Econometrics

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An Alternative to Using Dummy Variables

February 21, 2011 in Econometrics | 1 comment

Oftentimes, researchers use dummy variables to determine how observations classified into different categorical groups affect the dependent variable of interest.  One drawback with this approach is using too many dummy variables can create small cell sizes, creating an identification problem.  Alternatively, using broad groupings for dummy variables may give the appearance that the effect of the covariate is homogenous within the category when this is not the case.

An alternative to using simple categorical dummy variables is to use overlap polynomials.  For instance, Lakdawalla,  Goldman, and  Bhattacharya have a working paper where they rely on the difference of normal cumulative density functions (CDF) to create a flexible form to build these overlapping polynomials.  In particular, they use the following specification:

  • g(age;β) = Σj=0 to K {Φ[(agei-kj+1)/σ]-Φ[(agei-kj)/σ]} * pj(agei;β)

Here is the equation from the paper in larger type.

Below I decribe how this function works in practice.

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Estimating lifetime or episode-of-illness costs under censoring

February 14, 2011 in Econometrics | 1 comment

How can you estimate an individual’s total lifetime cost of medical care?  For people who die in your sample, this is simple.  In most data sets, however, not all individuals will die during the period of observation.  Thus, the data set is censored for those who do not die.

In addition, many standard hazard models do not allow for researchers to disaggregate the effects of covariates on survival and the intensity of utilization.  Both factors have an effect on cost.

Assuming that censoring is random, Basu and Manning (2010) describe a method to calculate expected lifetime costs for each individual as follows:

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Problems with Risk Adjustment

November 30, 2010 in Econometrics | 1 comment

To evaluate providers based on the health outcomes or the cost of care, one must attempt to evaluate dimensions of care which are strictly within the providers control. For instance, if a physicians treats two patients with breast cancer, but one patient has a more advanced form of breast cancer, one should take this difference into account. Patient comorbidities also affect the prognosis for a successful recovery from illness, as well.

One method to take into account the patient’s health conditions upon presentation at a provider’s facility is to use risk adjustment methods. Risk adjustment methods take into account factors such as patient demographics (e.g., age, gender), health status (e.g., prior diagnosis, current illness severity), prior utilizations (e.g., previous hospitalizations) and other factors to predict the expected outcome for a typical patient. Risk adjustment, however, is never perfect. A paper by Garber, MaCurdy and McClellen (1998) review some of the problems with using risk adjustment in the health care setting.

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Duan’s Smearing Estimator

November 16, 2010 in Econometrics | 1 comment

The Problem

Many times, researchers wish to transform the dependent variable of a regression in order to estimate parameter values.  Performing the transformation, however, complicates the calculation of the expected value of the dependent variable on the untransformed scale.  Assume, the Yi is the dependent variable. Assume the function g is used to transform the dependent variable as follows:

  • ηi = g(Yi)
  • Yi = h(ηi)
  • h=g-1

The easiest way to image this functions is think of g as the ln function and h as the exp function. In health economics, researchers often use a log transformation to attenuate problems related to a heavily right-skewed distribution. In this case, one would estimate the following regression:

  • ηi = xiβ + εi
  • εi~F(iid), E(εi)=0; Var(εi)=σ2

One can estimate β consistently as follows:

  • β = (X’X)-1X’η

What is the predicted value of the dependent variable? Calculating this is not as easy as it seems:

  • E(Y0) = E[h(x0β + ε)] ≠ E[h(x0β)]

For instance, it is well known that using the log transformation, the expected value of the depended variable is equal to: exp(x0β + σ2/2). In cases where we do not know the true distribution of the error term, however, then calculating the expected value of the error term is more difficult. The solution is Duan’s Smearing Estimate.

The Solution

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Risk Adjustment: Predicting Future Expenditures

November 2, 2010 in Econometrics, Medicaid | No comments

Many states rely on managed care organizations (MCOs) to provide medical services for their Medicaid beneficiaries.  Contracting out medical services to private providers relies on the government’s capacity to accurately predict expected cost of care for each beneficiary.  This is typically done through risk-adjusted capitation rates.

Which risk adjustment strategy works best?  The answer of course depends on the context.  A paper by Yu and Dick (2010) examines 5 predictors specifications to predict future expenditures for Medicaid eligible children.  I list each of the five specifications and their performance (measured as the R2) below:

  • Age/Gender only: 0.2%
  • Age/Gender + subjective health status measure: 3.9%
  • Age/Gender + CSHCN: 7.3%
  • Age/Gender + HCC: 12.1%
  • Age/Gender + prior year expenditure: 43.5%

One can clearly see that the best predictor of a child’s current year expenditures is the child’s prior year’s expenditures.

