Tuning parameter selection for a penalized estimator of species richness

Abstract

Our goal is to estimate the true number of classes in a population, called the species richness. We consider the case where multiple frequency count tables have been collected from a homogeneous population and investigate a penalized maximum likelihood estimator under a negative binomial model. Because high probabilities of unobserved classes increase the variance of species richness estimates, our method penalizes the probability of a class being unobserved. Tuning the penalization parameter is challenging because the true species richness is never known, and so we propose and validate four novel methods for tuning the penalization parameter. We illustrate and contrast the performance of the proposed methods by estimating the strain-level microbial diversity of Lake Champlain over three consecutive years, and global human host-associated species-level microbial richness.

1. Introduction

The species problem concerns estimating C, the number of classes that are present in a population. n individuals from the population can be sampled to find which classes they belong to, but only c classes are observed ( $c\le C$). The problem is named for its origins in biological ecosystems, where C is species richness, or the total number of species. However, methods developed for the species problem can be applied to applications far removed from biology. For example, Efron and Thisted [9] estimated the number of words Shakespeare truly knew by modeling the frequencies of words in his published work, and Fegatelli and Tardella [10] estimated the number of cars covered by an insurer based on accident data where c cars had at least one accident.

In ecology, species richness is a quantitative measurement of ecosystem diversity. We focus on the specific application of estimating microbial diversity, that is, the number of strains of bacteria present in a population (e.g. a lake microenvironment or in an individual's oral cavity). Microbial diversity is often linked with ecosystem health, such as in the vaginal microbiome (where high diversity is associated with infection [20]) and in the gut microbiome (where high diversity is associated with healthy metabolism [17,18]). Microbial abundance data typically contain many species observed infrequently (rare species) and some which are observed a large number of times (abundant species). In data with both many rare and many abundant species, richness estimation methods often have high variance [28,30], motivating our regularized approach to estimation.

In this paper, we consider the penalized maximum likelihood approach of Wang and Lindsay [28] and investigate the open problem of selecting the penalization parameter. Because the true species richness C is never observed for any sample, common approaches to tuning parameter selection (e.g. cross-validation) cannot be employed. We propose using biological replicates to aid tuning. We find that replicate data is advantageous for tuning the required penalization parameter and provide a comparison of the performance of four proposed methods for tuning parameter selection.

This paper is organized as follows: We review species richness estimation in Section 2. In Section 3, we describe our extension to biological replicates, and in Section 4 we establish the utility of penalization via simulations. In Section 5, we propose several novel methods for tuning the penalization parameter, and in Section 6 we evaluate our proposals. In Section 7, we apply our methods to estimate microbial diversity in Lake Champlain, and Section 8 closes with a discussion of our results and suggested directions for future work. Software implementing the methods is available in the R package rre (regularized richness estimation), available at github.com/statdivlab/rre, and code to reproduce the simulation results is available at github.com/statdivlab/rre_sims.

2. Literature review

2.1. Poisson-mixture models

The classical model for estimating species abundance is a Poisson-mixture model. Under this model, the number of times we observe species i is $X_i\sim \mathrm{Poisson}(\mathrm{\Lambda }),$ where Λ is a random variable distributed according to some mixing distribution. Many mixing distributions have been proposed, including by [4,19,21] (see [5] for a comprehensive review). While our proposal may be easily generalized to other mixing distributions, in this paper, we consider the Gamma distribution for Λ, first considered by Greenwood and Yule [12] and Fisher et al. [11]. Note that we only observe $X_i|X_i>0$, and therefore we are interested in estimating the parameters of a Poisson-mixture model using zero-truncated Poisson-mixed data.

Let $f_k=\mathrm{\#}\{i:X_i=k\}$ be the number of classes observed k times, called the frequency counts. Then the set $\{f_k{\}}_{k\ge 1}$ is a frequency count table, a common way to represent species abundance data. Because $C=f_0+f_1+f_2+\cdots =f_0+c$, the species problem can be framed as predicting $f_0$ given $f_1,f_2,\dots$ to obtain a species richness estimate $\hat{C}={\hat{f}}_0+c$.

Let $\mathrm{\Lambda }\sim \mathrm{Gamma}(\alpha ,\delta )$ with distribution function $f(\lambda )={\delta }^{\alpha }\mathrm{\Gamma }(\alpha {)}^{-1}{\lambda }^{\alpha -1}{e}^{-\delta \lambda }$, and write $\eta =(\alpha ,\delta )$. There exist both frequentist [6,11] and Bayesian [2,9] approaches to parameter estimation under this model. We will focus on frequentist maximum likelihood (penalized and unpenalized) in this paper.

2.2. Estimation and computation

The most straightforward maximum likelihood approach is to simultaneously maximize C and η using the full likelihood, which is known as the direct approach. A complication is that we have continuous parameters η as well as one discrete parameter C, and so derivative-based optimization methods are not appropriate. This issue has been studied by Lindsay and Roeder [14], who provide a discrete analog to the score function for this model. Two other approaches to likelihood maximization are the conditional and profile approaches. The conditional approach [25] involves writing the likelihood as the product

$$L(C,\eta )=L_b(C,\eta )L_c(\eta ),$$ (1)

where

$$\begin{array}{rl}{L}_b(C,\eta )& =\frac{C!}{(C-c)!c!}{[1-p_{\eta }(0)]}^c{[p_{\eta }(0)]}^{C-c},\end{array}$$ (2)

$$\begin{array}{rl}{L}_c(\eta )& =\frac{c!}{{\displaystyle \prod _{k\ge 1}{f}_k!}}\prod _{k\ge 1}{\left[\frac{p_{\eta }(k)}{1-p_{\eta }(0)}\right]}^{f_k}.\end{array}$$ (3)

We see that the likelihood is the product of a binomial probability mass function $c\sim$ Binomial $(C,1-p_{\eta }(0))$, and a multinomial probability mass function $(f_1,f_2,\dots )|c\sim$ Multinomial $(c,(p_{\eta }(1),p_{\eta }(2),\dots )/1-p_{\eta }(0))$. This decomposition is convenient because $L_c$ is a function of η alone. Sanathanan [25]'s conditional approach to optimization involves first maximizing $L_c(\eta )$ to obtain a conditional abundance estimate ${\hat{\eta }}_c$, then maximizing $L_b(C,{\hat{\eta }}_c)$ over C. The conditional estimate of C is then

$${\hat{C}}_c=\lfloor \frac{c}{1-p_{{\hat{\eta }}_c}(0)}\rfloor ,$$ (4)

where $\lfloor a\rfloor$ is the largest integer less than or equal to a. In the profile likelihood approach the expression in Equation (4) is substituted into the full likelihood, which gives us a function of η alone (see Wang and Lindsay [28]). The conditional and profile approaches have been shown to be asymptotically equivalent to the direct approach.

