DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCEUniversity of Waterloo, 200 University Avenue West Waterloo, Ontario, Canada, N2L 3G1519-888-4567, ext. 00000 | Fax: 519-746-1875 | www.stats.uwaterloo.ca

UNIVERSITY OF WATERLOO

DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE

WORKING PAPER SERIES

2009-04

Atmospheric Concentration ofChlorofluorocarbons: Addressing the

Global Concern with the LongitudinalBent-Cable Model

Shahedul Ahsan Khan,University of Waterloo, e-mail:sa4khan@math.uwaterloo.ca

Grace Chiu,University of Waterloo, e-mail:

gchiu@math.uwaterloo.ca

Joel A. Dubin,University of Waterloo, e-mail:

jdubin@math.uwaterloo.ca

Abstract

The recent steady decline in atmospheric chlorofluorocarbon (CFC) con-centrations could be a direct result of the Montreal Protocol on SubstancesThat Deplete the Ozone Layer, in effect since 1987. To study the extent of thedecline, we introduce the longitudinal bent-cable model to describe CFC con-centrations observed over a global detection network. The bent cable is a para-metric regression model to study data that exhibit a trend change. It comprisestwo linear segments to describe the incoming and outgoing phases, joined bya quadratic bend to model the transition period. For longitudinal data, mea-surements taken over time are nested within observational units drawn fromsome population of interest. Here, it is useful to develop a mixed-effects modelextension of existing (frequentist) bent-cable methodology for a single time se-ries. We do so in a hierarchical Bayesian framework, where each observationalunit is associated with a random bent cable and within-unit serial correlation.Our analysis using the longitudinal bent-cable model reveals a global decreasein atmospheric levels of CFC-11. The global drop took place between January,1989 and September, 1994 approximately, with November, 1993 being the esti-mated time point at which CFC-11 levels went from increasing to decreasing.

keywords: Longitudinal data; Bent-cable regression; Changepoint model; Bayesianhierarchical model; Random effects.

1 Introduction

Many biological consequences such as skin cancer and cataracts, irreversible dam-age to plants, and reduction of drifting organisms (animals, plants, archaea, bacte-ria) in the ocean’s photic zone may result from the increased ultraviolet (UV) ex-posure due to ozone depletion. According to “Ozone Science: The Facts Behind thePhaseout” by the U.S. Environmental Protection Agency (U.S. EPA, “http://www.epa.gov/ozone/science/sc fact.html”), each natural reduction in ozone levels has beenfollowed by a recovery, though there is convincing scientific evidence that theozone shield is being depleted well beyond changes due to natural processes. Inparticular, ozone depletion due to human activities is a major concern, and may becontrolled. One such human activity is the use of chloroflurocarbons (CFCs). Ascited in “The Ozone Hole Tour” by the University of Cambridge (“http://www.atm.ch.cam.ac.uk/tour/part3.html”), the catalytic destruction of ozone by atomic chlo-rine and bromine is the major cause of the forming of polar ozone holes, and pho-todissociation of CFC compounds is the main reason for these atoms to be in thestratosphere.

CFCs are nontoxic, nonflammable chemicals containing atoms of carbon, chlo-rine and fluorine. CFC-11 is one such compound. CFCs were extensively usedin air conditioning/cooling units, and as aerosol propellants prior to the 1980’s.While CFCs are safe to use in most applications and are inert in the lower atmo-sphere, they do undergo significant reaction in the upper atmosphere. As cited inThe Columbia Encyclopedia, CFC molecules take an average of fifteen years to travel

1

from the ground to the upper atmosphere, and can stay there for about a century.Chlorine inside the CFCs is one of the most important free radical catalysts to de-stroy ozone. The destruction process continues over the atmospheric lifetime ofthe chlorine atom (one or two years), during which an average of 100,000 ozonemolecules are broken down. Because of this, CFCs were banned globally by the1987 Montreal Protocol on Substances That Deplete the Ozone Layer (“MontrealProtocol” in The Columbia Encyclopedia). Since this protocol came into effect, theatmospheric concentration of CFCs has either leveled off or decreased. For ex-ample, Figure 1 shows the monthly average concentrations of CFC-11 monitoredfrom a station in Mauna Loa, Hawaii (Global Monitoring Division of the NationalOceanic and Atmospheric Administration (NOAA)/Earth System Research Labo-ratory (ESRL)). We see roughly three phases: an initial increasing trend (incomingphase), a gradual transition period, and a decreasing trend after the transition pe-riod (outgoing phase). This trend is representative of CFC-11 measurements takenby stations across the world and illustrated in this paper.

Time

CFC

-11

(in p

pt)

1988 1990 1992 1994 1996 1998 2000

250

255

260

265

270

275

Figure 1: Trend of the monthly average concentrations of CFC-11 in Mauna Loa,Hawaii. (Data source: NOAA/ESRL global monitoring division)

The effects of CFCs in ozone depletion is a global concern. Although exploratorydata analyses reveal a decrease of CFCs in the earth’s atmosphere since the early1990’s, so far no sophisticated statistical analysis has been conducted to evalu-ate the global trend. In addition, there are several other important questions re-garding the CFC concentration in the atmosphere that could be useful not only topolicy makers, but also for human awareness. For example, (1) How long did ittake for the CFC concentration to show an obvious decline? (2) What were therates of change (increase/decrease) in CFCs before and after the transition pe-riod? (3) What was the critical time point (CTP) at which the CFC trend wentfrom increasing to decreasing? In this article, we will address these questions

2

statistically by fitting a special changepoint model for CFC-11 data. We focuson CFC-11, because it is considered one of the most dangerous CFCs to reducethe ozone layer in the atmosphere. In fact, it has the shortest lifetime of com-mon CFCs, and is regarded as a reference substance in the definition of the ozonedepletion potential (ODP). The ODP of a chemical is the ratio of its impact onozone compared to the impact of a similar mass of CFC-11. Thus, the ODP is 1for CFC-11 and ranges from 0.6 to 1 for other CFCs. (These facts about CFCs aretaken from the U.S. EPA websites, “http://www.epa.gov/ozone/defns.html” and“http://www.epa.gov/ozone/science/ods/classone.html”.)

In a broader sense, we will comment in this article on (i) the global trend ofCFC-11, and (ii) the effectiveness of the Montreal Protocol on preserving the ozonelevel by reducing the use of CFC-11. Our findings will also provide a rough ideaof how long it may take to diminish CFC-11 from the earth’s atmosphere. Notethat this article is associated with that of Khan, Chiu and Dubin (2009, to appear inCHANCE) with more details on our proposed methodology.

2 Data

CFCs are monitored from different stations all over the globe by the NOAA/ESRLglobal monitoring division (“ftp://ftp.cmdl.noaa.gov/hats/cfcs/cfc11/insituGCs/”),and by the Atmospheric Lifetime Experiment/Global Atmospheric Gases Exper-iment/Advanced Global Atmospheric Gases Experiment (ALE/GAGE/AGAGE)global network program (“http://cdiac.esd.ornl.gov/ndps/alegage.html”). Hence-forth, we will refer to these two programs simply as NOAA and AGAGE.

