ABSTRACT
Longitudinal binary or count functional data are common in neuroscience, but are often too large to analyze with existing functional regression methods. We propose one-step penalized generalized estimating equations that support generalized functional outcomes (e.g., count, binary, proportion, continuous-valued) and is fast even when datasets have a large number of clusters and large cluster sizes. The method applies to functional and scalar covariates and the one-step estimation framework enables efficient smoothing parameter selection and joint confidence interval construction. Importantly, this semi-parametric approach yields coefficient confidence intervals that are provably valid asymptotically even under working correlation misspecification. By developing a general theory for adaptive one-step M-estimation, we prove that the coefficient estimates are asymptotically normal and as efficient as the fully-iterated estimator; we verify these theoretical properties in simulations. We illustrate the benefits of our approach for analyzing large-scale neural recordings by applying it to a recent calcium imaging dataset published in Nature. We show that our method reveals important timing effects obscured in non-functional analyses. In doing so, we also demonstrate scaling to common neuroscience dataset sizes: the one-step estimator fits to a dataset with 150,000 (binary) functional outcomes, each observed at 120 functional domain points, in only 
minutes on a laptop without parallelization. We release our methods in the 
package 
, which supports a wide range of link functions and working covariances.
Keywords: calcium imaging, functional data analysis, generalized estimating equations, longitudinal data analysis, one-step estimators
1. Introduction
Neuroscience studies in animal models provide an invaluable tool to identify the neural mechanisms underpinning psychiatric disorders. Researchers can estimate moment-by-moment associations between experimental covariates (e.g. behavior) and the activity of hundreds of neurons per animal, with widely used in vivo brain recording techniques like calcium imaging (Grienberger et al., 2022) and Neuropixels (Steinmetz et al., 2018). A neuroscientist might study brain-behavior associations on, for example, a learning task in which an animal learns to press a lever for a food reward. These tasks are often performed over hundreds of experimental replicates called “trials” (longitudinal observations akin to “patient visits”). Each trial might be defined as a five second interval starting at extension of the lever and ending with delivery of the food reward. To test whether, for example, mean neural activity is higher on trials when animals press the lever, a common strategy is to analyze scalar summaries of each trial’s neuronal firing activity. For instance, analysts might calculate a firing rate of neuron i on trial j by averaging the response, 
, across within-trial timepoints indexed by s: 
, where 
denotes a grid of timepoints at which the outcome is observed. One might then test whether 
, where 
is an indicator that the animal on which neuron i was recorded pressed the lever on trial j. This discards important temporal information, however, by summarizing across timepoints.
Alternatively, the neural response of each five second trial can be conceptualized as a functional outcome, with within-trial timepoints, s, representing locations along the functional domain. This allows one to apply functional data analysis (FDA) techniques to test how brain–behavior relationships evolve within and across trials (Loewinger et al., 2025) as both the responses and the covariates (e.g., behavior) can be modeled as functional variables. The size and complexity of our dataset, however, require specialized FDA methods. First, analyses must account for the longitudinal correlation, since each neuron’s activity is collected across many trials. Second, the large number of clusters (500 neurons), large cluster sizes (300 trials/neuron), and densely sampled functional outcomes (
timepoints/trial) render many longitudinal methods for generalized functional outcomes computationally impractical. Our semiparametric approach models correlation in functional and longitudinal directions to improve statistical efficiency, and adopts a variance estimator robust to covariance misspecification to ensure valid inference.
To conduct inference in longitudinal FDA with large datasets, we propose a one-step estimator for functional generalized estimating equations (fGEE). Procedurally, we first fit a function-on-scalar regression with a working independence correlation structure to obtain a consistent but potentially inefficient initial estimate of the functional coefficients. We then update the initial estimate with one Newton-Raphson update step, derived from an estimating equation that models intra-cluster correlation. This approach can scale to large datasets and has desirable statistical properties. The initial estimate can be formed quickly because it ignores correlation; using only ‘one step’ in the update is fast because it greatly reduces the number of times potentially large working covariance matrices are inverted. Importantly, our approach still captures much of the statistical efficiency afforded by modeling intra-cluster correlation in longitudinal and/or functional directions with a fully-iterated fGEE. In fact, we prove the one-step fGEE is asymptotically as efficient as the fully-iterated version.
Our implementation supports functional data observed on regular, irregular, dense and sparse grids with functional and scalar covariates (see Appendix A.10). We propose fast strategies for smoothing parameter tuning, cluster bootstrapping, and joint confidence interval construction. Our general theory for adaptive one-step M-estimation may be of independent interest.
Our methods build on the rich literature for conditional and marginal methods for longitudinal function-on-scalar regression (Guo, 2002; Morris and Carroll, 2006; Ramsay and Silverman, 2005; Scheipl et al., 2016; Wang et al., 2016; Zhu et al., 2019). Functional mixed models are a versatile conditional strategy for longitudinal FDA. For non-Gaussian functional outcomes, many existing approaches do not scale well to large cluster sizes or cluster numbers. For example, Functional Additive Mixed Models, implemented in 
(Scheipl et al., 2016), can become computationally intensive when the number of clusters is in the low hundreds, even when specifying random functional intercept-only models: each participant adds 
columns to the design matrix, where 
is the number of spline bases used for the random intercept (Cui et al., 2022). Cui et al. (2022); Loewinger et al. (2025) proposed a fast functional mixed models approach based on univariate mixed models fits at each functional domain point. For non-Gaussian outcomes this relies, however, on a cluster bootstrap for inference, which can be slow for large datasets. Moreover, for non-Gaussian outcomes these approaches yield coefficient estimates that are only interpretable as conditional on the random effects. In many applications, estimates with marginal interpretations are desirable.
Functional GEE and Quadratic Inference Functions (QIF) are marginal methods for longitudinal function-on-scalar regression. Qu and Li (2005) and Guha Niyogi and Zhong (2025) proposed QIF-based methods applicable to FDA, but, to the best of our understanding, these works focused on a single observation of a functional outcome per subject. Chen et al. (2013) proposed a penalized GEE for longitudinal FDA that serves as part of the inspiration for our work. The method, however, requires inverting an 
matrix at each step of model fitting, where 
is the size of cluster i, and L is the number of points in the functional domain. Li et al. (2022) proposed a marginal estimator for continuous data, but it has not been extended to generalized outcomes. Taken together, marginal approaches for longitudinal functional regression with generalized outcomes do not scale well, limiting their widespread adoption.
Our work builds on existing one-step approaches to scale GEE to large problem sizes. Cheng et al. (2013) establish that a one-step GEE cluster bootstrap yields asymptotically equivalent inference as a fully-iterated GEE cluster bootstrap (see Section 2.2). This provides a basis for the computational steps we use for cross-validation and confidence interval construction. Lipsitz et al. (2017) use a non-adaptive, unpenalized one-step estimator to scale non-functional GEE to large cluster sizes when adopting an exchangeable working correlation. By contrast, our theory and methods are for adaptive penalized one-step M-estimation and can be used for functional data with many working correlations. One-step estimators are also used to enable inference for GEE in high dimensional, sparse settings. For example, Xia and Shojaie (2022) propose a one-step estimator using a projected estimating equation to enable inference of linear functionals of high dimensional GEE coefficients in non-functional data settings. In contrast, we focus on functional data with low-dimensional covariates where we do not encourage sparsity.
Our paper is organized as follows. We present our methodology in Section 2, provide theoretical results in Section 3, simulations in Section 4, and a data application in Section 5.
2. Methods
We begin by introducing notation, adopting that used in Li et al. (2022) where possible. We suppose that we observe the functional outcome 
at point s for cluster 
, at longitudinal observation (e.g. trial or visit) 
.
We express grids as regular (i.e. 
) and evenly spaced for ease of notation, but our methods also apply to irregular and unevenly spaced grids.
We denote 
as the functional outcome vector at point s for cluster i, concatenating all observations 
for 
, and write 
. We denote covariate vector 
for cluster i on observation j, and 
. We write covariates as scalar for ease of notation, though our method and theory applies to functional covariates.
2.1. Functional generalized estimating equations
We consider the marginal function-on-scalar regression with link function g,
 |
(1) |
where 
is a (smooth) coefficient function for covariate 
. We let 
denote the (assumed) mean function; 
represents the true within-cluster covariance, which is arbitrary and unknown.
We now discuss estimation of 
with spline basis expansions of the 
, although our methods can be used for other basis functions. For example, denoting 
as a set of m B-spline basis functions, we can represent the functional coefficients 
. In practice, the dimension m can differ across r. We denote 
as an unknown parameter vector associated with covariate r, 
, and the linear predictor for a full observation of the functional outcome as 
. We further define 
, where 
.
We then have that 
, where 
. Thus, we can estimate the functional coefficient vector, 
, by estimating 
and calculating 
.