2.3. Challenges with Poisson-mixture models

A perennial issue in the species problem, especially for microbial datasets, is the instability of species richness estimators [24,30]. A proposal which encourages stability of estimates under Poisson-mixture models is due to Wang and Lindsay [28]. Their proposal is to add a penalty term to the log-likelihood that penalizes the probability of observing a species zero times. They consider the penalized log-likelihood

$${\ell }_{\lambda }(C,\eta )=\ell (C,\eta )-\lambda \log p_{\eta }(0),$$ (5)

where $\ell (C,\eta )=\log L(C,\eta )$ is the log-likelihood and $\lambda >0$ is a penalization parameter. Let ${\hat{C}}_{\lambda }=\arg \underset{C}{max}{\ell }_{\lambda }(C,\eta )$. Then, for $\lambda \ge {\lambda }^{\prime },$ ${\hat{C}}_{\lambda }\le {\hat{C}}_{{\lambda }^{\prime }}$ [28, Theorem 1]. In particular, for $\lambda >0,$ the penalized maximum likelihood solution ${\hat{C}}_{\lambda }$ is less than or equal to ${\hat{C}}_0$, the maximum likelihood estimate. Furthermore, for a large enough λ, ${\hat{C}}_{\lambda }=c$; that is, the penalized maximum likelihood estimate shrinks to the observed richness c.

We note that $\log p_{\eta }(0)<0,$ and so the addition of the ‘penalty’ term in fact increases the objective function (5) over the (unpenalized) likelihood $\ell (C,\eta )$. While technically smaller $p_{\eta }(0)$ adds a larger reward to the objective function (for a fixed λ), we refer to the term $-\lambda \log p_{\eta }(0)$ as a penalty to be consistent with the terminology of Wang and Lindsay [28]. Wang and Lindsay [28] also consider two other penalty functions, which we do not discuss here except to note that our tuning parameter selection methods would equally apply to these other penalty functions.

Wang and Lindsay [28] show that a trade-off exists: greater values of λ correspond to a more stable estimator, but at the potential cost of negative bias. The choice of penalization parameter implies a preference for lower variance or lower bias, adding subjectivity to the estimation procedure. Wang and Lindsay [28] note that a limitation of their proposal is that ‘one must select a penalty function and a tuning parameter, and it is nigh impossible to make convincing statements about why one choice should be uniformly superior to another'. Furthermore, we expect different data sets to require different λ values for optimal mean squared error. The goal of this paper is to propose and investigate data-adaptive methods to select λ.

3. Extension to biological replicates

We focus on the gamma-Poisson model and penalized log-likelihood of the form (5), and consider the case where we observe r independent frequency count tables from the population under study. We will assume that these r frequency count tables are biological replicates drawn independently from the same population, i.e. they have a common structure specified by the parameters C and η. A set of r frequency count tables from the same population will be called a sample, and a single frequency count table will be referred to as a replicate. Let the number of species observed k times in replicate j be $f_{kj}$, $j\in \{1,\dots ,r\}$. Let $c_j$ be the observed richness in replicate j.

In Section 5, we will require an objective function defined in terms of a subset of indices of the frequency count tables, so we define it that way now. Let $J\subseteq \{1,\dots ,r\}$. J indicates the data being used in the evaluation of the objective function, with $J=\{1,\dots r\}$ meaning we use all replicates. Let $\ell (C,\eta ;\{f_{kj}{\}}_{k\ge 1})$ be the unpenalized log-likelihood for replicate j. Using the fact that the draws are independent, our objective function for a fixed $\lambda \ge 0$ is the sum of the log-likelihoods for each individual replicate. We define the penalized log-likelihood for multiple samples as

$$\begin{array}{rl}{\mathcal{O}}_{\lambda }(C,\eta ;J)& :=\sum _{j\in J}\ell (C,\eta ;\{f_{kj}{\}}_{k\ge 1})-\lambda \log p_{\eta }(0)\end{array}$$ (6)

$$\begin{array}{rl}& =\sum _{j\in J}[\log C!-\log (C-c)!+(C-c)\log p_{\eta }(0)\\ & -\sum _{k\ge 1}\log f_{kj}!+\sum _{k\ge 1}{f}_{kj}\log p_{\eta }(k)]-\lambda \log p_{\eta }(0).\end{array}$$ (7)

By fixing $\lambda =0$ we obtain the unpenalized log-likelihood for the set J. We define ${\hat{C}}_{\lambda }$ to be the penalized maximum likelihood estimate of C based on tuning parameter λ, that is,

$$({\hat{C}}_{\lambda },{\hat{\eta }}_{\lambda })={\arg \max }_{C,\eta }{\mathcal{O}}_{\lambda }(C,\eta ;\{1,\dots ,r\}),$$ (8)

where ${\hat{\eta }}_{\lambda }$ is a nuisance parameter.

To maximize the likelihood, we will use the direct approach, rather than the profile and conditional approaches reviewed in Section 2. While it may be faster to use the profile or conditional approach, the results are not always the same in finite samples. Since our objective is to evaluate approaches to tuning, we place a priority on the accuracy of optimization over speed of optimization.

To implement a direct optimization approach, we search over candidate C values in a grid. A gradient-based search over η values is used at each C in the grid to find the maximum penalized likelihood $(C,\eta )$.

4. Evaluating penalized species richness estimates

Before evaluating potential tuning parameter selection methods, we first establish that penalization can improve richness estimation. While Wang and Lindsay [28] established that penalization can improve estimates when r = 1, this needs to be investigated when r>1.

We use square root mean square error (RMSE) as the primary criterion for evaluating the performance of estimators. Let $n_{\mathrm{sim}}$ be the number of simulations completed for one choice of $(C,\eta ,r)$, and ${\hat{C}}_{\lambda ⟨s⟩}$ be the solution of Equation (8) obtained in simulation number s. We define

$$\mathrm{RMSE}\left({\hat{C}}_{\lambda }\right)=\sqrt{\frac{1}{n_{\mathrm{sim}}}\sum _{s=1}^{n_{\mathrm{sim}}}{({\hat{C}}_{\lambda ⟨s⟩}-C)}^2}.$$ (9)

In this simulation, if there exists some $\lambda >0$ for which RMSE( ${\hat{C}}_{\lambda }$) is less than RMSE( ${\hat{C}}_0)$, we conclude that penalization is effective.