Under the Radiatively Important Trace Species (RITS) program, NOAA beganmeasuring CFCs using in situ gas chromatographs at their four baseline obser-vatories — Pt. Barrow (Alaska), Cape Matatula (American Samoa), Mauna Loa(Hawaii) and South Pole (Antarctica) — and at Niwot Ridge (Colorado) in collabo-ration with the University of Colorado. We will label these five stations from 1 to 5respectively. During the period of 1988-1999, a new generation of gas chromatogra-phy called Chromatograph for Atmospheric Trace Species (CATS) was developedand has been used to measure CFC concentrations ever since.

The AGAGE program consists of three stages corresponding to advances andupgrades in instrumentation. The first stage (ALE) began in 1978, the second(GAGE) began during 1981-1985, and the third (AGAGE) began during 1993-1996.The current AGAGE stations are located in Mace Head (Ireland), Cape Grim (Tas-mania), Ragged Point (Barbados), Cape Matatula (American Samoa), and TrinidadHead (California). These five stations will be labeled from 6 to 10 respectively.

We consider monthly mean data for our statistical analysis. Ideally, we wishto have (a) full data for all stations, (b) a long enough period to capture all threephases of the CFC trend, and (c) no change in instrumentation to avoid the el-ements of non-stationarity and biased measurement, if any. However, we do nothave the same duration of consecutive observations for all stations. Moreover, datawere recorded by instrumentation that switched from one type to another. Table 1

3

summarizes the availability of the consecutive observations, and the instrumenta-tions used to record data.

Table 1: CFC-11 data summaryStation Available consecutive observations Instrumentation

Barrow (Station 1) Nov, 1987 - Feb, 1999 RITS

Jun, 1998 - Aug, 2008 CATS

Cape Matatula (Station 2) May, 1989 - Apr, 2000 RITS

Dec, 1998 - Aug, 2008 CATS

Mauna Loa (Station 3) Jul, 1987 - Aug, 2000 RITS

Jun, 1999 - Aug, 2008 CATS

South Pole (Station 4) Jun, 1990 - Nov, 2000 RITS

Feb, 1998 - Aug, 2008 CATS

Niwot Ridge (Station 5) Feb, 1990 - Apr, 2000 RITS

May, 2001 - Jul, 2006 CATS

Mace Head (Station 6) Feb, 1987 - Jun, 1994 GAGE

Mar, 1994 - Sep, 2007 AGAGE

Cape Grim (Station 7) Dec, 1981 - Dec, 1994 GAGE

Aug, 1993 - Sep, 2007 AGAGE

Ragged Point (Station 8) Aug, 1985 - Jun, 1996 GAGE

Jun, 1996 - Sep, 2007 AGAGE

Cap Matatula (Station 9) Jun, 1991 - Sep, 1996 GAGE

Aug, 1996 - Sep, 2007 AGAGE

Trinidad Head (Station 10) – GAGE

Oct, 1995 - Sep, 2007 AGAGE

Thus, ideal statistical conditions are not achievable in this case. As a compro-mise, we remove Stations 9 and 10 from our analysis due to insufficient data, andchoose a study period in such a way that it can reflect the changing behavior of theCFC-11 concentration in the atmosphere. The Montreal Protocol came into force onJanuary 1, 1989. So, we expect an increasing trend in CFC-11 prior to 1989 becauseof its extensive use during that period. After the implementation of the proto-col, we expect a change (either decreasing or leveling off) in the CFC-11 trend. Tocharacterize this change, we wish to have a study period starting from some pointbefore the implementation of the protocol. Moreover, we must have sufficient datato observe the CFC-11 trend, if any. Thus, we settle for a relatively long study pe-riod of 152 months from January, 1988 to August, 2000, which is perhaps the mostreasonable to satisfy (a)-(c) as much as possible. In particular, it covers Stations 3-4with a single measuring device, RITS. Stations 2 and 5 have RITS data until April,2000, at which point we truncate their data so that only RITS is present for all of Sta-

4

tions 2-5 throughout the study period. Data for the remaining four stations duringthis period were recorded by two measuring devices – RITS and CATS for Station1, and GAGE and AGAGE for Stations 6-8 – each device occupying a substantialrange of the 152 months. Figure 2 shows the eight profiles of the correspondingCFC-11 data. Specifically, each station constitutes an individual curve, which isdifferent from the others due to actual CFC-11 levels during measurement, expo-sure to wind and other environmental variables, sampling techniques, and so on.Our objective is to assess the global CFC-11 concentration in the atmosphere, aswell as station-specific characterization of the trends.

Time

CF

C-1

1 in

pa

rts-

pe

r-tr

illio

n (

pp

t)

1988 1990 1992 1994 1996 1998 2000

23

02

40

25

02

60

27

02

80

Barrow (Station 1)Cape Matatula (Station 2)Mauna Loa (Station 3)South Pole (Station 4)Niwot Ridge (Station 5)Mace Head (Station 6)Cape Grim (Station 7)Ragged Point (Station 8)

 Figure 2: CFC-11 profiles of eight stations (monthly mean data)

Note that in this paper, we do not take into consideration the effects of changein instrumentation (RITS/CATS and GAGE/AGAGE). Modeling these effects iscurrently in progress (see the Conclusion section), and a preliminary analysis re-veals statistically insignificant results. However, between-station differences areaccounted for, at least partially, by the random components introduced in our mod-eling (see the Methods section).

3 Methods

We wish to unify information from each station to aid the understanding of theglobal as well as station-specific behavior of CFC-11 in the atmosphere. We willindex the stations by i = 1, 2, . . . , 8, and the months from January, 1988 to August,2000 by j = 1, 2, . . . , 152. Let tij denote the jth measurement occasion of the ith

station. We model the CFC-11 measurement for the ith station at time tij , denoted

5

by yij , by the relationshipyij = f(tij,θi) + εij (1)

where θi is a vector of regression coefficients for the ith station, f(·) is a functionof tij and θi characterizing the trend of the station-specific data over time, and εijrepresents the random error component.

Some remarks are required to explain tij . Recall that some stations do not havedata for all 152 months. We employ the following system for defining tij . Forexample, the first and last months with recorded data by Station 3 are January, 1988and August, 2000, respectively; thus, t3,1 = 1, t3,2 = 2, . . . , t3,152 = 152. In contrast,Station 2 had its first and last recordings in May, 1989 and April, 2000, respectively;hence, t2,1 = 17, t2,2 = 18, . . . , t2,132 = 148. The same approach is used to define tij’sfor other i’s. Note that a few yij’s (from 1 to 5) were missing between the first andlast months for a given station. We replace them by observations from anotherdata set (e.g. CATS or AGAGE) or by mean imputation based on neighboring timepoints if not available from another data set. Though mean imputation can beproblematic resulting in biased estimates, if just a few missing values are replacedby the mean, the deleterious effect of mean substitution is reduced (McKnight,Figueredo and Sidani, 2007). So, we expect our findings to be minimally affectedby this replacement for so few time points.