We semi-parametrically estimate the 
with the penalized spline-based fGEE proposed in Chen et al. (2013). This assumes no likelihood and, if 
is correctly specified, yields valid inference for 
even if 
is misspecified. Specifically, the mean model parameters 
are estimated as the root of the penalized estimating equation
 |
(2) |
where 
is the working covariance matrix for cluster i (whose true covariance matrix is 
), 
, 
, and 
. The pre-specified penalty matrix, 
, is associated with the diagonal matrix of smoothing parameters 
(see below for details). Although no likelihood is adopted, the estimating equation (2) can be derived from the score equations from, for example, an exponential dispersion family (Liang and Zeger, 1986); we add the penalty term for improved estimation in finite samples. Compared to a working independence matrix (e.g., 
), estimation of 
can be made more efficient and accurate by exploiting correlation, in functional and longitudinal directions, by choosing the working 
to estimate 
. Although such choices for 
in this fGEE model yield desirable statistical properties for longitudinal FDA, estimation is computationally intensive: estimating 
based on equation (2) requires inversion of the 
covariance matrix 
for each cluster i, at each step in an optimization procedure.
2.2. One-step fGEE
To scale fGEE to large datasets, we propose a one-step estimator of the form 
,
where 
is an initial estimate fit with smoothing parameters 
. We denote 
, 
as the working covariance estimated with 
, and 
with 
applied component-wise. Our proposed estimator is formulated as follows
 |
(3) |
where the component of the gradient 
corresponding to the derivative of 
is evaluated at the population 
and can be ignored in expectation. Any consistent estimator for 
can be used for 
; we use a parametric function-on-scalar regression that ignores within-cluster correlation of outcome values across s and j, and uses the same penalty and spline bases as the fGEE (similar to solving (2) with a working independence correlation). In practice, we estimate 
with the 
function (Scheipl et al., 2016) (see Appendix A.9 for details). We formalize the necessary consistency properties of 
for the population parameter, 
in Section 3.
The one-step can be conceptualized as a de-biasing of, or equivalently a single Newton-Raphson step from, the initial estimate 
. It is much faster than the fully-iterated fGEE, because it requires inversion of a working covariance matrix only twice per cluster: 1) 
to estimate 
, and 2) 
to estimate 
(see Section 2.2.3). We summarize the estimation steps as an algorithm in Appendix A.1.
2.2.1. Working correlations
Although fGEE yields valid inference regardless of the 
structure used (under correct mean model specification), the statistical and computational efficiency of fGEE depends heavily on the 
form adopted. Our one-step estimator is often far faster than the fully-iterated fGEE in Chen et al. (2013) but, if 
is large, it still may not scale. Thus, while our theory applies to a one-step with general 
, we focus on forms that can be inverted quickly and improve statistical efficiency. To see this, we first rewrite 
, where 
, 
models 
, and the elements of 
model 
for 
.
For scalability, we focus on two classes of working covariance structures: block diagonal and Kronecker product. Block diagonal structures model correlation in either the longitudinal direction 
, where 
models 
, or the functional direction 
, where 
models 
. Each 
or 
can be parameterized by a unique 
or 
, respectively. If functional outcomes are observed on the same grid for each longitudinal measurement of a cluster, a Kronecker product-based 
can model correlation in both directions: 
, where 
models 
for all 
, and 
models 
for all 
; we suppress cluster indices i in 
and 
, when possible, for ease of notation. The 
and 
are parameterized by 
and 
correlation parameters.
These block and Kronecker structures are computationally efficient to invert. First, they avoid inversion of the full 
by reducing it to inversions of smaller blocks (block diagonal) or smaller Kronecker factors (Kronecker). For example, if we adopt 
, then 
. If we adopt 
, then 
. This can be made more computationally efficient by using Kronecker product identities. For example, denoting 
, we calculate 
in (3), with the identity 
, where 
denotes forming 
into a matrix. Writing this as 
emphasizes how one avoids forming the 
matrix 
by calculating 
and then 
. Thus, if 
and 
can be inverted efficiently, 
can be inverted efficiently for large L and 
. In fact, if one adopts a working correlation such that the sub-matrices (e.g., 
, 
, 
) have Toeplitz structure (e.g., exchangeable, AR1 with 
, moving average MA(q), Matern/RBF kernel Gaussian Process covariances), then one can calculate, for example, 
without having to ever construct (or store in memory) the sub-matrices 
, 
, 
(or their inverses); the corresponding systems of linear equations can be efficiently solved for large L and 
with, for example, the generalized Schur algorithm (Ammar and Gragg, 1988) (discussed in Appendices A.3–A.4). Similarly, our implementation also allows for FPCA-based estimation of 
which can be inverted efficiently (see Appendix A.5).
2.2.2. Tuning 
To calculate an initial estimate of 
, we select the smoothing parameters, denoted as 
, with fast restricted maximum likelihood (Wood, 2011). We found, however, that calculating the one-step estimate with the same 
values (i.e. 
) tends to produce inaccurate coefficient estimates as the inclusion of 
in the estimation equation changes the relative smoothing parameter magnitudes. Therefore, we tune the smoothing parameters for the one-step, denoted as 
, based on the cross-validated prediction performance of the one-step estimator. We develop a fast cluster CV for large datasets, by exploiting the fact that the form of the one-step estimator enables calculation of fold-specific estimates with pre-computable terms (see Appendix A.6 for further details and a summary of the steps as an algorithm): rewriting the update as 
, illustrates that we can pre-compute each cluster’s 
, and 
. Moreover, we only need to estimate 
once. We can then use that 
, calculated on the full sample, as the initial estimate for all folds and 
values. This is because any consistent initial estimate, 
, is sufficient to ensure that the one-step estimator of a given fold is consistent for the population 
. Finally, assuming 
, we can (heuristically, by Slutsky’s theorem) calculate consistent one-step estimates in fold k as
 |
(4) |
where 
and 
denotes the set of cluster index sets for fold k. By using the full sample estimate 
, we only need to invert this 
matrix once for each value of 
, instead of inverting a fold-specific 
matrix for each unique 
pair. The strategy of keeping 
and 
fixed across folds is motivated by an analogous strategy for cluster bootstrapping of unpenalized one-step GEE (see Remark and Theorem 3.3 in Cheng et al. (2013)). Specifically, Cheng et al. (2013) showed that a cluster bootstrap that fixes these two quantities (at the full-sample estimates) across replicates enjoys the same theoretical guarantees asymptotically as an approach that re-estimates these quantities in each replicate-specific sample. In our simulations, our adaptation of this strategy for cluster CV was often dramatically faster than, and performed nearly identically to, a CV strategy that calculates 
using the fold-specific estimate 
.
2.2.3. Uncertainty quantification
The one-step variance can be estimated with sandwich estimator 
, where 
, and 
. We estimate 
via 
, i.e., by plugging in 
, 
, 
, and 
. For fixed basis 
, 
, where 
is a block diagonal matrix, with 
in each block. An asymptotically valid (
-level pointwise CI is given by 
, where 
, and 
is diagonal entry s of 
. We found, however, that smoothing and small-sample bias sometimes led to undercoverage when constructing CIs with Gaussian quantiles, 
. We therefore constructed pointwise and joint CIs as 
and 
, respectively, where 
and 
, 
and 
are quantiles obtained from a fast score-based wild cluster bootstrap, and 
is a small-N effective degree of freedom inflation factor (see Appendix A.7 for the algorithm).
3. Theory
For a fixed sequence 
with 
, let 
be the solution to the population estimating equation 
, where 
has components denoted 
for 
, and let 
, where 
. We define 
, and 
. Note that implicit to these definitions, and to the ensuing theory, is that we treat the working covariance matrices 
as fixed or computed with the “true” limiting correlation parameters 
throughout. In practice, we can replace these with their estimated counterparts 
under the assumption that 
for all s, for some limiting parameters 
—see Corollary 1 of Chen et al. (2013).
Theorem 1:
Suppose that one-step estimators,
and
, are constructed using initial estimate,
, and that conditions (i)-(vi) in Appendix B.1 hold: then the one-step estimator satisfies
, where
. Consequently,
is asymptotically normal, centered around the sequence
, with large sample variance given by
.
We develop a more general result for adaptive one-step M-estimation that may be of independent interest in Appendix B. Theorem 1 provides a basis for asymptotically valid inference, akin to seminal results in non-functional settings (Zeger and Liang, 1986). The result assumes that the true mean function lies in the span of the basis functions, and holds for arbitrary 
and fixed L, q and p. Technical conditions (i)-(vi) are standard and hold if typical link functions are used, and the sample, true, and limiting 
and 
matrices are full rank, and if the initial estimator converges with sufficient rates. These are discussed further in Appendix B.1.
Remark 1:
Our result shows that the one-step is asymptotically equivalent to the fully-iterated fGEE. Moreover, our result extends to non-linear link functions, while existing fGEE theory is restricted to the linear case (Chen et al., 2013). In the linear case, the one-step shares the same properties as those characterized in Chen et al. (2013), such as the convergence rates in small knot and large knot regimes in terms of the smoothing parameter,
. We discuss the convergence rates of the coefficient estimates in terms of the parameters
further in Appendix B.
4. Simulations
We conducted simulations to assess 95% CI coverage and width, coefficient estimate error, and algorithm timing. We ran 
simulation replicates. We used 
, a maximum likelihood estimator, for an initial 
and benchmark, using penalized P-splines with 15 knots per functional coefficient (see Appendix A.9 for details). We set 
with 
evenly-spaced points, 
, and 
. Denoting 
as the functional coefficient r for simulation replicate t at s, we report 
. Denoting 
as the pointwise CI for replicate t for functional coefficient r at point s, we report average empirical pointwise coverage as: 
. We report empirical joint (uniform) coverage as: 
, where 
denotes the joint CI at s.
