Abstract
Background
Results from clinical trials are usually summarized in the form of sampling distributions. When full information (mean, SEM) about these distributions is given, performing metaanalysis is straightforward. However, when some of the sampling distributions only have mean values, a challenging issue is to decide how to use such distributions in metaanalysis. Currently, the most common approaches are either ignoring such trials or for each trial with a missing SEM, finding a similar trial and taking its SEM value as the missing SEM. Both approaches have drawbacks. As an alternative, this paper develops and tests two new methods, the first being the prognostic method and the second being the interval method, to estimate any missing SEMs from a set of sampling distributions with full information. A merging method is also proposed to handle clinical trials with partial information to simulate metaanalysis.
Methods
Both of our methods use the assumption that the samples for which the sampling distributions will be merged are randomly selected from the same population. In the prognostic method, we predict the missing SEMs from the given SEMs. In the interval method, we define intervals that we believe will contain the missing SEMs and then we use these intervals in the merging process.
Results
Two sets of clinical trials are used to verify our methods. One family of trials is on comparing different drugs for reduction of low density lipprotein cholesterol (LDL) for Type2 diabetes, and the other is about the effectiveness of drugs for lowering intraocular pressure (IOP). Both methods are shown to be useful for approximating the conventional metaanalysis including trials with incomplete information. For example, the metaanalysis result of Latanoprost versus Timolol on IOP reduction for six months provided in [1] was 5.05 ± 1.15 (Mean ± SEM) with full information. If the last trial in this study is assumed to be with partial information, the traditional analysis method for dealing with incomplete information that ignores this trial would give 6.49 ± 1.36 while our prognostic method gives 5.02 ± 1.15, and our interval method provides two intervals as Mean ∈ [4.25, 5.63] and SEM ∈ [1.01, 1.24].
Conclusion
Both the prognostic and the interval methods are useful alternatives for dealing with missing data in metaanalysis. We recommend clinicians to use the prognostic method to predict the missing SEMs in order to perform metaanalysis and the interval method for obtaining a more cautious result.
Background
Clinical trials are widely used to test new drugs or to compare the effect of different drugs [2]. For example, many clinical trials have been carried out to investigate the efficacy of drugs for lowering intraocular pressure, such as travoprost, bimatoprost, timolol, and latanoprost, [311], and many to investigate the oral diabetes medication for adults with Type2 diabetes [1217]. Given that there is a huge number of trials available and reports on trials are very time consuming to read and understand, a systematic review of related trials is useful for medical practioners or other health care professionals to obtain an overall estimation about drugs/therapies of interest. Metaanalysis is the technique commonly used to summarize related trials results. Statistically, metaanalysis is in effect a process for merging sampling distributions (see its explanation in Preliminaries) into a single distribution.
When the full information of sampling distributions used in the trials is available, merging the results from these trials is usually a matter of systematic use of established techniques from statistics. However, in reality, some trial results reported in the literature are statistically incomplete. For instance, the standard error of mean (SEM) may be missing from a sampling distribution. A clinical trial with incomplete information is often abandoned, or, an SEM from a similar trial is used to substitute the missing data. Obviously, abandoning trials will decrease the power of a metaanalysis since trials that are eligible for analysis cannot be taken into account. On the other hand, if these trials are considered and a missing SEM is replaced by a substitute SEM, two major difficulties of this approach are inevitable. First finding a similar trial can be very difficult. Second there is no guarantee that there exists a similar trial with full information. Furthermore, measuring the similarity between trials is another issue that must be addressed. Therefore, an appropriate method is urgently needed to deal with sampling distributions with missing information when considering metaanalysis.
In this paper, we propose a prognostic method where missing SEMs are predicted from known information (SEMs) and an interval method where an interval is calculated in which the missing SEMs are assumed to be. These two methods are based on the assumption that the sampling distributions to be analyzed and merged are from the same population by random selection. In a sense, the two methods can be viewed as extremes. The prognostic method uses an estimate of the missing SEM, while the interval method aims at considering the best and worse cases. Of course, there are other reasonable intermediate procedures, for example, a Bayesian predictive approach is a very natural tool for dealing with missing information.
Significantly, our methods can also be used in the process of merging between group differences (almost all metaanalysis performs merging of between group differences). In fact, our methods can be applied to any results obtained from clinical trials, as long as these results (regarded as random variables) theoretically follow a sampling distribution.
