Ideal observations

In practice, it is impossible to observe the number of times an individual is infected by each lineage, i.e., the values corresponding to m_k cannot be observed in an infection (see Fig 1 “unobservable information”). However, the absence/presence of lineages can be observed in a clinical sample. Let y_k ∈ {0, 1} denote the absence/presence of lineage A_k in a blood sample. Therefore, an observation (sample) represented by the configuration (3) where 0 is the n-dimensional vector with all zero components, and represents a clinical sample in which no lineage is detected (this configuration is excluded, because we only consider disease-positive individuals). The probability of a sample having configuration y is given by (4a) or (4b) (cf. [9]). This model is identifiable, i.e., different sets of parameters lead to different distributions of y (cf. [4]). Moreover, observations are “ideal” because we assume that all lineages actually present in an infection will be detectable and lineages are not erroneously detected.

Incomplete observations

In practice, a lineage within an infection might remain undetected. To accommodate this important factor, we extend the model. We assume that lineages remain undetected independently of each other and detection of each lineage is independent of the true number of super-infections by that lineage, m_k. We assume further that each lineage is equally likely to be detected. Hence, we let ε be the probability that a lineage, present in the infection, remains undetected in the blood sample. Consequently, the probability of a lineage being detected correctly is 1 − ε. Additionally, we ignore that lineages, which are absent from an infection, can be detected by mistake. Under these assumptions, the true absence/presence of lineages y remains unobserved, and the corresponding actual observation (denoted x) might not correctly indicate the presence of all lineages being actually present, i.e., (5)

Therefore, for an observation x, it remains unclear which y reflects the true absence/presence of lineages (see Fig 1 “actual observable information”). Thus, the probability of observing x has to take into account that any y with x ≼ y can lead to observing x with lineage A_k missing if y_k = 1 but x_k = 0. If all lineages fail to be detected, then x = 0. We refer to 0 as the empty record. According to the above assumptions, given that y reflects the true absence/presence of lineages in an infection, the probability of observing x is (6) for x ≼ y and zero otherwise. As for every observation x each x ≼ y potentially reflects the true absence/presence of lineages, according to the law of total probability, the probability of observing configuration x = (x₁, …, x_n) thus becomes (7a) where Q_y is specified in (4b). The sum runs over all possible ideal observations that can lead to the actual observations, i.e., for all x ≼ y (i.e., x_k ≤ y_k for all k = 1, …, n). It is shown in section Prevalence in S1 Appendix Mathematical appendix that can be expressed as (7b)

By accounting for incomplete data, the empty record 0 does not correspond to a disease-free host, as only samples from infected hosts are considered. The empty record occurs with probability (7c)

We refer to the new model (7) as the incomplete-data model (IDM). It includes the additional parameter ε compared with the original model (OM). The sample space of the IDM is denoted by .

Jointly, we denote the model parameters by θ = (λ, p, ε). Clearly, the MOI parameter λ is strictly positive. The frequency vector p satisfies and 0 < p_k < 1 for k = 1, …, n, and therefore is an element of the interior of the (n − 1)-dimensional simplex. Additionally, we assume ε ∈ (0, 1). Hence, the parameter space of the new model becomes (8) where denotes the interior of the (n − 1)-dimensional simplex. Note that, if ε = 0, no lineage remains undetected and all lineages present in an infection will be observed almost surely. In this case, the IDM coincides with the OM. Additionally, if ε = 1, (6) implies that and if for any choice of λ and p, and hence, the model is not identifiable.

Prevalence

It is important to point out that p_k is the relative frequency of lineage A_k and not its prevalence. The former refers to the relative abundance of the lineage in the pathogen population, while the latter refers to the probability of the lineage’s presence in an infection (within the population of disease-positive individuals). The (actual) prevalence of lineage A_k, i.e., the probability that the lineage occurs in a sample, irrespective of incomplete observations, is (9)

Accounting for incomplete data, the observable prevalence is not just mediated by MOI, but also by the probability of lineages not being detected (ε).

Remark 1. Under the IDM, the observable prevalence of lineage A_k, i.e., the probability of observing A_k in a blood sample, is (10)

The observable prevalence accounts for the possibility that a lineage remains unobserved because of incomplete information.

The probability of observing lineage A_k in samples, excluding empty records, i.e., the observable prevalence condition on not observing empty records (referred to as conditional prevalence), is given by (11)

A proof is provided in section Prevalence in S1 Appendix Mathematical appendix. If super-infections rarely occur, the prevalence of lineages is expected to be close to their corresponding frequencies. In fact, in the limit as λ → 0 these quantities coincide (see section Prevalence in S1 Appendix Mathematical appendix).

