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Methods for Cost Estimation in CEA: the GLM Approach Henry Glick University of Pennsylvania

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Methods for Cost Estimation in CEA: the GLM Approach

Henry Glick

University of Pennsylvania

AcademyHealth

Issues in Cost-Effectiveness Analysis

Washington, DC

06/10/2008

- Policy-relevant parameter for cost-effectiveness
- Problems posed by nonnormality of cost data
- Generalized linear models as a response to the problems
- Identifying links and families (gets a little technical)

- General comments
My objective is to provide practical advice for ways implement GLM models. Slides available at:

http://www.uphs.upenn.edu/dgimhsr/presentations.htm

- Policy relevant parameter: differences in the arithmetic, or sample, mean
- In welfare economics, a project is cost-beneficial if the winners from any policy gain enough to be able to compensate the losers and still be better off themselves
- Thus, we need a parameter that allows us to determine how much the losers lose, or cost, and how much the winners win, or benefit

- From a budgetary perspective, decision makers can use the arithmetic mean to determine how much they will spend on a program

- In welfare economics, a project is cost-beneficial if the winners from any policy gain enough to be able to compensate the losers and still be better off themselves

- In both cases, substitution of some other parameter for the sample mean can be justified only if it provides a better estimate of gains and losses or spending

- In simulation, when cost data are sufficiently nonnormal, the relative bias (truth - observed)2 for other parameters such as the median or adjusted geometric mean can sometimes be lower than the relative bias observed for the arithmetic mean
- HOWEVER,
- Distribution required to be sufficiently nonnormality that ln(cost) is also substantially nonnormally distributed
- In actual data, since we never know truth, it is difficult to determine whether other parameters will have lower relative bias than sample mean

- Common feature of cost data is right-skewness (i.e., long, heavy, right tails)

- Distributions with long, heavy, right tails will have a mean that differs from the median, independent of “outliers”
- Cost data also can’t be negative, and can have large fractions of observations with 0 cost
- Nonnormality of cost data can pose problems for common parametric tests such as t-test, ANOVA, and OLS regression

- Adopt nonparametric tests of other characteristics of the distribution that are not as affected by the nonnormality of the distribution (“biostatistical” approach)
- Transform the data so they approximate a normal distribution (“classic econometric” approach)

- Generalized Linear Models (GLM)
- Have the advantages of the log models, but
- don’t require normality or homoscedasticity
- and evaluate a direct of the difference in cost and don’t raise problems related to retransformation from the scale of estimation to the raw scale

- To build them, one must identify a "link function" and a "family“ (based on the data)

- Have the advantages of the log models, but

- STATA code:
glm y x, link(linkname) family (familyname)

- General SAS code (not appropriate for gamma family / log link):
proc genmod;

model y=x/ link=linkname dist=familyname;

run;

- When running gamma/log models, the general SAS code drops observations with an outcome of 0
- If you want to maintain these observations and are predicting y as a function of x (M Buntin):
proc genmod;

a = _mean_;

b = _resp_;

d = b/a + log(a)

variance var = a2

deviance dev =d;

model y = x / link = log;

run;

- Link function directly characterizes how the linear combination of the predictors is related to the prediction on the original scale
- e.g., predictions from the identity link -- which is used in OLS -- equal:

- Log link is most commonly used in literature
- When we adopt the log link, we are assuming:
ln(E(y/x))=Xβ

- GLM with a log link differs from log OLS in part because in log OLS, one is assuming:
E(ln(y)/x)=Xβ

- ln(E(y/x) ≠ E(ln(y)/x)
i.e. log of the mean mean of the log costs

* Difference = 0; † Difference = 0.179047

- Stata’s power link provides a flexible link function
- It allows generation of a wide variety of named and unnamed links, e.g.,
- power 1 = Identity link; = BiXi
- power .5 = Square root link; = (BiXi)2
- power .25: = (BiXi)4
- power 0 = log link; = exp(BiXi)
- power -1 = reciprocal link; = 1/(BiXi)
- power -2 = inverse quadratic; = 1/(BiXi)0.5

- Retranslation of negative power links to the raw scale:
- When using a negative power link, negative coefficients yield larger estimates on the raw scale; positive coefficients yield smaller estimates

- There is no single test that identifies the appropriate link
- Instead can employ multiple tests of fit
- :Pregibon link test checks linearity of response on scale of estimation
- Modified Hosmer and Lemeshow test checks for systematic bias in fit on raw scale
- Pearson’s correlation test checks for systematic bias in fit on raw scale
- Ideally, all 3 tests will yield nonsignificant p-values

- Others (e.g., Hardin and Hilbe) have proposed use of (larger) log likelihood, (smaller) deviance, AIC and BIC statistics

