Recent Advances in Methods for Quantiles Matteo Bottai, Sc.D. Many Thanks to Advisees
Collaborators
Andrew Ortaglia Huiling Zhen Joe Holbrook Junlong Wu Li Zhou Marco Geraci Nicola Orsini Paolo Frumento Yuan Liu Ane Johannessen Bo Cai Jiajia Zheng Jonathan Mitchell Lorenzo Maragoni Monica Chiogna Nicola Salvati Nikos Tzavidis Renee Gardner Robert McKeown International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 2 Example 1: Do We Think Percentiles? International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 3 Example 2: What’s in a Mean? Drug
0
5
10
15
Log(cholesterol)
Placebo
20 0
5
10
15
Log(cholesterol)
Log(cholesterol) t‐test p‐value = 0.0000 Cholesterol t‐test p‐value = 0.8869 Mean log(cholesterol) is significantly smaller on drug than on placebo. We detect no significant difference for mean cholesterol. International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 20
4 Example 3: Logistic Quantile Regression for Bounded Outcomes The Center for Epidemiologic Studies depression scale ranges from 0 to 60. Shape and location of its distribution change across family cohesion groups. CES - Depression Scale
60
40
20
0
16-
41-
49-
56-
62-
68-
International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 5 Example 3: Logistic Quantile Regression for Bounded Outcomes We apply a logit transform (Bottai et al 2010). 60
CES - Depression Scale
95%
75%
40
50%
25%
20
5%
0
10
20
30
40
50
Family Cohesion
60
International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 70
80
6 Example 4: Logistic Quantile Regression for Bounded Outcomes (Miniati et al 2010) International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 7 Example 5: Quantiles Can Be Robust to Outliers 0
20
40
60
Body Mass Index (kg/m-squared)
80
Age
P‐value
Median Regression
All sample
14<BMI<60
n = 90,244 n = 90,232 0.802
0.801
0.000
0.000
Linear Regression
All sample 14<BMI<60 n = 90,244 n = 90,232 0.316
0.257
0.367
0.035
Removing outliers affects the inference on the mean not on the median. International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 8 Example 6: Quantiles Can Be Efficient Bootstrap standard errors for mean and quantiles of IgE levels. Standard Error
Ratio
Mean 13.00 –
0.01 quantile
0.53
24.5
0.10 quantile
0.64
20.3
0.50 quantile
3.04
4.3
0.75 quantile
9.57
1.4
0.90 quantile
34.00 0.4
The 0.1‐quantile with 100 observations has the same power as the mean with 41,200. International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 9 Asymptotic Relative Efficiency of Mean vs. Quantiles Quantile 0.10
0.20
0.30
0.40 0.50
0.60
0.70
0.80
0.90 Normal(0,1) 0.12
0.31
0.48
0.60 0.64
0.60
0.48
0.31
0.12 Uniform(0,1) 0.33
0.33
0.33
0.33 0.33
0.33
0.33
0.33
0.33 t‐Student(3) 0.13
0.54
1.05
1.47 1.62
1.47
1.05
0.54
0.13 Exponential(1) 3.24
2.56
1.96
1.44 1.00
0.64
0.36
0.16
0.04 79.37 18.60
7.39
3.52 1.78
0.89
0.40
0.15
0.03 Chi‐square(1) Chi‐square(4) 0.98
1.31
1.34
1.21 0.98
0.72
0.45
0.22
0.06 Lognormal(0,1) 7.47
7.88
6.45
4.63 2.97
1.68
0.79
0.27
0.04 International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 10 The Asymmetric Laplace Distribution The standard asymmetric Laplace variable has probability density function ;
1
exp
The parameter 0,1 drives the asymmetry and 0
. The variable has The likelihood maximizer is the least‐weighted‐absolute‐residual estimator ̂
argmax
The parameters may depend on covariates, and International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden . 11 Linear Quantile Mixed Effects Models Consider the regression model ~
~
is an assumed multivariate zero‐location distribution. |
,
International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 12 Example 7: Linear Quantile Mixed Effects Model with Longitudinal Data (Liu and Bottai 2009) International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 13 Important Related Work This is an incomplete list of related publications in the last few years. Distribution‐free methods Fixed effects Koenker 2004; Lamarche 2010; Galvao and Montes‐Rojas 2010; Galvao 2011 Weighted methods Lipsitz et al. 1997; Karlsson 2008; Fu and Wang 2012 Likelihood‐based approaches Asymmetric Laplace Geraci and Bottai 2007, 2013; Liu and Bottai 2009; Yuan and Yin 2010; Lee and Neocleous 2010; Farcomeni 2012 Bayesian approaches Yu and Moyeed 2001; Yuan and Yin 2010; Reich et al. 2010, 2011; Yu et al (2013) Other approaches Canay 2011; Rigby and Stasinopoulos 2005 International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 14 Example 8: Survival in Metastatic Renal Carcinoma “There was a 28% reduction in the risk of death (hazard ratio 0.72).” The Lancet 1999 International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 15 Example 9: Laplace Regression with Censored Data Percentiles of survival Surgery
Therapy
0
2
4
6
8
10
12
14
Years
5%
25%
50%
75%
95%
Patients respond to surgery differently. The frailest 5% die within 6 months but the strongest 25% live at least 9.5 years.
International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 16 Laplace Regression with Censored Data Consider the regression model ~
We observe min
,
and
If 1 then the model is equal to ordinary quantile regression. The maximum likelihood estimator for is biased but has small mean squared error. (Bottai and Zhang 2010) International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 17 Important Related Work Early works Uncensored data Koenker and Geling 2001 Fixed censoring Powell 1986 Unconditionally independent censoring Semi‐parametric estimation Ying et al 1995 Estimating equations Bang and Tsiatis 2002 Random Censoring Global linearity assumption Portnoy 2003; Peng and Huang 2008 Nonparametric smoothing Wang and Wang 2009 Laplace regression Bottai and Zhang 2010 Applications of Laplace Regression Orsini et al 2012; Rizzuto et al 2012; Gigante 2012; Johanssen 2013 International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 18 Summary Quantiles − can be of research interest − provide a full description of continuous distributions − are invariant to monotone transformations − may be efficient and robust to outliers The asymmetric Laplace distribution offers a convenient likelihood framework. Current research − Laplace regression with frailty terms, competing risks − Flexible Laplace regression − Missing data imputation − Robust meta‐analysis − Computation aspects More information at www.imm.ki.se/biostatistics International Workshop, University of Padova, March 21‐23, 2013 – Matteo Bottai, Sc.D., Karolinska Institutet, Stockholm, Sweden 19