Description of Data

The Agincourt Health and Socio-Demographic Surveillance System (HDSS) is located in Bushbuckridge in northeast South Africa and was established in 1992. Bushbuckridge is a poor rural sub-district that is made up of South African and former Mozambican refugees (approximately a third of the population) [13, 14]. There have been annual updates of births, deaths, in- and out-migrations of individuals identified as members of households, as well as regular special modules (e.g. household asset ownership) used to derive a socio-economic status index.

Recall our purpose and questions given in (Eq 1) and (Eq 2). Regarding absolute poverty status (APS), it is independent of the household variables associated to questions in (Eq 2). Here, households above the absolute poverty line was defined using the definition proposed in [15] for a sub-Saharan African setting, namely ownership of a radio and bicycle, a cement floor in the house, and access to public water and a pit latrine (toilet). Absolute poverty classification is thus independent of the 6 explanatory variables used in our orbit analysis. APS of households below the poverty line are coded 0, while APS above the poverty line is coded 1. Because APS is gathered only in 4 out of the 7 observation years (i.e. at t = 2001, 2003, 2005, and 2007), we use the mean APS of a household over 7 years, which we denote by $\overline{APS}$. If $\overline{APS}\in$[0, 0.5], then $\overline{APS}$ is coded 0. Otherwise, $\overline{APS}$ is coded 1. Our sample population consists of 7715 household units, 4158 of which are either always above or below the absolute poverty line for all four years that APS was gathered. For these households, APS information not gathered for the three years 2002, 2004, and 2006 will not affect the coding of their $\overline{APS}$. Households with $\overline{APS}$ = 0 are referred to as Poor households, and households with $\overline{APS}$ = 1 as Rich households. Our analysis will only consider these 4158 households. Ethical clearance for the primary study was given by the University of the Witwatersrand Human Research Ethics Committee (Medical). The data used in this study does not contain clinical records (nor does the core Agincourt HDSS database). Individual and household identifiers are anonymized/de-identified by the data managers prior to handing it over to researchers for analysis to ensure confidentiality.

Aside from $\overline{APS}$, household head nationality is also constant throughout the survey period. In addition, former Mozambican (MOZ) refugees experience significantly higher levels of poverty compared to their South African (SA) counterparts and this gap has persisted over time [12, 16, 17]. It is then useful to extract Q₅ and Q₆ and analyse by these subpopulations of households where both poverty status and household head nationality are unchanging. We divide our population into four subpopulations, namely SA Rich, SA Poor, MOZ Rich, and MOZ Poor. Each subpopulation is analysed using p = 5 variables associated to Q₀ to Q₄. Binary data of the four subpopulations is given in S1 Dataset. From [12, 16], a ‘yes’ answer to questions Q₀ to Q₄ is assumed to be favourable to APS so we code all yes = 1, and all no = 0. Table 1 gives the favourable and unfavourable code for each of the five questions. Table 2 gives the number of households by constant variables (i.e. nationality and $\overline{APS}$).

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Table 1. Favourable(= 1) and unfavourable(= 0) answer to questions Q₀ to Q₄.

https://doi.org/10.1371/journal.pone.0123812.t001

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Table 2. Distribution of households by nationality and mean APS.

https://doi.org/10.1371/journal.pone.0123812.t002

The Method of Orbits

Given the number of variables p ≥ 1, denote by the space of binary strings (responses) of length p. For a subject ℓ observed at times t = 0, 1, …, T, we define the binary multivariate longitudinal data in p ≥ 1 variables of subject ℓ. The binary longitudinal data in p variables from a population of n ≥ 1 subjects observed over time T is the set We will only consider subjects with complete data.

Analysis of BMLD always involves a fixed variable order where one can use the summary measure of the frequency of response pattern (elements of M_p) and perform factor analysis on the longitudinal data [1] or construct Markov models using information of change of time encoded in the matrix of transition probabilities [3]. The method of orbits [9] uses the fundamental information of frequency of change of variables and order of variables for analysis. The information of change is used to define a non-autonomous dynamical system from data of each subject, dynamically rearranging order of variables so that most stable least changing variable is eventually placed to the left, but keeping the full information in the original data. Mathematical properties of the map are discussed in [9].