Methods
Study examples
The study examples were drawn from two systematic reviews [1,17]. [17] reported a metaanalysis on drugs for patients with Type2 diabetes and [1] reported a metaanalysis on lowering intraocular pressure for patients with open angle glaucoma or ocular hypertension.
In order to compare the results obtained from our methods with that from [17], we first attempted to collect all the original information on sampling distributions from each of the clinical trials papers used in [17]. After going through these papers, we found that we could not get the sampling distributions with full information from every single paper, as some of these papers did not provide the SEM values. We then applied our two methods to merge these (including partial) distributions and compared the merged result with that obtained in [17]. In contrast to [17], in [1] every sampling distribution used has full statistical information. To test the adequacy of our methods for dealing with missing information, we randomly selected a trial from [1] and deleted its SEM value. We then applied our two methods to recover this missing SEM value before merging this trial with the rest. Finally, we compared our results with that from [1].
Preliminaries
In statistics, a normal distribution associated with a random variable is denoted as X ~ N (μ, σ^{2}). For the convenience of further calculations in the rest of the paper, we use notation X ~ N (μ, σ) instead of X ~ N (μ, σ^{2}) for a normal distribution of variable X.
In statistics, random samples of individuals are often used as the representatives of the entire group of individuals (often denoted as a population) to estimate the values of some parameters of the population. The mean of variable X of the samples, when the sample size is reasonably large, follows a normal distribution. This distribution is typically referred to as a sampling distribution.
We use X ~ N (μ, SEM) to denote a sampling distribution with mean value μ and standard error of mean SEM.
The basic merging rule for sampling distributions is as follows.
Let X_{1 }~ N (μ_{1}, SEM_{1}) and X_{2 }~ N (μ_{1}, SEM_{2}) be two sampling distributions, then the merged sampling distribution X ~ N (μ, SEM) is given by
Equation 1 can be easily induced from σ_{1 }= σ_{2 }where σ_{i }= SEM_{i} and n_{i }is the number of samples of the ith trial (for i = 1, 2).
Conventionally, clinicians usually use notation , thus Equation 1 can be rewritten as:
It is easy to see that the merging rule is associative, thus this rule can be straightforwardly used to merge multiple sampling distributions. Let X_{i }~ N (μ_{i}, SEM_{i}), 1 ≤ i ≤ k, then for the merged sampling distribution X ~ N (μ, SEM), we have
This is the classical merging method (i.e., metaanalysis) used by clinicians.
The Prognostic Method
We propose a method to predict missing SEMs for sampling distributions from other distributions that have SEM values. Hereafter, we assume that there are k + l trials altogether where k trials are with full information, i.e.,
and l trials with partial information, i.e.,
We want to get the merging result of those k + l trials.
If we have both the SEM value and the sample size n from a trial, then we can obtain σ by equation SEM = . Thus for k trials with
we get σ_{i }= SEM_{i}, 1 ≤ i ≤ k. Because it is assumed that these trials randomly select samples from the same population, these σ_{i}s suggest what the real value σ could be. In fact, from the wellknown Error Theory, we can use
to replace the real σ. When k gets larger, σ* gets closer to the real σ. Therefore, for a sampling distribution without a standard error, we can use as its standard error where is the sample size of this sampling distribution.
To summarize, the prognostic method uses the following equation to predict the missing value for trial j with sample size , given that for k trials, each of which has the SEM_{i }value and the sample size n_{i}.
Then we are able to use the standard metaanalysis method to merge trials with predicted SEM values and trials with full information.
The Interval Method
In contrast to estimating a single value for a missing SEM as discussed above, in this section, we establish a method to estimate a reliable interval for a missing SEM. Then we propose the corresponding merging rule for this case.
For the k trials with
we have σ_{i }= SEM_{i }, 1 ≤ i ≤ k. Let
It is intuitive to consider that the real σ value is in between σ_{min }and σ_{max}. Therefore for a sampling distribution without the standard error SEM*, we have where n* is the sample size of this trial.
It remains for us to investigate how to use this interval value for merging. For simplicity and illustration, first let us consider the case where there is only one trial with a missing SEM. Let μ^{k }denote the weighted average of μ_{1 }to μ_{k }(the k trials with known SEMs), i.e.,
μ^{k }is equivalent to the merged mean value for the k trials in metaanalysis. Let
and
then we have the following result.
