Nomenclature

We use the term phenotype to refer to a continuous outcome variable (or dependent variable) with the error in the phenotype approximately following a Gaussian distribution; we use the term covariates to refer to independent variables, inclusive of covariates of interest and covariates of no interest; and the term random effects refers to various known sources of nesting and similarity between observations in the data. For longitudinal data, the term subject or participant refers to a unique individual followed over time while the term observation refers to one measurement for a given subject. Visit or timepoint corresponds to an ordered set of repeated measurements for the same subject; for example, visit 1 means the first measurement (first observation) on the subject, with increasing visit numbers indicating subsequent measurements. For mathematical notation, we use boldface for matrices or vectors, and regular font weight for scalar quantities.

FEMA-Long algorithm overview

Given a set of continuous phenotype(s), covariates, and random effects, the goal is to estimate the fixed effects (beta coefficients for the covariates) while accounting for random effects. In the original FEMA framework [10], we first calculate the ordinary least squares (OLS) residuals of the phenotype; then, we use them to estimate the variance components of the random effects using method of moments (MoM) or a regression estimator with a non-negativity constraint [11]. Then, the fixed effects are estimated using a generalized least squares (GLS) solution which incorporates a covariance matrix using the estimated variance components of the random effects. For scalability to large number of variables (such as for voxel-wise or vertex-wise data derived from brain imaging data), we use a binning approach – for all outcome variables that have similar variance components for the random effects, the GLS estimation is performed using a common covariance matrix that relies on the average of the variance components of all the outcome variables within that bin.

Here, we present FEMA-Long, an extension of FEMA that is adapted and optimized for large longitudinal datasets. The FEMA-Long framework includes i) time-varying covariances in the random effects using unstructured covariances; ii) modeling non-linear effects using splines; iii) examining time-dependent effects using spline interactions; and iv) performing GWAS and other similar omics analyses, where we test the association of a large number of variables of interest (e.g., genotyping matrix with millions of SNPs) with outcome(s), independently (i.e., marginal rather than joint effect), after controlling for covariates and random effects.

In FEMA-Long, we use unstructured covariance to model time-varying covariance of random effects. The elements of the unstructured covariance matrix are estimated by using the MoM estimation for every pair of visits that are nested within subjects. This results in a matrix of covariance terms for each random effect, which is then used for GLS estimation of the fixed effects. Importantly, this method does not assume a balanced design of all subjects having data for all the visits.

FEMA-Long also includes the option to model continuous outcome variables as a smooth function (for example, smooth function of age) using splines, represented by a set of basis functions. We also allow interaction terms between covariates, including those covariates that are modeled via splines. While the estimation of the unstructured covariance matrix uses pre-defined set of timepoints (visits), the spline modeling uses a continuous variable (such as age). This, therefore, brings the framework of generalized additive mixed models to FEMA, allowing flexible modeling of outcome variables as well as performing non-linear interaction analyses.

Finally, while the original FEMA was designed for many outcome variables and a common model (mass-univariate analyses), FEMA-Long includes a variant of FEMA that is intended for GWAS, where we fit the outcome(s) repeatedly on many models, each differing only in their genetic covariate. For this, we leverage the Frisch–Waugh–Lovell theorem [12,13]: first, use stage-1 FEMA to estimate the random effects variance components and fixed effects (omitting genetic variants); then, we GLS residualize genotype vector for the covariates using the stage-1 variance components. Finally, we perform a GLS estimation using the GLS residuals of the phenotype and the GLS residuals of the genotype vector. This framework also allows modeling the interaction of genetic variants with spline basis functions, thereby allowing discovery of genetic variants that may show a nonlinear, time-dependent effect on the phenotype.

Further details on each of these steps is described below:

General FEMA model set up (stage-1)

Let a dataset have observations indexed with . Let be a full-rank matrix of covariates, with each column of being a different covariate. are the values of the covariates for observation ( is a scalar if and a vector if ). The phenotype is indicated by , thus indicates the value of the phenotype for observation . For brevity, we present the framework for a single phenotype (i.e., is a vector), although FEMA fits the same model for each phenotype.