To explain the orbit method, we illustrate for p = 3 variables. Let Q = {Q₀, Q₁, Q₂} be a questionnaire and assign index i to Q_i, i = 0, 1, 2. Table 3 illustrates concatenated coded answers to Q of three subjects from a sample population. To Q₀, subject ℓ has constant answer 0 while ℓ′ has constant answer 1. On the other hand, ℓ″ has constant answer 1 to Q₂. Observe that this property of subjects having constant answers to certain questions is not trivially illustrated in the time series of the three subjects given in Fig 1.

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Table 3. Concatenated coded answers of three subjects to questionnaire Q = {Q₀, Q₁, Q₂}.

https://doi.org/10.1371/journal.pone.0123812.t003

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Fig 1. Time series of the three subjects in Table 3.

https://doi.org/10.1371/journal.pone.0123812.g001

Suppose we order questions and give more weight to those that least frequently change. As in numbers, we let the digit in the left-most position of the question order be most significant, and digit in the right-most be least significant. Observe that for both ℓ and ℓ′, answer to Q₀ is the most stable (i.e. it is constant), followed by Q₁ (changes once in ℓ and twice in ℓ′), and finally Q₂ as most frequently changing. Then question order for both ℓ and ℓ′ is chosen as Q₀, Q₁, Q₂, which we will denote by 012. Now position lexicographically in increasing order as binary integers the states (responses) (3) along an axis, and denote this by X₃. For fixed question order 012, a one-dimensional dynamics on the states in X₃ arises, where answers of subjects ℓ and ℓ′ are visualised jumping from one state to another, particularly staying in the distinct regions 0** and 1**, the left and right half of X₃, respectively. However, different subjects may have different frequencies of change in answer values. Because ℓ″ has constant answer to Q₂, question order for ℓ″ is chosen such that Q₂ is given more weight. In particular, question order for ℓ″ is set to 210.

We recall terms and notations as introduced in [9]. Let

Definition 1 Given p ≥ 1, the spaces of sequences and both with the lexicographic ordering of sequences (i.e. as increasing integers) are the fitness axis and significance axis for p variables, respectively. An element x ∈ X_p is called a fitness state, and y ∈ Y_p a significance state. The space is the change space in p variables composed of P = 2^p p! states.

For convenience, states in S_p are labeled from 1 to P = 2^p p! starting from left to right, top to bottom. The labeled space S_p for p = 3 is illustrated in Fig 2. The space X₃ is the sequences in (Eq 3), Y₃ = {012, 021, 102, 120, 201, 210}, and the cardinality of ∣S₃∣ = 2³3! = 48.

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Fig 2.

(a) Orbit of subject ℓ staying in subset of S₃ where variable 1 is favourable. (b) Time series of the orbit of ℓ.

https://doi.org/10.1371/journal.pone.0123812.g002

Definition 2 1. Consider subject ℓ. Given a set Q of p ≥ 1 questions, i_j ∈ {0, 1, 2, …, p−1}, and Q_ij ∈ Q let (4) Suppose Then the initial question order of ℓ is (5) If $f_{i_j}^{\ell }=f_{i_{j+1}}^{\ell },$ use population frequencies $f_{i_j}^n,f_{i_{j+1}}^n$ to determine order between i_j and i_j+1. If $f_{i_j}^n=f_{i_{j+1}}^n$ and i_j < i_j+1, choose question order i_j i_j+1. Otherwise, choose i_j+1 i_j.

2. Given initial question order $y_0^{\ell }=i_0{i}_1\cdots i_j\cdots i_{p-1}$, the initial fitness state of ℓ is where each x_j is the answer to question i_j in $y_0^{\ell }$.

3. The initial state of ℓ is $s_0^{\ell }=(x_0^{\ell },y_0^{\ell })\in S_p.$

The algorithm for determining the next states $s_t^{\ell }$ (t > 0) is as follows:

• Step 1:. [initial state $s_0^{\ell }$] For t = 0 and subject ℓ, determine the initial significance state $y_0^{\ell }$, followed by the initial fitness state $x_0^{\ell }$.
• Step 2:. [state $s_1^{\ell }$] Given initial state $s_0^{\ell }=(x_0^{\ell },y_0^{\ell })$ of ℓ, identify the questions that change answer values at t = 1. If there are none, then the next state $s_1^{\ell }=s_0^{\ell }$. Let If both Q_ij and Q_ij′ change answers at t = 1 and j < j′, then sequentially swap to the right i_j and i_j′ (resp. x_j and x_j′) of the question order (resp. answer order), starting with i_j′ (resp. x_j′). Change x_j to $x_j^{*}$ and x_j′ to $x_{j\prime }^{*}$, i.e. This new answer order and question order is the next state $s_1^{\ell }=(x_1^{\ell },y_1^{\ell }).$
• Step 3:. [edge color] Draw an edge from $s_0^{\ell }$ to $s_1^{\ell }$. To show direction of transitions between states, color the edge red if transition is from right to left, green if transition is from left to right, and blue otherwise (i.e. same x-coordinate).
• Step 4:. [state $s_t^{\ell },t\ge 2]$ Update state $s_0^{\ell }$ as $s_1^{\ell }$ and time t as t = 2 in Step 2. Repeat Steps 2 and 3, and iterate until t = T−1.

Definition 3 Let $x_t^{\ell }$, $y_t^{\ell }$, and $s_t^{\ell }=(x_t^{\ell },y_t^{\ell })$ be the fitness, significance, and state of subject ℓ at time t, respectively. The orbit of ℓ is the sequence of points

Example 1 Table 4 gives coded data of a subject ℓ to p = 3 questions. Recall that coding of answer is 0 = unfavourable and 1 = favourable according to purpose. The coded answer of ℓ to Q_i at time t is denoted by $a_{i,t}^{\ell }$. From (Eq 4), we have $f_0^{\ell }=3$, $f_1^{\ell }=0,$ and $f_2^{\ell }=2$, so initial significance of ℓ is $y_0^{\ell }=120$, with corresponding initial fitness $x_0^{\ell }=111$. This corresponds to state index 24 in Fig 2(a). No answer changes at t = 1 so $y_1^{\ell }=y_0^{\ell }$ and $x_1^{\ell }=x_0^{\ell }$ and state transition from t = 0 to t = 1 is denoted by 24 → 24. Now at t = 2, Q₀ changes answer so we swap 0 and 1 to the right of $y_1^{\ell }$ and $x_1^{\ell }$ respectively (note that both are already on the right), and then change answer 1 to 0. Hence, $y_2^{\ell }=120$ and $x_2^{\ell }=110$, which corresponds to state 23. Columns 3 and 4 give the rest of the fitness and significance states respectively of the orbit. Observe that ℓ has favourable answer to Q₁ for all times so its orbit 𝓞(ℓ) stays in the subset (6) where question i₀ = 1 is favourable. The longitudinal data of ℓ in S₃ is visualised as the orbit in Fig 2(a) with its time series illustrated in Fig 2(b).

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Table 4. Coded data and orbit of a subject ℓ. The number a_{i, t} is answer to Q_i at time t.

https://doi.org/10.1371/journal.pone.0123812.t004

Example 2 The orbits of the three subjects in Table 3 in S₃ and over time are illustrated in Fig 3(a) and 3(b) respectively. Observe that orbit of ℓ stays strictly on the left half of S₃, and the other two on the right half. Subject ℓ is unfavourable in stable variable 0, while ℓ′ and ℓ″ are favourable in stable variable 0 and variable 2, respectively.

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Fig 3. Orbit of the three subjects given in Table 3 (a) in S₃ and (b) over time.

https://doi.org/10.1371/journal.pone.0123812.g003

Remark 1 By the 0/1 coding of data, it is reasonable to suppose that the (concatenated) answers composed of unfavourable values 00⋯0 is ‘less fit’ than the answer composed of favourable values 11⋯1. By the weighting of variables, ‘relative fitness’ is made precise so that ordering of elements from the space M₃ = {0, 1}³ of multivariate binary responses has meaning. Because more weight (significance) is given to the left-most position, we can then, for a fixed question order, write x = 010 < x′ = 100, where most significant variable is unfavourable in x, and favourable in x′. For a fixed question order, we say that 100 is fitter than 010 (or 000, 001). The same argument holds in stating that 110 is less fit that 111.

Using orbits, the complete p-dimensional information of each subject at any moment is coded to a point in the 2-dimensional discrete space S_p. No information in data is lost nor approximated as each subject’s orbit has a one-to-one correspondence with the subject’s original data. Clearly, question order of each subject at each time step is monitored. The ordering is selected as frequently changing variables are swapped to the far right (less significant digit of y), thus pushing slow changing variables to the left (significant digit of y). This ‘swapping-changing-variable-to-the-right’ process exposes clusters associated to stable variables.