Next, we will describe the complete formulation of the model given by (1). Wewould like an expression for f that not only describes the CFC-11 profiles as shownin Figure 2, but also gives useful information regarding the rates of change and thetransition. Although a simple quadratic model such as f(tij,θi) = β0i+β1itij+β2it

2ij ,

where θi = (β0i, β1i, β2i)′, might be appealing to characterize the overall convexity

of the trend, it would not be expected to fit the observed data all that well, inlight of the apparent three phases: incoming and outgoing, joined by the curvedtransition. A characterization of such a trend can be well-accomplished by theso-called bent-cable function (Chiu, Lockhart and Routledge, 2006), given by

f(tij,θi) = β0i + β1itij + β2iq(tij, γi, τi) (2)

where

q(tij, γi, τi) =(tij − τi + γi)

2

4γi1{|tij − τi| ≤ γi}+ (tij − τi)1{tij − τi > γi}. (3)

A graphical description of this function is given in Figure 3. The model is parsi-monious in that it has only five regression coefficients, and is appealing due to thegreatly interpretable regression coefficients. We will extend their model to accountfor the between-station heterogeneity as suggested by the different profiles in Fig-ure 2. Henceforth, we will denote f(tij,θi) and q(tij, γi, τi) by simply fij and qij ,respectively.

Overall, the bent-cable function, f , represents a smooth trend over time. Therandom error components, εij’s in (1), account for within-station variability in themeasurements around the regression f . Many standard assumptions for regres-sion analyses do not hold for longitudinal data, including independence between

6

Time

Ben

t-cab

le fu

nctio

n, f

Incoming phase

Outgoing phase

𝜏𝜏𝑖𝑖 − 𝛾𝛾𝑖𝑖 𝜏𝜏𝑖𝑖 + 𝛾𝛾𝑖𝑖

𝛾𝛾𝑖𝑖 𝛾𝛾𝑖𝑖

𝜏𝜏𝑖𝑖

Figure 3: The bent-cable function, comprising two linear segments (incoming andoutgoing) joined by a quadratic bend. There are three linear parameters β0i, β1i

and β2i, and two transition parameters γi and τi, with θi = (β0i, β1i, β2i, γi, τi)′. The

intercept and slope for the incoming phase are β0i and β1i, respectively; the slopefor the outgoing phase is β1i + β2i; and the center and half-width of the bend are τiand γi, respectively. The critical time point (at the green line) is the point at whichthe response takes either an upturn from a decreasing trend or a downturn froman increasing trend.

all measurements (Fitzmaurice, Laird and Ware, 2004). Instead, it is necessaryto account for correlation between repeated measurements within the same unit(here, station) over time. Though, for some data, the unit-specific coefficients θimay adequately account for this correlation, quite often there is additional serialcorrelation remaining that can be accounted for by these εij’s. An order-p (p > 0)autoregressive (AR(p)) serial correlation structure (Box and Jenkins, 1994) may bewell-suited for many types of longitudinal data. Thus, we consider here an AR(p),p > 0, model for the εij’s, that is,

εij = φ1εi,j−1 + φ2εi,j−2 + . . .+ φpεi,j−p + uij (4)

where φ1, φ2, . . . , φp are the AR(p) parameters, which we denote by the vector φ,that is, φ = (φ1, φ2, . . . , φp)

′. We assume that uij’s in (4) are independently andidentically normally distributed with mean 0 and variance σ2

ui. Furthermore, weassume that the initial p observations for each i are known. We develop a Bayesianmethodology conditional on these initial observations, that is, the posterior (Section3.1) is conditioned on these initial observations. This assumption is customary inthe frequentist’s estimation procedures, and was considered by Chib (1993) in aBayesian approach. To analyze a single profile (measurements from a single stationover time) using a bent-cable model with AR(p) noise, the reader may refer to Chiuand Lockhart ([3] and [4]).

7

In particular, an AR(1) model, being very parsimonious, is characterized byan exponential decay of correlations as the separation between a pair of measure-ments increases, a feature that is common in many longitudinal data sets. Forour CFC-11 analysis, we assume an AR(1) structure with correlation parameterρ = corr(εi,j, εi,j±1), and variance parameters σ2

i = var(εij). To define a populationlevel of CFC-11, we wish to relate the station-specific coefficients θi to the popu-lation coefficients in some meaningful way. Before describing such a relation, wemention here one more assumption we make in developing the model: the linearcoefficients βi = (β0i, β1i, β2i)

′ and transition coefficients αi = (γi, τi)′ are inde-

pendent a priori (see Section 3.1). Practically, βi characterizes the intercepts andthe rates of change, while αi characterizes the structure of the transition. Thereis generally no way to infer about αi from prior knowledge of βi, without firstseeing the data. For example, suppose the two linear phases are very steep. Thisinformation alone is not sufficient to answer the question “what is the chance of αi

being, say, (30, 40)′ versus (35, 50)′?” Hence, it is reasonable to assume that βi andαi are a priori independent. Now, we relate βi to the population coefficient vectorβ = (β0, β1, β2)

′ plus a random component η1i, that is, βi = β + η1i, where η1i isa 3 × 1 column vector. Under the assumption that η1i has mean 0 and a 3 × 3 co-variance matrix D1, conceptually we can regard stations as having their own linearcoefficients, and the population coefficients as the average across stations. Then,the covariance matrix D1 provides information on the variability of the deviation ofthe station-specific coefficient βi from the population coefficient β. As an extremeexample, a zero variation of the deviation between β1i and β1 indicates that thestation-specific and global incoming slopes are identical. In other words, rates ofchange in the incoming phase are identical across stations. To complete the modelformulation for the linear coefficients, an assumption on the distribution of η1i, orequivalently βi, is necessary. Since (i) the dependence structure can be fully spec-ified by the covariance matrix through an assumption of the multivariate normaldistribution, and (ii) it is convenient theoretically and computationally, we assumethat η1i ∼MVN (0,D1), or equivalently, βi ∼MVN (β,D1).

Let us turn now to a model for the transition coefficients. Note that γi and τiare positive. We assume that α∗i = (log (γi), log (τi))

′ = α∗ + η2i, α∗ = (γ∗, τ ∗)′,and η2i is a 2 × 1 column vector with η2i ∼ MVN (0,D2), so that E(α∗i ) = α∗.This assumption is equivalent to αi ∼ MVLN (α∗,D2), i.e. a bivariate lognormaldistribution with α∗ and D2 being the mean vector and the covariance matrix ofα∗i , respectively. We may then define a population effect as (eγ

∗, eτ

∗)′, which is

the median of αi under the lognormal distribution. Statistical assumptions forεij , βi and αi as discussed above, together with Equations (1)-(4), constitute ourlongitudinal bent-cable.

3.1 Statistical Inference

We employ a Bayesian approach for statistical inference. The main idea of Bayesianinference is to combine data and prior knowledge on a parameter to determine itsposterior distribution (the conditional density of the parameter given the data). The

8

prior knowledge is supplied in the form of a prior distribution of the parameter,which quantifies information about the parameter prior to any data being gath-ered. For example, recall the argument for the a priori independence of βi andαi. When little is reliably known about the parameter, it is reasonable to choose afairly vague, minimally informative prior. For example, in analyzing the CFC-11data, we assumed a multivariate normal prior for β with mean 0 and a covariancematrix with very large variance parameters. This leads to a non-informative priorfor β, meaning the data, assuming a sufficient amount of data is collected, willprimarily dictate β’s resulting posterior distribution.