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Finite Population Correction

October 12, 2010 in Econometrics, P4P, Quality | No comments

Asymptotic theory has played a large role in the development of many recent econometric methods. For instance, the central limit theorem states that distribution of the mean drawn from any large samples is approximately normally distributed. Asymptotic theory, however, generally assumes that sampling occurs infinitely and with replacement. In the real world, populations are not infinite and sampling does not occur with replacement.

To take into account these real-world challenges, a finite population correction (fpc) factor is needed. One can express the fpc mathematically as:

  • fpc={(N-n)/(N-1)}1/2

where n is the sample size and N is the population size. For instance, one can calculate the standard error for the mean for finite populations as:

  • σX_bar=σ*n1/2 * {(N-n)/(N-1)}1/2
  • σp_bar={[p(1-p)/(n)}1/2 * {(N-n)/(N-1)}1/2

This website has some examples of how to apply fpc‘s in practice.

One potential application for fpc is physician ratings. Physicians who treat lots of patients eligible for a given quality measure certainly can have an accurate score. Some physicians, however, treat only a handful of patients eligible for any given quality metric. In this case, should the physician be punished if he happens to have one bad quality score among the very few observations? Can fpc correct for this problem?

A paper by Elliott, Zaslavsky and Cleary argues that fpc is not appropriate for adjusting physicians scores or confidence intervals to take into account the physicians small sample size. They cite work by Birnbaum who argues that profiling of hospitals is essentially an attempt to make inferences about future performance at the same facility if nothing changes, and involves “a theoretically infinite population.” The authors give the following example to explain why the fpc is not appropriate for rating physicians.

“…suppose we had 50 responses out of 50 total patients at a small hospital, and 300 responses out of 1000 total patients at a larger hospital. Under the finite population model, there is no sampling variability at the smaller hospital (because we have information for all patients) but considerable sampling variability at the larger hospital, which is the appropriate inference if all we care about is the experience of those 50 and 1000 patients. However, to tell a new group of patients what their experiences are likely to be like at each hospital, we have much more information about the large hospital.

Our concern, then, is that FPSM-based approaches would under-represent the uncertainty in data for small facilities with high (possibly 100%) sampling rates, misleading users into thinking that such a facility would be likely to provide below-average (or above-average) care to them.”

To correct for the small sample size problem for providers treating few patients eligible for quality scores each year, the authors believe using a moving average score over multiple years would be more appropriate. The benefit of this method is that the sample size increases, but the drawback is that provider quality improvements will not be fully reflected in the provider’s score for a number of years.

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Shrinkage and the James-Stein Estimator

October 5, 2010 in Econometrics | No comments

Is an average always the best estimate?  Let us say that we are evaluating physician quality.  Does a physician’s average score across patients (or episodes of care) best represent their true quality level?  Stein’s paradox says that when we are evaluating the true quality value for a number of doctors, we can do better than the average.  To do this, we can use a shrinkage methodology.

Shrinkage methodologies in essence estimate the physician’s true quality score as a weighted average of the individual physicians average score and the average score for all physicians.  The intuition behind this is that if we observe a high quality doctor, there is some probability that the are actually a high quality doctor and some probability that they are an average doctor, but they just happened to perform above average when treating the patients in the sample.  Similarly, for low quality doctors, there is some probability that the individual is actually a low quality doctor and some probability that they are an average doctor who just scored poorly on the patients in the sample.

How to calculate the James-Stein Shrinkage Estimator

Because of an abundance of statistics, let us move from evaluating physician quality to evaluating the quality of quarterbacks (QB) in the NFL.  Let yi be the average QB rating for an individual quarterback i, and Y be the average QB rating for all quarterbacks. Then in this case, given we observe yi and Y, we can calculate the James-Stein estimator as:

  • yJS=Y+c(yi-Y)

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Risk Adjustment Models

September 9, 2010 in Econometrics | 1 comment

Risk adjustment is important for many aspects of health care.  Medicare uses risk adjustment to modify payments to Medicare Advantage (Part C) plans based on the health of the beneficiaries they cover.  Private insurance companies can use risk adjustment to fine-tune capitation payments to physicians or determine a potential enrollee’s premium.  Providers can use risk adjustment to identify likely high cost patients and how to adjust their likely treatment pattern accordingly.

