...
ln 1 1 1 2 2 3 3
Where, p is the probability of occurrence of multi-morbidity, p(у=1);1, 2, 3,…
i refers to the beta coefficients; x1, x2, x , …3 x refers to the independent variables i and e is the error term.
2.5.2 Generalized Linear Regression Models (GLRM)
In recent years general linear model (GLM) is a common method of estimating ANOVA and MANOVA models. The GLM is composed of three elements.
Variate: the independent variables are defined at first then they are assigned estimated weights showing the variables contribution to the predicted value.
Random component: The probability distribution is assumed to underline the dependent variables which may be normal distribution, binomial distribution, and passion distribution. Random component is selected based on the type of responsible variable.
Link function: It gives the theoretical connect between the variate and random component to accommodate the differing model formulation. The link function
33 specifies the type of transformation needed to specify the desired model. The three most common link functions are the identity, logit and log links.
The GLM approach gives a single estimated model within which any number of differing statistical models can be accommodated. The special advantages of GLM are its flexibility and simplicity in model design
In its simplest form, a linear model specifies the (linear) relationship between a dependent (or response) variable Y, and a set of predictor variables, the X's, so that
Y = b0 + b1X1 + b2X2 + ... + bkXk
In this equation b0 is the regression coefficient for the intercept and the bi values are the regression coefficients (for variables 1 through k) computed from the data. The generalized linear model can be used to predict responses both for dependent variables with discrete distributions and for dependent variables which are nonlinearly related to the predictors.
The generalized linear regression model (GLRM) differs from the general linear model (of which, for example, multiple regression is a special case) in two major respects: First, the distribution of the dependent or response variable can be (explicitly) non-normal, and does not have to be continuous, i.e., it can be binomial, multinomial, or ordinal multinomial (i.e. contain information on ranks only); second, the dependent variable values are predicted from a linear combination of predictor variables, which are
"connected" to the dependent variable via a link function.
This method is applied to examine the covariates of Out of Pocket Expenditure (OOPE) among rural elderly in chapter VI. The outcome variable OOP health expenditure was typically non-parametric and positively skewed with influential outliers. Traditional ordinary least square (OLS) regressions with log-transformation and retransformations are too inconsistent to handle skewness in the data and provide inferences in natural units of mean expenditures (Manning, 1998). Generalized linear regression models (GLRM) are flexible to handle skewed expenditure data and avoid the issue of outcome transformation (Basu, 2009, Gregori et al., 2011). Survey GLM with gamma distribution and log link function (Kilian et al., 2002), was employed to assess various determinants
34 of OOPE and account for the complex survey design. The equations for gamma distribution as follow:
f(y) = | exp | for < y <
1 Scale = Var(Y) = /
Application of GLM for healthcare financing: A study was conducted among 1202 COPD patients at Kaiser Permanente at USA (Omachi et al., 2013).Using a general linear model (GLM) with a gamma response probability distribution and a log-link function gamma response probability distributions were chosen for analysis.
They analyzed the substantial impact of disease-specific clinical measures have on predicted costs among COPD patients, even after risk-adjustment using diagnosis codes.
Although clinical measures need to be made more easily accessible for cost prediction, they convincingly showed that COPD severity measures, in absolute dollar terms, meaningfully impact costs. Incorporating disease-specific measures into risk models is important to encourage providers to accept responsibility for sicker COPD patients.
Additionally, incorporating these measures may reduce the financial disadvantages faced by organizations that care for lower SES populations. This is likely to become increasingly important with the growth of accountable care organizations ACOs, expected under the Affordable Care Act, and the application of risk-adjustment more broadly.4 Simultaneously, it is likely to become easier to implement with the growth of electronic health records EHRs. Caution must therefore be taken about the extent to which currently-employed risk adjustment methods may adequately control for disease severity while further work to incorporate clinical measures is conducted.
Another study where two models have been used to address the econometric problems caused by skewness in data commonly encountered in health care applications first was transformation to deal with skewness (e.g., OLS on ln(y)); and second was an alternative weighting approaches based on exponential conditional models (ECM) and generalized linear model (GLM) approaches. This study encompass these two classes of models using the three parameter generalized gamma (GGM) distribution, which includes several of
35 the standard alternatives as special cases OLS with a normal error, OLS for the log normal, the standard gamma and exponential with a log link, and the Weibull. Using simulation methods, they tested and identified distribution to be robust. The GGM also provides a potentially more robust alternative estimator to the standard alternatives. An example using inpatient expenditures is also analyzed (Manning et al., 2005).
The result of GLRM model was presented in terms of β coefficients. The β co-efficients can be negative or positive. If the β coefficient is not statistically significant (based on p values), no statistical significance can be interpreted from that predictor. If the regression beta coefficient is positive, the interpretation is that for every 1-unit increase in the predictor variable, the dependent variable will increase by the standardized beta coefficient value. For example, if the beta coefficient is .80 and statistically significant, then for each unit increase in the predictor variable, the outcome variable will increase by .80 units.