We simulated using C = 1000, and $r\in \{6,10,14\}$. We found that $\eta =(\alpha ,\delta )=({10}^{-1},{10}^{-1})$ provided a good fit to the data of Walsh et al. [27], and $\eta =({10}^{-2},{10}^{-5})$ provided a good fit to the data of Tromas et al. [26] (see Section 7). For completeness, we also investigated nearby η values: $\eta =({10}^{-1},{10}^{-3})$ and $\eta =({10}^{-1},{10}^{-5})$. In Table 1, we provide summary measures for frequency count tables generated under each choice of η. This characterizes the η values by the proportion of unobserved, rare or abundant species we would expect to see. We return to a discussion of the different data structures in Section 6. For each combination of η and r, we performed 100 simulations. We investigated the grid $\lambda \in \{0,5,\dots ,120\}$, and found that this grid was sufficiently expansive to ensure that the RMSE-minimizing λ was not on the boundary of the grid (see Figures 1 and 2).

Table 1. The expected proportion of unobserved (k = 0), singleton (k = 1), rare (k = 1, 2, 3), and abundant ( $k\ge 10$) species for 4 choices of η.

η	( ${10}^{-1}$, ${10}^{-1}$)	( ${10}^{-1}$, ${10}^{-3}$)	( ${10}^{-1}$, ${10}^{-5}$)	( ${10}^{-2}$, ${10}^{-5}$)	( ${10}^{-2}$, ${10}^{-3}$)
Proportion unobserved ( $p_{\eta }(0)$)	0.787	0.501	0.316	0.891	0.933
Proportion singletons ( $p_{\eta }(1)$)	0.072	0.050	0.032	0.009	0.009
Proportion rare ( $\sum _{k=1}^3{p}_{\eta }(k)$)	0.130	0.097	0.061	0.016	0.017
Proportion abundant ( $\sum _{k=10}^{\mathrm{\infty }}{p}_{\eta }(k)$)	0.028	0.340	0.583	0.083	0.040
Expected max abundance ( $\mathbb{E}[k_{max}]$)	$4.1\times {10}^1$	$4.0\times {10}^3$	$3.9\times {10}^5$	$2.0\times {10}^5$	$1.9\times {10}^3$

We also give the expected frequency count of the most abundant species when $C=1000$.

Figure 1.

Estimates of C and their root-MSE over λ when $\eta =({10}^{-1},{10}^{-1})$ and C = 1000. Results are based on 100 simulations per λ.

Figure 2.

Estimates of C and their root-MSE over λ when $\eta =({10}^{-2},{10}^{-5})$ and C = 1000. Results are based on 100 simulations per λ.

The results of the simulation are shown in Table 2. For every parameter choice a reduction in RMSE( ${\hat{C}}_{\lambda }$) was found for some $\lambda >0$. Depending on simulation parameters the reduction varied from 22% to 70% compared to RMSE( ${\hat{C}}_0$) for the best λ. These results show that for a variety of $(\eta ,r)$ choices, penalization improves species richness estimation. We also see that the optimal λ value varies considerably over $(\eta ,r)$, ranging from 10 to 70. This suggests that there is not a universally appropriate λ choice. Therefore, tuning λ to the sample appears desirable. We note that the optimal choice of λ is not consistent either for fixed r and variable η, nor for fixed η and variable r.

Table 2. The penalized maximum likelihood estimate of C has lower RMSE for all investigated choices of η and r under a zero-truncated Gamma-mixed Poisson model for species abundances based on 100 simulations for each choice of η and r.

$\eta =(\alpha ,\delta )$	r	RMSE ( ${\hat{C}}_0$)	${\lambda }_{\mathrm{opt}}$	RMSE ( ${\hat{C}}_{{\lambda }_{\mathrm{opt}}}$)
$({10}^{-1},{10}^{-1})$	6	796.90	10	326.50
$({10}^{-1},{10}^{-1})$	10	527.53	15	286.27
$({10}^{-1},{10}^{-1})$	14	337.31	10	235.51
$({10}^{-2},{10}^{-5})$	6	735.95	15	516.37
$({10}^{-2},{10}^{-5})$	10	700.21	20	456.04
$({10}^{-2},{10}^{-5})$	14	666.55	20	470.35
$({10}^{-1},{10}^{-3})$	6	200.09	20	156.29
$({10}^{-1},{10}^{-3})$	10	213.22	55	142.64
$({10}^{-1},{10}^{-3})$	14	243.42	55	148.26
$({10}^{-1},{10}^{-5})$	6	401.17	55	147.13
$({10}^{-1},{10}^{-5})$	10	283.91	25	137.04
$({10}^{-1},{10}^{-5})$	14	415.19	70	126.72

${\lambda }_{\mathrm{opt}}$ is the value of λ which produced the lowest RMSE. ${\hat{C}}_0$ is the estimate of C when $\lambda =0$, and ${\hat{C}}_{{\lambda }_{\mathrm{opt}}}$ is the estimate of C when $\lambda ={\lambda }_{\mathrm{opt}}$.

To illustrate the bias-variance trade-off in this problem, we show the effect on ${\hat{C}}_{\lambda }$ of increasing λ in Figure 1 (when $\eta =({10}^{-1},{10}^{-1})$) and in Figure 2 (when $\eta =({10}^{-2},{10}^{-5})$). When $\eta =({10}^{-2},{10}^{-5})$, for small values of λ we observe lower variance in the estimates while incurring some positive bias, and the RMSE improves. However, as λ continues to increase, the bias term becomes large and negative and the RMSE increases. Similarly, when $\eta =({10}^{-1},{10}^{-1})$, we observe that the bias becomes large and negative and the RMSE increases. However, even for small values of λ, the estimate is not positively biased, and the larger RMSE is attributable to higher variance.

5. Methods for tuning λ

We have established that penalization can improve richness estimates, but different abundance structures (η) require different values of λ to minimize RMSE. We therefore develop methods to tune λ based on a sample. We propose several novel methods in this section. Each is evaluated in Section 6.

5.1. Method 0: no penalization

We compare all proposed methods to the unpenalized MLE using $J=\{1,\dots r\}$:

$$({\hat{C}}_{[0]},{\hat{\eta }}_{[0]})={\arg \max }_{C,\eta }{\mathcal{O}}_0(C,\eta ;\{1,\dots ,r\}).$$ (10)

This method is fast and simple, making it an ideal baseline for comparison.