We simulated outcomes with mean model 
, where 
, 
, 
, and 
denotes the standard normal density function. We drew Gaussian (identity), binary (logit), Poisson (log) and gamma (log) outcomes (link functions, g) with a copula method (Nelsen, 2006). Briefly, we drew 
, transformed it into a vector of uniform random variables with the probability integral transform 
, and set 
where we applied the inverse CDF 
of the target distribution, and the standard normal CDF 
, element-wise. We used 
, with 
for all 
for Gaussian data and 
for other families. We set 
for all 
. We drew 
, and 
, where 
, and 
. We plot the true functional coefficients and average fGEE estimates in simulations in Appendix C.1.
We compared performance of the one-step and a fully-iterated fGEE in a setting where the outcome was simulated to be correlated in both longitudinal and functional directions. Namely, we set 
, where the 
entry of 
contains 
for 
, and the 
entry of 
contains 
for 
. We set 
to have an AR1 structure 
. We simulated 
with both an AR1 
, and exchangeable structure 
. We set 
. We then measured how the fGEE performs under varying degrees of working correlation misspecification: working correlations that are 1) correctly specified; specified correctly in the 2) functional or 3) longitudinal direction; or 4) a working independence structure: 
.
We compare performance of two fully-iterated fGEEs: “Full-1Step” which selects 
with our fast cluster CV, and “Full-Full” which instead selects 
with a cluster CV that fits each fold-specific coefficient estimate, 
, with a fully-iterated fGEE. The Full-Full approach does not use one-step methods and thus shows how slow fGEE is without our one-step framework. In fact, Full-Full simulations finished only with small sample sizes even on a high performance cluster (see Appendix C.2 for Full-Full results). We describe implementation details in Appendix A.2.
Table 1 shows that, relative to a 
fit, the one-step with correctly specified working correlation (
) reduces functional coefficient estimation RMSE by as much as 35% for certain combinations of 
and outcome distributions. Interestingly, an fGEE that adopts a working correlation that is correctly specified in just the longitudinal direction (e.g., 
) often performs comparably to an fGEE that correctly specifies correlation in both directions (e.g., 
). Table 1 also shows how the one-step performs nearly identically to a fully-iterated fGEE that uses the same smoothing parameter values: Full-1Step and one-step coefficients with a given working correlation are estimated with the same 
value because they apply the same cluster CV procedure. Table 1 also shows how the fast cluster CV reduces estimation error compared to 
. The 
and 
are fit with a full working independence correlation structure, but the one-step is fit with 
values selected with fast cluster CV, whereas 
selects 
with a fast REML strategy. Interestingly, when N is small, the fully-iterated fGEE sometimes performs far worse than the one-step, potentially because re-estimating correlation parameters at each iteration causes instability.
Table 1.
Functional Coefficient Estimation Performance (RMSE) of each method relative to the 
fit (
). Cells contain the average of 300 replicates 
SE (SE
indicates a value 
). Outcomes were simulated with an 
, where 
had an AR1 structure and the table columns indicate results where 
had exchangeable or AR1 correlation. The “1-Step” label indicates a one-step was used for tuning and final coefficient estimation, and “Full-1Step” indicates one-step tuning and a fully-iterated fGEE for final coefficient estimation, with the indicated working correlation. 
indicates that values 
were removed from that cell to avoid skewing the mean.
| Exchangeable | AR(1) | |||||
|---|---|---|---|---|---|---|
| Method | Gaussian | Poisson | Binomial | Gaussian | Poisson | Binomial |
| ||||||
|
0.82  0.01 |
0.83  0.01 |
0.88  0.01 |
0.85  0.01 |
0.85  0.01 |
0.92  0.01 |
|
0.82  0.01 |
0.85  0.01 |
0.88  0.01 |
0.85  0.01 |
0.83  0.01 |
0.92  0.01 |
|
0.86  0.01 |
0.91  0.01 |
0.93  0.01 |
0.90  0.01 |
0.92  0.01 |
0.95  0.01 |
|
0.87  0.01 |
0.88  0.00 |
0.93  0.01 |
0.90  0.01 |
0.90  0.00 |
0.95  0.01 |
|
0.82  0.01 |
0.83  0.01 |
1.00  0.09
|
0.86  0.01 |
0.85  0.01 |
0.94  0.01 |
|
0.81  0.01 |
1.02  0.09
|
1.29  0.21
|
0.85  0.01 |
0.83  0.01 |
0.93  0.01 |
|
0.87  0.01 |
0.91  0.01 |
0.96  0.02 |
0.90  0.01 |
0.92  0.01 |
0.97  0.01 |
|
0.87  0.01 |
0.89  0.01 |
0.95  0.01 |
0.90  0.01 |
0.90  0.00 |
0.96  0.01 |
| ||||||
|
0.78  0.01 |
0.65  0.01 |
0.82  0.01 |
0.87  0.01 |
0.87  0.01 |
0.89  0.01 |
|
0.77  0.01 |
0.65  0.01 |
0.81  0.01 |
0.85  0.01 |
0.86  0.01 |
0.88  0.01 |
|
0.80  0.01 |
0.87  0.01 |
0.79  0.01 |
0.93  0.01 |
0.95  0.00 |
0.93  0.01 |
|
0.79  0.01 |
0.85  0.01 |
0.79  0.01 |
0.90  0.01 |
0.94  0.00 |
0.92  0.01 |
|
0.78  0.01 |
0.66  0.01 |
4.75  0.69
|
0.87  0.01 |
0.87  0.01 |
0.90  0.01 |
|
0.77  0.01 |
0.86  0.07
|
7.42  0.95
|
0.85  0.01 |
0.86  0.01 |
0.88  0.01 |
|
0.81  0.01 |
0.87  0.01 |
0.83  0.01 |
0.93  0.01 |
0.95  0.00 |
0.94  0.01 |
|
0.79  0.01 |
0.85  0.01 |
0.90  0.09 |
0.90  0.01 |
0.94  0.00 |
0.92  0.01 |
| ||||||
|
0.88  0.01 |
0.84  0.01 |
0.88  0.01 |
0.89  0.01 |
0.87  0.01 |
0.89  0.01 |
|
0.88  0.01 |
0.84  0.01 |
0.87  0.01 |
0.88  0.01 |
0.85  0.01 |
0.89  0.01 |
|
0.93  0.01 |
0.93  0.00 |
0.92  0.01 |
0.94  0.01 |
0.93  0.00 |
0.93  0.01 |
|
0.93  0.01 |
0.91  0.01 |
0.92  0.01 |
0.94  0.01 |
0.91  0.00 |
0.92  0.01 |
|
0.88  0.01 |
0.84  0.01 |
0.90  0.01 |
0.89  0.01 |
0.87  0.01 |
0.91  0.01 |
|
0.88  0.01 |
1.01  0.07 |
0.96  0.08
|
0.88  0.01 |
0.85  0.01 |
0.90  0.01 |
|
0.94  0.01 |
0.93  0.01 |
0.94  0.01 |
0.95  0.01 |
0.93  0.00 |
0.95  0.01 |
|
0.93  0.01 |
0.91  0.01 |
0.93  0.01 |
0.94  0.01 |
0.91  0.00 |
0.93  0.01 |
| ||||||
|
0.85  0.01 |
0.65  0.01 |
0.83  0.01 |
0.90  0.01 |
0.87  0.00 |
0.89  0.01 |
|
0.85  0.01 |
0.65  0.01 |
0.81  0.01 |
0.86  0.01 |
0.87  0.00 |
0.88  0.01 |
|
0.85  0.01 |
0.89  0.01 |
0.82  0.01 |
0.95  0.01 |
0.97  0.00 |
0.93  0.01 |
|
0.84  0.01 |
0.87  0.01 |
0.81  0.01 |
0.92  0.00 |
0.96  0.00 |
0.92  0.01 |
|
0.86  0.01 |
0.66  0.01 |
2.63  0.52
|
0.90  0.01 |
0.87  0.00 |
0.89  0.01 |
|
0.85  0.01 |
0.76  0.03 |
4.33  0.85
|
0.86  0.01 |
0.87  0.00 |
0.88  0.01 |
|
0.86  0.01 |
0.90  0.01 |
0.85  0.01 |
0.95  0.01 |
0.97  0.00 |
0.93  0.01 |
|