Any conclusions about the plausible value of the parameter are to be drawnfrom the posterior distribution. For example, the posterior mean (median if theposterior density is noticeably asymmetric) and variance can be considered an esti-mate of the parameter and the variance of the estimate, respectively. A 100(1−2p)%Bayesian interval, or credible interval, is [c1, c2], where c1 and c2 are the pth and(1−p)th quantiles of the posterior density, respectively. This credible interval has aprobabilistic interpretation. For example, for p = 0.025, the conditional probabilitythat the parameter falls in the interval [c1, c2] given the data is 0.95.

The Bayesian model formulation of our longitudinal bent-cable model derivedfrom the above assumptions involves three levels (see Appendix 1 for details of thefirst level):

Y(2)i |y

(1)i ,βi,αi,φ, σ

2ui ∼MVN

(µi(βi,αi,φ), σ2

uiIi), (5)

βi|β,D1 ∼MVN (β,D1)

αi|α∗,D2 ∼MVLN (α∗,D2)

, (6)

β|h1,H1 ∼MVN (h1,H1), α∗|h2,H2 ∼MVN (h2,H2),

D−11 ∼ W

(ν1, (ν1A1)

−1), D−1

2 ∼ W(ν2, (ν2A2)

−1),

φ|h3,H3 ∼MVN (h3,H3), σ−2ui |a, b ∼ G(a2 ,

ab2)

, (7)

where Y(2)i = (Yi,p+1, . . . , Yi,ni)

′, y(1)i = (yi1, . . . , yip)

′, µi(βi,αi,φ) ≡ µi = (µi,p+1,. . . , µi,ni)

′, µij = β0i (1 −∑p

k=1 φk) + β1i xij + β2i rij +∑p

k=1 φk yi,j−k, xij = tij −∑pk=1 φk ti,j−k, rij = qij−

∑pk=1 φk qi,j−k (j = p+1, . . . , ni), andW and G stand for

Wishart and gamma distributions, respectively. The first two levels are (5) and (6)which represent, respectively, the within-station and between-station characteris-tics, and the third level is (7), which is the specification of the prior distributionswith fixed, prespecified hyperparameters.

Bayesian inference is carried out by the Markov Chain Monte Carlo (MCMC)method (Gilks, Richardson and Spiegelhalter, 1996). More specifically, we gener-ate data by the Hybrid Gibbs (Smith and Roberts, 1993) (Metropolis within Gibbs)

9

algorithm to summarize the posterior distributions. We prefer conjugate priorsat level 3 to facilitate the straightforward implementation of the Gibbs samplerbecause of the posterior’s closed form. While occasionally criticized to be too re-strictive, conjugate priors are preferred here to reduce the extensiveness of compu-tations involved in the MCMC method. Specification of the priors in Equation (7)leads to the following full conditionals (see Appendix 2 for derivations):

βi|. ∼MVN(Mi

(σ−2ui X′i zi + D−1

1 β),Mi

), (8)

β|. ∼MVN(U1(D−1

1 β + H−11 h1),U1

), (9)

α∗|. ∼MVN(U2(D−1

2 α + H−12 h2),U2

), (10)

D−11 |. ∼ W

(m+ ν1,

[ m∑i=1

(βi − β) (βi − β)′ + ν1A1

]−1), (11)

D−12 |. ∼ W

(m+ ν2,

[ m∑i=1

(α∗i −α∗) (α∗i −α∗)′ + ν2A2

]−1), (12)

σ−2ui |. ∼ G

(ni − p+ a

2,(zi − Xi βi

)′(zi − Xi βi

)+ ab

2

), (13)

φ|. ∼MVN(

V( m∑i=1

σ−2ui W′

i εi + H−13 h3

),V), (14)

where zi = (zi,p+1, . . . , zi,ni)′ with zij = yij −

∑pk=1 φk yi,j−k, εi = (εi,p+1, . . . , εi,ni)

′

with εij = yij − fij , β =∑m

i=1 βi, α =∑m

i=1 α∗i , M−1i = σ−2

ui X′i Xi + D−11 ,

U−11 = mD−1

1 + H−11 , U−1

2 = mD−12 + H−1

2 , V−1 =∑m

i=1 σ−2ui W′

i Wi + H−13 , and

Xi =

(1−

∑pk=1 φk

)xi,p+1 ri,p+1(

1−∑p

k=1 φk

)xi,p+2 ri,p+2

......

...(1−

∑pk=1 φk

)xi,ni ri,ni

, Wi =

εi,p εi,p−1 εi,1

εi,p+1 εi,p εi,2...

......

εi,ni−1 εi,ni−2 εi,ni−p

. (15)

The full conditional for αi cannot be expressed in a closed form. It is proportionalto the following expression

1

γiτiexp

{− 1

2σ2ui

(zi − Xiβi)′(zi − Xiβi)

}exp

{− 1

2(α∗i −α∗)′ D−1

2 (α∗i −α∗)}.

(16)

10

We use a random walk Metropolis (Givens, Hoeting and Sidani, 2005) for αi withinthe Gibbs because of its nonstandard full conditional.

As little is known reliably on the station-specific profiles of the CFC-11 dataprior to data collection, values of the hyper-parameters in (7) are chosen to reflectweak knowledge. We have written our own code in C to generate Monte Carlosamples, and analyzed these in R using the coda package.

4 Results

Figure 4 presents the estimated global CFC-11 concentrations using longitudinalbent-cable regression assuming the AR(1) structure for within-station dependence.The global drop (gradual) in CFC-11 took place between January, 1989 and Septem-ber, 1994 approximately. Estimated incoming and outgoing slopes were 0.65 and−0.12, respectively. Thus, the average increase in CFC-11 was about 0.65 ppt for aone-month increase during the incoming phase (January, 1988 - December, 1988),and the average decrease was about 0.12 ppt during the outgoing phase (October,1994 - August, 2000). The 95% credible intervals indicated significant slopes forthe incoming/outgoing phases. Specifically, these intervals were (0.50, 0.80) and(−0.22,−0.01), respectively, for the two linear phases, neither interval including0. The estimated population critical time point (CTP) was November, 1993, whichimplies that, on average, CFC-11 went from increasing to decreasing around thistime. The corresponding 95% credible interval ranged from December, 1992 toNovember, 1994.

 

 

Time

CF

C-1

1 (

in p

pt)

1988 1990 1992 1994 1996 1998 2000

24

02

45

25

02

55

26

02

65

27

02

75

 

CTP = Nov, 1993Slope = ‐0.12Slope = 

0.65 

End of transition = Sep, 1994 

Beginning of transition = Jan, 1989 

Figure 4: Global fit of the CFC-11 data using longitudinal bent-cable regressionassuming AR(1)

11

The station-specific fits, as well as the population fit are displayed in Figure 5.It shows that the model fits the data well, with the observed data and individualfits agreeing quite closely. Table 2 summarizes the fits numerically. We see sig-nificant increase/decrease of CFC-11 in the incoming/outgoing phases for all thestations separately, as well as globally. The rates at which these changes occurred(Columns 2 and 3) agree closely for the stations. This phenomenon was also evi-dent in the estimate of D, showing small variation of the deviations between theglobal and station-specific incoming/outgoing slope parameters (variance ≈ 0.02for both cases).