For all of the remaining methods (1)–(4), we generate estimates ${\hat{C}}_{\lambda }$ over $\lambda \in {\lambda }^{\mathrm{grid}}$, where ${\lambda }^{\mathrm{grid}}$ is user-specified (e.g., ${\lambda }^{\mathrm{grid}}=\{0,5,10,\dots 120\}$ in Section 4).

5.2. Method 1: minimum subset variance

Since large variance in C is a major concern in species richness estimation, ideally an estimator will have low variance. If this is the case, there should be low variance in estimates from equally sized subsets of J. For Method 1, we exploit the fact that we have replicate data by repeatedly partitioning the replicates into two subsets and calculating two estimates. We then select the λ, which yields the lowest between-subset variance. This partitioning is repeated p times to average out the arbitrary choice of subsets.

Let $T_1(l)$ be the first subset of the lth partition and $T_2(l)$ be the second subset of the lth partition, for $l\in \{1,\dots ,p\}$. That is, $T_i(l)\subseteq \{1,\dots ,r\}$, $T_1(l)\cap T_2(l)=\mathrm{\varnothing }$ and $T_1(l)\cup T_2(l)=\{1,\dots ,r\}$. For each $\lambda \in {\lambda }^{\mathrm{grid}}$, $l\in \{1,\dots ,p\},$ let

$$\begin{array}{rl}({\hat{C}}_{\lambda }^{T_1(l)},{\hat{\eta }}_{\lambda }^{T_1(l)})& ={\arg \max }_{C,\eta }{\mathcal{O}}_{\lambda }(C,\eta ;T_1(l)),\end{array}$$ (11)

$$\begin{array}{rl}({\hat{C}}_{\lambda }^{T_2(l)},{\hat{\eta }}_{\lambda }^{T_2(l)})& ={\arg \max }_{C,\eta }{\mathcal{O}}_{\lambda }(C,\eta ;T_2(l)).\end{array}$$ (12)

We now have a $\hat{C}$ corresponding to each $\lambda \in {\lambda }^{\mathrm{grid}}$ in each subset of each partition. Our goal in Method 1 is to use these estimates to select the λ value which gave us the lowest average variance over all partitions, denoted by ${\tilde{\lambda }}_{[1]}$. The overall estimate for Method 1, ${\hat{C}}_{[1]}$, is a simple average of the estimates produced under ${\tilde{\lambda }}_{[1]}$:

$$\begin{array}{rl}{\tilde{\lambda }}_{[1]}& ={\arg \min }_{\lambda }\frac{1}{p}\sum _{l=1}^p\mathrm{Var}[{\hat{C}}_{\lambda }^{T_1(l)},{\hat{C}}_{\lambda }^{T_2(l)}],\end{array}$$ (13)

$$\begin{array}{rl}{\hat{C}}_{[1]}& =\frac{1}{p}\sum _{l=1}^p\left[\frac{{\hat{C}}_{{\tilde{\lambda }}_{[1]}}^{T_1(l)}+{\hat{C}}_{{\tilde{\lambda }}_{[1]}}^{T_2(l)}}{2}\right].\end{array}$$ (14)

In our simulations, we chose an equal split of the indices for each partition: $|T_1(l)|=|T_2(l)|$. The subsets of each partition are selected at random, and we sample with replacement. We partition a total 10 times (p = 10). For example, if r = 4, then $T_1(1)=\{1,4\}$ and $T_2(1)=\{2,3\}$ would be valid subsets. This approach to partitioning is also used in Methods 2 and 4.

5.3. Method 2: cross-validated likelihood

In Method 2, we propose to repeatedly partition the data into subsets and evaluate the estimates based on the ‘training’ subset using the likelihood based on the ‘evaluation’ subset. We partition the data into two subsets p times, calling them $T(l)$ for the training subset of the lth partition, and $E(l)$ for the evaluation subset of the lth partition. For each $\lambda \in {\lambda }^{\mathrm{grid}}$, $l\in \{1,\dots ,p\},$ let

$$({\hat{C}}_{\lambda }^{T(l)},{\hat{\eta }}_{\lambda }^{T(l)})={\arg \max }_{C,\eta }{\mathcal{O}}_{\lambda }(C,\eta ;T(l)),$$ (15)

that is, $({\hat{C}}_{\lambda }^{T(l)},{\hat{\eta }}_{\lambda }^{T(l)})$ is the penalized maximum likelihood estimate evaluated on the training subset. The λ value which maximizes the unpenalized likelihood calculated using the evaluation subset is selected, and the average species richness estimate at ${\tilde{\lambda }}_{[2]}$ is the estimated richness from Method 2:

$$\begin{array}{rl}{\tilde{\lambda }}_{[2]}& ={\arg \max }_{\lambda }\sum _{l=1}^p{\mathcal{O}}_0({\hat{C}}_{\lambda }^{T(l)},{\hat{\eta }}_{\lambda }^{T(l)};E(l)),\end{array}$$ (16)

$$\begin{array}{rl}{\hat{C}}_{[2]}& =\frac{1}{p}\sum _{l=1}^p{\hat{C}}_{{\tilde{\lambda }}_{[2]}}^{T(l)}.\end{array}$$ (17)

5.4. Method 3: goodness of fit

Method 3 uses goodness of fit of the fitted frequency counts to select an optimal tuning parameter. Given that $c\sim$ Binomial $(C,1-p_{\eta }(0))$ and $(f_1,f_2,\dots )|c\sim$ Multinomial $(c,(p_{\eta }(1),p_{\eta }(2),\dots )/1-p_{\eta }(0))$, we have that

$$\mathbb{E}\left[f_k\right]=\mathbb{E}\left[\mathbb{E}\left[f_k|c\right]\right]=\mathbb{E}\left[c\frac{p_{\eta }(k)}{1-p_{\eta }(0)}\right]=C(1-p_{\eta }(0))\frac{p_{\eta }(k)}{1-p_{\eta }(0)}=C{p}_{\eta }(k).$$ (18)

Therefore, given estimates $\hat{C}$ and $\hat{\eta }$, we consider a plug-in estimate for the expected frequency counts: ${\hat{f}}_k=\hat{C}{p}_{\hat{\eta }}(k)$. The usual ${\chi }^2$ goodness of fit statistic is then

$$\sum _{k=1}^{\mathrm{\infty }}\frac{{(f_k-{\hat{f}}_k)}^2}{{\hat{f}}_k}=\sum _{k=1}^{k_{\max }}\frac{{(f_k-\hat{C}{p}_{\hat{\eta }}(k))}^2}{\hat{C}{p}_{\hat{\eta }(k)}}+\hat{C}\sum _{k=k_{max}+1}^{\mathrm{\infty }}{p}_{\hat{\eta }}(k),$$ (19)

where $k_{\max }$ is the largest k such that $f_k>0$. Equation (19) is useful in software implementation as we can make use of precomputed tail probabilities $\sum _{k_{max}+1}^{\mathrm{\infty }}{p}_{\hat{\eta }}(k)$.

