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

Outline

• 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 for CEA

• 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

Policy Relevant Parameter for CEA (2)

• 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

Are Sample Means Always the Best Estimator?

• 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

The Problem

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

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The Problem (cont.)

• 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

Common (Relatively Bad) Responses To Violation Of Normality

• 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)

Recommended Response: Adopt More Flexible Models

• 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)

Stata and SAS Code

• 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;

SAS Code for a Gamma Family / Log Link

The Link Function

• 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:

ˆi i iY = β X

The Log Link

• 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

ln(E(y/x) ≠ E(ln(y)/x)

Variable Group 1 Group 2

Observations

1 15 35

2 45 45

3 75 55

Arithmetic mean 45 45

Log, arith mean cost 3.806662 3.806662 *

Natural log

1 2.70805 3.555348

2 3.806662 3.806662

3 4.317488 4.007333

Arith mean, log cost 3.610734 3.789781 †

* Difference = 0; † Difference = 0.179047

Comparison of Results of GLM Gamma/Log and log OLS Regression

Variable Coefficient SE z/T p value

GLM, gamma family, log link

Group 2 0.000000 0.405730 0.00 1.000

Constant 3.806662 0.286894 13.27 0.000

Log OLS

Group 2 0.179048 0.492494 0.36 0.74

Constant 3.610734 0.348246 10.32 0.000

The Power Link Function

• 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

ˆiu

ˆiu

ˆiu

ˆiu

ˆiu

ˆiu

Negative Power Links

• 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

i 1

abs(power)i i

1y =

B X

Selecting a Link Function

• 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

The Family

• 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• If weights or clustering are used in the original GLM,

same weights and clustering should be used for modified Park test

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

Recommended Family, Modified Park Test

Variance function: V(u) = u^2Link function: g(u) = ln(u)

[Gamma]

[Log]

Deviance = 214.8893Log likelihood = -20782.51

AIC 20.7895BIC -14933.71

cost Coef Std Err z P>|t| 95% CI

treat -.0361495 .015123 -2.39 0.017 -.065790 -.006510

dissev1 .1250814 .101609 1.23 0.218 -.074696 .324232

dissev2 -.6837741 .076062 -8.99 0.000 -.832853 -.534696

blcost .0069198 .023707 0.29 0.770 -.039545 .053385

blqscore -.1993532 .046650 -4.27 0.000 -.290786 -.107921

bledvis .0158549 .005082 3.12 0.002 .005894 .025816

_cons 9.756198 .090169 108.20 0.000 9.57947 9.93293

Example, GLM gamma/log

miscand1.dta

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

FITTED MODEL: Link = Log ; Family = Gamma

Results, Modified Park Test (for Family)

Coefficient: 1.1680

Family, Chi2, and p-value in descending order of likelihood

Family Chi2 P-value

Poisson: 0.0675 0.7951

Gamma: 1.6539 0.1984

Gaussian NLLS: 3.2599 0.0710

Inverse Gaussian or Wald: 8.0193 0.0046

Results of tests of GLM Identity link

Pearson Correlation Test: 0.9688

Pregibon Link Test: 0.8529

Modified Hosmer and Lemeshow: 0.8818

GLM Diagnostics, log/gamma

miscand1.dta

Variance function: V(u) = uLink function: g(u) = ln(u)

[Poisson]

[Log]

Deviance = 25171259Log likelihood = -1296805

AIC = 1296.812BIC = 2556110

cost Coef Std Err z P>|t| 95% CI

treat -.0342075 .00041 -83.64 0.000 -.035010 -.033406

dissev1 .1209046 .00274 44.12 0.000 .115533 .126276

dissev2 -.6857563 .00206 -333.4 0.000 -.689788 -.681725

blcost .0072812 .00064 11.41 0.000 .006031 .008532

blqscore -.1970875 .00125 -157.6 0.000 -.199538 -.194637

bledvis .0159274 .00014 116.7 0.000 .015660 .016195

_cons 9.756579 .00243 4014 0.000 9.75182 9.76134

Change Family to Poisson and Rerun Model

miscand1.dta

FITTED MODEL: Link = Log ; Family = Poisson

Results, Modified Park Test (for Family)