The linear model setup is:

(1)

where is a vector of weights associated with each covariate, and is the unmodeled measurement error for each observation . In a mixed modeling framework, we assume that the error term, , is normally distributed with zero mean and variance-covariance :

(2)

Then, the total residual variance is . This normalized residual variance can be decomposed as a combination of random effects:

(3)

where, generically, is a random effects indicator matrix with entries of indicating membership of observation ; for example, in equation , represents family with a value of 1 indicating observations that belong to the same family, entries of represent the genetic relationship between a pair of observations, represents subject effect with a value of 1 indicating repeated measurements for the same subject, and represents the independent unmodeled variance across each observation; each random effect is associated with a corresponding variance component . The choice of which random effects should be included depends on the data.

Within FEMA, we first use OLS to get an estimate of the fixed effects:

(4)

Then, the OLS residuals are:

(5)

and the total residual variance estimated as:

(6)

Following , for each pair of observations , the expected value of the product of corresponding residuals and is:

(7)

where the operator is the expectation with respect to the random effects and generically denotes the element of the random effects design matrix. Solving results in the estimated values of random effects variance components, which can be used to construct a covariance matrix :

(8)

Finally, the stage-1 fixed effects can be estimated using the GLS solution:

(9)

with variance

(10)

More details of these steps can be found in (10). We refer to this overall procedure as stage-1, which includes estimation of the random effects parameters as well as GLS estimation of the fixed effects coefficients.

Estimating unstructured covariance

The original framework in FEMA assumed compound symmetry (i.e., constant covariance over time; sometimes also referred as diagonal covariance patterns since only the variance components are being modeled), an assumption which may not hold for longitudinal studies. In FEMA-Long, we allow modeling arbitrary unstructured covariance, where the parameters of the unstructured covariance matrix are estimated using the available data. Specifically, using the OLS residuals from equation (5), we solve equation (11) for every pair of visits :

(11)

When solving , if , we use non-negativity constrained solution to estimate the variance component for that visit number ; if , we drop the non-negativity constraint and estimate the covariance of the random effects between pairs of visits and . Then, the estimated variance-covariance components can be put together to get variance-covariance matrix for each random effect. Note that when estimating the variance components (i.e., the diagonal entries of the variance-covariance matrices), the subject and error terms are colinear, and therefore these variance parameters ( and ) cannot be uniquely identified, and thus are estimated as one. Similarly, when estimating the covariance components (i.e., the off-diagonal entries of the variance-covariance matrices), the system of equations constructed from 11 makes the error term unidentifiable (see S1 and S2 Figs for a demonstration of why these terms cannot be identified). The implication of this is that the estimated subject covariance matrix should be interpreted as describing the longitudinal stability of the phenotype within individuals, rather than as separate subject and noise variance components: the off-diagonal covariances between visits reflect the degree to which measurements remain stable within individuals over time, while diagonal elements capture the total variance at each visit, including both stable subject differences and visit-specific noise. If the estimated subject covariance matrix were purely diagonal, this would indicate little longitudinal stability and variability dominated by visit-specific noise; conversely, substantial off-diagonal covariance would indicate that measurements are correlated across visits, consistent with persistent subject-level differences over time. This overall formulation does not require a balanced dataset – i.e., subjects with differing number of visits are allowed and the covariances are estimated according to the available data, with no observations being removed.

After estimating the elements of the unstructured covariance matrix, we construct the covariance matrix following equation , with appropriate visit-wise reordering (such that the correct variance or covariance term for each random effect is used). Then, we use this covariance matrix to perform GLS estimation for the fixed effects (equation ). When computing the standard errors using equation , we use a nearest symmetric positive semidefinite algorithm (based on [14] and implemented in [15]) to ensure that the covariance matrix is positive semidefinite; if this process fails, we generate a convergence warning message.

Modeling spline basis functions and omnibus test

FEMA-Long allows modeling the outcome as a set of smooth basis functions (typically splines; indexed by ) of continuous variable :

(12)

This leads to representing the variable by derived columns in the design matrix . Then, the set of coefficients, , define the non-linear function , which represents the trajectory of the outcome over, for example, time. These basis functions can also be used to model interactions with another covariate.

In FEMA, the default option is to create natural cubic splines with unit height at knots, similar to the nsk option in the splines2 package [16,17] in R (with intercept set as TRUE). Users can specify the knot placement based on domain knowledge, experiment design, or choose to place the knots at the quartiles of the variable. We also support creating other spline basis functions, namely natural cubic splines or B-splines which internally rely on a call to ns or bs functions in the splines package [18] in R.

