Communications in Statistics—Theory and Methods, 34: 229–244, 2005 Copyright © Taylor & Francis, Inc. ISSN: 0361-0926 print/1532-415X online DOI: 10.1081/STA-200045817
Statistical Quality Control
On Statistical Design of the S 2 Control Chart LINGYUN ZHANG, M. S. BEBBINGTON, C. D. LAI, AND K. GOVINDARAJU Institute of Information Sciences and Technology, Massey University, Palmerston North, New Zealand The usual S 2 chart is constructed with equal false alarm probabilities for the lower and upper control limits. An alternative is the ARL-unbiased S 2 chart, where the ARL (average run length) curve attains its maximum when the common-cause variance is at its in-control value. We examine the ARL properties of these S 2 charts, and show that neither the usual S 2 chart nor the ARL-unbiased S 2 chart is optimal in a natural sense. We propose two criteria, under which optimal S 2 charts can be constructed. The effects of parameter estimation on our designs are also discussed, and suggestions are made regarding practical implementation. Keywords ARL-unbiased; Average run length; S 2 chart; Shewhart S chart. Mathematics Subject Classification 62P30.
1. Introduction The Shewhart S chart based on three-sigma control limits for monitoring the standard deviation of the process quality characteristic is commonly used in industry. The statistical disadvantages of this standard S chart are well known. For example, the central line is not an unbiased estimate of the true process . The use of three-sigma limits is approximate and often leads to a lower control limit of zero when the subgroup size (also called sample size) is small, of the order of five. Hence the standard S chart cannot detect any quality improvement or reduction of process variation. Statisticians advocate the use of probability limits in control chart design, and recommend the use of the S 2 chart in place of the Shewhart S chart. One may also design an S chart with probability limits (see, for example, Ryan, 1989). Unfortunately, the exact evaluation of the average run length (ARL) and derivation of other results for the S chart are less tractable than for the S 2 chart. In this article, we consider in detail the statistical properties of the S 2 chart. Received November 20, 2002; Accepted April 24, 2004 Address correspondence to Lingyun Zhang, Institute of Information Sciences and Technology, Massey University, Palmerston North, New Zealand; E-mail:
[email protected]
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Let X be a characteristic of the production process that is of interest. To monitor changes in the variance of X, we have sequential samples, xt1 xt2 xtn , where t = 1 2 and n is the sample size, e.g., n = 5. We assume that all the observations in and between samples are independent, and that xti has a normal distribution, for t ≥ 1 i = 1 2 n. Let x¯ t = n1 ni=1 xti be the sample mean at stage t. If we further assume that the variance of xti is equal to 02 , which is known (later we will discuss the estimation of 02 and its effects), the S 2 control chart can be constructed by plotting sample variances St2 =
n 1 x − x¯ t 2 t = 1 2 n − 1 i=1 ti
with lower control limit (LCL), center line, and upper control limit (UCL) set as 02 h n−1 1 Center line = 02 and LCL =
UCL =
02 h n−1 2
where 0 < h1 < h2 are control limit constants that must be specified, such that n − 1St2 (1.1) < h2 = 1 − Pr h1 < 02 in which is a prefixed false alarm rate, e.g., 00027. Statistical design of the S 2 chart denotes the choice of h1 and h2 that ensure the chart performance meets the statistical criteria: (C1) When the process variance is 02 , the ARL of the S 2 chart is equal to (or not less than) some prefixed value, e.g., 370, and (C2) If the variance has shifted to another value, the ARL should be small. Note that satisfying (1.1) is equivalent to the first criterion (C1). Because there are two variables, (1.1) alone is not sufficient to identify h1 and h2 . In the literature (see, for example, Montgomery, 2001), h1 and h2 are determined through the equations n − 1St2 = < h (1.2) Pr 1 2 02 and
n − 1St2 > h2 Pr 02
=
2
(1.3)
In the following, the design given by (1.2) and (1.3) will be called the standard design. Obviously, the h1 and h2 in the standard design satisfy (1.1). However, we are dealing with variances, and n−1 n − 1St2 2 ∼ n−1 = 2 (1.4) 2 02