In Method 3, we make use of this goodness of fit metric by first generating estimates using all replicates. For each $\lambda \in {\lambda }^{\mathrm{grid}}$, let

$$({\hat{C}}_{\lambda },{\hat{\eta }}_{\lambda })={\arg \max }_{C,\eta }{\mathcal{O}}_{\lambda }(C,\eta ;\{1,\dots ,r\}).$$ (20)

${\tilde{\lambda }}_{[3]}$ is the λ value with the best-fitting estimates of C and η:

$${\tilde{\lambda }}_{[3]}={\arg \min }_{\lambda }\sum _{j=1}^r\sum _{k=1}^{\mathrm{\infty }}\frac{{(f_{kj}-{\hat{C}}_{\lambda }{p}_{{\hat{\eta }}_{\lambda }}(k))}^2}{{\hat{C}}_{\lambda }{p}_{{\hat{\eta }}_{\lambda }}(k)}$$ (21)

and ${\hat{C}}_{[3]}$ is the estimated value of C at this choice of λ:

$${\hat{C}}_{[3]}={\hat{C}}_{{\tilde{\lambda }}_{[3]}}.$$ (22)

An advantage of Method 3 that is not shared by Methods 1, 2 and 4 is that it can be used when r = 1, that is, when no repeated measurements are available.

5.5. Method 4: cross-validated goodness of fit

In Method 4, we return to the partitioning scheme of Method 2, but rather than using the likelihood in the evaluation step, we hypothesize that the goodness of fit metric may be a better choice. We partition the data p times, indexing the partitions by l. For each partition we have a training set $T(l)$ and a evaluation set $E(l)$. For all $\lambda \in {\lambda }^{\mathrm{grid}}$ and $l\in \{1,\dots ,p\}$, we generate estimates exactly as in Method 2:

$$({\hat{C}}_{\lambda }^{T(l)},{\hat{\eta }}_{\lambda }^{T(l)})={\arg \max }_{C,\eta }{\mathcal{O}}_{\lambda }(C,\eta ;T(l)).$$ (23)

To select λ we evaluate the goodness of fit metric using only the evaluation subset data, $j\in E(l)$. The λ which produces the best fitting estimates on the evaluation subset is ${\tilde{\lambda }}_{[4]}$:

$${\tilde{\lambda }}_{[4]}={\arg \min }_{\lambda }\sum _{l=1}^p\sum _{j\in E(l)}\sum _{k=1}^{\mathrm{\infty }}\frac{{(f_{kj}-{\hat{C}}_{\lambda }^{T(l)}{p}_{{\hat{\eta }}_{\lambda }^{T(l)}}(k))}^2}{{\hat{C}}_{\lambda }^{T(l)}{p}_{{\hat{\eta }}_{\lambda }^{T(l)}}(k)}$$ (24)

and ${\hat{C}}_{[4]}$ is the mean estimate of C at ${\tilde{\lambda }}_{[4]}$, averaged all partitions l:

$${\hat{C}}_{[4]}=\frac{1}{p}\sum _{l=1}^p{\hat{C}}_{{\tilde{\lambda }}_{[4]}}^{T(l)}.$$ (25)

Similar to Method 2, this method generates training set-based estimates ${\hat{C}}_{\lambda }^{T(l)}$, however, it evaluates these estimates using goodness of fit rather than likelihood maximization. Compared to Method 3, this method uses each replicate to either generate an estimate or evaluate the fit, while in Method 3 all replicates are used in both steps.

6. Comparison of methods for selecting λ

We have proposed four tuning methods motivated by properties which would be desirable in an estimator of C. In this section, we compare the performance of each estimator. We simulate zero-truncated gamma-mixed Poisson data using the same parameters used in Section 4. Based on the results of Section 4, we know that estimation can be improved through penalization, at least for the choices of η that we propose to simulate from. The purpose of this section is to determine if any method can reliably select a λ that reduces RMSE compared to unpenalized maximum likelihood estimation. In each simulation below, we select the largest value in ${\lambda }^{\mathrm{grid}}$ by doubling the optimal value of λ found in Table 2.

6.1. Initial comparison of methods' performance

In this simulation, we let C = 1000, and simulate 100 times over all combinations of $\eta \in \{({10}^{-1},{10}^{-1}),({10}^{-2},{10}^{-5})\}$, $r\in \{6,10,14\}$. Recall these η are the values chosen based on our motivating examples. We know that the optimal λ choice for these r and η are between 10 and 20, so we chose ${\lambda }^{\mathrm{grid}}=\{0,5,10,\dots 60\}$ for this simulation. Methods 0–4 are all evaluated over the same random draws for each parameter choice using the R package simulator [3].

In Table 3, we show the RMSE over all simulations for each method and each combination of η and r. Over all parameter choices, Method 3 performs at least as well as Method 0, with an RMSE which was between 0 and 23% lower. Method 3 is the only method which performs at least as well as Method 0 for all parameter choices.

Table 3. RMSE $(\hat{C})$ for Methods 0–4 based on a zero-truncated gamma-mixed Poisson data generating process.

	$\eta =({10}^{-1},{10}^{-1})$			$\eta =({10}^{-2},{10}^{-5})$
	r = 6	r = 10	r = 14	r = 6	r = 10	r = 14
Method 0: MLE (no penalization)	709	326	339	787	775	716
Method 1: Minimum subset variance	689	630	578	763	821	796
Method 2: Cross-validated likelihood	797	521	492	602	658	617
Method 3: Goodness of fit	707	326	339	781	663	554
Method 4: Cross-validated g.o.f.	812	571	533	738	787	679

C = 1000 is constant for all simulations. Under each $\eta ,r$ combination, 100 simulations were run. Methods with RMSE better than Method 0 have a grey highlighting, and the best method for each $\eta ,r$ combination is bolded.

We note some differences between the performance of the methods for different η. When $\eta =({10}^{-2},{10}^{-5})$, Method 2 also outperforms Method 0 for $r\in \{6,10,14\}$, while Method 2 does not outperform Method 0 for any r when $\eta =({10}^{-1},{10}^{-1})$. Method 4 outperforms Method 0 for $r\in \{6,14\}$ when $\eta =({10}^{-2},{10}^{-5})$, but never when $\eta =({10}^{-1},{10}^{-1})$. We conjecture that the differing performance across η is due to the differing rare species structures implied by the different η's. For example, $\eta =({10}^{-2},{10}^{-5})$ has more abundant species and larger expected maximum abundance $\mathbb{E}(k_{max})$, but relatively few rare species (Table 1).

