Thinking Beyond the Mean: a Practical Guide for Using Quantile Regression Methods for Health Services Research

Total Page:16

File Type:pdf, Size:1020Kb

Thinking Beyond the Mean: a Practical Guide for Using Quantile Regression Methods for Health Services Research Thinking beyond the mean: a practical guide for using quantile regression methods for health services research The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Lê Cook, Benjamin, and Willard G. Manning. 2013. “Thinking beyond the mean: a practical guide for using quantile regression methods for health services research.” Shanghai Archives of Psychiatry 25 (1): 55-59. doi:10.3969/j.issn.1002-0829.2013.01.011. http:// dx.doi.org/10.3969/j.issn.1002-0829.2013.01.011. Published Version doi:10.3969/j.issn.1002-0829.2013.01.011 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:12406692 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Shanghai Archives of Psychiatry, 2013, Vol.25, No.1 ·55· • Biostatistics in psychiatry (13) • Thinking beyond the mean: a practical guide for using quantile regression methods for health services research Benjamin Lê COOK1*, Willard G. MANNING2 1. Introduction moderate drinkers; that is, higher taxes did not reduce consumption nearly as much for light and heavy drinkers Health services and health economics research ar- as it did for moderate drinkers. The policy implication ticles commonly use multivariate regression techniques is that increasing alcohol taxes might bring in revenue to measure the relationship of health service utilization (and reduce alcohol-related accidents among moderate and health outcomes (the outcomes of interest) with drinkers) but will have limited success in reducing the clinical characteristics, sociodemographic factors, and prevalence of heavy drinking and its sequelae. policy changes (usually treated as explanatory variables). Common regression methods measure differences in Another example is that associations of interest outcome variables between populations at the mean explaining health care and health outcomes may be (i.e., ordinary least squares regression), or a population very different among the highest utilizers of health average effect (i.e., logistic regression models), after care, compared to individuals at the bottom or middle adjustment for other explanatory variables of interest. of the distribution of health care utilization. As a simple These are often done assuming that the regression illustration, Figure 1 plots the relationship between coefficients are constant across the population – in the number of hours attended of a hypothetical other words, the relationships between the outcomes psychotherapy intervention (x-axis) and a fictitious of interest and the explanatory variables remain the scale of post-intervention mental health (higher score same across different values of the variables. There are indicates better mental health on the y-axis) for a group times, however, when researchers, policymakers, and of 400 individuals. In this example, the regression line clinicians may be interested in group differences across from an ordinary least squares (OLS) regression model the distribution of a given dependent variable rather is essentially flat, suggesting that there is no relationship than only at the mean. between number of psychotherapy session-hours and mental health at follow-up. To describe the association Taking a more concrete example from the literature, between number of session-hours and mental health for research on individuals’ consumption of alcohol individuals with low and high post-treatment scores on consistently reported that higher alcohol prices were the mental health scale using OLS, the analyst extends [1] associated with lower alcohol consumption. This the line up or down to the 90th and 10th quantiles led to a call for increases in taxes as a policy lever to in a parallel fashion, as the OLS model assumes the reduce alcohol consumption and the subsequent social association between hours of psychotherapy and mental costs of alcoholism and alcohol abuse. However, these health outcome remains the same at different levels of studies did not provide any information about whether the mental health scale. increased price decreased alcohol use similarly for light drinkers, moderate drinkers, and heavy drinkers. Because In contrast, in Figure 2, we use quantile regression there are positive social benefits for light drinkers and to allow slopes of the regression line to vary across negative health and social consequences for heavy quantiles of the mental health scale. Although the me- drinkers, analyzing the demand response of different dian line is flat as before, the 90th quantile prediction types of drinkers was important to understanding line is significantly increasing whereas the 10th quantile who was most likely to modify their behavior due to prediction line is significantly decreasing. This suggests increasing alcohol taxes. A subsequent study[2] found that the association between the hypothetical inter- light and heavy drinkers were much less price elastic than vention and post-intervention mental health is positive doi: 10.3969/j.issn.1002-0829.2013.01.011 1Center for Multicultural Mental Health Research, Cambridge Health Alliance/Harvard Medical School, Boston, MA, United States 2Harris School of Public Policy Studies, University of Chicago, Chicago, IL, United States *correspondence: [email protected] ·56· Shanghai Archives of Psychiatry, 2013, Vol.25, No.1 for those with better post-intervention mental health but provides greater flexibility