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which is not a symmetric distribution. Determining h1 and h2 through (1.2) and (1.3) allocates equal tail probabilities. Thus the asymmetry translates into very different performance in detecting variance increases and decreases. In the light of (C2), this is not the best strategy. This opportunity to “improve” on the standard design is our major motivation to study the construction of the S 2 chart. A control chart is said to be ARL-unbiased (Pignatiello et al., 1995) if its ARL curve attains its maximum when the process parameter is equal to its in-control value. Pignatiello et al. (1995) showed that the standard S chart is ARL-biased, and presented a method of designing an ARL-unbiased S chart, arguing that the ARLunbiased S chart has a better overall ARL performance in detecting increases and decreases in the process variance. The initial object of this article was to design ARL-unbiased S 2 charts, and our study prompted us to suggest two criteria for measuring chart performance, under which optimal designs of the S 2 chart are feasible. In the next section we investigate the ARL properties of the S 2 chart. Based on this study, we will propose the smallest area criterion and the two points criterion, under which we will construct optimal designs of the S 2 chart. The ‘effects of estimation of parameter(s)’ is a general and, in practice, an important problem in the study of control charts (Woodall, 1997; Woodall and Montgomery, 1999). Recent work on the effects for various control charts includes Quesenberry (1993), Chen (1997, 1998), Jones et al. (2001), and Jones (2002). In Sec. 3 we will discuss the effects of estimation of 02 on the design of the S 2 chart. In our concluding remarks we will touch on the performance of the chart in a nonidealized setting. The Appendix contains proofs of the results and other relevant technical details.
2. Optimal Designs In order to deal with both in-control and out-of-control cases, we will write the variance of X as 2 = 02 , where > 0. If = 1, the process is in control, otherwise the process has shifted. Thus the average run length, ARL , is a function of . ARL1 is the in-control ARL, and ARL , = 1, is the out-of-control ARL. Let us first define what is meant by a design. We say that h1 h2 is a design, if: (D1) 0 < h1 < h2 < ; and (D2) the equation
n − 1St2 Pr h1 < < h2 = 1 = 1 − 02
(2.1)
holds. Note that (1.4) implies that (2.1) is equivalent to
h2 /2
h1 /2
1 s−1 −u u e du = 1 − s
(2.2)
where s ≡ n − 1/2, and n is the sample size. We are now able to identify certain properties of the ARL curve for a given design.
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Theorem 1. Let h1 h2 be a design and set ∗ =
h1 − h2 2s lnh1 − lnh2
(2.3)
Then ARL has the following properties: (i) ARL is monotonically increasing, if < ∗ ; (ii) ARL is monotonically decreasing, if > ∗ ; and (iii) ARL attains its maximum at ∗ . We have thus arrived at two equations that determine the design of the ARLunbiased S 2 chart: h2 /2 1 s−1 −u u e du = 1 − h1 /2 s (2.4) h1 − h 2 = 1 2s lnh1 − lnh2 Discussion of the solution of (2.4) is deferred until the Appendix, but we will demonstrate that a unique solution exists. Lemma 1. Suppose that h1 h2 is a design and that ∗ is defined as in (2.3). Then (i) ∗ increases as h1 increases; (ii) ∗ → 0, if h1 → 0; and (iii) ∗ → +, if h2 → +. Because of the monotonicity over the range of ∗ stated in the Lemma, the system of equations (2.4) has a unique solution, h1 h2 , which gives the ARLunbiased design of the S 2 chart. Due to the constraint (2.2), all designs, including the standard design, ARLunbiased design, and any other possible designs, give the chart the same in-control ARL. The “performance” of the chart therefore derives from differences in the outof-control ARL performance of charts resulting from different designs. The next theorem encapsulates the relative performance of different charts. Theorem 2. Suppose we have two designs h1 h2 , and k1 k2 . Let ARL h1 h2 denote the ARL function of the design h1 h2 , and ARL k1 k2 that of the design
k1 k2 . If h1 < k1 , then ARL h1 h2 > ARL k1 k2 for 0 < < 1 and ARL h1 h2 < ARL k1 k2 for > 1 Theorem 2 shows that there is no design that provides the S 2 chart with uniformly smaller out-of-control ARLs than any other S 2 chart. If h1 h2 is a generic design, then a small h1 (note that by (2.2), h2 is determined once h1 is given)