Remark 2 The time complexity of computation of orbits for n subjects and time T scales like pnT and is feasible for large data. We also note that there are admissible and non-admissible state transitions in S_p[9], e.g. in Fig 2, a transition that starts at 23 can only end at 23, 24, 29, and 31.

Remark 3 The tendency of a subject to favour a particular state, or subset of states, is clustering in S_p. The strategy for choosing the initial question order in (Eq 5) places an orbit in its most likely position. This facilitates clustering and is useful for short data sets.

Note that many households may share an edge (or orbit) in S_p. The following definitions are of interest regarding transitions in S_p.

Definition 4 a. The accumulated number of transitions from state s = (x, y) to s′ = (x′ y′) is called the density from s to s′, denoted by d(s, s′).

b. The number of orbits at state s at time t is called the capacity of state s at time t and is denoted by c_{s, t}.

Remark 4 We can deduce correlation among variables from orbits in S_p. We explain for p = 3. Using Fig 4, if state transition of orbits most of the time stay in states 1 = (000, 210) and 48 = (111, 012) then we can test for positive correlation among the three variables 0, 1, and 2. If orbits spend most (if not all) of the time in a subset L of S_p such that L ≅ S_m for some m < p, then there is strong correlation between the first p−m variables constant in S_m. In Fig 4 for example, if orbits stay strictly in the subset then we can check for (positive) correlation between the first two variables 0 and 1. The asterisk * in x (resp. in y) can take any binary value (resp. any question index except for i).

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Fig 4. Orbits of (a) SA Rich, (b) SA Poor, (c) MOZ Rich, and (d) MOZ Poor households.

Observe clusters formed in regions of each subpopulation. variable 4 (influx) is most frequently changing in all four subpopulations so orbits do not stay in the region where y = 4****. Not all 5! significance states are shown.

https://doi.org/10.1371/journal.pone.0123812.g004

Remark 5 The orbit method is not limited to binary data in p variables. For m-ary valued data, the space S_p is composed of same number p! of significance states but now with m^p fitness states. For the continuous case, data of each observation are binned, where bins are labelled from 0 to m−1 [18, 19]. For instance, binary coding can be done by assigning 0/1 if variable is above or below a given value, tertiary coding if the variable is in a good/neutral/bad range of values, and so on.

Orbit Results

Household orbits in S₅ for each of the four subpopulations are illustrated in Fig 4. The x-axis is composed of 2⁵ = 32 states but not all the 5! = 120 states in the y-axis are shown. There are no transitions between the four subpopulations as they are associated to constant variables. Recall that a red edge is used to denote a transition that goes from right to left on the next time step, a green edge for a transition that goes from left to right, and a blue edge a transition that goes to the same fitness state. The percentage of unfavourable answers for each question in each of the four subpopulations is given in Table 5 while the frequencies of answer change are given in Table 6. The frequencies of change for Q₀ (HH) and Q₁ (HS) are low, while Q₄ (IF) is the highest. This means that there is stability in the variables HH and HS in that most subjects will stay in the region where significance is either y = 01i₂ i₃ i₄ or y = 10i₂ i₃ i₄, i_j ∈ {2, 3, 4}. In addition, there is high activity of the IF variable, which means few transitions where y = 4i₁ i₂ i₃ i₄, i_j ∈ {0, 1, 2, 3}. All of this is recognized in Fig 4.

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Table 5. Percentage of unfavourable = 0 responses in Q_i for each of the four subpopulations.

https://doi.org/10.1371/journal.pone.0123812.t005

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Table 6. Questions with corresponding frequency of answer change in each of the four subpopulations.

https://doi.org/10.1371/journal.pone.0123812.t006

There are immediately regions of interest in Fig 4. As an aide in interpreting regions in S₅, we present in Fig 5 the subsets of S₅ determined by the first significant variable i₀. A subject ℓ that spends most of its time in the region where x = j****, y = i**** means that answer of ℓ to Q_i is least frequently changing, with answer = j. The initial state of ℓ is chosen to lie in this region (Definition 5). For example, a subject that stays in the region of Fig 5 frequently experiences younger (<40) household head age.

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Fig 5. Regions in S₅ determined by the first significant variable.