than other regression there is a negative association among those with poorer methods to identify differing relationships at different post-intervention mental health. Quantile regression parts of the distribution of the dependent variable. Figure 1. Figure 2. Prediction lines at 10th quantile, mean, and 90th Prediction lines at 10th quantile, mean, and 90th quantile using ordinary least squares (OLS) regression quantile using quantile regression Same slopes - Parallel quantiles Differing slopes - Not parallel quantiles 90th quantile prediction line 90th quantile prediction line Mental health scale score health scale Mental score health scale Mental 10th quantile prediction line 10th quantile prediction line Number of hours of psychotheraphy Number of hours of psychotheraphy Health care expenditures are another area impor- same at all levels. Quantile regression is not a regression tant to policy that is amenable to an analytical strategy estimated on a quantile, or subsample of data as the that measures differences across the distribution. The name may suggest. Quantile methods allow the analyst average user of health care is obviously very different to relax the common regression slope assumption. In from the heavy user in terms of health status, but what OLS regression, the goal is to minimize the distances about other factors such as race/ethnicity, gender, between the values predicted by the regression line and employment, insurance status and other factors of the observed values. In contrast, quantile regression policy interest? Quantile regression allows for analysis differentially weights the distances between the values of these other differences that exist among heavy predicted by the regression line and the observed health care users in a way that is not possible with com- values, then tries to minimize the weighted distances.[4-6] monly used regression methods. Referring to Figure 2 above, estimating a 75th quantile regression fits a regression line through the data so that In previous applications, we have used quantile 90 percent of the observations are below the regression regression methods to assess racial and ethnic disparities line and 25 percent are above. Alternatively, this can be in health care expenditures and in mental health care viewed as weighting the distances between the values expenditures across different quantiles of expenditures, predicted by the regression line and the observed adjusting for covariates.[3] In the United States, dispari- values below the line (negative residuals) by 0.5, and ties in the distribution of health care expenditures weighting the distances between the values predicted between Blacks and Whites, and between Hispanics and by the regression line and the observed values above the Whites diminish in the upper quantiles of expenditure, line (positive residuals) by 1.75. Doing so ensures that but remain significant throughout the distribution. This minimization occurs when 75 percent of the residuals same pattern of persistent disparities was still evident in are negative. the highest education and income categories. 2.1 Describing differences across the distribution of 2. What is quantile regression? health care expenditures Quantile regression provides an alternative to To demonstrate basic SAS software package code ordinary least squares (OLS) regression and related for implementing descriptive statistics and quantile methods, which typically assume that associations regression, we use a sample of Black and White adults between independent and dependent variables are the Shanghai Archives of Psychiatry, 2013, Vol.25, No.1 ·57· of at least 18 years of age taken from the United 2.2 Example of quantile regression to measure racial States 2009 Medical Expenditure Panel Survey (MEPS). and ethnic differences across the distribution of This survey is conducted in a nationally representative health care expenditures sample of
Recommended publications
  • QUANTILE REGRESSION for CLIMATE DATA Dilhani Marasinghe Clemson University, [email protected]
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Clemson University: TigerPrints Clemson University TigerPrints All Theses Theses 8-2014 QUANTILE REGRESSION FOR CLIMATE DATA Dilhani Marasinghe Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_theses Part of the Statistics and Probability Commons Recommended Citation Marasinghe, Dilhani, "QUANTILE REGRESSION FOR CLIMATE DATA" (2014). All Theses. 1909. https://tigerprints.clemson.edu/all_theses/1909 This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. QUANTILE REGRESSION FOR CLIMATE DATA A Master Thesis Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE Mathematical Sciences by DILHANI SHALIKA MARASINGHE August 2014 Accepted by: Dr. Collin Gallagher, Committee Chair Dr. Christoper McMahan Dr. Robert Lund Abstract Quantile regression is a developing statistical tool which is used to explain the relationship between response and predictor variables. This thesis describes two examples of climatology using quantile re- gression. Our main goal is to estimate derivatives of a conditional mean and/or conditional quantile function. We introduce a method to handle autocorrelation in the framework of quantile regression and used it with the temperature data. Also we explain some properties of the tornado data which is non-normally distributed. Even though quantile regression provides a more comprehensive view, when talking about residuals with the normality and the constant variance assumption, we would prefer least square regression for our temperature analysis.