The likelihood function

Let be a dataset containing the information obtained from N samples. That is, consists of N 0–1 vectors, where each vector corresponds to an observation indicating the detected lineages. Let denote the observed data corresponding to the j-th sample. Furthermore, let n_x denote the number of samples with configuration x in the dataset . Thus, if x^(j) ≠ x for j = 1, …, N, we have n_x = 0. In particular, n₀ is the number of empty records in . Clearly, (12)

The log-likelihood function for the observed dataset is given by (13)

Let (14) be the number of samples for which lineage A_k is observed. It is shown in section Log-likelihood function in S1 Appendix Mathematical appendix that the log-likelihood function (13) can be simplified to (15)

The sample size N can be written as N = n₀ + N₊, where (16) is the number of samples for which at least one lineage is observed—or the number of non-empty records. Clearly, the quantities n₀, N₁, …, N_n form a sufficient statistic for the IDM. More importantly, is the (unconditional) observable prevalence, and is the conditional observable prevalence of lineage A_k in the pathogen population.

Assessing the performance of the model extension

The advantage of the extended model (7) in comparison to the OM (4) is that it allows to incorporate empty records, i.e., all samples in a dataset can be included in the parameter estimation. When parameters are estimated from the OM, empty records have to be excluded from the data, because such observations are not accounted for in the model. The advantage of the model extension (7) is assessed here numerically.

Numerical investigations are conducted for a representative range of model parameters (summarized in Table 1) as follows. For a set of parameters (N, λ, p, ε) a total number of B = 100000 regular datasets (cf. Pathological Cases) of size N are created according to the model (7). Each dataset might contain empty records. Let and be the number of empty and non-empty records of . Let further be the dataset with all empty records removed. Thus, the sample size of is . For each dataset , the maximum-likelihood estimate (MLE), , based on the extended model (2), is calculated (see Deriving the maximum-likelihood estimate). Furthermore, from the MLE the average MOI is calculated according to (2).

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Table 1. Summary of model parameters.

Shown are the parameter values used to generate simulated datasets. A total number of B datasets were generated for each combination of model parameters and sample size (N).

https://doi.org/10.1371/journal.pone.0287161.t001

Furthermore, for each dataset the MLE of the parameter vector θ^* is calculated based on the methods described in [9], i.e., based on the OM. From the average MOI was calculated according to (2).

Bias and variance of the estimators for both models (IDM and OM) were estimated as follows. Denoting a model parameter generically by θ (i.e., θ is λ, ψ, p_k, or ε), the bias is estimated as (17a) where (17b) and θ_b is the corresponding MLE from the b-th dataset. The variance of the estimator for θ is calculated as (17c)

Because bias and variance of the estimators depend on the range of the true parameters, the relative bias and coefficient of variation, i.e., (18a) and (18b) are more appropriate for comparison of the bias and variance across different parameter ranges.

Additionally, we derived the empirical mean-squared error to measure the overall quality of the estimators. Similarly, to be able to compare the approximated mean-squared error of an estimator across a range of ψ values, we used the normalized root-mean-square error (NRMSE), i.e., (19)

Clearly, (i) sample size N, (ii) the true value of the MOI parameter λ, (iii) the number of lineages n, (iv) the lineage-frequency distribution p, and (v) the probability that a lineage remains undetected ε affect bias and variance of the MLEs. For the numerical investigations, we choose the following parameters.

Maximum likelihood estimate and Fisher information

Here, we discuss approaches to obtain the maximum-likelihood estimate (MLE) for the model parameters θ = (λ, p, ε). The first approach is to maximize the likelihood function (15) directly. Since the frequencies p are an element of the simplex , the log-likelihood function (15) needs to be maximized over the parameter space Θ, either by using a Lagrange multiplier or by eliminating one of the redundant variables, e.g., by setting . Using a Lagrange multiplier, the log-likelihood function (15) is replaced by (21) where . The MLEs of the model parameters are found by equating the score function to zero, i.e., by solving the following system of equations (22)

Unfortunately, the system has no explicit solutions and needs to be solved recursively, e.g., by a multidimensional Newton-Raphson method, i.e., by iterating (23) where is the observed information (negative of the Hessian matrix of ) evaluated at . The score function and the observed information are given in section Inverse Fisher information in S1 Appendix Mathematical appendix.

As seen in section Inverse Fisher information in S1 Appendix Mathematical appendix, the data (N₁, …, N_n, n₀) (which are random variables) occurs only in linear terms in the observed information. Because the Fisher information is the expected observed information, it can be easily calculated because and follow readily. Namely, is the expected number of infections in which lineage k is present. Because the probability of observing lineage k in a sample is its prevalence , N_k is a binomially distributed random variable with parameters N and . Similarly, n₀ is binomially distributed with parameters N and . Therefore, (24) where and Q₀ are given by (10) and (7c), respectively. The resulting entries of the Fisher information are presented in section Inverse Fisher information in S1 Appendix Mathematical appendix.