0.84  0.01 |
0.87  0.01 |
0.82  0.01 |
0.92  0.00 |
0.96  0.00 |
0.92  0.01 |
| ||||||
|
0.89  0.01 |
0.87  0.01 |
0.91  0.01 |
0.89  0.01 |
0.89  0.01 |
0.92  0.01 |
|
0.85  0.01 |
0.88  0.01 |
0.89  0.01 |
0.86  0.01 |
0.89  0.01 |
0.90  0.01 |
|
0.95  0.00 |
0.96  0.00 |
0.95  0.01 |
0.96  0.00 |
0.97  0.00 |
0.96  0.01 |
|
0.91  0.00 |
0.95  0.00 |
0.93  0.01 |
0.92  0.00 |
0.96  0.00 |
0.94  0.01 |
|
0.89  0.01 |
0.87  0.01 |
0.92  0.01 |
0.89  0.01 |
0.89  0.01 |
0.92  0.01 |
|
0.85  0.01 |
0.88  0.01 |
0.89  0.01 |
0.86  0.01 |
0.89  0.01 |
0.90  0.01 |
|
0.95  0.00 |
0.96  0.00 |
0.95  0.01 |
0.96  0.00 |
0.97  0.00 |
0.96  0.01 |
|
0.91  0.00 |
0.95  0.00 |
0.94  0.01 |
0.92  0.00 |
0.96  0.00 |
0.94  0.01 |
| ||||||
|
0.90  0.00 |
0.67  0.01 |
0.96  0.01 |
0.93  0.00 |
0.88  0.01 |
0.94  0.01 |
|
0.87  0.00 |
0.67  0.01 |
0.94  0.01 |
0.90  0.00 |
0.89  0.00 |
0.93  0.01 |
|
0.92  0.00 |
0.96  0.00 |
0.90  0.01 |
0.99  0.00 |
0.98  0.00 |
0.98  0.00 |
|
0.88  0.00 |
0.95  0.00 |
0.89  0.01 |
0.96  0.00 |
0.99  0.00 |
0.97  0.00 |
|
0.90  0.00 |
0.67  0.01 |
0.98  0.01 |
0.93  0.00 |
0.88  0.01 |
0.94  0.01 |
|
0.87  0.00 |
0.67  0.01 |
0.95  0.01 |
0.90  0.00 |
0.89  0.00 |
0.93  0.01 |
|
0.92  0.00 |
0.96  0.00 |
0.91  0.01 |
0.99  0.00 |
0.98  0.00 |
0.98  0.00 |
|
0.88  0.00 |
0.95  0.00 |
0.89  0.01 |
0.96  0.00 |
0.99  0.00 |
0.97  0.00 |
Table 2 shows that the one-step achieves reasonable joint (uniform) CI coverage in large samples, but sometimes exhibits over/under-coverage. Under-coverage was more common in small N settings. Table 3 shows that the one-step achieves pointwise CI coverage at roughly nominal levels for all settings. Importantly, these results show how the “fast/simple” approach of constructing CIs with a sandwich estimator based on a 
fit can yield coverage far below the nominal levels. Thus, we also show CI coverage from 
CIs constructed with our wild cluster bootstrap approach, which achieve the nominal coverage but can be conservative. In fact, compared to these 
(wild) CIs, the one-step substantially reduces pointwise CI width (Table 4) in most simulation settings. This shows how the fGEE can substantially improve statistical power. Finally, Table 5 shows that the one-step is relatively fast for large datasets.
Table 2.
Joint 95% CI coverage from 300 replicates 
SE (SE
indicates a value 
). Table columns indicate if 
had exchangeable or AR1 correlation. pffr (Wild) indicates CIs constructed for 
coefficient estimates with quantiles obtained from a wild cluster bootstrap. pffr (
) are standard Wald CIs constructed with Gaussian quantiles.
| Exchangeable | AR(1) | |||||
|---|---|---|---|---|---|---|
| Method | Gaussian | Poisson | Binomial | Gaussian | Poisson | Binomial |
| ||||||
|
0.90  0.02 |
0.90  0.02 |
0.95  0.01 |
0.90  0.02 |
0.88  0.02 |
0.95  0.01 |
|
0.92  0.02 |
0.92  0.02 |
0.95  0.01 |
0.91  0.02 |
0.90  0.02 |
0.94  0.01 |
|
0.89  0.02 |
0.84  0.02 |
0.94  0.01 |
0.87  0.02 |
0.85  0.02 |
0.93  0.01 |
|
0.91  0.02 |
0.88  0.02 |
0.95  0.01 |
0.90  0.02 |
0.86  0.02 |
0.94  0.01 |
|
0.90  0.02 |
0.90  0.02 |
0.94  0.01 |
0.89  0.02 |
0.88  0.02 |
0.93  0.01 |
|
0.93  0.02 |
0.91  0.02 |
0.90  0.02 |
0.91  0.02 |
0.90  0.02 |
0.93  0.01 |
|
0.89  0.02 |
0.84  0.02 |
0.93  0.01 |
0.86  0.02 |
0.84  0.02 |
0.92  0.02 |
|
0.91  0.02 |
0.87  0.02 |
0.94  0.01 |
0.90  0.02 |
0.86  0.02 |
0.93  0.02 |
| pffr (Wild) | 0.92  0.02 |
0.85  0.02 |
0.95  0.01 |
0.93  0.01 |
0.86  0.02 |
0.96  0.01 |
|
0.06  0.01 |
0.55  0.03 |
0.32  0.03 |
0.06  0.01 |
0.56  0.03 |
0.32  0.03 |
| ||||||
|
0.96  0.01 |
0.98  0.01 |
0.99  0.01 |
0.89  0.02 |
0.88  0.02 |
0.93  0.02 |
|
0.97  0.01 |
0.98  0.01 |
0.99  0.01 |
0.92  0.02 |
0.88  0.02 |
0.93  0.01 |
|
0.97  0.01 |
0.91  0.02 |
0.99  0.01 |
0.88  0.02 |
0.82  0.02 |
0.93  0.01 |
|
0.97  0.01 |
0.91  0.02 |
0.98  0.01 |
0.92  0.02 |
0.83  0.02 |
0.93  0.01 |
|
0.96  0.01 |
0.97  0.01 |
0.81  0.02 |
0.89  0.02 |
0.88  0.02 |
0.93  0.01 |
|
0.98  0.01 |
0.98  0.01 |
0.61  0.03 |
0.92  0.02 |
0.88  0.02 |
0.93  0.01 |
|
0.96  0.01 |
0.91  0.02 |
0.99  0.01 |
0.88  0.02 |
0.82  0.02 |
0.93  0.01 |
|
0.97  0.01 |
0.91  0.02 |
0.98  0.01 |
0.92  0.02 |
0.83  0.02 |
0.93  0.01 |
| pffr (Wild) | 0.79  0.02 |
0.78  0.02 |
0.75  0.03 |
0.99  0.01 |
0.78  0.02 |
0.93  0.01 |
|
0.49  0.03 |
0.81  0.02 |
0.64  0.03 |
0.23  0.02 |
0.79  0.02 |
0.62  0.03 |
| ||||||
|
0.90  0.02 |
0.90  0.02 |
0.95  0.01 |
0.89  0.02 |
0.87  0.02 |
0.94  0.01 |
|
0.92  0.02 |
0.90  0.02 |
0.95  0.01 |
0.92  0.02 |
0.90  0.02 |
0.95  0.01 |
|
0.89  0.02 |
0.85  0.02 |
0.95  0.01 |
0.89  0.02 |
0.85  0.02 |
0.94  0.01 |
|
0.92  0.02 |
0.88  0.02 |
0.95  0.01 |
0.90  0.02 |
0.86  0.02 |
0.94  0.01 |
|
0.90  0.02 |
0.90  0.02 |
0.94  0.01 |
0.89  0.02 |
0.87  0.02 |
0.94  0.01 |
|
0.92  0.02 |
0.90  0.02 |
0.94  0.01 |
0.92  0.02 |
0.90  0.02 |
0.94  0.01 |
|
0.89  0.02 |
0.85  0.02 |
0.94  0.01 |
0.88  0.02 |
0.85  0.02 |
0.93  0.01 |
|
0.92  0.02 |
0.87  0.02 |
0.94  0.01 |
0.90  0.02 |
0.86  0.02 |
0.94  0.01 |
| pffr (Wild) | 0.97  0.01 |
0.91  0.02 |
0.99  0.01 |
0.97  0.01 |
0.91  0.02 |
0.99  0.01 |
|
0.04  0.01 |
0.59  0.03 |
0.23  0.02 |
0.04  0.01 |
0.57  0.03 |
0.26  0.03 |
| ||||||
|
0.96  0.01 |
0.97  0.01 |
0.98  0.01 |
0.93  0.02 |
0.90  0.02 |
0.95  0.01 |
|
0.97  0.01 |
0.98  0.01 |
0.98  0.01 |
0.95  0.01 |
0.91  0.02 |
0.95  0.01 |
|
0.98  0.01 |
0.91  0.02 |
0.99  0.01 |
0.92  0.02 |
0.84  0.02 |
0.94  0.01 |
|
0.98  0.01 |
0.94  0.01 |
0.99  0.01 |
0.95  0.01 |
0.85  0.02 |
0.95  0.01 |
|
0.96  0.01 |
0.98  0.01 |
0.89  0.02 |
0.93  0.02 |
0.90  0.02 |
0.95  0.01 |
|
0.97  0.01 |
0.98  0.01 |
0.76  0.03 |
0.95  0.01 |
0.91  0.02 |
0.95  0.01 |
|
0.98  0.01 |
0.91  0.02 |
0.98  0.01 |
0.92  0.02 |
0.84  0.02 |
0.94  0.01 |
|
0.98  0.01 |
0.94  0.01 |
0.98  0.01 |
0.95  0.01 |
0.85  0.02 |
0.95  0.01 |
| pffr (Wild) | 0.89  0.02 |
0.82  0.02 |
0.85  0.02 |
1.00  0.00 |
0.85  0.02 |
0.99  0.01 |
|
0.50  0.03 |
0.82  0.02 |
0.70  0.03 |
0.19  0.02 |
0.81  0.02 |
0.65  0.03 |
| ||||||
|
0.94  0.01 |
0.93  0.01 |
0.96  0.01 |
0.95  0.01 |
0.93  0.02 |
0.96  0.01 |
|
0.97  0.01 |
0.94  0.01 |
0.96  0.01 |
0.96  0.01 |
0.93  0.01 |
0.97  0.01 |
|
0.95  0.01 |
0.93  0.02 |
0.96  0.01 |
0.95  0.01 |
0.93  0.02 |
0.96  0.01 |
|
0.97  0.01 |
0.94  0.01 |
0.96  0.01 |
0.96  0.01 |
0.93  0.01 |
0.97  0.01 |
|
0.94  0.01 |
0.93  0.01 |
0.96  0.01 |
0.95  0.01 |