In Figure 3, we display the C estimates for each method. Recall that as we increase λ, ${\hat{C}}_{\lambda }$ monotonically decreases. Therefore, we can discuss whether a method is on average over-penalizing (selecting $\tilde{\lambda }$ which is larger than optimal) or under-penalizing (selecting $\tilde{\lambda }$ which is smaller than optimal): if $\hat{C}>C$, then the method has under-penalized while if $\hat{C}<C$ the method has over-penalized.

Figure 3.

Simulation results for all proposed methods when $\eta =({10}^{-1},{10}^{-1})$, and when $\eta =({10}^{-2},{10}^{-5})$.

We see from Figure 3 that Method 1 over-penalizes for all parameter choices, as all of the estimates are far below the truth. We can understand this result given Figure 1: for very high values of λ the estimates are equal to $\underset{j}{max}{c}_j$, and so have low variance. However, the observed richness is severely negatively biased for C. Given its large and consistent negative bias and high RMSE, we do not discuss Method 1 further.

Method 2 has the opposite behavior: the estimates tend to be too high, especially when $\eta =({10}^{-2},{10}^{-5})$. This is similarly the case for Method 0, though we see that Method 2 has a slightly lower bias compared to Method 0. Even for $\eta =({10}^{-1},{10}^{-1})$, where the Method 0 bias is smaller, we see that Method 2 has poor performance (Table 3). Recall that in each partition of Method 2, we use only half the data to generate the estimates ${\hat{C}}_{\lambda }^{T(l)}$. As a consequence Method 2 has slightly higher variance when compared with Method 0, as evidenced by the interquartile range in Figure 3 for $\eta =({10}^{-1},{10}^{-1})$. In conclusion, Method 2 does not appear to be effectively tuning λ, and sample splitting to generate estimates may be leading to high variance.

Methods 3 and 4, which are both based on goodness of fit criteria, display different patterns depending on the η and r values of the simulation. Method 4 outperforms Method 0 when $\eta =({10}^{-2},{10}^{-5})$ (Table 3), but under $\eta =({10}^{-1},{10}^{-1})$ the RMSE for Method 4 is 15–75% greater than for Method 0. In comparison, Method 3 is at least as good as Method 0 with respect to RMSE for all parameter combinations tested. In addition, Method 3 is simpler and faster than Method 4. We conclude that Method 3 is the most promising method based on this simulation. We propose to conduct a secondary simulation to further investigate Method 3, testing whether it remains superior to Method 0 over a wider range of η values.

6.2. Performance over wider range of parameter values

In this simulation, we will vary $C\in \{500,1000,2000\}$, $r\in \{6,10,14,30,50\}$ and perform 100 simulations for three additional choices of η. We let $\eta \in \{({10}^{-1},{10}^{-3}),({10}^{-1},{10}^{-5}),({10}^{-2},{10}^{-3})\}$ and only consider Methods 0 and 3, based on the results of the previous section. To account for the fact that a larger λ was optimal for these η values (see Table 2) we use ${\lambda }^{\mathrm{grid}}=\{0,10,20,\dots 140\}$. In Table 4, we display the RMSE for both methods and each combination of C, r and η.

Table 4. RMSE for Methods 0 and 3 when counts are drawn from a gamma-mixed Poisson distribution with parameter η.

Method	C	η	r = 6	r = 10	r = 14	r = 30	r = 50
Method 0: MLE (no penalization)	500	$({10}^{-1},{10}^{-3})$	287	244	216	133	73
Method 3: Goodness of fit	500	$({10}^{-1},{10}^{-3})$	312	249	248	139	83
Method 0: MLE (no penalization)	1000	$({10}^{-1},{10}^{-3})$	418	266	277	158	127
Method 3: Goodness of fit	1000	$({10}^{-1},{10}^{-3})$	427	268	273	187	153
Method 0: MLE (no penalization)	2000	$({10}^{-1},{10}^{-3})$	463	419	372	367	346
Method 3: Goodness of fit	2000	$({10}^{-1},{10}^{-3})$	513	439	338	413	386
Method 0: MLE (no penalization)	500	$({10}^{-2},{10}^{-3})$	266	230	211	208	173
Method 3: Goodness of fit	500	$({10}^{-2},{10}^{-3})$	301	241	229	230	192
Method 0: MLE (no penalization)	1000	$({10}^{-2},{10}^{-3})$	485	430	429	365	372
Method 3: Goodness of fit	1000	$({10}^{-2},{10}^{-3})$	498	446	445	393	455
Method 0: MLE (no penalization)	2000	$({10}^{-2},{10}^{-3})$	908	775	781	719	787
Method 3: Goodness of fit	2000	$({10}^{-2},{10}^{-3})$	999	863	833	906	945
Method 0: MLE (no penalization)	500	$({10}^{-1},{10}^{-5})$	314	207	305	295	84
Method 3: Goodness of fit	500	$({10}^{-1},{10}^{-5})$	41	89	9	8	7
Method 0: MLE (no penalization)	1000	$({10}^{-1},{10}^{-5})$	701	438	500	375	227
Method 3: Goodness of fit	1000	$({10}^{-1},{10}^{-5})$	35	218	387	246	16
Method 0: MLE (no penalization)	2000	$({10}^{-1},{10}^{-5})$	1648	1143	766	743	998
Method 3: Goodness of fit	2000	$({10}^{-1},{10}^{-5})$	796	690	542	538	74

Results are based on 100 draws.

We find that for $\eta =({10}^{-1},{10}^{-3})$, Method 0 has a lower RMSE for 28 out of 30 combinations of C and r. However, on average over these 30 combinations, the RMSE is only 8% lower. Similarly, when $\eta =({10}^{-2},{10}^{-3})$, we also observe that Method 0 has a lower RMSE for 30 out of 30 combinations of C and r, but the RMSE is only 11% lower. We find that ${\tilde{\lambda }}_{[3]}$ in 59% of simulations when $\eta =({10}^{-2},{10}^{-3})$, and ${\tilde{\lambda }}_{[3]}$ in 22% of simulations when $\eta =({10}^{-1},{10}^{-3})$. We conclude that for these choices of η, maximum likelihood estimates outperform the goodness of fit-based penalization method, but that the advantage of not penalizing is marginal and the chosen value of λ is often zero.

