Semiparametric Mixed Models for Increment-Averaged Data with Application to Carbon Sequestration in Agricultural Soils

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  • Semiparametric Mixed Models for Increment-Averaged Data with Application to CarbonSequestration in Agricultural SoilsAuthor(s): F. Jay Breidt, Nan-Jung Hsu and Stephen OgleSource: Journal of the American Statistical Association, Vol. 102, No. 479 (Sep., 2007), pp. 803-812Published by: American Statistical AssociationStable URL: .Accessed: 14/06/2014 07:44

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  • Semiparametric Mixed Models for

    Increment-Averaged Data With Application to Carbon Sequestration in Agricultural Soils

    F. Jay Breidt, Nan-Jung Hsu, and Stephen Ogle

    Adoption of conservation tillage practice in agriculture offers the potential to mitigate greenhouse gas emissions. Studies comparing conser

    vation tillage methods to traditional tillage pair fields under the two management systems and obtain soil core samples from each treatment.

    Cores are divided into multiple increments, and matching increments from one or more cores are aggregated and analyzed for carbon stock. These data represent not the actual value at a specific depth, but rather the total or average over a depth increment. A semiparametric mixed model is developed for such increment-averaged data. The model uses parametric fixed effects to represent covariate effects, random effects to capture correlation within studies, and an integrated smooth function to describe effects of depth. The depth function is specified as an

    additive model, estimated with penalized splines using standard mixed model software. Smoothing parameters are automatically selected

    using restricted maximum likelihood. The methodology is applied to the problem of estimating a change in carbon stock due to a change in

    tillage practice.

    KEY WORDS: Core sample; Greenhouse gas; Nonparametric regression; Ornstein-Uhlenbeck process; Penalized spline; Restricted max

    imum likelihood; Varying-coefficient model.


    Traditional agricultural management uses tillage to turn over

    the soil and bury postharvest crop residues, often several times

    before planting. Recently, "no-till" production systems that do

    not use tillage have become economically feasible due to new

    techniques and equipment. No-till, in which crop residues are

    left on the soil surface, reduces soil losses due to wind and wa

    ter erosion (Lindstrom, Schumacher, Cogo, and Blecha 1998). This in turn reduces the flow of sediments, nutrients, and pes

    ticides into surface waters. In addition, no-till enhances soil or

    ganic matter due to reduced soil disturbance (Six, Elliot, Paus

    tian, and Doran 1998) and over time may improve soil fertility. Furthermore, no-till may result in lower production costs, due

    to fewer management steps and lower machinery costs. (Con

    ventional tillage requires more expensive, higher horsepower

    tractors.) "Reduced-till" systems limit tillage and other soil

    disturbing activities and leave substantial residue on the soil

    surface, but to a lesser extent than no-till. Reduced-till systems

    offer many of the same advantages as no-till. Together, these

    systems are known as "conservation tillage" methods (Kern and

    Johnson 1993; U.S. Department of Agriculture 1994). Recent interest in conservation tillage has focused on its

    potential for reducing greenhouse gas (GHG) emissions, be cause of reduced soil disturbance that leads to more carbon

    storage in the profile, particularly in no-till systems (Kern and Johnson 1993; Paustian et al. 1997; Lai, Kimble, Fol

    lett, and Cole 1998; Smith, Powlson, Smith, Falloon, and Coleman 2000). The amount of carbon sequestered due to a

    change in tillage system is economically as well as environmen

    tally important; for example, the Chicago Climate Exchange

    F. Jay Breidt is Professor, Department of Statistics, Colorado State Uni

    versity, Fort Collins, CO 80523 (E-mail: Nan-Jung Hsu is Associate Professor, Institute of Statistics, National Tsing-Hua Univer

    sity, Hsin-Chu, Taiwan 30043 (E-mail: Stephen Ogle is Research Scientist, Natural Resource Ecology Laboratory, Colorado State

    University, Fort Collins, CO 80523 (E-mail: The work reported here was developed under STAR Research Assistance Agree ment CR-829095 awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This report has not been formally reviewed by the

    EPA, and the EPA does not endorse any products or commercial services men tioned in this report.

    ( lists agricultural soil sequestration as a means of obtaining tradable carbon credits.

    Note that there are three major biogenic GHGs (CO2, N2O, and CH4) that determine the overall net change in radiative

    forcing to the atmosphere. Few studies have considered the ef

    fect of all three GHGs (e.g., Robertson, Paul, and Harwood

    2000), an essential question because the GHGs differ greatly in their global warming potential (computed by converting kg per hectare of each gas to CO2 equivalents). In particular, N2O has about 300 times the global warming potential of CO2.

    In this article, however, we study the effect of tillage practice on emissions of CO2, with the cautionary note from the forego

    ing discussion that this is only part of the GHG story on agri cultural soils. We consider all available studies reporting differ ences in soil-mediated carbon fluxes between traditional tillage and conservation tillage systems. These studies pair fields man

    aged with traditional tillage with fields managed with conser vation tillage (or in some cases plots within fields) and track carbon storage over time. From these data, we select those stud

    ies with complete information on conservation tillage type (no till or reduced till), soil type (aquic or nonaquic), climate (wet or dry), years since management change, carbon stock under

    traditional tillage, and carbon stock under conservation tillage. Because the methods described in this article are similar for ei ther no-till or reduced till comparisons, from this point on we focus on no-till exclusively. The basic measures of interest are

    then changes in carbon stock after 1 or more years since man

    agement change from traditional tillage to no-till, with positive values indicating more carbon sequestered under no-till.

    A special challenge of these data is that they are collected from studies in which one or more soil cores are divided into

    depth increments, with matching increments across cores ag

    gregated for carbon stock analysis. There are 63 paired studies in these data, with a total of 211 increments. The increment

    averaged data are displayed in Figure 1. The upper and lower

    endpoints of these increments vary from study to study. For ex

    ample, one study may report Y\\ = total change in carbon stock

    ? 2007 American Statistical Association Journal of the American Statistical Association

    September 2007, Vol. 102, No. 479, Applications and Case Studies DOI 10.1198/016214506000001167


    This content downloaded from on Sat, 14 Jun 2014 07:44:10 AMAll use subject to JSTOR Terms and Conditions

  • 804 Journal of the American Statistical Association, September 2007

    Depth (cm)

    Figure 1. Increment-averaged carbon differences between no-till and traditional tillage versus depth. Fitted curves for wet ( ) and dry (??) climates at 20 years since management change are superimposed.

    over the increment 0-15 cm and Y12 = total change in car

    bon stock over the increment 15-30 cm. A second study may

    report only I21 = total change over the increment 0-50 cm.

    Soil scientists have used a variety of ad hoc methods to deal with this challenge. They might drop studies with nonmatching increments, or even "adjust" the Y values to make the incre

    ments match. In the foregoing example, they might form the new variables Y?

    = YU + Y\2 and Y* = (30/50)Y2u each rep

    resenting the increment 0-30 cm. Clearly, these ad hoc methods run the risk of losing information or relying heavily on implicit assumptions.

    Another technique that might seem quite natural would be to ignore the nonmatching problem by assigning Y values to

    the midpoints of the increments. Such midpoint assignment can lead to substantial bias, as the following numerical exper iment illustrates. In what follows, we convert totals to aver

    ages by dividing by the increment width. We estimated a simple parametric mode


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