Let X_{i }~ N (μ_{i}, SEM_{i}), 1 ≤ i ≤ k, denote the ith sampling distribution with sample size n_{i }and X_{i+1 }~ N (μ_{i+1}, SEM_{i+1}) denote the (k + 1)th sampling distribution with sample size n_{i+1 }where value SEM_{i+1 }is unknown. Then the merged result N (μ, SEM) of the interval method is:
1. If μ_{k+1}≤ μ^{k}, then μ ∈ [, ]
2. If μ_{k+1 }> μ^{k}, then μ ∈ [, ]
and
The corresponding bounds of interval for μ depend on whether μ_{k+1 }is larger or smaller than μ^{k}. If μ_{k+1 }is not bigger than μ^{k}, then the lower (resp. upper) bound of its interval (resp. ) is obtained from the result of merging k + 1 trials by the traditional metaanalysis method where the (k + 1)th SEM is assumed to be . On the other hand, if μ_{k+1 }is larger than μ^{k}, then the lower (resp. upper) bound is obtained from the result of merging k + 1 trials by the traditional metaanalysis method where the (k + 1)th SEM is assumed to be
Similarly, for l trials with incomplete statistical information, we define
and
∀i, k + 1 ≤ i ≤ k + l, we let
and
then we have the following result.
Let X_{i }~ N (μ_{i}, SEM_{i}), 1 ≤ i ≤ k + l, denote the ith sampling distribution with sample size n_{i }such that SEM_{i }is assumed missing when i > k, then the merged result N (μ, SEM) applying the interval method to these k + l trials is:
and
Merging of the Differences between Groups
In clinical trials, the between group difference of two drugs/therapies about two groups is frequently used. Suppose that the numbers of patients in the two groups are the same (majority of clinical trials allocate (approximately) the same number of patients in two contrast groups) and we denote this number as n. Assume that the first group gives a sampling distribution of the effect of one drug as X_{1 }~ N (μ_{1}, SEM_{1}) and the second group about another drug as X_{2 }~ N (μ_{2}, SEM_{2}). Conventionally, the between group difference (of the two drugs involved) is calculated as
Thus we get
Let be a fixed value since σ_{1 }and σ_{2 }are fixed by the selected populations, we get SEM = . Therefore, the two methods proposed in the previous sections are also suitable in the process of merging the between group differences.
Results and discussion
Case Study of Drugs for Type2 Diabetes
In this section, we use the data from clinical trials on medication for adults with Type2 diabetes as our first case study.
Many research papers and reports have been published to show the effectiveness of various oral medication for Type2 diabetes ([1216], etc.). For oral medication of Type2 diabetes, the metaanalysis in [17] compared each pair of drugs on systolic blood pressure (SBP for short), diastolic blood pressure (DBP), low density lipoprotein cholesterol (LDLC) and high density lipoprotein cholesterol (HDLC), etc.
In this section, we consider the between group difference on the effectiveness of pairs of drugs for lowering LDLC. In [17], there were 14 sets of results comparing between group difference of 14 pairs of drugs on lowering LDLC. We selected four sets among these 14 sets for our case study, because each of them contains more than four trials so that our merging method can be applied more adequately. Our data were obtained from the original papers, most of which are the same as the data used in [17], however some data used in [17] are different from what we have found in the original papers. When this happens, we will state it clearly in the examples.
The drugs being compared in the selected four groups are
1. Thiazolidinedione versus Metformin (Example 1),
2. Triazolidinedione versus second generation Sulfonylureas (Example 2),
3. Metformin versus Metformin plus second generation Sulfonylureas (Example 3),
4. Second generation Sulfonylureas versus Metformin plus second generation Sulfonylureas (Example 4).
Example 1 Drugs for reduction of low density lipoprotein (LDL for short) were studied by many papers. They compared the LDLC reduction between different trial groups. For example, to compare drugs Thiazolidinedione and Metformin, we obtained the following (in mg/dL) sampling distributions from four trials.
[18]: X_{Pa }~ N (13.3, SEM_{Pa}) with n = 102.
[12]: X_{LT }~ N (7.8, 20.5) with n = 20.
[19]: X_{Han }~ N (10.1, 2.7) with n = 320.