    [Show full text]
  • Gretl User's Guide
    Gretl User’s Guide Gnu Regression, Econometrics and Time-series Allin Cottrell Department of Economics Wake Forest university Riccardo “Jack” Lucchetti Dipartimento di Economia Università Politecnica delle Marche December, 2008 Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation (see http://www.gnu.org/licenses/fdl.html). Contents 1 Introduction 1 1.1 Features at a glance ......................................... 1 1.2 Acknowledgements ......................................... 1 1.3 Installing the programs ....................................... 2 I Running the program 4 2 Getting started 5 2.1 Let’s run a regression ........................................ 5 2.2 Estimation output .......................................... 7 2.3 The main window menus ...................................... 8 2.4 Keyboard shortcuts ......................................... 11 2.5 The gretl toolbar ........................................... 11 3 Modes of working 13 3.1 Command scripts ........................................... 13 3.2 Saving script objects ......................................... 15 3.3 The gretl console ........................................... 15 3.4 The Session concept ......................................... 16 4 Data files 19 4.1 Native format ............................................. 19 4.2 Other data file formats ....................................... 19 4.3 Binary databases ..........................................
    [Show full text]
  • Quantreg: Quantile Regression
    Package ‘quantreg’ June 6, 2021 Title Quantile Regression Description Estimation and inference methods for models of conditional quantiles: Linear and nonlinear parametric and non-parametric (total variation penalized) models for conditional quantiles of a univariate response and several methods for handling censored survival data. Portfolio selection methods based on expected shortfall risk are also now included. See Koenker (2006) <doi:10.1017/CBO9780511754098> and Koenker et al. (2017) <doi:10.1201/9781315120256>. Version 5.86 Maintainer Roger Koenker <[email protected]> Repository CRAN Depends R (>= 2.6), stats, SparseM Imports methods, graphics, Matrix, MatrixModels, conquer Suggests tripack, akima, MASS, survival, rgl, logspline, nor1mix, Formula, zoo, R.rsp License GPL (>= 2) URL https://www.r-project.org NeedsCompilation yes VignetteBuilder R.rsp Author Roger Koenker [cre, aut], Stephen Portnoy [ctb] (Contributions to Censored QR code), Pin Tian Ng [ctb] (Contributions to Sparse QR code), Blaise Melly [ctb] (Contributions to preprocessing code), Achim Zeileis [ctb] (Contributions to dynrq code essentially identical to his dynlm code), Philip Grosjean [ctb] (Contributions to nlrq code), Cleve Moler [ctb] (author of several linpack routines), Yousef Saad [ctb] (author of sparskit2), Victor Chernozhukov [ctb] (contributions to extreme value inference code), Ivan Fernandez-Val [ctb] (contributions to extreme value inference code), Brian D Ripley [trl, ctb] (Initial (2001) R port from S (to my 1 2 R topics documented: everlasting shame -- how could I have been so slow to adopt R!) and for numerous other suggestions and useful advice) Date/Publication 2021-06-06 17:10:02 UTC R topics documented: akj..............................................3 anova.rq . .5 bandwidth.rq .
    [Show full text]
  • 柏際股份有限公司 Bockytech, Inc. 9F-3,No 70, Yanping S
    柏際股份有限公司 BockyTech, Inc. 9F-3,No 70, YanPing S. Rd., Taipei 10042, Taiwan, R.O.C. 10042 台北市延平南路 70 號 9F-3 Http://www.bockytech.com.tw Tel:886-2-23618050 Fax:886-2-23619803 Version 10 LIMDEP Version 10 is an integrated program for estimation and analysis of linear and nonlinear models, with cross section, time series and panel data. LIMDEP has long been a leader in the field of econometric analysis and has provided many recent innovations including cutting edge techniques in panel data analysis, frontier and efficiency estimation and discrete choice modeling. The collection of techniques and procedures for analyzing panel data is without parallel in any other computer program available anywhere. Recognized for years as the standard software for the estimation and manipulation of discrete and limited dependent variable models, LIMDEP10 is now unsurpassed in the breadth and variety of its estimation tools. The main feature of the package is a suite of more than 100 built-in estimators for all forms of the linear regression model, and stochastic frontier, discrete choice and limited dependent variable models, including models for binary, censored, truncated, survival, count, discrete and continuous variables and a variety of sample selection models. No other program offers a wider range of single and multiple equation linear and nonlinear models. LIMDEP is a true state-of-the-art program that is used for teaching and research at thousands of universities, government agencies, research institutes, businesses and industries around the world. LIMDEP is a Complete Econometrics Package LIMDEP takes the form of an econometrics studio.