By default, we modify the created basis functions such that the linear (or main) effect of can be retained in the model. First, we create a linearly spaced vector of values over the range of values in . Then, we create basis functions using these values. At this point, the user has an option of regressing out powers of – for example, regressing out the zeroth and first powers of would result in demeaned basis functions that are orthogonal to the linear effect of . Next, we perform a singular value decomposition on the demeaned basis functions. We then rescale the orthonormal basis functions such that each basis function has a bounded value between . Finally, we linearly interpolate the modified basis functions to get corresponding basis function values for . These modified orthogonal basis functions are added as columns of . If the design matrix of the covariates already includes an intercept, then, the intercept taken together with these orthogonal basis functions covers the span of the variable.

Once the coefficients have been estimated, we can perform an omnibus Wald test to test the null hypothesis that the linear combination of the estimated coefficients is zero. Generically, given a vector or matrix of weights with rank equal to number of linearly independent rows, for a vector of estimated coefficients the multivariate Wald statistics is [19]:

(13)

where is the estimated variance-covariance matrix of the fixed effects. The Wald statistic follows a distribution with degrees of freedom. Note that equation can be used to perform Wald tests on any combination of coefficients estimated in ; i.e., not just limited to the estimated spline coefficients. Therefore, for an omnibus test of the null hypothesis that the linear combination of estimated coefficients of the basis functions is zero, , i.e., an identity matrix of size (the number of basis functions). Alternatively, it is possible to use an distribution to calculate the value with two degrees of freedom: the numerator degree of freedom is the rank of and the denominator degree of freedom is (see [20] for a discussion on the similarity and differences in these approaches).

Scaling FEMA-Long for GWAS-like analyses

Within GWAS, the goal is to estimate the effect of genetic variants on the phenotype(s), considering one variant at a time. Let represent the time-invariant genotyping matrix, in which entry denotes the best-guess genotype (imputed genotypes which have a posterior probability above a threshold, followed by conversion to discrete count indicating the most probable genotype [21] for the single nucleotide polymorphism (SNP) for observation ; 0 represents homozygous for the reference allele, 1 represents heterozygous, and 2 represents homozygous for the other allele. Let be the weight associated with this SNP . Since we are only interested in the marginal effect of each SNP in , instead of fitting the full model (with covariates) for every SNP, we employ a two-stage procedure.

The Frisch-Waugh-Lovell theorem [12,13] states that the estimate of coefficients from a full regression model are equivalent to regression done in parts (or partial regressions) if both the and variables are residualized [22]. This implies that we can estimate by first fitting a reduced model without genetic effect (stage-1), then residualizing each genetic variant for the covariates, and finally fitting a second model using the residuals from stage 1 as the dependent variable and the residualized genotype as the independent variable (see [23] for a proof of the Frisch-Waugh-Lovell theorem in the GLS case).

Concretely, we first fit a reduced model excluding the genotyping vector. We estimate the variance components (compound symmetry or unstructured covariance) and the GLS estimates for the covariates using the stage-1 model as described previously. Then, the GLS residuals can be calculated as:

(14)

Next, we residualize for the covariates. The coefficients of on , , can be calculated as:

(15)

resulting in the GLS residuals:

(16)

It is important to note that this residualized genotype vector is phenotype specific. This follows from being composed from phenotype specific variance components (equation ). Now, using the stage-1 residuals and the residualized genotype vector , we can estimate as:

(17)

Then, for each genetic variant (and for each phenotype), the GLS residuals can be calculated as:

(18)

We use the notation to distinguish these residuals (which are specific to each genetic variant) from which are the stage-1 GLS residuals obtained from FEMA after accounting for the covariates (without the genetic variant). Therefore, the residuals from are the residualized phenotype for which are estimated using the residualized genotype (from .

The mean squared error is given by:

(19)

where the denominator is normalized by to account for the genotyping vector as an additional independent variable [13,24]. The variance for each can be estimated as:

(20)

Since the term has been normalized by , the standard error is scaled by .

The above framework can be applied to other GWAS-like analyses where one is interested in examining the effect of many predictors, taken one at a time (for example, given protein assays, the effect of each protein on a phenotype). Note, that in this framework, we assume that the individual contribution of each SNP is small and unlikely to impact the estimation of the covariance components in stage-1 (see S1 Text and S46–S51 Figs for a comparison of the estimates between the two approaches).