[20]: X_{Sch }~ N (15.2, SEM_{Sch}) with n = 597.
Here n is the size of samples (number of patients) in each group of the trial, and SEM_{Pa }and SEM_{Sch }stand for the missing values (SEM values) from their respective trials data.
The metaanalysis results obtained by using prognostic method, interval method, and traditional method for incomplete trials information are provided in Table 1.
Table 1. Thiazolidinedione vs. Metformin
In Table 1, the first column shows the SEMs of X_{Pa }and X_{Sch }are missing. The second column shows that the result obtained by the prognostic (P for short) method is X_{p }~ N (12.56, 1.91) (denoted as 12.56 ± 1.91 in Table 1). The third column stands for the result obtained by the interval method (Int for short) and the last column stands for the result given by the traditional method on metaanalysis with incomplete trials information (Inc for short), i.e., removing trials with incomplete information. The rest of the tables in this case study follow the same notations and explanations.
Compared to X_{BW }~ N (12.5,1.89) [17]which is the metaanalysis result when all information is complete, obviously, with two SEM values missing, X_{P }is still very close to X_{BW }. Therefore, X_{P }can be used to replace X_{BW } for this set of trials. In addition, we also have 12.5 ∈ [11.94, 13.39] and 1.89 ∈ [1.54, 2.14] which indicate that X_{I }is a good estimation.
Example 2 To compare Thiazolidinedione and second generation Sulfonylureas, we obtained the following sampling distributions (in mg/dL) from five trials.
[12]: X_{LT }~ N (10.5, 14.44) with n = 20.
[13]: X_{CM }~ N (11.31, 1.59) with n = 620.
[14]: X_{TJ }~ N (6.63, SEM_{TJ}) with n = 100.
[15]: X_{PM }~ N (5, 6.04) with n = 86.
[16]: X_{MC }~ N (14.6, SEM_{MC}) with n = 315.
There are two missing SEM values.
The metaanalysis results for this example are provided in Table 2.
Table 2. Triazolidinedione vs. second generation Sulfonylureas
In this example, X_{P }is reasonably close to X_{BW }where X_{BW }~ N (10.4,1.61) [17]. However X_{Inc }is closer.
Example 3 To compare Metformin and Metformin plus second generation Sulfonylureas, we looked at another set containing six trials with sampling distributions as follows (in mg/dL).
[21]: X_{Ga }~ N (10.2, SEM_{Ga}) with n = 168.
[22]: X_{Go }~ N (7.0, 5.18) with n = 60.
[23]: X_{Ma }~ N (3.9, 4.11) with n = 103.
[24]: X_{DeF }~ N (2.0, 2.8) with n = 186.
[25]: X_{H94 }~ N (3.1, 3.6) with n = 30.
[26]: X_{H91 }~ N (8.2, 5.8) with n = 15.
There is only one missing SEM value. The metaanalysis result in [17] is X_{BW }~ N (1.6, 2.53) while we obtained the following results.
The metaanalysis results for this example are provided in Table 3.
Table 3. Metformin vs. Metformin plus second generation Sulfonylureas
Some data (sampling distributions of [23,25,26]) used in [17] are somehow different from the data we found from the original papers. Naturally, there is a margin between the merged sampling distribution from our methods and that from the metaanalysis. Therefore, it is difficult to compare these results.
Example 4 A set of seven trials was analyzed in [17] to compare the second generation Sulfonylureas versus Metformin plus second generation Sulfonylureas. We obtained the following seven sampling distributions (in mg/dL) from them.
[21]: X_{Ga }~ N (2.2, SEM_{Ga}) with n = 168.
[22]: X_{Go }~ N (0.2, 4.6) with n = 60.
[23]: X_{Ma }~ N (3.9, 4.72) with n = 103.
[27]: X_{Gr }~ N (10.14, 10.17) with n = 87.
[24]: X_{DeF }~ N (11.0, 2.83) with n = 186.
[25]: X_{H94 }~ N (7.41, 4.22) with n = 30.
[26]: X_{H91 }~ N (19.89, 5.54) with n = 15.
There is one SEM value missing. The results from our methods are given in Table 4.
Table 4. Second generation Sulfonylureas vs. Metformin plus second generation Sulfonylureas
It appears that X_{P }is not so close to X_{BW }where X_{BW }~ N (8.1,2.55) [17]. Once again, some data (sampling distributions of [23,25,26]) used in [17] are not the same as given in the original papers. So effectively, we had to use different data to perform the merging of these several trials.