Observed units that often stay in a region determined by one significant variable often experience the property of that region.

https://doi.org/10.1371/journal.pone.0123812.g005

Regions in Fig 5 may be further analysed. A more detailed description of the regions HH≥40, HH<40, HS≥3, and HS<3 is illustrated in Fig 6. In each sub region, the two variables i₀ and i₁ are significant (i.e. less frequently changing answers), with i₀ being more significant. Of course these sub regions may be further subdivided.

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Fig 6. Regions in S₅ determined by the first and second significant variables.

https://doi.org/10.1371/journal.pone.0123812.g006

In general, we say that a variable i is stable if orbits cluster in a subset of S_p determined by first significant variable i. Regions that are never visited (e.g. those associated to variables IF⁺, IF⁻, and A<M in Fig 4) are termed holes. Clusters are contained in regions where the leading significant variable is stable while holes are contained in regions with high activity of the leading variable. By visual inspection of orbits in S_p, we can immediately detect stable variables (via clusters) and unstable variables (via holes).

Clusters in the right half regions of S_p are fitter than clusters located on the left half of S_p as they are associated to stable leading variable with favourable condition. From the orbits of subpopulations in Fig 4, observe that there are no transitions between the left and right half of S₅ in both SA Poor and MOZ Poor subpopulations. This is verified by Fig 7, the orbits of the four subpopulations, in time. In addition, columnar structures over clusters correspond to variables that are stable over the survey period. Although there are few transitions between clusters, there is considerable activity within each. Household orbits in one cluster may then reasonably be analysed independently of households in other clusters. The time series representation of orbits reveals idle behaviour (sequence of vertical blue edges) that are not always visible in orbits in S₅.

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Fig 7. Column structures over clusters of (a) SA Rich, (b) SA Poor, (c) MOZ Rich, and (d) MOZ Poor household orbits in S₅.

https://doi.org/10.1371/journal.pone.0123812.g007

As observed in Fig 4, some regions in a subpopulation appear denser than those of the other subpopulations (e.g. the region HH<40 appears to be more dense in MOZ poor than in MOZ rich, with the opposite phenomenon for the SA population). We use histograms to denote the accumulated number of visits (i.e. capacity, Defn. 4) to each state in S₅. Fig 8 gives the accumulated capacity in states of S₅, represented by the height of bars. It is immediately noted that there are regions of high and low numbers. Density at each state at each time step can also be computed, and can be represented by bubbles. Fig 9 illustrates this case for the SA Rich subpopulation.

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Fig 8. Accumulated number of visits (height of bars) in S₅ of (a) SA Rich, (b) SA Poor, (c) MOZ Rich, and (d) MOZ Poor household orbits.

https://doi.org/10.1371/journal.pone.0123812.g008

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Fig 9. Capacity at each state and each time step in the SA Rich, represented by bubbles.

https://doi.org/10.1371/journal.pone.0123812.g009

Fig 10 gives the percentages of visits in regions determined by one and two significant variables. The regions with no percentages are holes. The largest percentage in SA Rich is in the subregion associated to older household head and larger household size (62%), with household head more stable. For the other three subpopulations, the largest percentage is in the subregion associated again to older household head and larger household size, but with household size more stable.

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Fig 10. Percentage of visits of (a) SA Rich, (b) SA Poor, (c) MOZ Rich, and (d) MOZ Poor household orbits to regions in S₅ determined by the first and second significant variables.

https://doi.org/10.1371/journal.pone.0123812.g010

Remark 6 Population-level information is visible, but detailed individual information may be lost in the cluster. We may zoom into regions of interest (e.g. regions of high percentage of visit) to unclutter the display, as in the SA Rich region R_{HH ≥ 40, HS ≥ 3} illustrated in Fig 11. As for individual orbits, of interest in Fig 4 are those that seem to be ‘outliers’. They can further be analysed e.g. by using interactive techniques such as focusing and brushing, as in dynamic parallel component plots [20].

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Fig 11. Orbits in SA rich population that cluster in the region R_{HH ≥ 40, HS ≥ 3} where both household head and household size are favourably constant.

This is the zoomed region in Fig 4(a) with high percentage of household visit.

https://doi.org/10.1371/journal.pone.0123812.g011

Regarding Remark 6, we can further analyse orbits from the SA Rich subpopulation. Fig 12 shows dominant accumulated number of transitions ≥ 100 from state s = (x, y) to s′ = (x′, y′) in SA Rich (i.e. density d(s, s′) ≥ 100). Most transitions ‘idle’ at state (11111, 01234) and correspond to household orbits that are constantly favourably in the five variables. As for non-idling transitions, it is dominant between states s = (11110, 01234) and s′ = (11111, 01234) and involve change in variable 4 (IF) where s → s′ indicate negative influx, and s′ → s is non-negative influx.

