School on Modelling Tools and Capacity Building in Climate and Public Health April 2013

Transcription

1 School on Modelling Tools and Capacity Building in Climate and Public Health April 2013 Expectile and Quantile Regression and Other Extensions KAZEMBE Lawrence University of Malawi Chancellor College Faculty of Science Department of Mathematical Sciences 18 Chirunga Road, 0000 Zomba MALAWI

2 Expectile and Quantile Regression and Other Extensions Lawrence Kazembe University of Namibia Windhoek, Namibia A presentation at School on Modelling Tools and Capacity Building in Climate and Public Health ICTP, Trieste, Italy 1

3 Objectives and Questions Objective: Introduce other regression beyond the mean Are there median regression models What about regression at the mode? Can we fit regression model at any other locations too? What of the variance, skewness and kurtosis? 2

4 Prelimaries Given a r.v. y f( ) then -E(Y )=μ defines the mean; -Var(Y )=σ 2 is the variance. General assumption: two measures completely define the distribution (stationarity concept) Classical regression is often characterized by relating to the mean -E(y x) =βx if x is the set of covariates. -y N(μ, σ 2 ), then μ = βx 3

5 -y Bern(π), then log ( ) π 1 π = βx -y P ois(λ), then log(λ) =βx. Statisticians are mean lovers (Friedman, Friedman & Amoo) It goes like: We are mean lovers. Deviation is considered normal. We are right 95% of the time. Statisticians do it discretely and continuously. We can legally comment on someone s posterior distribution. Why the mean? -Mean regression reduces complexity -However, the mean is not sufficient to describe a distribution.

6 Examples Plot (Rent in Munich, Kneib et al) 4

7 Examples Quantile Plot (Rent in Munich, Kneib et al) 5

8 Example (Malaria vs Altitude, Kazembe et al) 6

9 Example (Botswana data) 7

10 Motivating Examples Epidemiology and Public Health: -Investigating height, weight and body mass index as a function of different covariates e.g. age (Wei et al. 2006) -Exploring stunting curves in African and Indian children (Yue et al. 2012) -Generating age-specific centile charts (Chitty et al. 1994). Economics: -Study of determinants of wages (Koenker, 2005). 8

11 Education: -The performance of students in public schools (Koenker and Hallock, 2001). Climate data: - Spatiotemporal analysis of Boston temperature (Reich 2012)

12 Double GLM Classical regression assumes a homogeneous variance y x,ε N(βx,σ 2 ) ε N(0,σ 2 ) However variance heterogeneity (heteroscedasticity) is an order in real-life problems The variance of the response may depend on the covariates 9

13 Normal regression example: -Regression for mean and variance of a normal distribution y i = η i1 + exp(η 2i )ε i, ε i N(0, 1) such that E(y i x i )=η 1i Var(y i x i ) = exp(η 2i ) 2

14 Regression for location, scale and shape In general: Specify a distribution for the response on all parameters and relate to the predictors. The generalized additive model location, scale and shape (GAMLSS) is a statistical model developed by Rigby and Stasinopoulos (2005). For a probability (density) function f(y i μ i,σ i,ν i,τ i ) conditional on (μ i,σ i,ν i,τ i ) a vector of four distribution parameters, each of which can be a function 10

15 to the explanatory variables (X) g 1 (μ) =η 1 = β 1 X 1 + g 2 (σ) =η 2 = β 2 X 2 + g 3 (ν) =η 3 = β 3 X 3 + g 4 (τ) =η 4 = β 4 X 4 + J 1 j=1 J 2 j=1 J 3 j=1 J 4 j=1 h j1 (x j1 ) (1) h j2 (x j2 ) (2) h j3 (x j3 ) (3) h j4 (x j4 ) (4)

16 GAMLSS assumes different models for each parameter -Model 1: Mean regression -Model 2: Dispersion regression -Model 3: Skewness regression -Model 4: Kurtosis regression Other summary measures are also permissible

17 Quantile Regression Quantile, Centile, and Percentile are terms that can be used interchangeably -A 0.5 quantile 50 percentile, which is a median. Related terms are quartiles, quintiles and deciles: divides distribution into 4, 5, and 10 parts Quantiles are related to the median. Suppose Y has a cumulative distribution F (y) = P (Y τ). Then τth quantile of Y is defined to be Q(τ) = τ f(y)dy =inf{y : τ F (y)} 11

18 for 0 <τ<1. The quantile regression drops the parametric assumption for the error/ response distribution. Fit separate models for different asymmetries τ [0, 1]: y i = η iτ + ε iτ Instead of assuming E(y i η iτ = 0, one assumes or P (ε iτ 0) = τ F yi (η iτ )=P (ε iτ 0) = τ

19 This gives a set of regression function at any assumed quantile value. Assumptions: -it is distribution-free since it does not make any specific assumption on the type of errors -it does not even require i.i.d errors -it allows for heterogeneity.

20 Expectile Regression Expectiles are related to the mean, as are quantiles related to the median. ni=1 y i η i min median regression ni=1 w τ (y i,η iτ ) y i η iτ min quantile regression ni=1 y i η i 2 min n i=1 w τ (y i,η iτ ) y i η iτ 2 min mean regression expectile regression where w τ is the check function defined by 12

21 w τ = τ if y i >μ(τ) 1 τ if y i μ(τ) for some population expectile μ(τ) for different values of an asymmetric parameter 0 <τ<1. Expectiles are obtained by solving eτ τ = y e τ f y (y)dy y e τ f y (y)dy = G y (e τ ) e τ F y (e τ ) 2(G y (e τ ) e τ F y (e τ ))+(e τ μ) where -f y ( ) andf y ( ) denote the density and cumulative distribution function of y. -G y (e) = e yf y (y)dy is the partial moment function of y and -G y ( ) =μ is the expectation of y.

22 Worked example - binomial data 13

23 ALT ALT ALT s(alt, df = c(2, 4, 2)): s(alt, df = c(2, 4, 2)): s(alt, df = c(2, 4, 2)): ALT ALT ALT s(alt, df = c(2, 4, 2)): s(alt, df = c(2, 4, 2)): s(alt, df = c(2, 4, 2)):

24 TMIN TMIN TMIN s(tmin, df = c(2, 4, 2)): s(tmin, df = c(2, 4, 2)): s(tmin, df = c(2, 4, 2)): TMIN TMIN TMIN 14 s(tmin, df = c(2, 4, 2)): s(tmin, df = c(2, 4, 2)): s(tmin, df = c(2, 4, 2)):

25 TMAX TMAX TMAX s(tmax, df = c(2, 4, 2)): s(tmax, df = c(2, 4, 2)): s(tmax, df = c(2, 4, 2)): TMAX TMAX TMAX 15 s(tmax, df = c(2, 4, 2)): s(tmax, df = c(2, 4, 2)): s(tmax, df = c(2, 4, 2)):

26 Reference Hudson, I. L., Rea, A., Dalrymple, M. L., Eilers, P. H. C (2008). Climate impacts on sudden infant death syndrome: a GAMLSS approach. Proceedings of the 23rd international workshop on statistical modelling pp Beyerlein, A., Fahrmeir, L., Mansmann, U., Toschke., A. M (2008). Alternative regression models to assess increase in childhood BM. BMC Medical Research Methodology, 8(59). Smyth, G. K (1989). Generalized linear models with varying dispersion. J. R. Statist. Soc. B, 51, Sobotka, F., and T. Kneib, Geoadditive Expectile Regression. Computational Statistics and Data Analysis, doi: /j.csda