Case Study of IOP Reduction
In the above subsection, we applied our methods to approximate the traditional metaanalysis when some SEM values are missing. Because some of the data we found in the original papers are different from that used in the metaanalysis paper [17], it is hard to judge our approaches in some cases. In order to further validate our approaches experimentally, in this section, we consider and apply our methods to a set of trials with full statistical information. In other words, every trial we consider will have both the mean and the SEM value in its sampling distribution. To validate how accurate our two approaches are for predicting an SEM value, we deleted an SEM value from a trial selected randomly from a set of trials, and applied our methods to predict it. We then applied the metaanalysis method to merge the trial with the predicted SEM value together with the rest of trials in the group to see how close this new result is to the original metaanalysis result. Furthermore, as the traditional method for trials with incomplete information always abandons trials with incomplete information, we also compared our methods with this traditional method.
We considered four sets of trials used in [1] on IOP reduction. [1] reported the metaanalysis of some trials about between group difference of IOP reduction from baselines using drugs Latanoprost and Timolol.
Example 5 (one week results) Three papers on clinical trials about IOP reduction gave the following three sampling distributions comparing Latanoprost and Timolol for oneweek trial duration.
[28]: X_{1 }~ N (8.5, 8.24) with n = 46.
[29]: X_{2 }~ N (5.62, 7.91) with n = 15.
[30]: X_{3 }~ N (6.85, 4.01) with n = 20.
The traditional metaanalysis in [1] produced X_{1w }~ N (6.90, 3.28). We deleted the SEM value from one of these trials in turn and applied our methods to predict it. Then we calculated the merged sampling distribution involving this predicted SEM value. The results are summarized in Table 5.
Table 5. Latanoprost vs. Timolol: one week
In Table 5, the second row starting with "X_{1}" means that if the SEM value of X_{1 }is missing, then from the prognostic method, we get ~ N (7.55, 2.53), from the interval method, we get ~ N ([7.33, 7.83], [2.12, 2.80]), and from the traditional method for trials with incomplete information, we get ~ N (6.60, 3.57). Other rows are explained similarly. We can see that s, 1 ≤ i ≤ 3, are close to the result obtained by the traditional method.
Example 6 (one month results) Three papers on clinical trials about IOP reduction gave the following three sampling distributions comparing Latanoprost and Timolol for one month duration.
[28]: X_{1 }~ N (11.0, 6.05) with n = 46.
[31]: X_{2 }~ N (5.2, 1.66) with n = 184.
[32]: X_{3 }~ N (0.4, 2.17) with n = 294.
The traditional metaanalysis gives X_{1m }~ N (3.77, 1.29). The results are summarized in Table 6. s, 1 ≤ i ≤ 3, are also very close to the result obtained by the traditional method.
Table 6. Latanoprost vs. Timolol: one month
Example 7 (three months results) Five papers on clinical trials about IOP reduction gave the following five sampling distributions comparing Latanoprost and Timolol for three months duration.
[33]: X_{1 }~ N (4.0, 3.11) with n = 267.
[34]: X_{2 }~ N (5.41, 5.56) with n = 60.
[35]: X_{3 }~ N (3.2, 4.03) with n = 36.
[31]: X_{4 }~ N (7.8, 3.98) with n = 184.
[32]: X_{5 }~ N (1.9, 2.12) with n = 294.
The traditional metaanalysis gives X_{3m }~ N (3.52, 1.44). The results are summarized in Table 7. Once again, s, 1 ≤ i ≤ 5, are close to the result obtained by the traditional method.
Table 7. Latanoprost vs. Timolol: three months
Example 8 (six months results) Four papers on clinical trials about IOP reduction gave the following four sampling distributions comparing Latanoprost and Timolol for six months duration.
[33]: X_{1 }~ N (4.8, 3.14) with n = 267.
[36]: X_{2 }~ N (7.1, 1.58) with n = 268.
[35]: X_{3 }~ N (4.5, 5.23) with n = 36.
[32]: X_{4 }~ N (1.5, 2.14) with n = 294.
The traditional meta analysis gives X_{3m }~ N (5.05, 1.15). The results are summarized in Table 8. Finally, s, 1 ≤ i ≤ 4, are also close to the result obtained by the traditional method.