Coefficient: 1.1627

Family, Chi2, and p-value in descending order of likelihood

Family Chi2 P-value

Poisson: 0.0621 0.8032

Gamma: 1.6460 0.1995

Gaussian NLLS: 3.1734 0.0748

Inverse Gaussian or Wald: 7.9249 0.0049

Results of tests of GLM Identity link

Pearson Correlation Test: 0.9882

Pregibon Link Test: 0.8136

Modified Hosmer and Lemeshow: 0.8928

GLM Diagnostics, log/poisson

miscand1.dta

Passes Tests, But Can We Improve the Link?

• 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

Results for Selected Power Links

Power Family Pearson Pregibon mH&M

-0.2 Poisson 0.9835 0.7089 0.8929

-0.1 Poisson 0.9865 0.761 0.8943

0.0 Poisson 0.9882 0.8136 0.8928

0.1 Poisson 0.9905 0.8669 0.9063

0.2 Poisson 0.9934 0.9209 0.9559

0.3 Poisson 0.9969 0.9737 0.9461

0.4 Poisson 0.9991 0.9730 0.9359

0.5 Poisson 0.9946 0.9201 0.9369

0.6 Poisson 0.9895 0.8678 0.9008

0.7 Poisson 0.9839 0.8164 0.8125

0.8 Poisson 0.9778 0.7661 0.7444

miscand1.dta

Individual Criteria Do Not Uniquely Identify Link, But…

Power Family Pearson Pregibon mH&M

-0.2 Poisson 0.9835 0.7089 0.8929

-0.1 Poisson 0.9865 0.761 0.8943

0.0 Poisson 0.9882 0.8136 0.8928

0.1 Poisson 0.9905 0.8669 0.9063

0.2 Poisson 0.9934 0.9209 0.9559

0.3 Poisson 0.9969 0.9737 0.9461

0.4 Poisson 0.9991 0.9730 0.9359

0.5 Poisson 0.9946 0.9201 0.9369

0.6 Poisson 0.9895 0.8678 0.9008

0.7 Poisson 0.9839 0.8164 0.8125

0.8 Poisson 0.9778 0.7661 0.7444

miscand1.dta

Can We Develop a Summary Measure?

Power Family Pearson Pregibon mH&M Summary

-0.2 Poisson 0.9835 0.7089 0.8929 .096426

-0.1 Poisson 0.9865 0.761 0.8943 .068476

0.0 Poisson 0.9882 0.8136 0.8928 .046376

0.1 Poisson 0.9905 0.8669 0.9063 .026586

0.2 Poisson 0.9934 0.9209 0.9559 .008356

0.3 Poisson 0.9969 0.9737 0.9461 .003607

0.4 Poisson 0.9991 0.9730 0.9359 .004839

0.5 Poisson 0.9946 0.9201 0.9369 .010395

0.6 Poisson 0.9895 0.8678 0.9008 .027428

0.7 Poisson 0.9839 0.8164 0.8125 .069124

0.8 Poisson 0.9778 0.7661 0.7444 .120533

miscand1.dta

LL, AIC, BIC, and Deviance

Power Family LL AIC BIC Deviance

-0.2 Poisson -1296851.7 1296.8587 2556204 2571353

-0.1 Poisson -1296825.5 1296.8325 2556152 2571300

0.0 Poisson -1296804.8 1296.8118 2556110 2571259

0.1 Poisson -1296789.7 1296.7967 2556080 2571229

0.2 Poisson -1296779.9 1296.7869 2556060 2571209

0.3 Poisson -1296775.5 1296.7825 2556052 2571200

0.4 Poisson -1296776.3 1296.7833 2556053 2571202

0.5 Poisson -1296782.3 1296.7893 2556065 2571214

0.6 Poisson -1296793.5 1296.8005 2556088 2571236

0.7 Poisson -1296809.8 1296.8168 2556120 2571269

0.8 Poisson -1296831.1 1296.8381 2556163 2571311

miscand1.dta

Why Not Simply Use AIC and BIC?