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Fig 12. Densities d(s, s′) ≥ 100 from state s = (x, y) to s′ = (x′, y′) in SA Rich households.

Highlighted lines are self-transition, i.e. s = s′.

https://doi.org/10.1371/journal.pone.0123812.g012

Fig 13 shows the corresponding state indices for the subset R_{HH ≥ 40, HS ≥ 3} of S₅ given in Fig 11. The capacity in SA Rich households at each time step for states with c_{s, t} ≥ 50 is illustrated in Fig 14. We have the following observations:

1. The capacity at state 48 = (11111, 01234) is dominant. This is expected as most orbits idle in this state, as given by the numbers in Fig 12.
2. The capacity graphs for state pairs 48 and 47 = (11110, 01234), and 39 = (11110, 01243) and 40 = (11111, 01243), behave inversely and are almost symmetrical. Note that transition between state pair 47 and 48, and 39 and 40, are associated to change in variable 4 (IF) and 3 (AM) respectively. It is expected that capacity increase in 48 (more individuals migrating into households) result in decrease of capacity in 47. The same argument goes for exchange in capacity of states 39 and 40.
3. Transition between 23 = (11110, 01342) and 24 = (11111, 01342) are associated to change in variable 2 (HD). The capacity graph of 23 (HD = 0) is always above 24, except at t = 2007. The sharp increase in 24 (low household death) at this time corresponds to a drop in 23.

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Fig 13. State indices associated to states s_i in the subset R_{HH ≥ 40, HS ≥ 3} of S₅ where both household head and household size are favourably constant.

https://doi.org/10.1371/journal.pone.0123812.g013

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Fig 14. Number of SA Rich household orbits (capacity) at each time step in states (a) 23, 24, 30, 39, 40, 46, 47, and 48, (b) 47 and 48 associated to variable 4 (IF), (c) 39 and 40 associated to variable 3 (AM); 30, and 46, and (d) 23 and 24 associated to variable 2 (HD).

https://doi.org/10.1371/journal.pone.0123812.g014

Results Regarding Purpose

We particularly use Fig 8 to draw conclusions with regard to our Purpose as stated in (Eq 1).

1. There is one dominant peak in SA Rich. This occurs at the fully fit state (11111, 01234), where it is most stable in Q₀ = 1 (HH ≥ 40), followed by Q₁ = 1 (HS ≥ 3), and so on. For SA Poor and MOZ Rich/Poor we find fully fit states at (11111, 10234) characterized by stability of HS ≥ 3, followed by HH ≥ 40. We then associate (i) households headed by older adults to larger $\overline{APS}$, and (ii) larger households with lower $\overline{APS}$.
2. The peaks for SA Poor, MOZ Rich, and MOZ Poor are at states (7) For spikes at states ii., iv., and vi., unfavourable answer is either in Q₀ or Q₃ (i.e. HH<40 or A<M). We then associate young household heads and less adults to minors to poorer (i.e. not Rich SA) households. Now spike at state vii. which is unfavourable in Q₁ and Q₄ (HS<3 and IF⁺) is also associated to poorer households. The condition of small households should be examined.
3. Spikes at states ii., iv., v., vi., and vii. are identified with relatively stable unfavourable states HH<40, A<M, and HS<3 with IF⁺. We then associate absence of visits to these states with SA Rich, and their presence with the other three subpopulations.
4. For the two dominant peaks at states i. and ii. in MOZ Rich in Fig 8(c), A ≥ M has a higher peak than A<M. This is reversed in MOZ Poor in Fig 8(d). We associate MOZ Poor with a stable, dominant population of households with small adult component.

Other Methods of Binary Multivariate Longitudinal Data Analysis

We discuss the use of other conventional methods in analysis of BMLD and mention the advantage of using orbits.

Generalized Estimating Equation Model.

To compare the performance of the conventional statistical model to the deterministic orbit approach we have adopted a generalized estimating equation (GEE) population modelling approach. In [21], the estimation-equation approach is proposed for population average models. It is argued that in general, mixed models involve unverifiable assumptions on the data-generating distribution resulting to potentially misleading estimates and biased inference. We use the quasi-information criterion (QIC) to identify the best working correlation structure to be used for our data [22]. Maximum likelihood based model selection methods, such as the widely used Akaike Information Criterion (AIC), are not directly applicable to the GEE approach [23]. The exchangeable correlation structure proved to be the best when fitted to our data.