- Specifies the distribution that reflects the mean-variance relationship
- Gaussian: Constant variance
- Poisson: Variance is proportional to mean
- Gamma: Variance is proportional to square of mean
- Inverse Gaussian or Wald: Variance is proportional to cube of mean

- Use of the poisson, gamma, and inverse Gausian families relax the assumption of homoscedasticity

- A “constructive” test that recommends a family given a particular link function
- Implemented after GLM regression that uses the particular link
- The test predicts the square of the residuals (res2) as a function of the log of the predictions (lnyhat) by use of a GLM with a log link and gamma family to
- Stata code
glm res2 lnyhat,link(log) family(gamma), robust

- Stata code
- If weights or clustering are used in the original GLM, same weights and clustering should be used for modified Park test

- Recommended family derived from the coefficient for lnyhat:
- If coefficient ~=0, Gaussian
- If coefficient ~=1, Poisson
- If coefficient ~=2, Gamma
- If coefficient ~=3, Inverse Gaussian or Wald

- Given the absence of families for negative coefficients:
- If coefficient < -0.5, consider subtracting all observations from maximum-valued observation and rerunning analysis

glm cost treat dis* bl*,link(log) family(gamma)

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- Iteratively evaluate power links (in 0.1 intervals) between -2 and 2
- Use the modified Park test to select a family
- Evaluate the fit statistics
- Don’t show you the results here, but I then fine tune the power link in 0.01 intervals within candidate regions of the power link

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- In the current example:
- AIC, BIC, log likelihood, and deviance all agreed
- They yielded an answer that was similar answer to that from the Pearson correlation test, Pregibon link test, and modified Hosmer and Lemeshow tests
- Power link 0.3

- AIC, BIC, log likelihood, (and deviance?) already commonly used for decisions about model fit
- Why do we need the new tests?

- There are at least 3 reasons why in the long run log likelihood, AIC, BIC, and deviance are unlikely to be the recommended tests for identifying the appropriate link function
- First, when there are a large fraction of observations with zero cost:
- The recommendations from log likelihood / AIC agree with each other
- The recommendations from BIC / deviance agree with each other
- But the log likelihood/AIC recommendations differ from the BIC/deviation recommendations

- Second, the 4 statistics aren’t stable across families, and the shifts in their magnitude across families do not provide information about which family/link is best
- For example, in a dataset where the modified Park test recommends a gamma family for power links < 0.4, but recommends a poisson family for power links > 0.5, the magnitude of the AIC statistic shifts from ~18 for the gamma family to ~454 for the poisson family
- Although the smaller AIC values associated with power links < 0.4 suggest that these links have the better fit, the Pearson, Pregibon, and H&M tests all suggest that the power links > 0.5 are actually superior

- Third, while this instability across families is less of a problem when our statistical packages offer 4 continuous families only, it will eliminate comparability across links when statistical packages begin to offer more flexible GLM power families
- i.e., when we don’t have to choose between Gaussian (0) and poisson (1) families, but instead can use a family of 0.7
- In this case, given that each power will be associated with a slightly different family, it will be impossible to compare the resulting AIC/BIC statistics

- i.e., when we don’t have to choose between Gaussian (0) and poisson (1) families, but instead can use a family of 0.7

- Basu and Rathouz (2005) have proposed use of extended estimating equations (EEE) which estimate the link function and family along with the coefficients and standard errors
- Tends to need a large number of observations (thousands not hundreds) to converge
- Currently can’t take the results and use them with a simple GLM command (makes bootstrapping of resulting models cumbersome)

- Disadvantages
- Can suffer substantial precision losses
- If heavy-tailed (log) error term, i.e., log-scale residuals have high kurtosis (>3)
- If family is misspecified

- Can suffer substantial precision losses

- The distribution of cost can pose problems for common parametric tests of cost
- Responses in the literature that suggest that we should evaluate something other than the difference in the sample mean (or a direct transformation of this difference) – e.g., nonparametric tests of other characteristics of the distribution or transformations of cost – generally create more problems than they solve
- Use of more flexible models that evaluate a direct transformation of the difference in cost generally pose fewer problems
- Does require we identify functional forms for the relationship between the predictors and the mean and for the variance structure

EXTRA SLIDES

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- (When there are a large fraction of 0s) log likelihood / AIC recommendations differ from BIC / deviance recommendations
- 4 statistics aren’t stable across families, and shifts in their magnitude across families do not provide information about which family/link is best

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- Generally not a big problem when we are limited to the 4 named continuous families
- Because, as in the example, within families we can look for the power at which the statistics reach a maximum (ll) or maximum (AIC, BIC, deviance)

- When a more flexible GLM family is added to our statistical packages that allows a family of 0.7 or 1.3, rather than forcing us to round to a poisson family (1.0), the change in the scale of the AIC, etc., will make these statistics difficult, if not impossible, to use