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Add to Your Skills Toolkit: Bootstrapping Confidence Intervals

March 6, 2012 in Econometrics | No comments

In previous posts, I have explained how to create bootstrap estimates for a variety of statistics.  Doing so is fairly simple and involves a 3 step procedure:

  • Step 1: Using the observe data, create m boostrap data sets by using  random resampling with replacement.
  • Step 2: Calculate the statistic of interest for each bootstrap data set.
  • Step 3: The bootstrap estimate of the statistic of interest is the average value from Step 2 across all bootstrap samples.

One question that has not yet been answered is how to calculate the confidence interval for the statistic of interest.  A paper by Haukoos and Lewis describes five methods for computing bootstrap confidence intervals: i) normal approximation, ii) percentile, iii) bias-corrected(BC), iv) bias-corrected and accelerated (BCa) and v) approximate bootstrap confidence (ABC) methods.

The normal approximation method is calculated as follows:

  • original statistic +/- Z* (standard Error)

For instance, for a 95% confidence interval, Z=1.96.  Another alternative is to use the percentile method.  To calculate, the percentile confidence intervals for a 95% CI, one simple takes calculates the 2.5 and 97.5 percentiles for the distribution of statistics calculated in Step 2 of the bootstrap procedures.

The other bootstrap CI methods are a bit more complex.

The BCa method adjusts for bias in the bootstrapped sampling distributions relative to the actual sampling distribution, and is thus considered a substantial improvement over the percentile method. The BCa confidence interval is an adjustment of the percentiles used in the percentile method based upon the calculation of two coefficients
called ‘‘bias correction’’ and ‘‘acceleration.’’ The bias correction coefficient adjusts for the skewness in the bootstrap sampling distribution. If the bootstrap sampling distribution is perfectly symmetric, then the bias correction will be zero. The acceleration coefficient adjusts for nonconstant variances within the resampled data sets. The ABC method is an approximation of the BCa method that requires fewer resampled data sets than the BCa method.

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Statistics from the Medicare Trustees Report

July 19, 2011 in HC Statistics, Medicare | 1 comment

Basic Statistics
In 2010, 47.5 million people were covered by Medicare: 39.6 million aged 65 and older, and 7.9 million disabled. About 25 percent of beneficiaries have chosen to enroll in Part C private health plans that contract with Medicare to provide Part A and Part B health services. Total benefits paid in 2010 were $516 billion. Income was $486 billion, expenditures were $523 billion, and assets held in special issue U.S. Treasury securities were $344 billion.

Total Medicare expenditures have risen by 9.0 percent per year since 2000. Enrollment growth during that time was only 1.8 percent per year; thus, a per capita growth rate of 7.0 percent is driving overall Medicare growth. For more statistics, see here.

Fiscal Solvency
The financial status of the HI [Hospital Insurance] trust fund was substantially improved by the lower expenditures and additional tax revenues instituted by the Affordable Care Act. However, the HI trust fund is now estimated to be exhausted in 2024, 5 years earlier than was shown in last year’s report, and the fund is not adequately financed over the next 10 years.

Office of the Actuary (OACT) Forecasts
Medicare expenditures represented 3.6 percent of GDP in 2010. Under current law, costs would increase to about 5.6 percent of GDP by 2035 under the intermediate assumptions and to 6.2 percent of GDP by the end of the 75-year period. However, it is important to note that Medicare expenditures are almost certainly understated because of unrealistic substantial reductions in physician payments scheduled under current law and may be further understated (and to a greater degree).

Source:

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Biases

June 28, 2011 in Econometrics | No comments

All economists are familiar with the problem of selection bias.  In non-randomized samples, patients may choose to be in either the treatment or control group based on factors which are also related to the outcome of interest.  Even if researchers can design a study that fully controls for selection bias, robust studies must also account for other biases.  These include:

  • Recall bias: Patients in one group have better or worse memory of a given event.  If one wishes to compare changes in income for individual who received certain workforce training, individuals who participated in the program may be more or less likely to inflate their income levels over time.
  • Interviewer bias: If new data is being collected and researchers use separate interviewers for the treatment and control groups, if one interviewer systematically over/understates the interviewee responses, the study results will be biased.
  • Observation bias: This problem is particularly problematic for medical studies.  Observation bias occurs when physicians (or patients) are more likely to detect a disease.  Thus, a study identifying how pollution affected disease rates may underestimate the impact of the pollution if those affected are less likely to detect any disease than those who are not.  For instance, if poor individuals are more likely to drink polluted water than rich individuals, but also less likely to go to the doctor, the disease incidence from polluted water would be underreported and the causal impact of water pollution would be underestimated.

Outside of purely statistical biases, the research community at large may suffer from other biases as well.  These include:

  • Funding bias: Researcher bias towards interpreting quantitative results in favor of the entity which funded their study.
  • Status quo bias: Survey respondents may base their opinions closer to the status quo or researchers can interpret their results in a fashion more likely to coincide with the existing academic literature.
  • Publication Bias: tendency of researchers, editors, and pharmaceutical companies to handle the reporting of experimental results that are positive (i.e. showing a significant finding) differently from results that are negative (i.e. supporting the null hypothesis) or inconclusive, leading to bias in the overall published literature.
  • Hindsight bias: is the inclination to see events that have already occurred as being more predictable than they were before they took place

 

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Medicare Dashboards

June 24, 2011 in HC Statistics, Medicare | 2 comments

For which drugs does Medicare spend the most money?  For which inpatient hospital treatments does Medicare have the highest expenses.  CMS’s new Dashboards provide an easy to use source to access these high level summary statistics.  You can find this information here:

For instance, some results from the Prescription Drug Benefit Dashboard include:

Drug Class 2008 Drug Cost ($)
ANTIHYPERLIPIDEMICS $6,165,831,884
ANTIPSYCHOTICS/ANTIMANIC AGENTS $5,698,011,103
ANTIDIABETICS $4,688,777,238
ULCER DRUGS $4,411,792,980
ANTIHYPERTENSIVES $4,177,531,157
ANTIASTHMATIC AND BRONCHODILATOR AGENTS $3,598,966,726
ANTICONVULSANTS $3,288,116,849
PSYCHOTHERAPEUTIC AND NEUROLOGICAL AGENTS – MISC. $3,112,790,209
ANTIDEPRESSANTS $2,835,474,451
ANALGESICS – OPIOID $2,578,161,329

 

Drug Class 2008 Drug Cost ($)
LIPITOR $2,397,843,000
PLAVIX $2,305,145,585
NEXIUM $1,487,052,730
SEROQUEL $1,462,338,499
ARICEPT $1,326,144,339
ZYPREXA $1,229,061,198
ADVAIR DISKUS $1,213,298,009
ACTOS $1,062,975,107
PREVACID $848,394,558
ABILIFY $837,090,968

More publicly available government statistics can also be found at Data.gov.

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Share of Federal Budget Spent on Health Care Jumped in 2009

June 20, 2011 in HC Statistics | 1 comment

In 2008, 38 percent of the federal government’s revenue was spent on health care.  In 2009, however, this figure jumped to 54 percent of total revenues.  Although federal health spending only increased by 17.9%, a decline in revenues of a similar magnitude caused this large change.  Surprisingly, state and local spending on healthcare barely budged.  In 2008, state an local spending was 26 percent of total revenues, and this figure only inched up to 27 percent in 2009.  In 2009, households still contributed 6% of their income (just like in 2008) and business’s health care expense remained constant at 8 percent of cost in 2009.

Other highlights from the California Health Care Foundation’s 2011 edition of Healthcare 101 include:

  • Health spending grew 4.0% in 2009, an all-time low, and the smallest annual increase on record.
  • While health spending by private insurers only grew 1.3% in 2009, Medicare spending grew by 7.9% and Medicaid by 9.0%.
  • Households contribute the largest share to the financing of health care (28%) followed closely by the federal government (27%).
  • Spending on home health care (10.0%) grew the fastest, while spending on the capital-intensive category, structures and equipment, declined (– 2.7%).
  • In 2009, spending growth on prescription drugs rose for the first time since 2006, to 5.3%.
  • Hospital care (31%) and physician and clinical services (20%) account for slightly more than half of all health spending.

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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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    Health Care Spending Distribution in 2008

    July 14, 2010 in HC Statistics | 1 comment

    National Health Expenditures reached $2.3 trillion, or $7,681 per person. This means that health care services made up 16.2% of the economy. Where did these dollars go? The CBO summarizes where America spends these funds.

    Source: CBO Long Term Budget Outlook, June 2010.

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    Homegrown Terror

    May 11, 2010 in Economics - General | 1 comment

    On 60 Minutes this week, I saw a piece called “Uncovering the Roots of Homegrown Terrorism” which documents the rise in the number of American citizens who are receiving training in Pakistan for Terrorists operations.  Many of these ‘homegrown terrorists’ are ethnically Pakistani, though certainly not all.  The story also documents a similar problem with some young Somalis in Minneapolis.

    This leads to the natural (often unasked) question: What are the odds that the Pakistani-looking individual walking down the street is in fact a suicide bomber or terrorist.  The answer, is very low.

    To illustrate, let’s assume that 70% of all terrorists are of Pakistani decent.  If there are 1000 American-born terrorists currently on U.S. soil, then this implies that there are 700 Pakistani-American terrorists.  [I made up the 1000 figure.  This figure might be too low, especially if we define a terrorists as a young men with anti-establishment beliefs; if one were to apply this definition, however, it would include almost every teenager in the country.  On the other hand, I doubt that there are 1000 American citizens who would actually blow themselves up to prove a point, despite the number of people on the internet claiming they would do so.] Since there are about 210,000 Pakistani-Americans in the U.S., the chance than any one of them is a terrorist is only 0.3% (700/210,000). This means that if you see 100 Pakistanis in a day, the chances even one of them is a terrorist is very low.

    To make a comparison, the incarceration rate in the U.S. is 0.7%.  In fact, 3.2% of all U.S. adult residents (or 1 in every 31 adults) will be on probation, in jail or prison, or on parole at year-end.  Thus, it is more likely that you know an American-born criminal than a Pakistani-American terrorist.

    While terrorism is a serious concern to American security, the chances the Pakistani man sitting next to you on the plane is a terrorist is in fact extremely low.

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    Health Spending Climbs to 16.2% of GDP

    January 25, 2010 in HC Statistics | 5 comments

    From the CMS Office of the Actuary:

    U.S. health care spending growth decelerated in 2008, increasing 4.4 percent compared to 6.0 percent in 2007, as spending growth slowed for nearly all health care goods and services, particularly for hospitals. Health spending growth for state and local and private sources of funds also slowed while federal health spending growth accelerated in 2008. Total health expenditures reached $2.3 trillion in 2008, which translates to $7,681 per person and 16.2 percent of the nation’s Gross Domestic Product (GDP). Despite slower growth in overall health expenditures, the share of GDP devoted to health care increased from 15.9 percent in 2007.

    A detailed table of health expenditures by service type (e.g., hospital, physician services) can be found here.

    Also interesting is the changes in spending by payor.  While overall Medicaid expenditure growth decelerated between 2007 and 2008, the composition of expenditures changed significantly.  Federal Medicaid spending increased by 8.4%, but state Medicaid spending actually decrease by 0.1%.  A table providing more information on the changes health expenditures by payer is available here.

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