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makes the chart sensitive to increases in process variance, while large h1 makes the chart sensitive to decreases in process variance. We can therefore ask whether the standard design or the ARL-unbiased design is optimal. And, if it is not optimal, what is an optimal design? However, in order to answer these questions, we need some criteria from which we can evaluate chart performance. First we need to do some preparation. Let us define L/2 1 s−1 −x −1 x e dx for 0 < ≤ 1 s ARL∗ = 0 R/2 −1 1 s−1 −x x e dx for > 1 1− s 0 where L and R are specified such that:
L/2
0
1 s−1 −x x e dx = s
and 0
R/2
1 s−1 −x x e dx = 1 − s
Thus ARL∗ has two components. The left ( < 1) part corresponds to the ARL function with the design h1 = L, which is a limiting case (detection only of variance decrease) — note that in such a case h2 → +. The right ( > 1) part corresponds to the ARL function with the design in which h1 = 0, which is also a limiting case (detection only of variance increase), since a two-sided S 2 chart does not allow h1 = 0. According to Theorem 2, for any design h1 h2 such that 0 < h1 < L, the inequality ARL h1 h2 > ARL∗ for > 0 and = 1 holds. Plotting ARL∗ against gives a curve which we call the limiting ARL curve. We can now suggest the following two testable criteria, that are consistent with (C1) and (C2). The first criterion is to derive the optimal design h1 h2 by minimizing ARL h1 h2 − ARL∗ d 0
with h1 and h2 restricted by (2.2). The idea is to use the area between the ARL curve for a possible design and the limiting ARL curve defined above to measure the out-of-control ARL performance. The smaller this area, the better the design performance. We call this criterion the smallest area criterion (SAC). The existence and finiteness of the area is attested by the following result. Lemma 2. If n ≥ 4, then 0
ARL h1 h2 − ARL∗ d <
(2.5)
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Zhang et al. Table 1 Various designs when n = 5, 10 and = 00027. The is for TPC designs n=5
Standard design
SAC design TPC design
005 010 015 020 025 030 040 050
ARL-unbiased design
n = 10
h1 0105767
h2 17800408
h1 1241253
h2 27093130
0126285 0128318 0128385 0128499 0128659 0128857 0129080 0129468 0129466 0136461
18992895 19169905 19176070 19186529 19201290 19219849 19240788 19277918 19277746 20107771
1328989 1335738 1336468 1337582 1338879 1340018 1340543 1337647 1325783 1369116
28072043 28174667 28186091 28203680 28224341 28242666 28251161 28204707 28025147 28778203
The second criterion is to determine the optimal design h1 h2 by minimizing ARL1 − h1 h2 + ARL1 + h1 h2 with h1 and h2 restricted by (2.2). The parameter 0 < < 1, representing a characteristic scale in the variance shift, must be provided by the practitioner, according to their specific
situation. The idea behind
the criterion is a natural one. We choose two points, 1 − ARL1 − and 1 + ARL1 + , and try to pull down the ARL curve from the two points. This criterion we call the two points criterion (TPC). Using SAC and TPC, we give the optimal designs for n = 5 10 and = 00027 in Table 1. We used MSIMSL routines DQDAGS and DQDAGI to handle the integral in the SAC and then approach the minimum in a manual way. The MSIMSL routine DUVMIF deals with the minimization in the TPC. We also show the ARL curves for the standard design, the SAC design, the TPC designs with = 005 and 05, and the ARL-unbiased design, as well as the limiting ARL curve, in Figs. 1 and 2. The figures show that the curves for the SAC and TPC designs are quite close to each other, and are visibly different from the curves for the standard design and the ARL-unbiased design. The closeness indicates that the SAC and the TPC are similar, while the differences indicate that the standard design and ARLunbiased design are not optimal according to our criteria.
3. Effects of Estimation of 02 The statistical properties of a control chart procedure are commonly evaluated using the ARL function. The concept of ARL is similar to that of the power or OC function in the statistical inference literature. Like the power function, the ARL is a function of the true process parameters (such as the mean and the process variance 2 ) and the control chart parameters (such as the subgroup size n, and the control limit constants). This theoretical notion of a function dependent on
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Figure 1. ARL curves for various designs when n = 5, = 00027. TPC 1 and TPC 8 are the TPC designs with = 005 and 05, respectively.