Modeling time-dependent genetic effects using spline interactions

This GWAS framework can also include interaction terms – linear interaction or smooth interaction. In case of linear interaction, if the main effect of the interacting variable has already been included as an variable in stage-1 regression, then, is a matrix with the first column containing the main effect of the genetic variant and the second column containing the elementwise (Hadamard) product of the genetic variant and the interacting variable. In this case, we residualize both columns of for (equation ), followed by estimation using equation , resulting in two coefficients: the coefficient for the main effect and the coefficient for the interaction term. The denominator of the mean squared error (equation ) is adjusted for two parameters, i.e., .

For modeling spline interactions, is a matrix having the elementwise product of the genetic variant and the smooth function . The estimation follows the same procedure, with the estimation of as many coefficients as the columns of . If the basis functions have been appropriately modified and a constant term added (see the section on Modeling spline basis functions), then both the main effect of the genetic variant as well as the interaction of the genetic variant with smooth functions can be estimated. Generically, if there are basis functions (with or without the constant term), the denominator in equation is penalized by . Having estimated the coefficients, we can use equation to either perform an omnibus test across the main and the time-dependent genetic effect, or perform an omnibus test for just the time-dependent effect and a separate test for just the main effect of the SNP (if desired). Depending on the number of tests performed, multiple comparison correction may be necessary.

Concretely, the GWAS interaction can be framed as modeling the effects of on the outcome by altering regression coefficients in :

(21)

where coefficients are equivalent to the interaction term between variable and the basis functions . Note that corresponds to the main effect of the SNP. This formulation is equivalent to:

(22)

For example, if the time was modeled as smooth functions , then after estimating the regression coefficients, FEMA-Long can be used to calculate the following quantities:

The estimated cross-sectional effect of the genotype on the outcome, at a specific time :

(23)

and the instantaneous effect of genotype on the rate of change of the outcome at time :

(24)

Both these quantities can be plotted as a function of time, , to assess the effect of on the outcome over time.

Verification and Comparison

Simulation 1: parameter recovery

In the first simulation, we created phenotypes with known variance-covariance matrices and known effect sizes. Then, using FEMA, we estimated the parameters and compared them with the ground truth. We created a grid of known sources of variances: fixed effects , family effect (i.e., variance attributed to common environment) , subject effect (i.e., variance attributed to repeated measurements) , and noise . The total phenotypic variance was the sum of these four variances with the total variance being 1; i.e., . This resulted in 84 combinations of variances.

For each of these combinations, we simulated data for 5,000 unique individuals with five repeated measurements; these individuals were grouped in families such that a family had no more than five members; then, we randomly subset the data to result in a total of 20,000 observations, reflecting repeated and missing measurements, as well as differing number of individuals across families. We simulated the age at each measurement using the mean and standard deviation of age as days – this scenario represents repeated measures data in infants with measurements conducted at birth, two months, six months, eight months, and one year.

For each of the 84 combinations of variances, we created an arbitrary covariance matrix for family effect ( and subject effect (. We simulated 15 covariates– intercept, spline basis functions of the simulated age, and other random variables drawn from a normal distribution. The ground truth beta coefficients were drawn from a uniform distribution such that , having mean and standard deviation ; we then mean centered and scaled to have variance. Then, using and covariance matrices, we drew samples from a multivariate normal (MVN) distribution with zero means and scaled them to have and variances each, where . In other words, and . Therefore, . Finally, we simulated noise drawn from a uniform distribution and scaled it to have zero mean and variance. Thus, the simulated phenotype was .

For this simulated phenotype, we fitted the model using FEMA, specifying the covariance type to be unstructured and the random effects to be family and subject effects. As an additional comparator, we fit the same model using the glmmTMB toolbox version 1.1.10 [26] in R version 4.2.1 [18]. The model in glmmTMB was specified as: , where indicates the intercept, indicates the remaining covariates (i.e., excluding the intercept), specifics an unstructured covariance matrix for family (FID) and subject (IID) terms, the denotes a factor variable indicating which visit the observation belonged to, and the component disables the addition of an additional error term as that would be an overparameterized model [27]. After preliminary testing, we noticed convergence issues with glmmTMB; therefore, we changed the default settings to have higher number of iterations, specifically setting and to 5,000.