Table 8. Latanoprost vs. Timolol: six months
By comparing these results, we can conclude that the more trials we have in an example, the closer our result is to the traditional metaanalysis method. This is because when the number of trials included in a metaanalysis gets larger, the error of metaanalysis gets smaller. Therefore, the predicted SEM value from the prognostic method gets closer to the real SEM value and the intervals used in the interval method have a better chance to contain the real values.
Discussion
Here we mainly analyze the results obtained from the IOP examples. Since we have stated that some data used in [17] are different from the data we found in the original papers, it would be difficult to compare those results objectively.
First, we found that the results obtained from the prognostic method are closer to the true results. It implies that the prognostic method is truly applicable. Although the interval method gives an interval instead of a single value, the interval provides a clear indication as to where the real value could be, and so it is an alternative to supplement the missing value.
Second, we found that the prognostic method is superior to the traditional method for trials with incomplete information which abandons trials with missing information. In fact, we can see that although in some cases, the results obtained by the traditional method for trials with incomplete information are also close to the true results, in some other cases, they deviate from the true results significantly, such as cases like in Table 6, when the SEM of X_{2 }is missing, in Table 7, that of X_{4 }or X_{5 }is missing, and in Table 8, that of X_{2 }or X_{4 }is missing. That is, the prognostic method is more stable and precise than the traditional method for incomplete trial information. This result is not surprising because the prognostic method takes into account all the eligible trials. On the contrary, if the SEM value of a trial with a large/small mean value is missing, then the traditional method for trials with incomplete information will abandon this trial and naturally gets a result with smaller/higher mean value than the true one. However, the prognostic method does not abandon this trial, and instead it predicts a value for the missing one. Therefore, the final combined result is closer to the real result of metaanalysis. This is also illustrated by Examples 5,6,7,8, etc, except the case of the missing SEM of X_{1 }in Example 5.
Dealing with missing data in statistics, especially in metaanalysis is a very important issue (e.g., [3739]). However, there are hardly any papers focusing on missing standard errors. This paper thus provides some fresh results about how to deal with this situation. Our assumption that the populations should be the same or similar intuitively follows the well known Missingatrandom (MAR) assumption, except that MAR mainly focuses on repeatedly measured data while ours are focusing on different trials. With this assumption, the prognostic and the interval methods are easily induced. Generally speaking, the prognostic method is related to the regression imputation and the interval method is to some extent a concrete implementation of the best/worst case analysis [40] with our assumption and situations.
A small caveat of the prognostic and the interval methods is that when the input data are imprecise (as shown in Examples 3 and 4), we may obtain worse results than that from the traditional method for incomplete information. This is because when some input data are imprecise, the prognostic method will get wrong predictions for missing SEMs, hence obtain worse metaanalysis results whilst the traditional method for incomplete trial information simply abandons trials with incomplete information, avoiding any wrong predictions and therefore may obtain better results. In addition, in Example 5, when the SEM value of X_{1 }is missing, we found that the result obtained by the prognostic method is not as accurate as that obtained by the traditional method for incomplete information. This is because the SEM value of X_{1 }is exceptionally larger than the others, i.e., as the sample size of X_{1 }is greater than those of X_{2 }and X_{3}, it is expected (according to our assumption of same population) that the SEM of X_{1 }should be smaller, however, it is not. So the prognostic method performs less effectively here.
Conclusion
Methods to deal with statistical information in clinical trials for which the SEMs values are missing is a neglected area in the literature. An obvious solution is to ignore a trial with missing SEM values. Our methods provide an application tool to solve this problem by effectively predicting missing SEMs or providing applicable intervals for the missing SEMs. The prognostic method can be used to obtain a result in a commonly accepted format, e.g., sampling distribution, whilst the interval method, although seemingly not so straightforward, provides a safer and also informative result.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
JM, WL and AH contributed to the analysis methods and the collection of materials on case studies. WZ contributed to the analysis methods and the evaluation of these new methods with respect to the traditional metaanalysis method and the traditional method for dealing with missing values. All authors read and approved the final manuscript.
Acknowledgements
We would like to thank the UK EPSRC for the project grant supports to Jianbing Ma, Weiru Liu and Anthony Hunter, and Joanna Rawmowski for her logistic support for this project.
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