unknown parameters poses practical problems in the actual implementation of the control chart, because the true process parameters are unknown and are estimated using retrospective in-control data. The run length random variable often involves more variability but unfortunately little research has been done on dealing with the variance of the run length for many control charts (see Ryan, 1997, for a discussion). So a practical compromise is to ensure that the chart has good ARL properties at the estimated values rather than at the true unknown values. We shall call this ARL the conditional ARL, which is the expected number of subgroups to a signal given the estimated process parameters. In practice, to set up an S 2 chart we must first estimate 02 . We shall review the details of this estimation, and then follow the approach of Chen (1998) in investigating the effect of the estimation on the ARL. We will assume that m subgroups of data, y11 y12 y1n y21 y22 y2n · · · ; ym1 ym2 ymn are available. Note that in order to distinguish this parameter estimation stage from the monitoring stage described above, we use yij instead of xij . In the literature (see, for example Montgomery, 2001), 02 is estimated by ˆ 02 =
m n 1 y − y¯ i 2 mn − 1 i=1 j=1 ij
(3.1)
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Figure 2. ARL curves for various designs when n = 10, = 00027. TPC 1 and TPC 8 are the TPC designs with = 005 and 05, respectively.
where y¯ i = n1 nj=1 yij . Boyles (1997) studied other estimation methods which provide protection against symptoms of special causes, such as outliers, mean shifts, and trends, with results that are useful in practice. In our context, we will assume that the data used to estimate 02 has been confirmed to be “clean”, i.e., the data are obtained when the process is in the state of control, and we focus our attention on the estimator (3.1). With our assumptions, mn − 1ˆ 02 /02 has a 2 distribution with mn − 1 degrees of freedom, which is equivalent to the distribution mn − 1/2 2. Thus, ˆ 02 2 mn − 1 ∼ 2 mn − 1 02 Using ˆ 02 , the control limits become ˆ 02 h and n−1 1 2 = ˆ 0 h2 UCL n−1 = LCL
and monitoring proceeds as usual.
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We will retain our definition that the process variance 2 = 02 , so that when = 1 the process is in control, otherwise it has shifted. In the previous section we dealt with n − 1St2 /02 , now we turn to n − 1St2 /ˆ 02 . We first observe that n − 1St2 n − 1St2 2 = 2 ˆ 02 ˆ 02 =
n − 1St2 2 /02 2 ˆ 02 /02
=
n − 1St2 2 W
where W ≡ ˆ 02 /02 ∼ mn − 1/2 2/mn − 1. Since n − 1St2 /2 ∼ n − 1/2 2, if we condition on W = w we have n − 1St2 n − 1 2 W = w ∼ (3.2) 2 w ˆ 02 Compare this with the distribution when 02 is known (without estimation), n − 1St2 n−1 ∼ 2 2 02
(3.3)
Let ARLc denote the ARL function conditioned on W = w. From (3.2) and (3.3), we see that ARLc = ARL w where ARL = 1 −
h2 /2 h1 /2
1 s−1 −x x e dx s
−1
is the ARL function when 02 is known, which has been discussed in detail in the previous section. Let ARLu be the unconditioned ARL function, that is to say the ARL function on which all possible estimates have been averaged out. Then ARLu = ARLc kwdw 0 = ARL kwdw (3.4) w 0 where kw =
aa a−1 −aw w e a
(a = mn − 1/2) is the density function of W . The integral (3.4) is analytically intractable, hence we use the MSIMSL routine DQDAGI to calculate the integral numerically.
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Chen (1998) studied the run length distributions of S 2 charts (using the standard design) when 02 is estimated. He reported that the traditional rule-of-thumb—20 to 30 samples of size 4 or 5—for estimation of 02 to set up a chart is not adequate, and suggested taking at least 75 samples of size 5. Using the SAC design, we plot ARLu against for n = 5, = 00027, and various m (m = is for the case 02 known), in Fig. 3. We see from Fig. 3 that, when the SAC design is used, the negative effects from the estimation of 02 are two-fold. Firstly, the estimation may result in more false alarms. Suppose that the target value of the process variance is one. If the true process variance lies in 09 11 , one might regard the process as being in a state of control. Equivalently, for ∈ 09 11 , we see from Fig. 3 that all the ARL curves, and in particular the curves for small m, lie below the ARL curve for m = , and thus more false alarms may occur. The second problem is that the chart may not signal as quickly that the process variance has shifted as the chart without estimation. This is particularly acute when attempting to detect upward shifts in variance of the order of 50%. We also note that the ARL values at = 1 in Fig. 3 are monotonic in m. Thus we can solve the problem of more false alarms in practice by choosing a larger nominal ARL, depending on m. However, this will exacerbate the problem of slow signaling. The choice of m requires balancing the responsiveness of the chart against the practicalities of collecting stable retrospective data. When using the SAC design,
Figure 3. ARL curves for SAC design when n = 5, = 00027. The case of known 02 is denoted by m = Infinity.