Barrow(Station 1)

Time

CF

C-1

1 (in

ppt

)

1988 1992 1996 2000

230

240

250

260

270

280

Cap Matatula(Station 2)

Time

CF

C-1

1 (in

ppt

)

1988 1992 1996 2000

230

240

250

260

270

280

Mauna Loa(Station 3)

Time

CF

C-1

1 (in

ppt

)

1988 1992 1996 2000

230

240

250

260

270

280

South Pole(Station 4)

Time

CF

C-1

1 (in

ppt

)

1988 1992 1996 2000

230

240

250

260

270

280

Niwot Ridge(Station 5)

Time

CF

C-1

1 (in

ppt

)

1988 1992 1996 2000

230

240

250

260

270

280

Mace Head(Station 6)

Time

CF

C-1

1 (in

ppt

)

1988 1992 1996 2000

230

240

250

260

270

280

Cape Grim(Station 7)

Time

CF

C-1

1 (in

ppt

)

1988 1992 1996 2000

230

240

250

260

270

280

Ragged Point(Station 8)

Time

CF

C-1

1 (in

ppt

)

1988 1992 1996 2000

230

240

250

260

270

280

Figure 5: Station-specific fits (red) and population fit (green) of the CFC-11 data,with estimated transition marked by vertical lines. The black line indicates ob-served data.

The above findings support the notion of constant rates of increase and de-crease, respectively, before and after the enforcement of the Montreal Protocol, ob-servable despite a geographically spread-out detection network. They also pointto the success of the widespread adoption and implementation of the MontrealProtocol across the globe. However, the rate by which CFC-11 has been decreasing(about 0.12 ppt per month, globally) suggests that it will remain in the atmospherethroughout the 21st century, should current conditions prevail.

Let us turn now to the behavior of the transition of CFC-11 over time. Thetransition periods and critical time points varied somewhat across stations (Ta-

12

Table 2: Estimated station-specific and global concentrtions of CFC-11 assumingAR(1)

Incoming slope Outgoing slope Transition period CTP(95% credible (95% credible (Duration) (99% credible

interval) interval) interval)Global 0.65 −0.12 Jan, 1989 - Sep, 1994 Nov, 1993

(0.50, 0.80) (−0.22,−0.01) (69 months) (Aug, 1992 toMay, 1995)

Barrow 0.55 −0.19 Jan, 1989 - Aug, 1994 Mar, 1993(Station 1) (0.39, 0.72) (−0.24,−0.15) (68 months) (Jul, 1992 toσ2

1 ≈ 2.97 Nov, 1993)Cap Matatula 0.74 −0.10 May, 1989 - Jan, 1995 May, 1994

(Station 2) (0.56, 0.94) (−0.13,−0.07) (69 months) (Oct, 1993 toσ2

2 ≈ 1.01 Feb, 1995)Mauna Loa 0.67 −0.12 Mar, 1989 - Jun, 1994 Aug, 1993(Station 3) (0.52, 0.83) (−0.16,−0.09) (64 months) (Dec, 1992 toσ2

3 ≈ 1.81 May, 1994)South Pole 0.60 −0.12 Dec, 1988 - Nov, 1995 Sep, 1994(Station 4) (0.42, 0.77) (−0.15,−0.10) (84 months) (Apr, 1994 toσ2

4 ≈ 0.30 Mar, 1995Niwot Ridge 0.56 −0.11 Nov, 1988 - Jul, 1994 Aug, 1993

(station 5) (0.34, 0.79) (−0.13,−0.08) (69 months) (Dec, 1992 toσ2

5 ≈ 0.82 May, 1994)Mace Head 0.59 −0.11 Sep, 1988 - Jan, 1994 Mar, 1993(Station 6) (0.44, 0.74) (−0.13,−0.08) (65 months) (Jul, 1992 toσ2

6 ≈ 1.20 Dec, 1993)Cape Grim 0.78 −0.07 Mar, 1989 - Nov, 1994 Jun, 1994(Station 7) (0.68, 0.93) (−0.09,−0.06) (69 months) (Jan, 1994 toσ2

7 ≈ 0.29 Oct, 1994)Ragged Point 0.70 −0.10 Jan, 1989 - Apr, 1994 Aug, 1993

(Station 8) (0.55, 0.86) (−0.14,−0.07) (64 months) (Nov, 1992 toσ2

8 ≈ 2.25 Jun, 1994)

ble 2). This may be due to the extended CFC-11 phase-out schedules containedin the Montreal Protocol – 1996 for developed countries and 2010 for developingcountries. Thus, many countries at various geographical locations continued tocontribute CFCs to the atmosphere during the 152 months in our study period,while those at other locations had stopped. Overall, the eight transitions began be-tween September, 1988 and May, 1989, a period of only nine months. This reflectsthe success and acceptability of the protocol all over the globe. Durations of the

13

transition periods are very similar among stations except for South Pole. Thus, ittook almost the same amount of time in different parts of the world for CFC-11to start dropping linearly with an average rate of about 0.12 ppt per month. Thelast column of the table indicates that the global estimate of the CTP is containedin all the station-specific 99% credible intervals except for those of South Pole andCape Grim, with the lower bound for South Pole coming three months later thanthat for Cape Grim, making South Pole a greater outlier. Specifically, the transitionfor South Pole was estimated to take place over 84 months, an extended periodcompared to the other stations. This could be due to the highly unusual weatherconditions specific to the location. CFCs are not disassociated during the longwinter nights in the South Pole. Only when sunlight returns in October does ul-traviolet light break the bond holding chlorine atoms to the CFC molecule (OzoneFacts, NASA, “http://ozonewatch.gsfc.nasa.gov/facts/hole.html”). For this rea-son, it may be expected for CFCs to remain in the atmosphere over the South Polefor a longer period of time, and hence, an extended transition period. Indeed,our findings for South Pole are very similar to those reported by Ghude, Jain andArya (2009). To evaluate the trend, the authors used the NASA EdGCM model –a deterministic global climate model wrapped in a graphical user interface. Theyfound the average growth rate to be 9 ppt per year for 1983-1992, and about −1.4ppt per year for 1993-2004, turning negative in the mid 1990’s. With our statisticalmodeling approach, we estimated a linear growth rate of 0.6 ppt per month (7.2ppt per year) prior to December, 1988, a transition between December, 1988 andNovember, 1995, and a negative linear phase (−0.12 ppt per month, or −1.44 pptper year) after November, 1995.