• 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?

AIC / BIC

• 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

AIC / BIC (2)

• 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

AIC / BIC (3)

• 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

• 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)

Extended Estimating Equations

GLM Disadvantages

• 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

• 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

Review

EXTRA SLIDES

Fine Tuning (1)

Power Family Pearson Pregibon mH&M Summary

0.31 Poisson 0.9973 0.9790 0.9399 .00406

0.32 Poisson 0.9760 0.9844 0.9394 .00392

0.33 Poisson 0.9980 0.9897 0.9390 .00383

0.34 Poisson 0.9984 0.9951 0.9386 .00379

0.35 Poisson 0.9988 0.9996 0.9382 .00382

0.36 Poisson 0.9992 0.9943 0.9378 .00390

0.37 Poisson 0.9996 0.9889 0.9373 .00405

0.38 Poisson 1.0000 0.9836 0.9367 .00427

0.39 Poisson 0.9995 0.9783 0.9363 .00452

miscand1.dta

Fine Tuning (2)

Power Family LL AIC BIC Deviance

0.31 Poisson -1296775.324 1296.7823 2556051.2 2571199.801

0.32 Poisson -1296775.224 1296.7822 2556051.0 2571199.600

0.33 Poisson -1296775.176 1296.7822 2556050.9 2571199.505

0.34 Poisson -1296775.181 1296.7822 2556050.9 2571199.514

0.35 Poisson -1296775.238 1296.7822 2556051.0 2571199.628

0.36 Poisson -1296775.337 1296.7823 2556051.2 2571199.847

0.37 Poisson -1296775.509 1296.7825 2556051.6 2571200.170

0.38 Poisson -1296775.723 1296.7827 2556052.0 2571200.598

0.39 Poisson -1296776.989 1296.7830 2556052.5 2571201.130

miscand1.dta

AIC / BIC

• (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

Different Recommendations

Power Family LL AIC BIC Deviance

-0.5 Gamma -6408.4 18.767 -3144.6 1094.8

-0.4 Gamma -6399.1 18.740 -3127.7 1111.8

-0.3 Gamma -6390.3 18.714 -3103.9 1135.6

-0.2 Gamma -6382.3 18.691 -3073.6 1165.9

-0.1 Gamma -6374.1 18.667 -3035.8 1203.7

0.0 Gamma -6359.1 18.623 -2979.5 1259.9

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4 Statistics Aren’t Stable Across Families

Power Family LL AIC BIC Deviance

-1.2 Gamma -4495.52 18.0061 -2985.78 84.2

-1.1 Gamma -4495.35 18.0054 -2986.12 83.9

-1.0 Gamma -4495.19 18.0048 -2986.44 83.6

. . . . . .

0.4 Gamma -4493.99 18.0000 -2988.85 81.2

0.5 Gamma -4493.97 17.9999 -2988.89 81.1

0.6 Poisson -113533 454.16 219123 222193

0.7 Poisson -113504 454.04 219066 222136

0.8 Poisson -113502 454.03 219061 222131

0.9 Poisson -113526 454.13 219109 222178

1.0 Poisson -113576 454.32 219210 222280

rchapter5.dta

4 Statistics Aren’t Stable Across Families

Power Family Pearson Pregibon mH&M

-1.2 Gamma .0102 .0004 .0275

-1.1 Gamma .0115 .0007 .0411

-1.0 Gamma .0144 .0010 .0673

. . . . .

0.4 Gamma .5259 .4715 .5783

0.5 Gamma .6181 .6064 .6607

0.6 Poisson .8783 .6736 .5934

0.7 Poisson .9286 .8231 .4050

0.8 Poisson .9853 .9812 .3191

0.9 Poisson .9514 .8582 .4205

1.0 Poisson .8818 .7021 .5134

rchapter5.dta

Limitations If Power Family Becomes Available

• 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