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in order to make the S 2 chart with estimated 02 perform essentially like that with known 02 in an average sense, we would make the following suggestions on the number of samples: Take at least 30 samples of size 5 to set up the trial control limits; take at least 50 samples of size 5 to set up the permanent control limits. Since the TPC is similar to the SAC, the same conclusion applies to the TPC designs.
4. Concluding Remarks We have shown that the standard design and the ARL-unbiased design are not optimal in a natural sense. By providing the smallest area and two points criteria, we enable optimal S 2 charts to be constructed, with results for n = 5 10 and = 00027 summarized in Table 1. We also indicate a suitable number of samples required, for calculation of control limits in the SAC and TPC designs, in order to control the effects of estimation of 02 . The results derived in the previous sections require that the process quality characteristic is exactly normally distributed. The distributional properties, such as S 2 following a scaled chi-square distribution, rely on this assumption of normality. The discussion on the effects of estimation of 02 also relies on clean retrospective data completely free from special causes. No production process will ever be exactly normal, and completely stable (free from special causes). The presence of unknown or undiscovered special causes in the data (that may include extra components of both within- and between-sample variation) will lead to a slightly larger value for the true unknown process variance for the common causes. Figures 1 and 2 establish that the standard design involves a larger ARL than the ARL-unbiased design when < 1 but has a smaller ARL when > 1. In Sec. 3, we have seen that estimation of 02 leads to a marginally smaller conditional ARL. Combining these observations, we anticipate that the S 2 chart will not perform that badly in the presence of contaminated retrospective data. In industrial applications, the Shewhart S chart with three-sigma control limits is more frequently used for monitoring the process standard deviation. The statistical disadvantages of the S chart are well known, and hence some authors advocate the use of probability limits control chart design for S charts. The ideas in this work provide a useful starting point for probability limits control chart design for S charts, with the possibility of deriving analogous results.
Appendix First, let us recall our notation. Let s ≡ n − 1/2, where n is the sample size. Since normality and independence are assumed, and the process variance is assumed to be 2 = 02 , > 0, n − 1St2 n − 1St2 2 n − 1St2 = = 2 2 2 2 0 0 has a Gamma distribution with shape parameter s and scale parameter 2 . Let Fx ≡
0
x
1 s−1 −u u e du s
and use fx = F x to denote the derivative of Fx.
(A.1)
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Proof of Theorem 1. Since the run length of an S 2 chart is geometrically distributed, ARL =
1 1 − g
where n − 1St2 < h g = Pr h1 < 2 02 h2 1 = xs−1 e−x/2 dx s h1 s2 h2 /2 1 = ys−1 e−y dy h1 /2 s h2 h1 =F −F 2 2 Noting that ARL and g have the same monotonicity and attain their maxima at the same value of , to prove Theorem 1 we only need to discuss the increasingdecreasing and maximum properties of g . The derivative of g is h h h1 h2 g = f − 22 − f − 12 2 2 2 2 =
1 l 2s s+1 s
where l ≡ hs1 e−h1 /2 − hs2 e−h2 /2 Setting g = 0 we get the solution ∗ =
h1 − h 2 2slnh1 − lnh2
Note that the sign (positive or negative) of g is determined by l , and that the sign of l is determined by
q ≡ ln hs1 e−h1 /2 − ln hs2 e−h2 /2
1 = s lnh1 − lnh2 + h − h1 2 2 Since h2 > h1 , it is easy to show that ∗ > 0 if < q = 0 if = ∗ < 0 if > ∗ With the sign of g available, the proof is complete.