    [Show full text]
  • Unconditional Quantile Regressions*
    UNCONDITIONAL QUANTILE REGRESSIONS* Sergio Firpo Nicole M. Fortin Departamento de Economia Department of Economics Catholic University of Rio de Janeiro (PUC-Rio) University of British Columbia R. Marques de S. Vicente, 225/210F #997-1873 East Mall Rio de Janeiro, RJ, Brasil, 22453-900 Vancouver, BC, Canada V6T 1Z1 [email protected] [email protected] Thomas Lemieux Department of Economics University of British Columbia #997-1873 East Mall Vancouver, Canada, BC V6T 1Z1 [email protected] July 2007 ABSTRACT We propose a new regression method to estimate the impact of explanatory variables on quantiles of the unconditional (marginal) distribution of an outcome variable. The proposed method consists of running a regression of the (recentered) influence function (RIF) of the unconditional quantile on the explanatory variables. The influence function is a widely used tool in robust estimation that can easily be computed for each quantile of interest. We show how standard partial effects, as well as policy effects, can be estimated using our regression approach. We propose three different regression estimators based on a standard OLS regression (RIF-OLS), a logit regression (RIF-Logit), and a nonparametric logit regression (RIF-OLS). We also discuss how our approach can be generalized to other distributional statistics besides quantiles. * We are indebted to Joe Altonji, Richard Blundell, David Card, Vinicius Carrasco, Marcelo Fernandes, Chuan Goh, Joel Horowitz, Shakeeb Khan, Roger Koenker, Thierry Magnac, Whitney Newey, Geert Ridder, Jean-Marc Robin, Hal White and seminar participants at Yale University, CAEN-UFC, CEDEPLARUFMG, PUC-Rio, IPEA-RJ and Econometrics in Rio 2006 for useful comments on this and earlier versions of the manuscript.
    [Show full text]
  • Stochastic Dominance Via Quantile Regression
    Abstract We derive a new way to test for stochastic dominance between the return of two assets using a quantile regression formulation. The test statistic is a variant of the one-sided Kolmogorov-Smirnoffstatistic and has a limiting distribution of the standard Brownian bridge. We also illustrate how the test statistic can be extended to test for stochastic dominance among k assets. This is useful when comparing the performance of individual assets in a portfolio against some market index. We show how the test statistic can be modified to test for stochastic dominance up to the α-quantile in situation where the return of one asset does not dominate another over the whole spectrum of the return distribution. Keywords: Quantile regression, stochastic dominance, Brownian bridge, test statis- tic. 1Introduction Stochastic dominance finds applications in many areas. In finance, it is used to assess portfolio diversification, capital structure, bankruptcy risk, and option’s price bound. In welfare economics, it is used to measure income distribution and income inequality In reinsurance coverage, the insured use it to select the best coverage option while the insurers use it to assess whether the options are consistently priced. It is also used to select effective treatment in medicine and selection of the best irrigation system in agriculture. There are two big classes of stochastic dominance tests. The first is based on the inf / sup statistics over the support of the distributions as in McFadden (1989), Klecan, McFadden and McFadden (1991), and Kaur, Rao and Singh (1994). The second class is based on comparison of the distributions over a set of grid points as in Anderson (1996), Dardanoni and Forcina (1998, 1999), and Davidson and Duclos (2000).