There is little doubt that risk adjustment is important, but determining which risk adjustment model is ideal is difficult.  A paper by the Society of Actuaries (2007) examines this topic.  The ideal risk adjustment method will depend on a number of factors.   These include:

  • Ease of use of the software;
  • Specificity of the model to the population to which it is being applied;
  • Cost of the software;
  • Transparency of the mechanics and results of the model;
  • Access to data of sufficient quality;
  • Underlying logic or perspective of a model that makes it best for a specific application;
  • Whether the model provides both useful clinical as well as financial information;
  • Whether the model will be used mostly for payment to providers and plans or for underwriting, rating and/or case management;
  • Reliability of the model across settings, over time or with imperfect data (models that are calibrated and tested on a single data set and population may or may not perform well on different data sets/populations);
  • Whether the model is currently in use in the market or organization; and
  • Susceptibility of the model to gaming or upcoding.

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Econometric Methods for Fractional Response Variables

August 31, 2010 in Econometrics | No comments

Oftentimes, people use the following rule of thumb: if the dependent variable is continuous, use OLS; if binary use a logit or probit.  But what should you do if your dependent variable is fraction between 0 and 1.  To use a logit or probit one would have to unnecessarily transform the dependent variable into binary form.  If one would use OLS, the estimation of the coefficients would likely be incorrect.  Because the dependent variable is bounded between 0 and 1, the effenct of any explanatory variably xj cannot be constant through its entire range. Additionally, the predicted values from an OLS regression often produce figures outside the range of 0 to 1.

A paper by Papke and Wooldridge (1996) examines potential econometric alternatives when your dependent variable is fractional.

LOG-ODDS RATIO

One option to estimate a fractional response variable is to transform the dependent variable into a a log-odds ratio.  For instance:

  • E(log[y/(1-y)]|x) = xβ

This model is simple and can be estimated with OLS techniques onces the depenent variable is transformed.  It only works, however, when the dependent variable is strictly between 0 and 1. [If y=0 the you have the log(0) and if y=1 then you get the log(1/0) which is ∞].   Additionally, using this framework, it is difficult to recover E(y|x).  Under the model specified above:

  • E(y|x)=∫ {exp(xβ+ν)/[1+exp(xβ+ν)]} * f(ν|x)dν

If the residuals are independent of the explanatory variables (i.e., ν⊥x), one can use Duan’s (1983) smearing technique to estimate f(•).   If not, one must make functional form assumptions regarding the distribution of the error terms.

QUASI-LIKELIHOOD METHODS

Papke and Wooldridge support using quasi-likelihood methods. Assume the following relationship:

  • E(y|x) = G(xβ)

where 0<1 for all z∈ℜ. The most popular choice for G(z) is the logistic function where G(z)=exp(z)/[1+exp(z)]. In this model, one can estimate the parameters β using the following Brenoulli log-likelihood function:

  • li(β) ≡ yilog[G(xiβ)] + (1-yi)log[1-G(xiβ)]

This method has several advantages.  First, it is fairly easy to estimate.  Secondly, the equation above is a member of the linear exponential family thus the quasi MLE method will produce a consistent estimator of β where β is normally distributed.  Assuming a logit function for G(z) produces the following variance:

  • Var(yi|xi) = σ2 * G(xiβ)[1-G(xiβ)]

The Papke and Wooldridge (1996) also describe how to compute the asymptotic variance of the estimator β.

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Statistical Power

August 3, 2010 in Econometrics | No comments

What is power?  Merriam Webster defines power as the “possession of control, authority, or influence over others.”  The power I will talk about today, however, is statistical power.  Statistical power measures the ability of a statistical test to determine whether the null hypothesis is false.  For instance, in the U.S. judicial system, the null hypothesis is that the defendant is innocent.  Trials that can more accurately determine when the defendant is in fact guilty have more power.

In statistics, there are two types of errors: Type I and Type II. The probability of a Type II error, a false negative, is represented by the symbol β.  Thus, the probability of correctly rejecting the null (i.e., the power) is 1-β.

The larger the magnitude of the hypothesized effect, the higher the power.  It is much easier to detect a large effect than a small effect.  Also, as the size of the sample increases, so does a test’s statistical power.

The more variation that exists in the data, however, the lower the power.  If there is a lot of variation in the data, it is difficult to determine if null hypothesis is false or if observing a phenomenon that contradicts the null is simply due to the excessive amount of variability in the data.  On the hand, if the variability (i.e., standard deviation) is low, then one can generally conclude that that the null hypothesis is false, since the low variability indicates that the anomaly is not caused by normal variation in the data.

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