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Proof of Lemma 1. Let F −1 · be the inverse of F· in (A.1). The constraint (2.2) yields
h F 2 2
h −F 1 2
= 1 −
from which we obtain h2 = 2F −1 1 − + Fh1 /2 ≡ Gh1 Since h1 h2 constitutes a design, h1 must be positive. The constraint (2.2) implies that h1 < 2F −1 , h1 < h2 , and that h2 increases as h1 increases, that is, h1 < Gh1 and G h1 > 0. Using Gh1 , we can rewrite ∗ as ∗ h1 =
h1 − Gh1 2s lnh1 − lnGh1
The derivative of ∗ h1 is then
∗ h1 =
h 1
1−G h1 lnh1 −lnGh1 − h1 −Gh1 h1 − GGh 1
1
2s lnh1 −lnGh1 2
the numerator of which,
Gh1 h1 Gh1 h1 −1 +G h1 −ln −ln −1 > 0 h1 h1 Gh1 Gh1
1 because G h1 > 0, Gh > 1, and the function Tu = u−lnu−1 > 0, h1 for u > 0 and u = 1 Therefore, ∗ increases as h1 increases. It is now easily shown that, if h1 → 0, then ∗ → 0; and if h2 → +, then h1 → 2F −1 , thus ∗ → +. The proof is complete.
Solution of Eq. (2.4). Let us define a new variable 0 < h1 < h2 0 < k < 1. The second equation in (2.4) gives h1 =
k = h1 /h2 .
Since
2sklnk k−1
and h2 =
2slnk k−1
Substituting these new expressions for h1 and h2 into the first equation in (2.4) gives an equation with respect to k, which can be solved numerically. We can then obtain h1 and h2 from k.
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Proof of Theorem 2. First let us introduce variables 1 and 2 , where 1 > 0, 2 > 0, and 1 +2 = , into which we will transform the result. So (2.2) is equivalent to h1 F 2 = 1 h2 = 1−2 = 1−+1 F 2 and thus
h1 = 2F −1 1 h2 = 2F −1 1−+1
Defining p1 = F
−1 F −1 1−+1 F 1 −F for 0 < 1 <
we note that p· is similar to the g· defined in the proof of Theorem 1. In fact, 1−p1 also gives the tail probability, and the ARL is the reciprocal of 1−p1 . Different 1 denote different designs, and a small 1 corresponds to a small h1 . To prove Theorem 2, it is equivalent to prove that: p1 is monotonically increasing as 1 increases, if > 1, and p1 is monotonically decreasing as 1 increases, if 0 < < 1. Again, we resort to differentiation. The derivative of p1 is
F −1 1−+1 1 1 p 1 = f −1 f F 1−+1 −1 F 1 1 1 −f f F −1 1 −1 −1 F 1−+1 F 1 −1 −f f F f F −1 1−+1 f 1 1 = f F −1 1 f F −1 1−+1
1 xs−1 e−x .) The numerator in p 1 becomes (Recall that fx = s
C
F −1 1 F −1 1−+1
s−1
e−I1 −e−I2
where C = 1/s2 , I1 = F −1 1 +
F −1 1−+1
and I2 =
F −1 1 +F −1 1−+1
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At this point we see that the sign of p 1 is determined by −I1 −−I2 = I2 −I1 . Now −1 F −1 1−+1 −F −1 1 I2 −I1 = since h2 > h1 , F −1 1−+1 −F −1 1 > 0. Thus the sign of I2 −I1 is determined by −1, which completes the proof. Proof of Lemma 2. Since ARL∗ has two parts, 0
ARL h1 h2 −ARL∗ d = +
1
1 0
ARL h1 h2 −ARL∗ d
ARL h1 h2 −ARL∗ d
To obtain (2.5), we need only prove that 1
ARL h1 h2 −ARL∗ d <
(A.2)
Note that, for > 1, ARL h1 h2 =
1 h2 h1 1− F 2 −F 2
and that ARL∗ =
1 R 1−F 2
Let > 0 be a constant to be specified, and consider I3 = lim →+ 1+ ARL h1 h2 −ARL∗ Using l’Hôpital’s rule, we can show that I3 = lim →+ D −s+1+ where D > 0 is a constant. Because n > 3, s = n−1/2 > 1, i.e., −s +1 < 0, and we can choose a small such that −s +1+ < 0, resulting in I3 = 0. Therefore (A.2) holds, and the proof is complete.
Acknowledgments The authors thank the referee for the suggestions for improving the presentation. The first author thanks Dr. Jinhong You for his assistance in software; his research is supported by a Massey University Postdoctoral Fellowship.
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