0.92  0.02 |
0.96  0.01 |
|
0.97  0.01 |
0.94  0.01 |
0.96  0.01 |
0.96  0.01 |
0.93  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.93  0.01 |
0.96  0.01 |
0.95  0.01 |
0.93  0.02 |
0.96  0.01 |
|
0.97  0.01 |
0.94  0.01 |
0.96  0.01 |
0.96  0.01 |
0.93  0.02 |
0.97  0.01 |
| pffr (Wild) | 1.00  0.00 |
1.00  0.00 |
1.00  0.00 |
1.00  0.00 |
1.00  0.00 |
1.00  0.00 |
|
0.01  0.00 |
0.63  0.03 |
0.09  0.02 |
0.01  0.01 |
0.61  0.03 |
0.07  0.01 |
| ||||||
|
0.94  0.01 |
0.92  0.02 |
0.97  0.01 |
0.94  0.01 |
0.89  0.02 |
0.95  0.01 |
|
0.96  0.01 |
0.92  0.02 |
0.97  0.01 |
0.95  0.01 |
0.91  0.02 |
0.96  0.01 |
|
0.95  0.01 |
0.92  0.02 |
0.98  0.01 |
0.95  0.01 |
0.91  0.02 |
0.95  0.01 |
|
0.97  0.01 |
0.93  0.01 |
0.98  0.01 |
0.96  0.01 |
0.91  0.02 |
0.95  0.01 |
|
0.94  0.01 |
0.92  0.02 |
0.97  0.01 |
0.94  0.01 |
0.89  0.02 |
0.95  0.01 |
|
0.96  0.01 |
0.92  0.02 |
0.98  0.01 |
0.95  0.01 |
0.90  0.02 |
0.96  0.01 |
|
0.95  0.01 |
0.92  0.02 |
0.98  0.01 |
0.95  0.01 |
0.91  0.02 |
0.95  0.01 |
|
0.97  0.01 |
0.93  0.01 |
0.98  0.01 |
0.96  0.01 |
0.91  0.02 |
0.95  0.01 |
| pffr (Wild) | 1.00  0.00 |
0.93  0.01 |
0.98  0.01 |
1.00  0.00 |
0.96  0.01 |
1.00  0.00 |
|
0.26  0.03 |
0.84  0.02 |
0.62  0.03 |
0.11  0.02 |
0.83  0.02 |
0.56  0.03 |
Table 3.
Pointwise 95% CI coverage from 300 replicates 
SE (SE
indicates a value 
). Table columns indicate if 
had exchangeable or AR1 correlation. pffr (Wild) indicates CIs constructed for 
coefficient estimates with quantiles obtained from a wild cluster bootstrap. pffr (
) are standard Wald CIs constructed with Gaussian quantiles.
| Exchangeable | AR(1) | |||||
|---|---|---|---|---|---|---|
| Method | Gaussian | Poisson | Binomial | Gaussian | Poisson | Binomial |
| ||||||
|
0.95  0.01 |
0.95  0.01 |
0.97  0.01 |
0.94  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.97  0.01 |
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.94  0.01 |
0.93  0.01 |
0.96  0.01 |
0.93  0.01 |
0.93  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
0.94  0.01 |
0.94  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
0.94  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.93  0.02 |
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.94  0.01 |
0.93  0.01 |
0.96  0.01 |
0.93  0.01 |
0.93  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
0.94  0.01 |
0.94  0.01 |
0.96  0.01 |
| pffr (Wild) | 0.96  0.01 |
0.93  0.01 |
0.97  0.01 |
0.97  0.01 |
0.94  0.01 |
0.97  0.01 |
|
0.67  0.03 |
0.87  0.02 |
0.80  0.02 |
0.66  0.03 |
0.87  0.02 |
0.80  0.02 |
| ||||||
|
0.97  0.01 |
0.98  0.01 |
0.98  0.01 |
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.98  0.01 |
0.99  0.01 |
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.95  0.01 |
0.99  0.01 |
0.94  0.01 |
0.94  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.95  0.01 |
0.98  0.01 |
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.98  0.01 |
0.84  0.02 |
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.98  0.01 |
0.98  0.01 |
0.67  0.03 |
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.95  0.01 |
0.98  0.01 |
0.94  0.01 |
0.94  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.95  0.01 |
0.98  0.01 |
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
| pffr (Wild) | 0.94  0.01 |
0.93  0.01 |
0.92  0.02 |
0.99  0.00 |
0.93  0.01 |
0.96  0.01 |
|
0.86  0.02 |
0.94  0.01 |
0.90  0.02 |
0.78  0.02 |
0.93  0.01 |
0.89  0.02 |
| ||||||
|
0.94  0.01 |
0.94  0.01 |
0.96  0.01 |
0.93  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
0.94  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.93  0.01 |
0.93  0.01 |
0.95  0.01 |
0.93  0.01 |
0.93  0.01 |
0.95  0.01 |
|
0.94  0.01 |
0.93  0.01 |
0.95  0.01 |
0.94  0.01 |
0.93  0.01 |
0.95  0.01 |
|
0.94  0.01 |
0.94  0.01 |
0.95  0.01 |
0.93  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.95  0.01 |
0.94  0.01 |
0.95  0.01 |
0.94  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.93  0.01 |
0.93  0.01 |
0.95  0.01 |
0.93  0.01 |
0.93  0.01 |
0.95  0.01 |
|
0.94  0.01 |
0.93  0.01 |
0.95  0.01 |
0.94  0.01 |
0.93  0.01 |
0.95  0.01 |
| pffr (Wild) | 0.98  0.01 |
0.95  0.01 |
0.99  0.01 |
0.99  0.01 |
0.96  0.01 |
0.99  0.01 |
|
0.65  0.03 |
0.87  0.02 |
0.77  0.02 |
0.65  0.03 |
0.86  0.02 |
0.77  0.02 |
| ||||||
|
0.96  0.01 |
0.98  0.01 |
0.98  0.01 |
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.98  0.01 |
0.98  0.01 |
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.95  0.01 |
0.98  0.01 |
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.95  0.01 |
0.98  0.01 |
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.97  0.01 |
0.90  0.02 |
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.98  0.01 |
0.79  0.02 |
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.96  0.01 |
0.95  0.01 |
0.98  0.01 |
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
|
0.97  0.01 |
0.95  0.01 |
0.98  0.01 |
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
| pffr (Wild) | 0.95  0.01 |
0.93  0.01 |
0.94  0.01 |
1.00  0.00 |
0.93  0.01 |
0.98  0.01 |
|
0.85  0.02 |
0.93  0.01 |
0.90  0.02 |
0.77  0.02 |
0.93  0.02 |
0.88  0.02 |
| ||||||
|
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
|
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
|
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
0.95  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
| pffr (Wild) | 1.00  0.00 |
1.00  0.00 |
1.00  0.00 |
1.00  0.00 |
1.00  0.00 |
1.00  0.00 |
|
0.60  0.03 |
0.88  0.02 |
0.71  0.03 |
0.60  0.03 |
0.88  0.02 |
0.71  0.03 |
| ||||||
|
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
0.95  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
0.95  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
0.95  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.96  0.01 |
0.94  0.01 |
0.96  0.01 |
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
|
0.95  0.01 |
0.95  0.01 |
0.96  0.01 |
0.95  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.96  0.01 |
0.95  0.01 |
0.96  0.01 |
0.95  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.95  0.01 |
0.94  0.01 |
0.96  0.01 |
0.95  0.01 |
0.94  0.01 |
0.95  0.01 |
|
0.96  0.01 |
0.94  0.01 |
0.96  0.01 |
0.95  0.01 |
0.94  0.01 |
0.95  0.01 |
| pffr (Wild) | 0.99  0.00 |
0.94  0.01 |
0.97  0.01 |
1.00  0.00 |
0.97  0.01 |
1.00  0.00 |
|
0.80  0.02 |
0.93  0.02 |
0.88  0.02 |
0.74  0.03 |
0.92  0.02 |
0.87  0.02 |
Table 4.