Both the Fisher information and the observed information contain the Lagrange parameter γ as a nuisance parameter. The reason is that the structure of the matrix is simpler. Namely, it contains a diagonal matrix corresponding to the second derivatives with respect to the lineage frequencies as a sub-matrix. This structure allows straightforward inversion by using block-wise inversion formula [11]. The Cramér-Rao lower bound (cf. [7]) is the inverse Fisher information matrix (with the entries corresponding to the nuisance parameter disregarded) and yields the covariance matrix of the estimator. The Cramér-Rao lower bound yields the minimum variance of any unbiased estimator, and will be asymptotically reached by the MLE. Although it is straightforward to invert the information matrix, the resulting formulae are lengthy. Consequently, we refrain from presenting analytical expressions here and refer S1 Appendix Mathematical appendix for the expressions of the information matrices.

In practice, the true parameter is unknown, so either the inverse Fisher information or the inverse observed information has to be estimated using the MLE as a plug-in estimate. It follows from Remark 2 and the derivations in section Inverse Fisher information in S1 Appendix Mathematical appendix that the estimates for the inverse Fisher information and the inverse observed information coincide. One might be more interested in the average MOI ψ (cf. Eq 2) rather than in the MOI parameter λ. To obtain the covariance matrix with respect to this estimator, the inverse Fisher information needs a slight adjustment (see Inverse Fisher information in S1 Appendix Mathematical appendix).

We can conclude that it is advisable to use the method of Lagrange multipliers to derive the covariance matrix of the estimator. To derive the estimate itself, the Newton method (23) is not advisable. Even if it is replaced by a quasi-Newton method or other modified versions of the Newton method, convergence of these methods is too sensitive on initial conditions. For such systems, usually sufficiently accurate initial values are required for convergence.

We will pursue with the expectation-maximization (EM) algorithm. The EM algorithm turns out to be convenient in our case. However, the EM algorithm has a tendency to approach the maximum fast, but convergence might slow down close to the maximum. As an improvement, a combination of the EM algorithm and a Newton method can be used (cf. [12] chapter 2).

Before doing so, we will rewrite the model into an exponential family. We will show that the model belongs to a regular and natural exponential family, which concludes that the EM algorithm finds the unique MLE for the model parameters. Additionally, rewriting the model as an exponential family helps us to understand the restrictions of the model, i.e., the cases for which the MLE does not exist.

Model as a natural exponential family

The model (7a) can be rewritten as a natural exponential family [13]. Consequently, the log-likelihood (15) also can be rewritten in this form, since the exponential families are preserved under repeated sampling, given that the observations are independent.

Remember, for the IDM we defined the sample space of all possible observations as . While retains some allelic information about the underlying data, even if the information is incomplete, the empty record 0 harbors no ‘direct’ information. This particular observation deserves special attention. We transform the sample space into a higher-dimensional space for which the first component specifies whether a record is empty. More precisely, a non-empty observation is identified with the vector (0, x) and the empty record with the vector (1, 0). Let (25)

We define the transformation by (26)

It is easy to see that h(x) is a bijection between and . Note, h₀(x) serves as an indicator for the empty record, i.e., h₀(0) = 1 and h₀(x) = 0 for all .

Let the vector-valued function , where , be defined as (27) with (28a) and (28b) for k = 1, …, n. Additionally, let (29) and (30)

It is shown in section Incomplete-data model (IDM) as a natural exponential family in S1 Appendix Mathematical appendix that the model (7a) can be rewritten as an exponential family in the form (31)

The function f₀(x) is the base density, which is a uniform density on the support of x. This representation is not in natural form. It is possible to rewrite (31) into the natural form by substituting since g is a 1–1 mapping (see section Incomplete-data model (IDM) as a natural exponential family in S1 Appendix Mathematical appendix).

Let z^T β denote the scalar product of z and β, i.e., (32)

Hence, the model (31) is rewritten into its natural form as (33) where (34) with β_k = g_k(θ) for k = 0, 1, …, n, (35) and is the base density of the natural exponential family (see section Incomplete-data model (IDM) as a natural exponential family in S1 Appendix Mathematical appendix).

Because h(x) is a bijection, the marginal distribution of z is equivalent to the distribution of x, i.e., both have the same structure function (base density). The closed convex hull of its support is (36) where is specified in (25). It is not difficult to see that h is also a bijection between and the closed convex hull of , which is an n-dimensional rectangular set in .

The function can be interpreted as the cumulant-generating function of z (cf. [7], chapter 5.2.1). The natural parameter space of the exponential family is the set for which is finite. From (35) it can be seen that this is . This implies that the natural exponential family (33) is regular. Additionally, the natural statistic z = h(x) and the natural parameter space are (n + 1)-dimensional, showing that the natural representation is in fact minimal (see section Minimality of the natural exponential family in S1 Appendix Mathematical appendix). The minimality of (33) ensures that the natural exponential family is full, not curved, and that the statistic z is minimal sufficient for β [13].

