithmetic means; therefore, log scale estimates may deliver biased estimated impacts of an explanatory variable on arithmetic mean. In the current literature, the complication of using log-transformed models is well documented. Therefore, simple exponentiation of predicted values will not create unbiased estimators of the transformed values as illustrated. 

Necessary adjustments are required for retransformation. 

2.2.4. Smearing Transformation 

Smearing factors are majorly used for transformation. When the variables are continuous due to heteroscedasticity, then transformation will not exist which provides unbiased estimates of the retransformed values. However, an alternative approach for retransformed is to use GLMs.Estimation of smearing transformation with homoskedasticity becomes applicable when a model has homoskedasticity and linearity in a model, but lacks normality on a known transformation. On the other hand, heteroskedasticity-adjusted smearing is used to correct heteroskedasticity caused by categorical variables and the ignorance of homoskedasticity postulation. Smearing transformation can be summarized using SAS example. 

SAS Example: Smearing Transformation 

Getting Model n in a Macro Variable 

Using Score to identify estimated Values 

Calculating smearing estimate