When $\eta =({10}^{-1},{10}^{-5})$, Method 3 significantly outperforms Method 0 in 30 out of 30 combinations of C and r. The RMSE is 914% lower for Method 3 on average over these combinations. We also compare these results with those from Table 3, where we found superior performance of Method 3 when $\eta =({10}^{-2},{10}^{-5})$. Taken together, these results suggest that Method 3 outperforms Method 0 for small values of δ, but that the methods become comparable as δ increases.

In Figure 4, we display boxplots of the C estimates over r when $\alpha ={10}^{-1}$ for $\delta \in \{{10}^{-3},{10}^{-5}\}$ and $C\in \{500,1000,2000\}$. When $\eta =({10}^{-1},{10}^{-5})$ we note that ${\hat{C}}_{[0]}$ is very large for a few simulations, especially for smaller values of r. We recall from Table 1 that this η choice has the highest proportion of abundant species. This is an example of a simulation structure which will cause the instability problem in maximum likelihood estimation discussed in Section 2. Method 3 outperforms Method 0 on this η by selecting lower estimates. In contrast, while Table 4 indicates that Method 3 has larger RMSE than Method 0 when $\eta =({10}^{-1},{10}^{-3})$, Figure 4 suggests that the estimates produced by the methods are generally very similar.

Figure 4.

Simulation results for Methods 0 and 3 when $\eta =({10}^{-1},{10}^{-3})$, $\eta =({10}^{-1},{10}^{-5})$ and $C\in \{500,1000,2000\}$. The distribution of $\hat{C}$ is shown over 100 draws. The true value of C is indicated with a solid horizontal line.

As a result of these simulations, we conclude that no method (including Method 0) is best for all simulation settings, but Method 3 has advantages in many settings. We found that when $\eta =({10}^{-1},{10}^{-3})$ and $\eta =({10}^{-2},{10}^{-3})$, Method 0 slightly outperformed Method 3 (Table 4). However, when $\eta =({10}^{-1},{10}^{-5})$ and $\eta =({10}^{-2},{10}^{-5})$, Method 3 outperformed Method 0 (Tables 3 and 4). Both methods were about the same when $\eta =({10}^{-1},{10}^{-1})$ (Table 3). This is consistent with Method 3 having improved performance compared to Method 0 when there are highly abundant species present in the data (see Table 1), in which case Method 0 can be unstable (see Figure 4).

6.3. Performance under model misspecification

We now investigate the performance of Methods 0 and 3 when the model is misspecified. Specifically, we simulate species frequencies under two additional (non-gamma–Poisson) models and compare the performance of the methods.

We first investigate the effect of zero-inflation on species richness estimation by simulating data from the mixture distribution $Pr(X_i=x)=p{\mathbb{1}}_{\{x=0\}}+(1-p)\times F_{\eta }(x),$ where $F_{\eta }(x)$ is the probability mass function of a gamma-mixed Poisson distribution with parameters η. We investigated $\eta =({10}^{-1},{10}^{-3})$ (where Method 0 outperformed Method 3) and $\eta =({10}^{-1},{10}^{-5})$ (where Method 3 outperformed Method 0) and fixed C = 1000. We investigate $p\in \{0.1,0.2,0.3\}$ and perform 100 simulations. We find that in 16 out of 18 combinations of p and r, Method 0 outperforms Method 3 when $\eta =({10}^{-1},{10}^{-3})$, and that Method 3 outperforms Method 0 in 16 out of 18 combinations of p and r when $\eta =({10}^{-1},{10}^{-5})$ (Table 5). Unsurprisingly, we find that increased zero-inflation adversely affects both methods. We conclude that neither method is robust to model misspecification via zero-inflation, and that the value of η is more important than the zero-inflation parameter in determining the relative performance of the two methods.

Table 5. RMSE for Methods 0 and 3 when counts are drawn according to a zero-inflated gamma-mixed Poisson distribution with C = 1000.

Method	η	p	r = 6	r = 10	r = 14	r = 30	r = 50
Method 0: MLE (no penalization)	$({10}^{-1},{10}^{-3})$	0.1	246	315	146	237	142
Method 3: Goodness of fit	$({10}^{-1},{10}^{-3})$	0.1	293	327	163	247	163
Method 0: MLE (no penalization)	$({10}^{-1},{10}^{-3})$	0.2	288	203	209	194	227
Method 3: Goodness of fit	$({10}^{-1},{10}^{-3})$	0.2	331	239	211	209	218
Method 0: MLE (no penalization)	$({10}^{-1},{10}^{-3})$	0.3	393	375	342	290	298
Method 3: Goodness of fit	$({10}^{-1},{10}^{-3})$	0.3	439	397	334	291	312
Method 0: MLE (no penalization)	$({10}^{-1},{10}^{-5})$	0.1	292	466	224	171	299
Method 3: Goodness of fit	$({10}^{-1},{10}^{-5})$	0.1	130	121	145	195	216
Method 0: MLE (no penalization)	$({10}^{-1},{10}^{-5})$	0.2	566	320	250	286	222
Method 3: Goodness of fit	$({10}^{-1},{10}^{-5})$	0.2	244	241	233	204	199
Method 0: MLE (no penalization)	$({10}^{-1},{10}^{-5})$	0.3	446	422	344	297	299
Method 3: Goodness of fit	$({10}^{-1},{10}^{-5})$	0.3	320	307	323	319	291

Results are based on 100 draws from the distribution $Pr(X_i=x)=p{\mathbb{1}}_{\{x=0\}}+(1-p)\times F_{\eta }(x)$ where $F_{\eta }(x)$ is a gamma-mixed Poisson distribution with parameters η.

We also investigate the effect of data draws from a different non-gamma-mixed Poisson distribution on the estimation error of Methods 0 and 3. We simulate data according to a shifted logarithmic distribution with probability mass function $Pr(X_i=x)=(-1/\ln (1-p))(p^{x+1}/x+1),$ for $x=0,1,2,\dots$ and $p\in (0,1)$. We investigate $C\in \{1000,2000\}$, $r\in \{6,14,30\}$, and $p\in \{0.99,0.9937,0.995\}$. Note that p = 0.9937 is the maximum likelihood estimate of p for the data described in Section 7.2. For each combination of C, r and p, we performed 50 simulations. We found that, without exception, ${\hat{C}}_{[3]}={\hat{C}}_{[0]}$ for every single draw. That is, for each of $2\times 3\times 3\times 50=900$ simulations from a logarithmic distribution, Methods 0 and 3 produced identical estimates. Correspondingly, both methods have identical RMSE for all C, r and p. We found that the median ${\tilde{\lambda }}_{[3]}$ was zero for 14 out of 18 combinations of C, r and p, and that ${\tilde{\lambda }}_{[3]}=0$ for 59% of the 900 simulations. Note that ${\tilde{\lambda }}_{[3]}$ was not always chosen to be zero, but even for a nonzero ${\tilde{\lambda }}_{[3]}$, the same value of C maximized the regularized likelihood function. We therefore conclude that regularization does not change the estimated species richness when the data generating process is misspecified and drawn according to a logarithmic distribution, and neither method has an advantage over the other in this setting.