Relative pointwise CI width (mean 
SE) vs. pffr (Wild). SE
indicates a value 
. We denote 
and 
and 
as the upper/lower bounds of the CIs (at s) of the indicated method and pffr, respectively. Below we report the average ratio 
across 300 simulation replicates. Values 
indicate narrower 95% CIs. Values with 
indicate extreme outliers (from poor estimates) were removed from the average of that cell to avoid skewing results.
| Exchangeable | AR(1) | |||||
|---|---|---|---|---|---|---|
| Method | Gaussian | Poisson | Binomial | Gaussian | Poisson | Binomial |
| ||||||
|
0.75  0.01 |
0.83  0.00 |
0.85  0.01 |
0.75  0.01 |
0.83  0.00 |
0.82  0.01 |
|
0.78  0.01 |
0.87  0.01 |
0.84  0.01 |
0.77  0.01 |
0.83  0.00 |
0.80  0.01 |
|
0.83  0.01 |
0.92  0.00 |
0.94  0.01 |
0.80  0.01 |
0.91  0.00 |
0.87  0.01 |
|
0.86  0.01 |
0.92  0.00 |
0.92  0.01 |
0.83  0.01 |
0.90  0.00 |
0.86  0.01 |
|
0.75  0.01 |
0.83  0.00 |
0.90  0.04 |
0.75  0.01 |
0.83  0.00 |
0.82  0.01 |
|
0.78  0.01 |
0.90  0.02 |
1.00  0.05 |
0.77  0.01 |
0.83  0.00 |
0.81  0.01 |
|
0.83  0.01 |
0.92  0.00 |
0.94  0.01 |
0.80  0.01 |
0.91  0.00 |
0.87  0.01 |
|
0.86  0.01 |
0.92  0.00 |
0.93  0.01 |
0.83  0.01 |
0.90  0.00 |
0.86  0.01 |
| ||||||
|
0.79  0.01 |
0.78  0.01 |
1.18  0.01 |
0.51  0.00 |
0.87  0.00 |
0.82  0.00 |
|
0.83  0.01 |
0.82  0.01 |
1.18  0.01 |
0.50  0.00 |
0.87  0.00 |
0.81  0.00 |
|
1.00  0.01 |
1.06  0.00 |
1.39  0.01 |
0.57  0.00 |
1.01  0.00 |
0.90  0.00 |
|
1.02  0.01 |
1.05  0.00 |
1.35  0.01 |
0.56  0.00 |
1.00  0.00 |
0.89  0.00 |
|
0.79  0.01 |
0.78  0.01 |
2.23  0.21 |
0.51  0.00 |
0.87  0.00 |
0.82  0.00 |
|
0.83  0.01 |
0.96  0.11 |
5.29  0.58
|
0.50  0.00 |
0.87  0.00 |
0.81  0.00 |
|
1.00  0.01 |
1.06  0.00 |
1.39  0.01 |
0.57  0.00 |
1.01  0.00 |
0.90  0.00 |
|
1.02  0.01 |
1.06  0.00 |
1.37  0.01 |
0.56  0.00 |
1.00  0.00 |
0.89  0.00 |
| ||||||
|
0.62  0.00 |
0.72  0.00 |
0.65  0.00 |
0.60  0.00 |
0.72  0.00 |
0.62  0.00 |
|
0.63  0.01 |
0.74  0.00 |
0.64  0.00 |
0.61  0.01 |
0.72  0.00 |
0.61  0.00 |
|
0.70  0.00 |
0.83  0.00 |
0.72  0.00 |
0.66  0.00 |
0.80  0.00 |
0.66  0.00 |
|
0.71  0.01 |
0.83  0.00 |
0.71  0.00 |
0.68  0.01 |
0.80  0.00 |
0.65  0.00 |
|
0.62  0.00 |
0.72  0.00 |
0.65  0.00 |
0.60  0.00 |
0.72  0.00 |
0.62  0.00 |
|
0.63  0.01 |
0.76  0.01 |
0.74  0.07 |
0.61  0.01 |
0.72  0.00 |
0.61  0.00 |
|
0.70  0.00 |
0.83  0.00 |
0.72  0.00 |
0.66  0.00 |
0.80  0.00 |
0.66  0.00 |
|
0.71  0.01 |
0.83  0.00 |
0.71  0.00 |
0.68  0.01 |
0.80  0.00 |
0.66  0.00 |
| ||||||
|
0.75  0.00 |
0.70  0.00 |
1.07  0.01 |
0.37  0.00 |
0.83  0.00 |
0.69  0.00 |
|
0.77  0.00 |
0.73  0.01 |
1.05  0.01 |
0.37  0.00 |
0.83  0.00 |
0.68  0.00 |
|
0.85  0.00 |
1.02  0.00 |
1.18  0.01 |
0.42  0.00 |
0.99  0.00 |
0.76  0.00 |
|
0.86  0.00 |
1.03  0.00 |
1.16  0.01 |
0.42  0.00 |
0.99  0.00 |
0.75  0.00 |
|
0.75  0.00 |
0.70  0.00 |
1.42  0.10 |
0.37  0.00 |
0.83  0.00 |
0.69  0.00 |
|
0.77  0.00 |
0.74  0.01 |
4.05  0.55 |
0.37  0.00 |
0.83  0.00 |
0.68  0.00 |
|
0.85  0.00 |
1.02  0.00 |
1.19  0.01 |
0.42  0.00 |
0.99  0.00 |
0.76  0.00 |
|
0.86  0.00 |
1.03  0.00 |
1.16  0.01 |
0.42  0.00 |
0.99  0.00 |
0.75  0.00 |
| ||||||
|
0.25  0.00 |
0.36  0.00 |
0.22  0.00 |
0.24  0.00 |
0.35  0.00 |
0.21  0.00 |
|
0.25  0.00 |
0.36  0.00 |
0.22  0.00 |
0.24  0.00 |
0.36  0.00 |
0.21  0.00 |
|
0.29  0.00 |
0.42  0.00 |
0.25  0.00 |
0.27  0.00 |
0.40  0.00 |
0.23  0.00 |
|
0.28  0.00 |
0.42  0.00 |
0.25  0.00 |
0.26  0.00 |
0.40  0.00 |
0.23  0.00 |
|
0.25  0.00 |
0.36  0.00 |
0.22  0.00 |
0.24  0.00 |
0.35  0.00 |
0.21  0.00 |
|
0.25  0.00 |
0.36  0.00 |
0.22  0.00 |
0.24  0.00 |
0.36  0.00 |
0.21  0.00 |
|
0.29  0.00 |
0.42  0.00 |
0.25  0.00 |
0.27  0.00 |
0.40  0.00 |
0.23  0.00 |
|
0.28  0.00 |
0.42  0.00 |
0.25  0.00 |
0.26  0.00 |
0.40  0.00 |
0.23  0.00 |
| ||||||
|
0.43  0.00 |
0.57  0.00 |
0.77  0.00 |
0.12  0.00 |
0.69  0.00 |
0.27  0.00 |
|
0.43  0.00 |
0.58  0.00 |
0.75  0.00 |
0.12  0.00 |
0.69  0.00 |
0.27  0.00 |
|
0.46  0.00 |
0.96  0.00 |
0.76  0.00 |
0.14  0.00 |
0.84  0.00 |
0.30  0.00 |
|
0.46  0.00 |
0.97  0.00 |
0.75  0.00 |
0.13  0.00 |
0.85  0.00 |
0.30  0.00 |
|
0.43  0.00 |
0.57  0.00 |
0.77  0.00 |
0.12  0.00 |
0.69  0.00 |
0.27  0.00 |
|
0.43  0.00 |
0.58  0.00 |
0.75  0.00 |
0.12  0.00 |
0.69  0.00 |
0.27  0.00 |
|
0.46  0.00 |
0.96  0.00 |
0.76  0.00 |
0.14  0.00 |
0.84  0.00 |
0.30  0.00 |
|
0.46  0.00 |
0.97  0.00 |
0.75  0.00 |
0.13  0.00 |
0.85  0.00 |
0.30  0.00 |
Table 5.