Now, consider a dataset consisting of N observations x⁽¹⁾, …, x^(N) obtained from N disease-positive samples, such that each x^(j) (for j = 1, …, N) is distributed according to the exponential family (31) with the common corresponding parameter space Θ. The corresponding natural statistics are the observations z⁽¹⁾, …, z^(N) with z^(j) = h(x^(j)), which follow the natural exponential family (33) with natural parameter space . The likelihood of observing as a function of β is (37) where (38)

Note that is the indicator of the j-th sample being an empty record, and for k > 0, indicates if lineage A_k was observed in sample j. Hence, (39a) and (39b) where n₀ is the number of empty records, and N_k is the number of samples infected by lineage A_k (one can easily conclude that this is an equivalent representation for Eq 14). Therefore, (38) becomes (40) which is a sufficient statistic for the likelihood (37). In fact, N is minimal sufficient, since the natural exponential family (33) is minimal [13]. Clearly, the likelihood has also the natural exponential family of the form (33).

Let (41) be the vector of empirical prevalences. The log-likelihood function is derived as (42)

It is easy to see that the support of U is (43)

The cumulant-generating function of U is the same as the cumulant-generating function of z. Because the family is regular, it is strictly convex and sufficiently many times differentiable [13]. Its first derivative with respect to the natural parameters is the mean vector of U, i.e., (44) (cf. [14]), where the partial derivatives are (45a) and (45b) for k = 1, …, n. The log-likelihood function (42) is a strictly concave function of θ (it differs from only by a linear term). Its first derivative (or the score function) is derived as (46)

The MLE of is then found as the unique root of (47) i.e., . Therefore, is derived by solving (48a) and (48b)

Since the natural model is regular, exists if U belongs to the interior of the closed convex hull of the support of U [15], which is the set (49)

We will treat all pathological cases in which the MLE does not exist in the admissible parameter space in the next section.

In general, the MLE yields a simple and intuitive interpretation, which is summarized in the following result.

Result 1. For the IDM (4b), the MLE is the choice of parameters , , and , for which (i) the probability of observing an empty record equals the observed relative frequency of the empty record, i.e., (50) (ii) the conditional prevalence of lineage k equals its relative frequency among non-empty records, (51) and (iii) the relative frequency of samples containing lineage k among all (empty and non-empty records) is the actual prevalence multiplied by the probability that a lineage being present in an infection is actually observed, which equals the observed prevalence, i.e., (52) where N_k, n₀, and N₊ = N − n₀ are considered as realizations of the corresponding random variables, i.e., as constants.

The proof is presented in section Interpretation of the MLE in S1 Appendix Mathematical appendix.

Remark 2. The data N_k, n₀, and N₊ = N − n₀ is a realization of the random variables , , and . From Result 1 and Remark 1 and (24) it follows that (53) where the expectation is taken with respect to the MLE . In other words, the MLE was the true parameter the expected number of samples containing lineage k equals the observed number and the expected number of empty records equals the observed number.

As a consequence, the estimates for the inverse Fisher information and the inverse observed information based on the MLE coincide (cf. section Maximum likelihood estimate and Fisher information and section Inverse Fisher information in S1 Appendix Mathematical appendix).

Pathological cases

The MLE exists and is unique if . Importantly, of primary interest is , the MLE of the original parameters. We must be more cautious in deriving the MLE of θ, since it only exists if . However, for some it is possible that . Therefore, the set of U for which exists is only a subset of (interior of the closed convex hull of ). On the contrary, if , does not exist. However, these cases are interpreted in a limit sense. We refer to data U for which does not exit as pathological data. These cases are fully classified in Result 2.

First, it is worthwhile mentioning that if N_k = 0 for one k the model can be reduced by eliminating the parameter p_k. Thus, we restrict our attention to N_k > 0 for all k.

Result 2. The following pathological cases occur.