Before presenting the GEE model, we note that with regards to the correlated indicators, there is potential co-linearity between the household size and certain other covariates. This is suggested by the marginally high variance inflation factor (VIF) for this variable (Q₁) of ∼ 10 in Table 7. Further, the spearman rank correlation coefficient of 0.68 between Q₁ (household size) and Q₄ (influx) in Table 8 would be cause for further concern. Removing the co-linear effect of Q₁, the GEE model for a binary outcome (APS = 0/1) using a binomial family, logit link function and an exchangeable correlation structure is given in Table 9. The VIF without Q₁ is given in Table 10.

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Table 7. Variance Inflation Factor.

https://doi.org/10.1371/journal.pone.0123812.t007

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Table 8. Spearman’s Rank Correlation Coefficient.

https://doi.org/10.1371/journal.pone.0123812.t008

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Table 9. GEE Model removing the co-linear effect of Q₁.

https://doi.org/10.1371/journal.pone.0123812.t009

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Table 10. Variance Inflation Factor (removing Q₁).

https://doi.org/10.1371/journal.pone.0123812.t010

Remark 7 The GEE model shows that HH ≥ 40, HS ≥ 3, HD low, A ≥ M, IF⁻, and HN = SA are more likely in the rich households. This is consistent with our favourable/unfavourable orbit coding to APS. In addition, the model also informs us that variables associated to holes (not just clusters) in S_p should also be analysed. In particular, the Q₄ (IF) variable (associated to holes) and Q₃ (AM) variable (associated to very few transitions) appear to be statistically significant and associated to households above the absolute poverty line.

Motion Charts and Heat Maps.

A motion chart is a dynamic bubble chart that enables the display of large multivariate data with large number of data points [6]. The central object in motion charts is a blob, or in general a 2-dimensional shape, which represents one entity from the dataset. This allows for visualization of the data by using additional dimensions (e.g. time, size and color of the blobs) to show different facets of the data. The dynamic appearance of the data in a motion chart facilitates visual inspection of associations, patterns and trends in multivariate datasets. The problem with motion charts is that for many variables, there is not enough dimensions (e.g. size, shape, color, etc.) to represent different entities. The advantage of using orbits is that adding more variables is easily accommodated by the increase in the number of fitness and significance states in the change space S_p. Fig 15 show the proportion of households (by nationality and fitness sequence) above poverty line over the survey period while Fig 16 shows the proportion of households above poverty line by nationality and time, i.e. (HN, t), where HN = 0 = SA, HN = 1 = MOZ, and t = 2001, 2003, 2005, 2007. The labeling of the fitness states along the x-axis is given in Table 11. While this gives a sense of where more households fall with regards to relative poverty probability (stratified by household nationality and then by nationality and poverty line status classification), it does not convey the changing trajectory of households with time.

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Table 11. Label for answer combination associated to Fig 15 and 16.

https://doi.org/10.1371/journal.pone.0123812.t011

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Fig 15. Proportion of households (by nationality and fitness sequence) above poverty line over the survey period 2001–2007.

https://doi.org/10.1371/journal.pone.0123812.g015

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Fig 16. Proportion of households above poverty line by nationality at time (HN, t), where HN = 0 = SA, HN = 1 = MOZ, and t = 2001, 2003, 2005, 2007.

https://doi.org/10.1371/journal.pone.0123812.g016

The heat map approach illustrated in Fig 17 reflects the observed proportion above the poverty line (by nationality) represented by the amplitude to graph at the point (x, y), where the fitness sequences are on the y axis and the 4 year time points (1 = 2001, 2 = 2003, 3 = 2005, 4 = 2007)are on the x-axis. While some differences can be observed by nationality, the clearer visualisation offered by the orbit approach is evident in our opinion. The heat map approach is not without merits (one being easy to implement) and would require more extensive and detailed application to longitudinal data such as ours to fully surmise its utility relative to the deterministic orbit approach.

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Fig 17. Heat map representing the observed proportion (density) above the poverty line (by nationality) and times 1 = 2001, 2 = 2003, 3 = 2005, 4 = 2007.

https://doi.org/10.1371/journal.pone.0123812.g017