    [Show full text]
  • A New Quantile Regression for Modeling Bounded Data Under a Unit Birnbaum–Saunders Distribution with Applications in Medicine and Politics
    S S symmetry Article A New Quantile Regression for Modeling Bounded Data under a Unit Birnbaum–Saunders Distribution with Applications in Medicine and Politics Josmar Mazucheli 1 , Víctor Leiva 2,* , Bruna Alves 1 and André F. B. Menezes 3 1 Departament of Statistics, Universidade Estadual de Maringá, Maringá 87020-900, Brazil; [email protected] (J.M.); [email protected] (B.A.) 2 School of Industrial Engineering, Pontificia Universidad Católica de Valparaíso, Valparaíso 2362807, Chile 3 Departament of Statistics, Universidade Estadual de Campinas, Campinas 13083-970, Brazil; [email protected] * Correspondence: [email protected] or [email protected] Abstract: Quantile regression provides a framework for modeling the relationship between a response variable and covariates using the quantile function. This work proposes a regression model for continuous variables bounded to the unit interval based on the unit Birnbaum–Saunders distribution as an alternative to the existing quantile regression models. By parameterizing the unit Birnbaum–Saunders distribution in terms of its quantile function allows us to model the effect of covariates across the entire response distribution, rather than only at the mean. Our proposal, especially useful for modeling quantiles using covariates, in general outperforms the other competing models available in the literature. These findings are supported by Monte Carlo simulations and applications using two real data sets. An R package, including parameter estimation, model checking Citation: Mazucheli, J.; Leiva, V.; as well as density, cumulative distribution, quantile and random number generating functions of the Alves, B.; Menezes, A.F.B. A New unit Birnbaum–Saunders distribution was developed and can be readily used to assess the suitability Quantile Regression for Modeling of our proposal.
    [Show full text]
  • Quantile Regressionregression
    QuantileQuantile RegressionRegression By Luyang Fu, Ph. D., FCAS, State Auto Insurance Company Cheng-sheng Peter Wu, FCAS, ASA, MAAA, Deloitte Consulting AgendaAgenda Overview of Predictive Modeling for P&C Applications Quantile Regression: Background and Theory Two Commercial Line Case Studies – Claim Severity Modeling and Loss Ratio Modeling Q&A Overview of Predictive Modeling for P&C Applications OverviewOverview ofof PMPM forfor P&CP&C ApplicationsApplications Modeling Techniques: ◦ Ordinary Least Square and Linear Regression: Normal distribution assumption Linear relationship between target and covariates Predict the mean of the target against the covariates ◦ GLM: Expansion of distribution assumptions with exponential family distributions,: frequency - Poisson, severity - Gamma, pure premium and loss ratio - Tweedie Linear relationship between the mean of the target and the covariates through the link function ◦ Minimum Bias/General Iteration Algorithm: Iterative algorithm Essentially derives the same results as GLM Categorical predictive variables only Linear relationship between target and covariates ◦ Neural Networks: ◦ Non-linear regression: a series of logit functions to approximate the nonlinear relationship between target and covariates ◦ Originated in the data mining field; a curve fitting technique; non parametric assumptions ◦ Different cost or error functions can be selected for minimization, for example, a least square error term or an absolute error function ◦ One popular algorithm is the backward propagation
    [Show full text]
  • Notes on Median and Quantile Regression
    Notes On Median and Quantile Regression James L. Powell Department of Economics University of California, Berkeley Conditional Median Restrictions and Least Absolute Deviations It is well-known that the expected value of a random variable Y minimizes the expected squared deviation between Y and a constant; that is, µ E[Y ] Y ≡ =argminE(Y c)2, c − assuming E Y 2 is finite. (In fact, it is only necessary to assume E Y is finite, if the minimand is normalized by|| subtracting|| Y 2, i.e., || || µ E[Y ] Y ≡ =argminE[(Y c)2 Y 2] c − − =argmin[c2 2cE[Y ]], c − and this normalization has no effect on the solution if the stronger condition holds.) It is less well-known that a median of Y, defined as any number ηY for which 1 Pr Y η and { ≤ Y } ≥ 2 1 Pr Y η , { ≥ Y } ≥ 2 minimizes an expected absolute deviation criterion, η =argminE[ Y c Y ], Y c | − | − | | though the solution to this minimization problem need not be unique. When the c.d.f. FY is strictly increasing everywhere (i.e., Y is continuously distributed with positive density), then uniqueness is not an issue, and 1 ηY = FY− (1/2). In this case, the first-order condition for minimization of E[ Y c Y ] is | − | − | | 0= E[sgn(Y c)], − − for sgn(u) the “sign” (or “signum”) function sgn(u) 1 2 1 u<0 , ≡ − · { } here defined to be right-continuous. 1 Thus, just as least squares (LS) estimation is the natural generalization of the sample mean to estimation of regression coefficients, least absolute deviations (LAD) estimation is the generalization of the sample median to the linear regression context.