The eight estimates of within-station variability (σ2i ) are given in the first col-

umn of Table 2. We noticed earlier from the profile plot (Figure 2) that Barrowmeasurements are more variable, whereas Cape Grim and South Pole show littlevariation over time. This is reflected in their within-station variance estimates of2.97, 0.29, and 0.30, respectively. This also explains, at least partially, as to whythe credible intervals of the South Pole and Cape Grim CTPs did not contain theestimate of the global CTP. As expected, we found a high estimated correlation be-tween consecutive within-station error terms (AR(1) parameter ρ ≈ 0.81 with 95%credible interval (0.77, 0.85)).

In summary, our methodology provides a satisfactory fit of the longitudinalbent-cable model to the CFC-11 data under the assumption of an AR(1) within-station correlation structure. We are currently investigating the goodness of thisfit in comparison to a higher order AR model. The generality of our methodologywith respect to the order of the AR process makes it a flexible tool in analyzingdifferent types of longitudinal changepoint data.

5 Conclusion

CFC-11 is a major source for the depletion of stratospheric ozone around the globe.Since the Montreal Protocol came into effect, a global decrease in the CFC-11 has

14

been observed, a finding confirmed by our analysis. Our analysis, using the longi-tudinal bent-cable model assuming an AR(1) within-station dependence structure,revealed a gradual rather than abrupt change, the latter of which is assumed bymost standard changepoint models. This makes scientific sense due to the factthat CFC molecules can stay in the upper atmosphere for about a century, andtheir breakdown does not take place instantaneously. The substantial decrease inglobal CFC-11 levels after the gradual change shown by our analysis suggests thatthe Montreal Protocol can be regarded as a successful international agreement toreduce the negative impact of CFCs on the ozone layer.

One possible extension of the proposed bent-cable model for longitudinal datais to incorporate individual-specific covariate(s) (e. g. effects of instrumentationsspecific to different stations in measuring the CFC-11 data) to see if they couldpartially explain the variations within and between the individual profiles. Onemay incorporate this by modeling the individual-specific coefficients θi’s to varysystematically with the covariate(s) plus a random component. This specificationmakes the estimation method more complicated, and is currently in progress.

Acknowledgement

This work is partially supported by NSERC through Discovery Grants to G. Chiu(RGPIN261806-05) and J.A. Dubin (RGPIN327093-06), and by the Government ofOntario through an Ontario Graduate Scholarship to S.A. Khan (000113006). Weextend our appreciation to the NOAA/ESRL and ALE/GAGE/AGAGE globalnetwork program, who made their CFC-11 data available to the public. We thankGeoffrey S. Dutton, NOAA/ESRL, and Derek Cunnold, ALE/GAGE/AGAGE globalnetwork program, for clarifications of their data used in this article. We also thankPeter X. Song, Professor, Department of Statistics and Actuarial Science, Universityof Waterloo, Waterloo, Ontario, for his temporary support through NSERC.

15

Appendix 1: Level 1 of the Hierarchical Model

We can write the first level of our hierarchical model as in Equation (5). This isformulated from the regression model represented by Equations (1)-(3), and underthe assumptions for εij’s, i.e.

yij = fij + εij, εij =

p∑k=1

φk εi,j−k + uij, uij ∼ N (0, σ2ui).

Here,

yij = fij + εij

= β0i + β1i tij + β2i qij + εij

= β0i + β1i tij + β2i qij + φ1 εi,j−1 + φ2 εi,j−2 + . . .+ φ1 εi,j−p + uij

= β0i + β1i tij + β2i qij + φ1 (yi,j−1 − β0i − β1i ti,j−1 − β2i qi,j−1)+

φ2 (yi,j−2 − β0i − β1i ti,j−2 − β2i qi,j−2) + . . .+

φp (yi,j−p − β0i − β1i ti,j−p − β2i qi,j−p) + uij

= β0i

(1−

p∑k=1

φk

)+ β1i

(tij −

p∑k=1

φk ti,j−k

)+ β2i

(qij −

p∑k=1

φk qi,j−k

)+

p∑k=1

φk yi,j−k + uij

= β0i

(1−

p∑k=1

φk

)+ β1i xij + β2i rij +

p∑k=1

φk yi,j−k + uij

= µij + uij.

Therefore, we specify the first level as

Y(2)i | y

(1)i ,βi,αi, σ

2ui,φ ∼MVN (µi, σ

2ui Ii).

Appendix 2: Full Conditionals

Full conditionals for βi, β, α∗, D−11 , D−1

2 , σ−2ui , φ and αi are given in Equations

(8)-(14) and (16), respectively. Here we will present the derivations for βi, β, D−11 ,

σ−2ui , φ and αi. Full conditionals for α∗ and D−1

2 can be derived in the same wayas for β and D−1

1 , respectively.In general, full conditionals are derived from the joint distribution of the vari-

ables of interest (Gilks, 1996). Letting Y(2) =(Y

(2)′

1 , . . . ,Y(2)′m

)′, the joint distribu-

16

tion of our hierarchical model (5)-(7) is given by

π(Y(2), β1, . . . , βm, α1, . . . , αm, σ−2u1 , . . . , σ

−2um, β, α∗, D−1

1 , D−12 , φ)

= π(Y(2)| β1, . . . , βm, α1, . . . , αm, σ

−2u1 , . . . , σ

−2um, β, α∗, D−1

1 , D−12 , φ) ×

π(β1, . . . , βm| α1, . . . , αm, σ−2u1 , . . . , σ

−2um, β, α∗, D−1

1 , D−12 , φ) ×

π(α1, . . . , αm| σ−2u1 , . . . , σ

−2um, β, α∗, D−1

1 , D−12 , φ) ×

π(σ2u1, . . . , σ

2um| β, α∗, D−1

1 , D−12 , φ) ×

π(β| α∗, D−11 , D−1

2 , φ) × π(α∗| D−11 , D−1

2 , φ) ×

π(D−11 | D−1

2 , φ) × π(D−12 | φ) × π(φ)

=[ m∏i=1

MVN (Y(2)i | µi, σ

2ui Ii)

]×[ m∏i=1

MVN (βi| β, D1)]×

[ m∏i=1

MVLN (αi| α∗, D2)]×[ m∏i=1

G(σ−2ui |

a

2,ab

2

)]×

[MVN

(β| h1, H1

)]×[MVN

(α∗| h2, H2

)]×[

W(D−1

1 | ν1, (ν1 A1)−1)]×[W(D−1

2 | ν2, (ν2 A2)−1)]×[MVN

(φ| h3, H3)

].

Now, to construct the full conditional of a variable, say βi, we need only to pickout the terms in the joint density which involve βi. Note that any term which doesnot depend on βi can be taken as a proportionality constant in the full conditional.

Full Conditional for βi

Picking out the terms in the joint density which involve βi, we get

π(βi| .) ∝ exp{− 1

2σ2ui

(y

(2)i − µi

)′(y

(2)i − µi

)− 1

2(βi − β)′ D−1

1 (βi − β)}.

Here,

yij − µij = yij − β0i

(1−

p∑k=1

φk

)− β1i xij − β2i rij −

p∑k=1

φk yi,j−k

= zij − β0i

(1−

p∑k=1

φk

)− β1i xij − β2i rij for j = p+ 1, . . . , ni.

