We propose a method for reducing the error of the prediction of a quantity of
interest when the outcome has missing values that are suspected to be nonignorable
and the data are correlated in space. We develop a maximum likelihood approach for
the parameter estimation of semi-parametric regressions in a mixed model framework.
We apply the proposed method to phytoplankton data collected at fixed stations in
the Chesapeake Bay, for which chlorophyll data coming from remote sensing are
available. A simulation study is also performed. The availability of a variable
correlated to the response allows us to achieve a substantial reduction of the
prediction error of the expected value of the smoother, without having to specify a
nonignorable model.
Baker SG
and
Laird NM
(1988)
Regression analysis for categorical variables with outcome subject to
nonignorable nonresponse
. Journal of the American Statistical Association83,
62-69
.
2.
Booth JG
and
Hobert JP
(1999)
Maximizing generalized linear mixed model likelihoods with an automated
Monte Carlo EM algorithm
. Journal of Royal Statistical Society B61,
265-285
.
3.
Breslow NE
and
Clayton DG
(1993)
Approximate inference in generalized linear mixed models
. Journal of the American Statistical Association88,
9-25
.
4.
Cowles MK
and
Carlin BP
(1996)
Markov chain Monte Carlo convergence diagnostics: a comparative review
. Journal of the American Statistical Association91,
883-904
.
5.
Craven P
and
Wahba G
(1979)
Smoothing noisy data with spline functions: estimating the correct degree
of smoothing by the method of generalized cross-validation
. Numerische Mathematik31,
377-403
.
6.
Desortová B
(1981)
Relationship between chlorophyll a concentration and phytoplankton biomass
in several reservoirs in Czechoslovakia
. Internationale Revue der Gesamten Hydrobiologie66,
153-169
.
7.
Felip M
and
Catalan J
(2000)
The relationship between phytoplankton biovolume and chlorophyll in a deep
oligotrophic lake: decoupling in their spatial and temporal maxima
. Journal of Plankton Research22,
91-105
.
8.
Gilks WR
,
Best NG
and
Tan KKC
(1995)
Adaptive rejection metropolis sampling within Gibbs sampling
. Applied Statistics44,
155-173
.
9.
Ibrahim JG
(1990)
Incomplete data in generalized linear models
. Journal of the American Statistical Association85,
765-769
.
10.
Ibrahim JG
,
Chen MH
and
Lipsitz SR
(1999a)
Monte Carlo EM for missing covariates in parametric regression models
. Biometrics99,
591-596
.
11.
Ibrahim JG
,
Chen MH
and
Lipsitz SR
(2001a)
Missing responses in generalized linear mixed models when the missing data
mechanism is nonignorable
. Biometrika88,
551-564
.
12.
Ibrahim JG
,
Lipsitz SR
and
Chen MH
(1999b)
Missing covariates in generalized linear models when the missing data
mechanism is nonignorableJournal of Royal Statistical Society B61,
173-190
.
13.
Ibrahim JG
,
Lipsitz SR
and
Horton N
(2001b)
Using auxiliary data for parameter estimation with non-ignorably missing outcomes
. Applied Statistics50,
361-373
.
14.
Kalchev RK
,
Beshkova MB
,
Boumbarova CS
,
Tsvetkova RL
and
Sais D
(1996)
Some allometric and non-allometric relationships between chlorophyll-a and
abundance variables of phytoplankton
. Hydrobiologia341,
235-245
.
15.
Kammann EE
and
Wand MP
(2003)
Geoadditive models
. Applied Statistics52,
1-18
.
16.
Lin X
and
Zhang D
(1999)
Inference in generalized additive mixed models by using smoothing splines
. Journal of the Royal Statistical Society B61,
381-400
.
17.
Little RJA
and
Rubin DB
(1987) Statistical analysis with missing data.
Wiley
.
18.
Nychka D
and
Saltzman N
(1998) Design of air quality monitoring networks.
In
Nychka D
,
Piegorsch WW
and
Cox LH
, editors, Case studies in environmental statistics. Volume 132, Lecture
Notes in Statistics.
Springer-Verlag
, 51-76.
19.
Nychka D
,
Haaland P
,
O’Connell M
and
Ellner S
(1998) Funfits, data analysis and statistical tools for
estimating functions. In
Nychka D
,
Piegorsch WW
and
Cox LH
, editors, Case studies in environmental statistics. Vol. 132, Lecture
Notes in Statistics.
Springer-Verlag
, 159-179.
20.
R Development Core Team
(2005) R: A language and environment for statistical
computing.
R Foundation for Statistical Computing, Vienna, Austria
.
21.
Rotnitzky A
,
Robins JM
and
Scharfstein DO
(1998)
Semiparametric regression for repeated outcomes with nonignorable nonresponse
. Journal of the American Statistical Association93,
1321-1339
.
22.
Rubin DB
(1987) Multiple imputation for nonresponse in surveys.
Wiley
.
23.
Ruppert D
,
Wand MP
and
Carroll RJ
(2003) Semiparametric regression.
Cambridge University Press
.
24.
Stubbendick AL
and
Ibrahim JG
(2003)
Maximum likelihood methods for nonignorable missing responses and
covariates in random effects models
. Biometrics59,
1140-1150
.
25.
Tolstoy A
(1979)
Chlorophyll-a in relation to phytoplankton volume in some Swedish lakes
. Archiv für hydrobiologie85,
133-151
.
26.
Vörös L
and
Padisak J
(1991)
Phytoplankton biomass and chlorophyll a in some shallow lakes in central Europe
. Hydrobiologia215,
111-119
.
27.
Wahba G
(1978)
Improper priors, spline smoothing and the problem of guarding against model
errors in regression
. Journal of the Royal Statistical Society B40,
364-372
.
28.
Zhang D
,
Lin X
,
Raz J
and
Sowers M
(1998)
Semi-parametric stochastic mixed models for longitudinal data
. Journal of the American Statistical Association93,
710-719
.