Computation time in seconds (mean 
SE) averaged across 300 simulation replicates. SE
indicates a value 
.
| Exchangeable | AR(1) | |||||
|---|---|---|---|---|---|---|
| Method | Gaussian | Poisson | Binomial | Gaussian | Poisson | Binomial |
| ||||||
|
2.22  0.05 |
2.47  0.05 |
2.49  0.05 |
2.12  0.01 |
2.25  0.01 |
2.40  0.02 |
|
4.57  0.12 |
4.71  0.08 |
4.75  0.08 |
4.32  0.02 |
4.67  0.04 |
5.01  0.06 |
|
1.16  0.02 |
1.37  0.03 |
1.38  0.03 |
1.08  0.00 |
1.24  0.01 |
1.33  0.01 |
|
0.66  0.01 |
0.83  0.02 |
0.84  0.02 |
0.64  0.00 |
0.74  0.00 |
0.80  0.01 |
|
7.74  0.17 |
7.94  0.18 |
12.56  0.53 |
6.96  0.05 |
8.07  0.28 |
11.95  0.50 |
|
12.98  0.28 |
35.67  2.85 |
32.97  2.17 |
11.99  0.07 |
17.72  0.72 |
24.82  0.83 |
|
3.37  0.07 |
3.22  0.07 |
4.60  0.20 |
3.08  0.02 |
4.37  0.37 |
6.17  0.52 |
|
0.87  0.02 |
1.08  0.02 |
1.33  0.04 |
0.84  0.00 |
4.08  0.70 |
5.03  0.90 |
| pffr (Wild) | 2.59  0.05 |
2.84  0.06 |
2.78  0.05 |
2.42  0.01 |
2.61  0.02 |
2.63  0.02 |
|
0.36  0.01 |
0.48  0.01 |
0.56  0.01 |
0.35  0.00 |
0.45  0.01 |
0.56  0.01 |
| ||||||
|
6.94  0.02 |
8.50  0.02 |
8.65  0.08 |
8.96  0.17 |
8.29  0.04 |
8.51  0.06 |
|
7.43  0.02 |
9.15  0.02 |
9.38  0.08 |
9.91  0.19 |
9.03  0.04 |
9.27  0.06 |
|
10.71  0.02 |
12.13  0.02 |
12.50  0.11 |
14.09  0.26 |
11.86  0.04 |
12.27  0.08 |
|
4.21  0.02 |
5.40  0.02 |
5.80  0.05 |
4.59  0.08 |
5.15  0.03 |
5.68  0.04 |
|
20.78  0.15 |
22.20  0.20 |
71.49  3.56 |
17.17  0.35 |
15.42  0.08 |
18.26  0.15 |
|
18.22  0.12 |
45.69  3.03 |
117.68  5.27 |
19.08  0.72 |
17.58  0.62 |
20.11  0.15 |
|
37.38  0.28 |
32.62  0.21 |
46.32  1.04 |
29.05  0.56 |
26.49  1.37 |
30.41  0.25 |
|
4.63  0.03 |
6.02  0.03 |
7.94  0.14 |
4.84  0.09 |
5.53  0.03 |
6.79  0.05 |
| pffr (Wild) | 6.69  0.01 |
7.11  0.02 |
6.90  0.06 |
8.24  0.14 |
6.95  0.03 |
6.74  0.04 |
|
2.66  0.02 |
3.00  0.01 |
2.72  0.02 |
3.03  0.06 |
2.92  0.02 |
2.70  0.02 |
| ||||||
|
3.57  0.06 |
3.73  0.03 |
3.50  0.01 |
3.40  0.01 |
3.83  0.02 |
3.83  0.02 |
|
7.71  0.12 |
7.54  0.10 |
7.55  0.02 |
7.51  0.03 |
7.98  0.06 |
7.87  0.04 |
|
1.80  0.03 |
1.97  0.02 |
2.03  0.01 |
1.71  0.01 |
1.93  0.02 |
1.89  0.01 |
|
0.84  0.01 |
0.94  0.01 |
1.00  0.00 |
0.80  0.00 |
0.96  0.00 |
1.20  0.01 |
|
11.94  0.22 |
11.80  0.16 |
16.00  0.28 |
10.61  0.06 |
11.65  0.09 |
16.18  0.30 |
|
20.36  0.31 |
43.17  3.27 |
37.27  1.85 |
19.67  0.13 |
24.21  0.86 |
31.17  0.28 |
|
5.23  0.09 |
4.68  0.06 |
7.16  0.09 |
4.53  0.03 |
4.63  0.03 |
6.99  0.10 |
|
1.09  0.02 |
1.20  0.01 |
1.43  0.01 |
1.02  0.00 |
1.17  0.00 |
1.42  0.01 |
| pffr (Wild) | 2.98  0.05 |
2.91  0.03 |
2.85  0.01 |
2.79  0.01 |
2.99  0.03 |
2.84  0.01 |
|
0.72  0.01 |
0.66  0.01 |
0.51  0.00 |
0.69  0.00 |
0.69  0.01 |
0.51  0.01 |
| ||||||
|
13.07  0.07 |
16.17  0.03 |
17.13  0.24 |
13.65  0.20 |
19.73  0.55 |
20.65  0.40 |
|
13.93  0.07 |
17.90  0.04 |
18.27  0.25 |
14.44  0.22 |
21.12  0.50 |
21.52  0.40 |
|
19.82  0.06 |
23.36  0.04 |
24.25  0.34 |
20.96  0.30 |
28.06  0.65 |
29.24  0.54 |
|
7.23  0.06 |
10.67  0.04 |
10.58  0.14 |
6.65  0.09 |
11.41  0.26 |
12.51  0.21 |
|
31.44  0.17 |
38.99  0.29 |
94.21  4.64 |
24.47  0.40 |
35.64  0.86 |
41.81  0.93 |
|
30.35  0.16 |
68.70  4.12 |
167.87  8.92 |
26.07  0.38 |
39.29  0.87 |
51.02  3.07 |
|
56.71  0.33 |
59.48  0.32 |
81.54  1.35 |
44.86  2.11 |
55.77  1.25 |
70.31  1.53 |
|
7.95  0.05 |
10.88  0.06 |
14.01  0.18 |
7.55  0.09 |
12.54  0.32 |
14.86  0.33 |
| pffr (Wild) | 9.67  0.03 |
11.32  0.02 |
10.10  0.12 |
9.97  0.13 |
12.77  0.36 |
11.94  0.20 |
|
4.67  0.02 |
5.62  0.02 |
4.56  0.06 |
4.68  0.06 |
6.34  0.16 |
5.47  0.10 |
| ||||||
|
30.28  0.27 |
32.41  0.43 |
31.90  0.26 |
29.85  0.31 |
38.94  0.79 |
32.75  0.07 |
|
56.86  0.46 |
60.19  0.71 |
59.45  0.44 |
57.88  0.50 |
71.17  1.27 |
61.01  0.14 |
|
12.59  0.10 |
14.28  0.17 |
14.55  0.11 |
12.47  0.11 |
16.52  0.30 |
14.49  0.03 |
|
3.42  0.03 |
5.11  0.06 |
5.38  0.04 |
3.39  0.03 |
5.79  0.11 |
5.36  0.01 |
|
70.57  0.65 |
89.43  1.20 |
104.10  1.00 |
70.65  0.73 |
102.98  2.20 |
104.37  0.49 |
|
122.42  1.26 |
162.84  2.16 |
200.21  1.59 |
131.24  8.02 |
183.09  3.39 |
193.70  0.87 |
|
25.21  0.21 |
29.30  0.38 |
37.51  0.29 |
24.35  0.23 |
33.84  0.63 |
36.19  0.16 |
|
4.14  0.03 |
6.16  0.07 |
7.24  0.05 |
4.11  0.03 |
6.94  0.13 |
6.98  0.02 |
| pffr (Wild) | 6.58  0.05 |
7.16  0.07 |
6.90  0.04 |
6.31  0.05 |
8.21  0.13 |
6.95  0.02 |
|
2.69  0.02 |
2.94  0.03 |
2.74  0.02 |
2.64  0.02 |
3.34  0.05 |
2.75  0.01 |
| ||||||
|
111.85  0.27 |
149.61  0.84 |
198.96  3.74 |
113.00  1.58 |
148.51  0.36 |
189.01  3.48 |
|
118.14  0.35 |
151.87  0.98 |
213.95  4.25 |
123.12  1.61 |
150.03  0.34 |
203.80  3.94 |
|
178.16  0.37 |
215.29  1.16 |
268.15  5.33 |
189.46  2.57 |
210.95  0.49 |
256.99  4.94 |
|
54.11  0.25 |
81.74  0.57 |
122.54  2.24 |
59.04  0.74 |
84.63  0.46 |
104.11  1.97 |
|
194.44  0.38 |
264.40  1.82 |
449.09  10.68 |
179.31  2.83 |
223.57  0.77 |
295.21  6.39 |
|
210.18  0.88 |
328.53  12.12 |
538.08  16.18 |
213.07  3.01 |
263.79  1.25 |
336.11  6.46 |
|
371.03  0.74 |
421.48  3.08 |
661.56  14.27 |
324.73  4.74 |
364.12  1.93 |
500.39  10.45 |
|
65.97  0.28 |
92.03  0.60 |
125.46  2.23 |
62.81  1.08 |
88.82  0.39 |
118.41  1.93 |
| pffr (Wild) | 49.35  0.22 |
56.25  0.32 |
61.14  1.00 |
51.59  0.53 |
54.28  0.18 |
59.47  0.90 |
|
37.16  0.10 |
44.27  0.23 |
49.47  0.87 |
39.71  0.44 |
43.32  0.14 |
48.18  0.78 |
In Appendix C.1 we show results for an fGEE that uses FPCA to estimate 
, and for simulations with gamma-distributed data. In Appendix C.3 we show results from additional simulations to compare our method with the approach of Li et al. (2022) for continuous outcomes. The outcomes are generated to be correlated in both longitudinal and functional directions with a simulation scheme that differs from the Kronecker product approach above. These results show that our one-step achieves comparable coefficient estimation RMSE, yields far better pointwise coverage with large cluster sizes, and scales to bigger datasets.