1. Case A. Assume at least one lineage is observed in every sample, i.e., n₀ = 0. The MLE coincides with those of the OM, which ignores incomplete data. The following four pathological cases occur.
   1. A.1. If at least one sample contains evidence of a multiple infection, but no lineage is observed in every sample, in technical terms and 0 < N_k < N for all k, then the MLE coincides with that of the OM. More precisely, and the estimates and are obtained as in [9].
   2. A. 2. If there is no indication of multiple infections in the data, and no lineage is present in all samples, i.e., if and 0 < N_k < N for all k, the MLE is attained at (54)
      Hence, the MOI parameter is estimated to be zero, and the lineage frequencies by their relative abundance in the data.
   3. A. 3 If at least one lineage occurs in all samples (and at least two lineages are observed in the data), i.e., if N_k = N and N_j > 0 for at least one j ≠ k, then and the MLE of the remaining parameters are reached in the limit λ → ∞, such that (55) subject to the constraint .
   4. A.4 If only one lineage is observed in the data, i.e., N_k ≠ 0 for only one k, the MLE is not unique and realized for any point (56)
2. Case B. Assume the fraction of empty observations is not “too large”, more specifically, . The only pathological case arises under this condition if there is no indication of multiple infections in the data, and no lineage is present in all samples. More precisely, if and 0 < N_k < N₊ for all k, the MLE is attained at (57)
   Hence, the probability that a lineage remains undetected is estimated as the fraction of empty records, the MOI parameter as zero, and the lineage frequencies by their relative abundance in the non-empty observations.
3. Case C. If the fraction of empty records is relatively large, specifically, the MLE is obtained in the limit case λ → ∞ such that (58a) and (58b) subject to the constraint . In this case, the supreme of the likelihood function is

For the sake of completeness, in the case where all samples failed to produce data, i.e., n₀ = N, the MLE is not well-defined. In particular, any point (59)

A proof of the results can be found in section Pathological cases in S1 Appendix Mathematical appendix.

Deriving the maximum-likelihood estimate

By rewriting the statistical model into a natural exponential family, we could classify all pathological cases in which the MLE does not exist. Moreover, it has enabled us to establish Result 1, which gives an intuitive interpretation of the MLE in terms of prevalence but not in terms of the actual parameters of interest, namely the θ = (λ, p, ε). These need to be derived numerically because (44) has no explicit solution. However, the exponential family form yields a much simpler algorithm to derive the MLE than the multidimensional Newton-Raphson method (23), which is too sensitive to initial conditions. In terms of the exponential family, two 1-dimensional Newton algorithms need to be iterated. However, the problem of sensitivity to initial conditions is not entirely remediated. The EM-algorithm is an alternative, with the convenient property that convergence to the MLE is guaranteed except in the pathological cases. It is described in the following result.

Result 3. Given regular data, the MLE of the model parameters is obtained by a two-step iterative procedure. Starting from initial values λ⁽⁰⁾, and ε⁽⁰⁾, in step t + 1 the parameters p_k and ε are updated by (60a) (60b) where (60c) (60d) and (60e)

The parameter λ is updated in step t + 1 by iterating the recursion (60f) starting from λ₀ = λ^(t), where (60g) until convergence is reached. This is the case if |λ_s+1 − λ_s| < δ, where δ is the numerical threshold for convergence. Then the parameter lambda is updated by λ^(t+1) = λ_s+1.

The EM algorithm is guaranteed to converge monotonically from any initial values θ⁽⁰⁾ = (λ⁽⁰⁾, p⁽⁰⁾, ε⁽⁰⁾). Numerical convergence is reached according to a specified criterion, e.g., (61)

A complete outline of the calculations is given in section The EM Algorithm in S1 Appendix Mathematical appendix.

Note that the EM algorithm needs a 1-dimensional Newton algorithm in the maximization step. This Newton algorithm finds a unique solution in every iteration if the data is regular. The MLE derived from the EM algorithm exists and it is unique since the log-likelihood function belongs to a natural and regular exponential family [16]. However, the EM algorithm does not converge (i.e., the MLE does not exist) in the pathological cases.

Finite sample properties

Concerning the finite sample size properties of the proposed method, we first concentrate on bias. To be able to compare the bias of the MOI parameter λ or rather the average MOI ψ across different parameter ranges, the relative bias (i.e., the bias in percent of the true underlying parameters) is reported. Regarding the lineage frequencies, we report the bias of the predominant lineage. Second, the variance of the estimators, in terms of the coefficient of variation, is reported, which again allows a comparison of the performance between different parameter ranges. The performance of the IDM is also compared to that of the OM.

Note that the bias and variance of the OM were already studied in [3, 4]. However, it was studied under the assumption of complete data, i.e., ε = 0. Once ε > 0, the OM is no longer the true model. Namely, the quality of observations is reduced, because of missing information, i.e., not all lineages in a sample are observable. As shown in Remark 1 if no empty records occur in the dataset, the IDM and the OM yield the same estimates, since they use the same data. In particular, the IDM wrongly estimates ε as . If empty records occur, the IDM and the OM yield different estimates and the OM is based on sample size N₊ rather than on N.

Bias of the average MOI.

In the case of perfect information ε = 0 the IDM and OM coincide, and the bias was described already in [3, 4], for which the OM and IDM coincide.