    [Show full text]
  • The Use of Logistic Regression and Quantile Regression in Medical Statistics
    The use of logistic regression and quantile regression in medical statistics Solveig Fosdal Master of Science Submission date: June 2017 Supervisor: Ingelin Steinsland, IMF Co-supervisor: Håkon Gjessing, Nasjonalt folkehelseinstitutt Norwegian University of Science and Technology Department of Mathematical Sciences Summary The main goal of this thesis is to compare and illustrate the use of logistic regression and quantile regression on continuous outcome variables. In medical statistics, logistic regres- sion is frequently applied to continuous outcomes by defining a cut-off value, whereas quantile regression can be applied directly to quantiles of the outcome distribution. The two approaches appear different, but are closely related. An approximate relation between the quantile effect and the log-odds ratio is derived. Practical examples and illustrations are shown through a case study concerning the effect of maternal smoking during preg- nancy and mother’s age on birth weight, where low birth weight is of special interest. Both maternal smoking during pregnancy and mother’s age are found to have a significant ef- fect on birth weight, and the effect of maternal smoking is found to have a slightly larger negative effect on low birth weight than for other quantiles. Trend in birth weight over years is also studied as a part of the case study. Further, the two approaches are tested on simulated data from known probability density functions, pdfs. We consider a population consisting of two groups, where one of the groups is exposed to a factor, and the effect of exposure is of interest. By this we illustrate the quantile effect and the odds ratio for several examples of location, scale and location-scale shift of the normal distribution and the Student t-distribution.
    [Show full text]
  • Chapter Quantile Regression
    A Handbook of Statistical Analyses Using R — 3rd Edition Torsten Hothorn and Brian S. Everitt CHAPTER 12 Quantile Regression: Head Circumference for Age 12.1 Introduction 12.2 Quantile Regression 12.3 Analysis Using R We begin with a graphical inspection of the influence of age on head circumfer- ence by means of a scatterplot. Plotting all pairs of age and head circumference in one panel gives more weight to the teens and 20s, so we produce one plot for younger boys between two and nine years old and one additional plot for boys older than nine years (or > 108 months, to be precise). The cut function is very convenient for constructing a factor representing these two groups R> summary(db) head age Min. :33.5 Min. : 0.03 1st Qu.:48.8 1st Qu.: 1.75 Median :53.0 Median : 9.99 Mean :51.7 Mean : 8.94 3rd Qu.:55.7 3rd Qu.:14.84 Max. :66.3 Max. :21.68 R> db$cut <- cut(db$age, breaks = c(2, 9, 23), + labels = c("2-9 yrs", "9-23 yrs")) which can then be used as a conditioning variable for conditional scatterplots produced with the xyplot function (Sarkar, 2014, package lattice). Because we draw 5101 points in total, we use transparent shading (via rgb(.1, .1, .1, .1)) in order to obtain a clearer picture for the more populated areas in the plot. Figure 12.1, as expected, shows that head circumference increases with age. It also shows that there is considerable variation and also quite a number of extremely large or small head circumferences in the respective age cohorts.
    [Show full text]
  • Logistic Quantile Regression to Evaluate Bounded Outcomes
    Logistic Quantile Regression to Evaluate Bounded Outcomes Vivi Wong Kandidatuppsats i matematisk statistik Bachelor Thesis in Mathematical Statistics Kandidatuppsats 2018:16 Matematisk statistik Juni 2018 www.math.su.se Matematisk statistik Matematiska institutionen Stockholms universitet 106 91 Stockholm Matematiska institutionen Mathematical Statistics Stockholm University Bachelor Thesis 2018:16 http://www.math.su.se Logistic Quantile Regression to Evaluate Bounded Outcomes Vivi Wong∗ June 2018 Abstract Lower urinary tract symptoms in men are common when men get older, and these symptoms can be measured with I-PSS (International Prostate Symptom Score), a scale between 0-35. The density function of the bounded outcome variable, I-PSS, is highly skewed to the right. It can therefore be difficult to analyze this type of variables with stan- dard regression methods such as OLS, since these methods give us the effect of the explanatory variables on the mean of the response variable. Epidemiological studies commonly study how lifestyle and several other factors affect health-related problems. We will therefore study the effect physical activity has on lower urinary tract symptoms by using logistic quantile regression, which is an appropriate method to use when we have bounded outcomes. The method works well because instead of the mean, it focuses on quantiles and it takes the bounded interval into account. The results show a negative relationship between total physical activity and lower urinary tract symptoms, so men who are more physical active will more likely have lower and milder symptoms. ∗Postal address: Mathematical Statistics, Stockholm University, SE-106 91, Sweden. E-mail: [email protected].
    [Show full text]