5. Application
We apply our framework to calcium imaging data to illustrate the benefits of longitudinal FDA in analyzing neural recordings. In Appendix D.1, we describe what we have observed are common strategies among experimentalists for hypothesis testing of covariate-neural activity associations, as they have similar goals to our proposed method. These approaches seem to vary largely in how 1) the target neural population is defined, 2) the longitudinal structure is accounted for, and 3) the trial-level neural time-series are modeled. To address limitations in these strategies, we apply our fGEE to a dataset that pools neurons across animals (akin to a “neural pseudo-population” strategy). This accounts for the 1) longitudinal and 2) functional nature of the response in each cluster (i.e., neuron), and 3) does not discard uncertainty in the animal-specific estimates in providing an overall neural pseudo population-level estimate. This assumes correlation between neurons within-animal is negligible. In some cases, it may be preferable to apply a fGEE to the neurons in each animal separately, but this necessitates an approach to construct a pooled estimate of animal-level fits that propagates uncertainty.
Application background
We apply our method on data from a recent Nature paper studying the role of pyramidal neurons in the primary somatosensory cortex (S1) in behavior and sensory input (Inácio et al., 2025). This study recorded neuronal activity in five mice, from 
(mean
SEM: 
) neurons per animal. Recording was done in head-fixed animals, running on a ball that tracked their movement speed. The authors were interested in identifying S1 neurons active during spontaneous movements. They tested whether each S1 neuron was associated with running speed, whisker movement, and whisker sensory input.
Identifying neural activity encoding speed information
The correlation analysis used in the original paper could only test the neural activity–speed association on average within trials. In contrast, an FDA approach can test how this association evolves across trial timepoints. To demonstrate that, we randomly sampled 
neurons (clusters) from the five animals, and identified five-second intervals when animals spontaneously began to run. Each running burst was considered a “trial” (experimental replicate) as typically analyzed in neuroscience. Thus the functional outcome for neuron i, on trial j was a binary timeseries vector across five seconds of neural activity measured at 30 Hz (starting at the onset of the running bout). This results in a functional outcome measured at an evenly spaced grid of 
points. Speed, 
was defined as a functional covariate and took the same value for all neurons recorded from the same animal. The cluster size was 
for all neurons 
. We fit the model 
, with an AR1 working correlation in both longitudinal and functional directions (see Figure 1a). The results were consistent across a range of working correlation structures (see Appendix D.2).
Figure 1.
Functional coefficient one-step estimates. fGEE models (one-step and fully-iterated) were fit with a working covariance 
where 
and 
have an AR1 structure each parameterized by separate 
and 
AR1 correlation parameters. Dark and light bands indicate pointwise and joint 95% CIs, respectively. CIs for both 
and the fGEE models were constructed with a sandwich variance estimator and CI quantiles obtained from a wild cluster bootstrap. The fGEE models (one-step and fully-iterated) show tighter confidence intervals than the 
fits. In fact, the 
fits do not show any time intervals in the 
or 
functional coefficient estimate plots that are statistically significant according to the joint CIs. Fit times (seconds) on a laptop without parallelization were (a) 
11.58; one-step 41.84; Full-1Step 255.12; and (b) 
97.47; one-step 349.84; Full-1Step 1131.92.
The most salient finding was that the speed–neural activity association does not become significant until a few seconds after the animals begin to run and that association becomes non-significant fairly quickly (see Figure 1a). The timing of the association suggests that these neurons are not driving the movement of the animal. This shows how the timing sensitivity of fGEE can help identify the type of cognition or behavior a brain region encodes, which is much harder to do with analyses of trial summary measures.
Whisker stimulation
The authors of Inácio et al. (2025) were also interested in how whisker stimulation affected S1 neural activity. The summary analyses they conducted only allowed estimation of the extent to which whisker stimulation changed neural activity on average across within-trial timepoints. To characterize the “temporal dynamics” of the neural response to this manipulation, we applied our method to activity from 
randomly selected neurons (clusters), each with 
observations of the functional outcome: 4 sec of neural activity, measured 1 sec before whisker stimulation to 3 sec after (
). We fit the model 
, where 
is an indicator that neuron i was recorded from an animal that was stimulated on trial j. We adopted AR1 working correlations in longitudinal and functional directions because longitudinal and functional domains are defined as between- and within-trial timepoints, respectively.
The one-step estimator, fit to 150,000 functional observations, took 
6.5 min to fit on a MacBook Pro with an Apple M1 Max chip with 64 GB of RAM, without parallelization, and maintained a reasonable memory footprint throughout. The coefficient associated with stimulation, 
, shows the estimated mean difference in neural activity between stimulated and non-stimulated trials. It appears stimulation leads to a rapid reduction in activity (see Figure 1b). The results were consistent across working correlation structures (see Appendix D.2). This analysis shows how the one-step makes it possible to identify a clear temporal profile for the effect of interest, and is scalable enough to allow estimation over a large sample of neurons.
The above analyses also demonstrate how fGEE yields more efficient inference than using a sandwich variance estimator to construct CIs centered at a 
fit. In contrast to the one-step, the 
fits do not show any time intervals in the 
or 
analyses that are statistically significant based on the joint CIs. In Figure 1b we also show estimates from a fully-iterated fGEE that uses the fast (one-step) cluster CV for smoothing parameter tuning (“Full-1Step”). The fully-iterated and one-step estimates are nearly indistinguishable, highlighting our theoretical result that the one-step is asymptotically equivalent to a fully-iterated fGEE.
6. Discussion
We hope that by developing a method that can scale to dataset sizes common in calcium imaging and electrophysiology research, we will encourage neuroscientists to apply longitudinal FDA in these settings. While FDA methods have been applied extensively in human neuroimaging (Cui et al., 2022; Guha Niyogi and Zhong, 2025; Zhu et al., 2019), little work, to our knowledge, has sought to apply these tools to neural recordings from animal model data. We believe, however, that FDA methods have great potential to help researchers answer a range of scientific questions common in animal model studies. For example, longitudinal FDA may be beneficial for estimating tuning curves that show how mean neural responding evolves depending on a tuning variable, like the volume of an auditory cue or the angle a visual stimulus is presented at. Since neuroscientists often analyze how tuning curves differ across covariate values (treatment vs. control arm), and data are collected across multiple experimental replicates within-subject (“trials”), longitudinal FDA provides a natural strategy for inference.
The one-step fGEE can be extended in a range of related settings. For example, our implementation can be adapted for flexible 
forms (e.g., 
with unstructured working covariance) and would likely be fast for moderate 
. Some 
forms (e.g., Kronecker product) for multivariate functional domains can also be scalable. In Appendix A.8, we also describe a fast one-step cluster bootstrap as an alternative estimator for 
. In preliminary simulations the bootstrap and sandwich 
estimates were nearly identical. This may therefore provide a fast and flexible strategy to construct CIs for functions/transformations 
of the coefficients (e.g., derivatives) without having to derive analytic forms of 
. Finally, future work might explore generalized cross-validation for fGEE smoothing parameter selection.
We release our methods in the 
package 
, which supports a wide range of link functions, implements fGEE based on quasi-likelihoods deriving from many distributions, and supports both functional and scalar outcomes and covariates. We hope our theoretical guarantees and efficient implementation encourage analysts to apply FDA methods more widely.
Supplementary Material
Web Appendices, Tables, Figures, proofs and implementation details, referenced in Sections 1, 2.2, 3, 4, 5, and 6, as well as code and example application data, are available with this paper at the Biometrics website on Oxford Academic.
Acknowledgements
The contributions of the NIH author(s) were made as part of their official duties as NIH federal employees, are in compliance with agency policy requirements, and are considered Works of the United States Government. However, the findings and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the NIH or the U.S. Department of Health and Human Services. This study utilized the high-performance computational capabilities of the Biowulf Linux cluster at the NIH, Bethesda, MD (http://biowulf.nih.gov). We thank Drs. Soohyun Lee for providing data for our application, Yuan Zhao for help with data pre-processing, and Keith Goldfeld for sharing his package code. We acknowledge the use of AI to extend an initial version of our package, modify existing code to generate tables, and generate one Appendix figure (see Appendix A.11).
Contributor Information
Gabriel Loewinger, Machine Learning Core, National Institute of Mental Health 3D41, Building 10, Bethesda, MD 20892, United States.
Alexander W Levis, Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania, Philadelphia, PA 19104-6021, USA.
Erjia Cui, Division of Biostatistics and Health Data Science, University of Minnesota, Minneapolis, MN 55414, USA.
Francisco Pereira, Machine Learning Core, National Institute of Mental Health 3D41, Building 10, Bethesda, MD 20892, United States.
Funding
G.L. and F.P. were supported by the Intramural Research Program of the National Institute of Mental Health, project ZIC-MH002968.
Conflicts of interest
None declared.
Data availability
For our code and 
package, see: https://github.com/gloewing/fgee_onestep. Application datasets are available at: https://osf.io/b93dc/files/osfstorage.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Web Appendices, Tables, Figures, proofs and implementation details, referenced in Sections 1, 2.2, 3, 4, 5, and 6, as well as code and example application data, are available with this paper at the Biometrics website on Oxford Academic.
Data Availability Statement
For our code and package, see: https://github.com/gloewing/fgee_onestep. Application datasets are available at: https://osf.io/b93dc/files/osfstorage.