The bias of the average MOI (Figs 2 and 3) estimated by the IDM is rather insensitive to the probability of lineages remaining undetected, i.e., to ε, and remains similar as described in [3, 4]. More precisely, if ε increases from 1.5% to 15%, the maximum bias increases only by 3%. The bias of is a non-linear function of the true average MOI ψ, which vanishes with increasing sample size N. In general, the estimator overestimates the true parameter. The reason for the overestimation is that the average MOI ranges from 1 to ∞, i.e., the amount by which the parameter can be underestimated is bounded, while it can be arbitrarily overestimated. The non-linear behavior has the following reason. For small λ, occasionally samples with two or more different lineages present are over-represented, and hence the true parameter is greatly overestimated. This is typically for MLEs, which are in general sensitive to outliers (cf. Section 5.5 in [17]). (This is particularly true for small sample sizes, where outliers have a stronger effect.) As the true λ increases, bias starts to decrease and is lowest for values of λ corresponding to intermediate transmission. For larger λ MOI within samples varies substantially due to the underlying Poisson distribution (which has equal mean and variance λ). Consequently, if samples with high MOI are over-represented in a dataset, the MOI parameter is also overestimated. In general, bias is larger for underlying unbalanced frequency distributions (cf. Figs 2 and 3A with 3B, 3C, with 3D and 3E with 3F). The reason is that with unbalanced frequency distributions, it is unlikely to observe infections with different lineages being present unless MOI is high. Hence, the method becomes more sensitive to outliers. In general, the bias is lower with an increase in the number of lineages (cf. Fig 2 with Fig 3). The reason is that there is a better resolution to represent super-infections.

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Fig 2. Bias of average MOI.

Each panel illustrates the bias of the average MOI in % of the true average MOI (relative bias) for different sample sizes N (colors). Solid lines and dashed lines show the bias of the IDM and OM, respectively. In all panels two lineages (n = 2) are assumed. Different panels correspond to different probabilities of missing information (ε) and frequency distributions (p). The left and right panels compare the bias for balanced and unbalanced frequency distributions for the same values of ε. In particular, ε = 0.015 in (A, B), ε = 0.1 in (C, D), and ε = 0.15 in (E, F).

https://doi.org/10.1371/journal.pone.0287161.g002

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Fig 3. Bias of average MOI.

See Fig 2 but for n = 4 lineages.

https://doi.org/10.1371/journal.pone.0287161.g003

If MOI is estimated by the OM, bias behaves quite differently. First, bias substantially increases with an increasing probability of undetected lineages. Particularly, the change in bias is strikingly high for large sample size, even if the probability of lineages remaining undetected is as low as ε = 1.5%. Second, the true parameter is underestimated, except for very small true λ (or equivalently ψ). Third, bias increases with increasing sample size (N). Fourth, bias almost linearly decreases as a function of the true parameter ψ. The reason is, that many lineages are not observed, which is not reflected at all by the model. This effect is stronger for unbalanced frequency distributions, because (i) infections with different lineages are rare, and (ii) due to incomplete data, it is likely that some of the lineages remain undetected. Hence, observations with no evidence of multiple infections become increasingly likely.

Bias of lineage frequency estimates.

For the IDM the lineage frequency estimates have very little bias, irrespective of the (i) probability of lineages remaining undetected (ε), (ii) sample size, (iii) MOI parameter λ (ψ), (iv) the number of lineages (n) and typically less than 1% (see Fig 4). For symmetric frequency distributions p the linage frequencies can be considered unbiased. For unbalanced distributions, the bias of the dominant lineages seems to increase linearly as a function of the parameter ψ. This is not surprising, since in the symmetric case all lineages are equally abundant, whereas in, the case of unbalanced frequencies, some lineages are over-represented. Note that bias in Fig 4 fluctuates due to the limited amount of simulation runs. Note further that the bias of minor lineage frequencies in relative terms can be much higher than for predominant lineages, but will be much smaller in absolute terms.

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Fig 4. Bias of lineage frequencies.

Similar to Fig 2 but for the bias of the dominant lineage in % of the true value as a function of the true average MOI (relative bias). Different panels correspond to different probabilities of missing information (ε) and frequency distributions (p). While the left panels (A, D, G) assume a balanced frequency distribution, the middle (B, E, H) and the right (C, F, I) panels assume unbalanced frequency distributions.

https://doi.org/10.1371/journal.pone.0287161.g004

The bias of lineage frequencies is based on the OM changes. Not surprisingly, for symmetric distributions, lineage frequencies are unbiased because all lineages are equivalent. Bias behaves in the same way as for the IDM (see Fig 4A, 4D and 4G). On the contrary, for unbalanced frequency distributions, the dominant lineage tends to be underestimated. Bias is negative and increases linearly with the average MOI ψ. For low transmission, the dominant lineage is unbiased, if the probability of lineages remaining undetected is small. However, the bias increases with increasing probability of lineages remaining undetected if transmission is high. Namely, for low transmission (almost only single infections, MOI = 1) lineages remain undetected approximately proportional to their frequencies. If transmission is high, the prevalence of rare lineages is high, and they tend to co-occur with predominant lineages. In such infections, both predominant and rare lineages have the same probability of remaining undetected, hence leading to an under-representation of the predominant lineages in the data.