7. Data analysis

7.1. Estimating microbial richness in Lake Champlain

To illustrate the performance of our methods on ecological data, we estimate strain-level microbial diversity in Lake Champlain, a large eutrophic lake in Canada. We analyze data from Tromas et al. [26], considering samples from the littoral zone in the summer season of the same year as replicates. This gives us 8 replicates from 2009, 6 replicates from 2010 and 6 replicates from 2011. Given our results from 6, we focus on Methods 0 and 3.

Method 3 produces lower estimates of C than Method 0, as expected. The 2009 and 2010 estimates were approximately 3.5 times lower for Method 3, while the 2011 estimate was approximately 1.5 times lower for Method 3. We see that the estimates of δ are comparable across the two methods, but that the estimates of α differ, and may be higher or lower depending on the dataset (Table 6).

Table 6. Diversity estimates from the Lake Champlain data analysis from 2009 (r = 8), 2010 (r = 6) and 2011 (r = 6) using our proposed methods.

	2009				2010				2011
Method	$\hat{C}$	$\tilde{\lambda }$	$\hat{\alpha }$	$\hat{\delta }$	$\hat{C}$	$\tilde{\lambda }$	$\hat{\alpha }$	$\hat{\delta }$	$\hat{C}$	$\tilde{\lambda }$	$\hat{\alpha }$	$\hat{\delta }$
[0] Unpenalized MLE	73,404	—	0.00088	0.00180	47,631	—	0.00185	0.00253	57,686	—	0.00161	0.00140
[3] Goodness of fit	20,160	550	0.00323	0.00174	13,156	225	0.00685	0.00257	40,040	230	0.00231	0.00137

7.2. Estimating global human host-associated microbial richness

We also applied our method to estimate the species-level diversity of human host-associated microbes. Pasolli et al. [22] assembled c = 4930 species-level genome bins (SGBs) using publicly available shotgun metagenomic data, but we expect that many SGBs were not observed due to undersampling and challenges in genome assembly. The frequency counts of each SGB are available at Pasolli et al. [22, Table S4].

In this dataset r = 1, and so it is not possible to sample split replicate frequency count tables. Therefore, only Methods 0 and Method 3 can be applied. We found that ${\hat{C}}_{[0]}=420,056$ with $(\hat{\alpha },\hat{\delta })=(0.00234,0.00641)$. In contrast, ${\hat{C}}_{[3]}=163,587$ with ${\tilde{\lambda }}_{[3]}=905$ and $(\hat{\alpha },\hat{\delta })=(0.00605,0.00635)$. We therefore find Method 3 to produce an estimate approximately 2.5 times lower than the estimate produced by Method 0. Similar to our analysis of the [26] dataset, we find comparable $\hat{\delta }$'s across the two methods but different $\hat{\alpha }$'s.

8. Discussion

8.1. Conclusions

In this paper, we outlined an extension of the penalized maximum likelihood procedure of Wang and Lindsay [28] for species richness estimation to data with biological replicates, and proposed several methods for tuning the penalization parameter. We demonstrated that penalization can reduce estimation error when analyzing replicate data. We found that tuning the penalization parameter is challenging, but that a tuning method based on goodness of fit (Method 3) has similar or better performance than the unpenalized MLE in many of the settings we analyzed. On two datasets, we found that it reduces the magnitude of species richness estimates. Since species richness estimates can be unstable, we find the reduction in estimates appealing. While we cannot conclude that our proposed goodness of fit tuning parameter selection method provides more reliable estimates than unregularized estimation, the performance of the goodness of fit approach on simulated data is encouraging.

Our investigation highlights the challenges of selecting tuning parameters in the absence of ground truth, since we never observe C. However, even in the absence of information with which to calibrate λ, we showed that with a parametric model for species abundance data we can use goodness of fit in conjunction with maximum likelihood to select λ. We conjecture that the goodness of fit method performs well because it employs a combination of likelihood and goodness of fit metrics calculated on the full dataset to select the tuning parameter, unlike methods that only rely on the likelihood (e.g. Method 2) or split the data (e.g. Method 4).

8.2. Limitations and future work

The approach of Wang and Lindsay [28] is considerably more general than the gamma–Poisson model. The gamma–Poisson model is common for modeling microbiome data [13,15], and for this reason, we focused on it in our investigation. However, our goodness of fit method is also amenable to other parametric models. We leave the investigation of tuning parameter selection under different models to future work.

Our results from Section 3 hint at a possible positive correlation between r and the optimal choice of λ. We investigated whether the optimal choice of λ remained constant for penalties of the form $-h(r)\lambda \log p_{\eta }(0)$ (instead of $-\lambda \log p_{\eta }(0)$; see Equation (7)). We tested $h(r)=r$ and $h(r)=\sqrt{r}$, but found that the trend in optimal λ is not so simple. We leave further investigation into how the optimal λ varies with r to future work. Understanding of $h(r)$ would allow us to consider unequally sized evaluation and training partitions for Methods 2 and 4.

For our investigations, we intentionally chose a likelihood optimization algorithm which was stable and exhaustive. We also did not construct standard errors for our estimates, and the long computation times precluded the consideration of a bootstrapping approach. Refining the optimization algorithm would be a valuable extension, and a faster optimization algorithm would facilitate a resample-based variance estimation procedure.

9. Code references

An R package implementing our methods is available at github.com/statdivlab/rre. Code to reproduce our figures and simulations can be found at github.com/statdivlab/rre_sims. We are also grateful to the R Core Team [23] and authors of the packages tidyverse [29], magrittr [1], breakaway [31], foreach [16], Rcpp [8] and data.table [7], which were used for constructing the figures and running the analyses in this paper.

Funding Statement

This work was supported in part by the National Institute of General Medical Sciences (NIGMS) of the NIH under grant number [R35 GM133420].

Disclosure statement

No potential conflict of interest was reported by the author(s).