17

In vector-matrix notation, y(2)i − µi = zi − Xi βi. Therefore,

π(βi| .) ∝ exp{− 1

2

[σ−2ui

(zi − Xi βi

)′(zi − Xi βi

)+ (βi − β)′ D−1

1 (βi − β)]}

= exp{− 1

2

[σ−2ui z′i zi − σ−2

ui z′i Xi βi − σ−2ui β′i X′i zi + σ−2

ui β′i X′i Xi βi+

β′i D−11 βi − β′i D−1

1 β − β′ D−11 βi + β′ D−1

1 β]}

∝ exp{− 1

2

[− σ−2

ui z′i Xi βi − σ−2ui β′i X′i zi + σ−2

ui β′i X′i Xi βi+

β′i D−11 βi − β′i D−1

1 β − β′ D−11 βi

]}[proportionality follows because σ−2

ui z′i zi and β′ D−11 β do not depend on βi

]= exp

{− 1

2

[− 2 σ−2

ui β′i X′i zi + σ−2ui β′i X′i Xi βi + β′i D−1

1 βi − 2 β′i D−11 β

]}[σ−2ui z′i Xi βi and β′ D−1

1 βi are scalars]

= exp{− 1

2

[− 2β′i

(σ−2ui X′i zi + D−1

1 β)

+ β′i(σ−2ui X′i Xi + D−1

1

)βi

]}= exp

{− 1

2

[− 2β′i

(σ−2ui X′i zi + D−1

1 β)

+ β′i M−1i βi

]}[where M−1

i = σ−2ui X′i Xi + D−1

1

]∝ exp

{− 1

2

[− β′i

(σ−2ui X′i zi + D−1

1 β)−(σ−2ui X′i zi + D−1

1 β)′

βi+

β′i M−1i βi +

(σ−2ui X′i zi + D−1

1 β)′ Mi

(σ−2ui X′i zi + D−1

1 β)]}

[β′i(σ−2ui X′i zi + D−1

1 β)

is a scalar, and so is β′i(σ−2ui X′i zi + D−1

1 β)

=(σ−2ui X′i zi + D−1

1 β)′

βi. Proportionality follows because(σ−2ui X′i zi + D−1

1 β)′ Mi(

σ−2ui X′i zi + D−1

1 β)

does not depend on βi

]= exp

{− 1

2

[βi −Mi

(σ−2ui X′i zi + D−1

1 β)]′ M−1

i

[βi −Mi

(σ−2ui X′i zi + D−1

1 β)]}

,

which is proportional to the pdf of a trivariate normal distribution with mean vec-tor Mi

(σ−2ui X′i y∗i + D−1

1 β)

and covariance matrix Mi. Therefore,

βi| . ∼MVN(Mi

(σ−2ui X′i zi + D−1

1 β), Mi

).

18

Full Conditional for β

Picking out the terms in the joint density which involve β, we get

π(β| .) ∝m∏i=1

exp{− 1

2(βi − β)′ D−1

1 (βi − β)}

exp{− 1

2(β − h1)

′ H−11 (β − h1)

}= exp

{− 1

2

[ m∑i=1

(βi − β)′ D−11 (βi − β) + (β − h1)

′ H−11 (β − h1)

]}= exp

{− 1

2

[ m∑i=1

(β′i D−11 βi − β′i D−1

1 β − β′ D−11 βi + β′ D−1

1 β)+

β′ H−11 β − β′ H−1

1 h1 − h′1 H−11 β + h′1 H−1

1 h1

]}∝ exp

{− 1

2

[ m∑i=1

(−β′i D−11 β − β′ D−1

1 βi + β′ D−11 β)+

β′ H−11 β − β′ H−1

1 h1 − h′1 H−11 β

]}[proportionality follows because β′i D−1

1 βi and h′1 H−11 h1 do not depend on β

]= exp

{− 1

2

[− 2β′ D−1

1

m∑i=1

βi +m β′ D−11 β + β′ H−1

1 β − 2β′ H−11 h1

]}[β′i D−1

1 β and h′1 H−11 β are scalars, and so are β′i D−1

1 β = β′ D−11 βi, h′1 H−1

1 β =

β′ H−11 h1

]= exp

{− 1

2

[− 2 β′ (D−1

1 β + H−11 h1) + β′ U−1

1 β]}

[where β =

m∑i=1

βi, and U−11 = m D−1

1 + H−11

]∝ exp

{− 1

2

[− β′ (D−1

1 β + H−11 h1)− (D−1

1 β + H−11 h1)

′ β + β′ U−11 β +

(D−11 β + H−1

1 h1)′ U1 (D−1

1 β + H−11 h1)

]}[β′ (D−1

1 β + H−11 h1) is a scalar, and so is β′(D−1

1 β + H−11 h1) =

(D−11 β + H−1

1 h1)′β. Proportionality follows because (D−1

1 β + H−11 h1)

′ U1

(D−11 β + H−1

1 h1) does not depend on β]

= exp{− 1

2

[β − U1(D−1

1 β + H−11 h1)]

′ U−11 [β − U1(D−1

1 β + H−11 h1)

]},

19

which is proportional to the pdf of a trivariate normal distribution with mean vec-tor U1

(D−1

1 β + H−11 h1

)and covariance matrix U1. Therefore,

β| . ∼MVN(U1

(D−1

1 β + H−11 h1

), U1

).

Full Conditional for D−11

Picking out the terms in the joint density which involve D1, we get

π(D−11 | .) ∝

m∏i=1

MVN (βi| β,D1)W(D−11 | ν1, (ν1A1)

−1)

∝m∏i=1

[1

|D1|1/2exp

{− 1

2(βi − β)′ D−1

1 (βi − β)}]

|D−11 |

ν1−3−12 exp

{− ν1

2tr(A1D−1

1 )}

=1

|D1|m/2exp

{− 1

2

m∑i=1

(βi − β)′ D−11 (βi − β)

}

|D−11 |

ν1−3−12 exp

{− ν1

2tr(A1D−1

1 )}

= |D−11 |

m+ν1−3−12 exp

{− 1

2

[ m∑i=1

(βi − β)′ D−11 (βi − β) + tr(ν1A1D−1

1 )]}

= |D−11 |

(m+ν1)−3−12 exp

{− 1

2

[ m∑i=1

tr((βi − β) (βi − β)′ D−1

1

)+ tr(ν1A1D−1

1 )]}

[if we let d1i = (βi − β)′ D−1

1 and d2i = (βi − β), then d1id2i = tr(di1di2) =

tr(d2id1i) by the property of the trace of a matrix]

= |D−11 |

(m+ν1)−3−12 exp

{− 1

2

[tr( m∑i=1

(βi − β) (βi − β)′ D−11

)+ tr(ν1A1D−1

1 )]}

= |D−11 |

(m+ν1)−3−12 exp

{− 1

2

[tr( m∑i=1

(βi − β) (βi − β)′ + ν1A1

)D−1

1

]},

which is proportional to a Wishart pdf with df m+ν1 and scale matrix[∑m

i=1 (βi − β)

(βi − β)′ + ν1A1

]−1. Therefore,

D−11 | . ∼ W

(m+ ν1,

[ m∑i=1

(βi − β) (βi − β)′ + ν1A1

]−1).