Bias of the probability of lineages to remain undetected.

The probability of lineages to remain undetected is only estimated by the IDM. In general, the parameter estimates have little bias, irrespective of the other parameters (Fig 5). The parameter tends to be less biased for larger true parameters ε. This is intuitive, as it is estimated to be in the absence of empty records, which is very likely if ε is small. For unbalanced frequency distributions bias tends to be smaller because empty records are more likely, leading to better estimates of the true parameter.

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Fig 5. Bias of the probability of lineages to remain undetected.

Similar to Fig 2, but for the relative bias of the probability of lineages remaining undetected (ε). This parameter is only estimated by the IDM.

https://doi.org/10.1371/journal.pone.0287161.g005

Variance of the estimator of average MOI.

The variance of the estimator based on the IDM in terms of the CV increases as a function of the true parameter. The reason is that for small true λ (or ψ) datasets tend to be more homogeneous, while the diversity of possible datasets increases with λ. This is particularly true for larger numbers of circulating lineages (larger n; cf. Fig 6 with Fig 7). The CV of the estimator based on the IDM increases substantially with the likelihood of incomplete observations (larger ε). The reason is the following. If no empty records are observed in a dataset, the IDM gives the same estimate as the OM, which tends to be an underestimate (cf. Bias of the average MOI). Particularly, if ε is large, the underestimate is substantial. However, once empty records occur in the dataset, the IDM tends to overestimate the true parameter. This creates the large variance of the estimator. Not surprisingly, the CV decreases with increasing sample size.

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Fig 6. CV of average MOI.

Similar to Fig 2, but for the CV of the average MOI in %. The dotted line shows the CV based on the Cramér-Rao lower bound, i.e., the CV is estimated as the square root of the Cramér-Rao lower, divided by the true parameter.

https://doi.org/10.1371/journal.pone.0287161.g006

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Fig 7. CV of average MOI.

See Fig 6 but for n = 4 lineages.

https://doi.org/10.1371/journal.pone.0287161.g007

For the OM the CV still increases with ε, but to a smaller extent, because the parameter is constantly underestimated, which leads to a generally smaller variance. If ε increases, empty records effectively lead to a depletion of sample size, which is associated with higher variance.

For small ψ the CV almost coincides with the Cramér-Rao lower bound, whereas the CV tends to lie above the lower bound as ψ increases (see Fig 6). However, the discrepancy improves for larger sample size N. On the contrary, the CV of the estimator based on the OM is lower than the Cramér-Rao lower bound, underlining that this estimator is biased. As expected, this becomes more pronounced as ε increases.

Variance of lineage frequency estimates.

Not surprisingly, the CV of the lineage frequencies decreases with increasing sample size, because datasets are more informative, leading to more accurate estimates, and hence less variation. Importantly, variance remains is hardly affected by the average MOI ψ. However, particularly for balanced frequency distributions, variance tends to be higher for lower transmission, because each lineage has lower prevalence, corresponding to less informative data and hence more variable estimates. This effect is stronger for symmetric frequency distributions (cf. Fig 8A–8C, 8E with Fig 8B, 8D).

The CV is similar for the estimates based on the IDM and OM, but it tends to be slightly lower for the OM. Again, there is a good agreement of the CV of the IDM with the Cramér-Rao lower bound.

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Fig 8. CV of lineage frequencies.

Similar to Fig 6 but for the CV of the dominant lineage frequency (p₁) in % as a function of the true average MOI (ψ).

https://doi.org/10.1371/journal.pone.0287161.g008

Variance of the probability of lineages to remain undetected.

The CV of the probability of lineages to remain undetected is high in general. The reason is that the parameter is estimated as in the absence of empty records, whereas it has a more meaningful estimate if empty records are observed. Consequently, the CV decreases with sample size (empty records are more likely to be observed in a dataset) and increases with MOI (which decreases the chance of observing empty records). For symmetric frequency distributions the CV is higher than for unbalanced distributions (cf. Fig 9A–9C, 9E with Fig 9B, 9D), because it is less likely to observe empty records. Again, there is a reasonable agreement between the CV and the Cramér-Rao lower bound. The fit improves as sample size N increases. For small sample size, the CV tends to be higher than the Cramér-Rao lower bound, which is not surprising since the estimator is sensitive to the number of empty records, which varies substantially for small sample size.

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Fig 9. CV of the probability of lineages to remain undetected.

Similar to Fig 6, but for the CV of the probability of lineages to remain undetected (ε). This parameter is only estimated by the IDM.

https://doi.org/10.1371/journal.pone.0287161.g009

Mean-squared error of the average MOI estimate.