20

Full Conditional for σ−2ui

Picking out the terms in the joint density which involve σ2ui, we get

π(σ−2ui | .) ∝MVN (Y

(2)i | µi, σ

2ui Ii) G

(σ−2ui

∣∣ a2,ab

2

)∝ 1

|σ2ui Ii|1/2

exp{− 1

2σ2ui

(y

(2)i − µi

)′(y

(2)i − µi

)}G(σ−2ui

∣∣ a2,ab

2

)=

1

|σ2ui Ii|1/2

exp{− 1

2σ2ui

(zi − Xi βi

)′(zi − Xi βi

)}G(σ−2ui

∣∣ a2,ab

2

)∝(σ−2ui

)ni−p2 exp

{− 1

2σ2ui

(zi − Xi βi

)′(zi − Xi βi

)}(σ−2ui

)a2−1 exp

(− ab

2σ−2ui

)=(σ−2ui

)ni−p+a2−1 exp

{−(zi − Xi βi

)′(zi − Xi βi

)+ ab

2σ−2ui

},

which is proportional to a gamma pdf with shape parameter (ni−p+a)/2 and rateparameter

[(zi − Xi βi

)′(zi − Xi βi

)+ ab

]/2. Therefore,

σ−2ui | . ∼ G

(ni − p+ a

2,

(zi − Xi βi

)′(zi − Xi βi

)+ ab

2

).

Full Conditional for φ

Picking out the terms in the joint density which involve φ, we get

π(φ| .) ∝m∏i=1

MVN (y(2)i | µi, σ

2ui Ii) MVN (φ| h3, H3)

∝m∏i=1

exp{− 1

2σ2ui

(y

(2)i − µi

)′ (y

(2)i − µi

)}exp

{− 1

2(φ− h3)

′ H−13 (φ− h3)

}.

Here, for j = p+ 1, p+ 2, . . . , ni,

yij − µij = yij − β0i

(1−

p∑k=1

φk

)− β1i xij − β2i rij −

p∑k=1

φ yi,j−k

= yij −p∑

k=1

φ yi,j−k − β0i

(1−

p∑k=1

φk

)− β1i

(tij −

p∑k=1

φk ti,j−k

)−

β2i

(qij −

p∑k=1

φk qi,j−k

)

21

= (yij − β0i − β1i tij − β2i qij)− φ1 (yi,j−1 − β0i − β1i ti,j−1 − β2i qi,j−1)−

φ2 (yi,j−2 − β0i − β1i ti,j−2 − β2i qi,j−2)− · · ·−

φp (yi,j−p − β0i − β1i ti,j−p − β2i qi,j−p)

= εij − φ1 εi,j−1 − φ2 εi,j−2 − . . . − φp εi,j−p

where εij = yij − β0i − β1i tij − β2i qij for j = p + 1, p + 2, . . . , ni. In vector-matrixnotation, y

(2)i − µi = εi −Wiφ, where εi = (εi,p+1, εi,p+2, . . . , εi,ni)

′, and Wi is a(ni − p)× 3 matrix as given in Equation (15). Thus,

π(φ| .) ∝m∏i=1

exp{− 1

2σ2ui

(y

(2)i − µi

)′ (y

(2)i − µi

)}exp

{− 1

2(φ− h3)

′ H−13 (φ− h3)

}=

m∏i=1

exp{− 1

2σ2ui

(εi −Wi φ

)′ (εi −Wi φ

)}exp

{− 1

2(φ− h3)

′ H−13 (φ− h3)

}= exp

{− 1

2

[ m∑i=1

σ−2ui

(εi −Wi φ

)′ (εi −Wi φ

)+ (φ− h3)

′ H−13 (φ− h3)

]}

= exp{− 1

2

[ m∑i=1

σ−2ui

(ε′i εi − ε′i Wi φ− φ′ W′

i εi + φ′ W′i Wi φ

)+ φ′ H−1

3 φ −

φ′ H−13 h3 − h′3 H−1

3 φ + h′3 H−13 h3

]}∝ exp

{− 1

2

[ m∑i=1

σ−2ui

(− 2 φ′ W′

i εi + φ′ W′i Wi φ

)+ φ′ H−1

3 φ− 2φ′ H−13 h3

]}[ε′i Wi φ and h′3 H−1

3 φ are sclars, and so are ε′i Wi φ = φ′ W′i εi, h′3 H−1

3 φ =

φ′ H−13 h3. Proportionality follows because σ−2

ui ε′i εi and h′3 H−1

3 h3 do not depend on φ]

= exp{− 1

2

[− 2 φ′

m∑i=1

σ−2ui W′

i εi + φ′( m∑i=1

σ−2ui W′

i Wi

)φ + φ′ H−1

3 φ−

2φ′ H−13 h3

]}= exp

{− 1

2

[− 2 φ′

( m∑i=1

σ−2ui W′

i εi + H−13 h3

)+ φ′

( m∑i=1

σ−2ui W′

i Wi + H−13

)φ]}

= exp{− 1

2

[− 2 φ′

( m∑i=1

σ−2ui W′

i εi + H−13 h3

)+ φ′ V−1 φ

]}

22

[where V−1 =

m∑i=1

σ−2ui W′

i Wi + H−13

]∝ exp

{− 1

2

[− φ′

( m∑i=1

σ−2ui W′

i εi + H−13 h3

)−( m∑i=1

σ−2ui W′

i εi + H−13 h3

)′φ +

φ′ V−1 φ +( m∑i=1

σ−2ui W′

i εi + H−13 h3

)′V( m∑i=1

σ−2ui W′

i εi + H−13 h3

)]}[φ′( m∑i=1

σ−2ui W′

i εi + H−13 h3

)=( m∑i=1

σ−2ui W′

i εi + H−13 h3

)′φ because

φ′( m∑i=1

σ−2ui W′

i εi + H−13 h3

)is a scalar. Proportionality follows because

( m∑i=1

σ−2ui W′

i εi + H−13 h3

)′V( m∑i=1

σ−2ui W′

i εi + H−13 h3

)does not depend on φ

]= exp

{− 1

2

[φ− V

( m∑i=1

σ−2ui W′

i εi + H−13 h3

)]′V−1

[φ− V

( m∑i=1

σ−2ui W′

i εi + H−13 h3

)]},

which is proportional to the pdf of a p-variate normal distribution with mean vec-tor V

(∑mi=1 σ

−2ui W′

i εi + H−13 h3

)and covariance matrix V. Therefore,

φ| . ∼MVN(

V( m∑i=1

σ−2ui W′

i εi + H−13 h3

), V).

Full Conditional for αi

Picking out the terms in the joint density which involve αi, we can express the fullconditional for αi only up to a proportionality constant, that is,

π(αi| .) ∝1

γiτiexp

{− 1

2σ2ui

(zi − Xi βi)′(zi − Xi βi)

}×

exp{− 1

2(α∗i −α∗)′ D−1

2 (α∗i −α∗)}.

References

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