When considering bias and CV of the MOI parameter, it is inconclusive whether the IDM or the OM is preferable. Whereas the estimate of λ based on the IDM has a lower bias, it has a higher variance. Therefore, the mean-squared error (MSE) should be considered (Fig 10). Also, no clear preference emerges from the MSE. The MSE of the IDM tends to be lower for large sample size and higher probability of lineages remaining undetected. This is particularly true for balanced frequency distributions (cf. Fig 10A–10C, 10E with Fig 10B, 10D, 10F). If ε is small, both models perform similarly well. However, if ε is large, the IDM performs only better for larger sample sizes (N ≥ 150).

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Fig 10. MSE of average MOI.

Similar to Fig 2 but for the normalized-root-mean-squared-error of the average MOI in % as a function of the true average MOI (ψ).

https://doi.org/10.1371/journal.pone.0287161.g010

Data application

To demonstrate the differences between the IDM and OM, they are applied to empirical data from Cameroon and Kenya. The first dataset, described in [18], consists of N = 166 P. falciparum samples collected in Yaoundé, Cameroon, between 2001 and 2002. It contains information on 6 microsatellite loci on chromosomes 4 and 8, far enough from Pfdhfr (C3, E1, Fr9, and Fr10 at positions -58kb, -30kb, 70kb, and 90kb Pfdhfr) and Pfdhps (O1 and Q4 at positions -50kb, and 71.6kb Pfdhps) to be considered neutral markers, and on 8 neutral microsatellite markers on chromosomes 2 and 3 (for details see [18]). The P. falciparum dataset from Kenya is described in [19]. Here, 9 neutral microsatellite markers at chromosomes 2 and 3 from N = 43 samples collected in Asembo Bay, Kenya, between April 1993 and March 1994 were included (see [19]). The data is provided as supporting information (S3 Data from Cameroon and Kenya.

For each marker in the two datasets, the maximum-likelihood estimate for the average MOI parameters was calculated alongside 95% asymptotic confidence intervals (CI). An asymptotic (1 − α)-CI is easily derived from the inverse Fisher information, which is implemented in the available R-script (S1 Data R-script; cf. S2 Data Description of R-script) and given by (62) where z_α is the α quantile of the standard normal distribution and is the first entry of the inverse Fisher information evaluated at the MLE (see Eq 25 in section Inverse Fisher information in S1 Appendix Mathematical appendix; for more details on the definition of the CIs for the OM see “Results” in [9] and Appendix E in [4]).

Hardly any empty records occurred in the data from Kenya (maximum 4.65%), in fact, none occurred for 5 markers. Hence, the estimates for the average MOI parameter and the CIs are either identical or in good agreement (Fig 11A). Overall, the CIs for the various markers are in good agreement. However, the CIs for markers U5 and U6 do not overlap with that of L1. Remember, that CIs are random intervals, which are constructed so that the true parameter lies in the intervals with probability 1 − α. Therefore, CIs do not necessarily overlap, apart from the fact that the actual coverage might be quite different from the nominal coverage, particularly for CIs based on an asymptotic normal distribution (cf. [20], chapters 12 and 13). Hence, also given the small sample size, it is not particularly worrisome that the CIs for some markers do not overlap. However, based on Fig 11A, one might decide to manually check the data for marker L1. In summary, the overall estimates from different markers agree well and there is no need to use the IDM given the few empty records.

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Fig 11. Data application.

Shown are the MLEs for average MOI () calculated from the IDM and OM for each marker of the dataset from Kenya (A) and Cameroon (B), alongside 95%-CIs. The number above the error bars are the number of empty records (n₀).

https://doi.org/10.1371/journal.pone.0287161.g011

The situation is different for the data from Cameroon. Although the sample size is much larger, empty records are common (between 9% and 41.6% are empty records). The OM leads to lower estimates of average MOI than the IDM for all markers (the fewer the number of empty records the better the estimates of the two models agree, cf. markers Fr9, Fr10, L1 in Fig 11B). The shortcomings of the OM are revealed by the CIs. Namely, CIs from the OM are much narrower than those for the IDM, yielding unjustified confidence in the estimates. The reason is that the OM does not properly account for the uncertainty in the data quality. As a result, it is not surprising that some CIs do not overlap. The CI of marker E1 does not overlap with the CIs of markers K6, Fr9, and Fr10. Moreover, some CIs barely overlap. While the OM seems to give strong evidence that the average MOI parameter lies in the range from 1.11 to 1.5, the results appear not very trustworthy because the estimates and CIs vary strongly across markers. On the contrary, the IDM yields CIs which all overlap, because uncertainty in the data is better reflected. In fact, from the results of the IDM, one would not exclude that the average MOI is larger than 1.5.