We compared the beta coefficients for the covariates and the estimated variance-covariance matrices for the family and subject effects from both FEMA-Long and glmmTMB with the ground truth and with each other. For the beta coefficients, , and for the remaining variables, ; for the family effect variance-covariance matrix, ; and for the subject effect variance-covariance matrix, , where was an identity matrix of size five (corresponding to number of visits). Then, we calculated the root mean squared error (RMSE) for the fixed effects and the family and subject variance-covariance matrices to assess how far the estimated values were from the true simulated parameters, and additionally to quantify the difference between the estimated parameters from FEMA-Long and glmmTMB. For each of the 84 combinations of variances explained by covariates and random effects, we repeated this entire process 50 times.

Simulation 2: false positive rate

To evaluate the false positive rate, we repeated the strategy from simulation one. The covariates included the intercept, age, and other random 98 variables drawn from a random normal distribution, resulting in 100 variables. However, the covariates had zero contribution to the simulated phenotype. The total phenotypic variance was , i.e., , resulting in 36 different combinations. For each simulation setting, we created 10 variables that have different amount of contribution from family, subject, and noise terms and repeated these simulations 1,000 times. Using these simulated data, we fitted the models using the unstructured covariance in FEMA-Long as well as repeated the model fitting using compound symmetry covariance type. Then, we examined the distribution of the -values from both the models.

Simulation 3: computational time and carbon footprint

To assess the computational performance of FEMA-Long with unstructured covariance, we simulated data with varying number of observations and varying number of outcome variables. We fixed , , and . We set the number of covariates to 50. We created 10 log-spaced values for the number of observations (rounded off), varying between and . Similarly, we created 10 log-spaced values for the number of outcome variables (rounded off), varying between and . For each of these combinations of number of observations and number of outcome variables, we estimated the time taken by FEMA. We repeated the time measurement five time and then took the median to get a robust estimate of the time taken to run the analysis. We also performed the time measurement using parallel computing with 32 parallel jobs and 2 threads per job. As comparators, we used the glmmTMB toolbox and the lme4 [28] toolbox (version 1.1-28); since these toolboxes are not designed for multiple outcome variables, we only timed it for one outcome variable (across the 10 values of number of observations) and then scaled the timing for larger number of outcome variables. For fitting the models with lme4, we used the lmer function, specifying the random effects as and; in addition, we specified a lmer control object that disabled the check for number of observations being greater than or equal to the number of random effects levels (i.e., we set to ). We ran the timing measurements on a compute cluster with two AMD EPYC 7702 64-core processors, with the job specification of 64 CPUs per task and 5GB memory per CPU. In addition, to estimate the carbon footprint of FEMA-Long, we used the formula from [29] and the data from the green-algorithms.org v3.0. We specified the number of cores as 64 and the memory usage as 320 GB; we set the CPU model to “Any” and the location to “World”. For comparison, we also estimated the carbon footprint for the glmmTMB and the lme4 toolboxes.

Simulation 4: impact of sample size and missingness on parameter recovery

Since FEMA-Long is designed for large sample sizes and the estimation of the covariance parameters uses pairs of timepoints, we performed an additional analysis to examine the impact of missing observations on parameter recovery. Specifically, we simulated data for 5,000 unique individuals with five repeated measurements; these individuals were assigned family numbers such that a family had no more than five members. The probability of a missing visit was between 0.1 and 0.8, while ensuring that subsequent visits had a higher missingness probability; since there are five repeated measurements, there are 15 possible visit combinations (five variances and ten covariances) – we varied the minimum number of observations between any pairs of visits between 500–800, in increments of 100; the resulting total number of observations were 12,000, 15,000, 18,000, and 20,000. To clarify, there were 16 scenarios: four and four . For simulating arbitrary covariance matrices (family and subject covariances), we allowed the within-visit variance to vary between 0.2 and 0.8, and the correlation between visits (scaled to covariances) to vary between -0.7 and 0.7; after ensuring that the simulated covariance matrix was positive semidefinite, we additionally ensured that the condition number of the matrices did not exceed 1000. Then, for these 16 scenarios, the rest of the simulation strategy was the same as simulation 1. We compared the estimated parameters with the ground truth and with glmmTMB as a comparator for 16 scenarios, each having 84 combinations of variance values, each of them having 15 covariates, one outcome variable, and the process repeated 50 times.

Simulation 5: impact of sample size and missingness on false positives

Similar to simulation 4, to examine the impact of missing observations on the number of false positives, we followed the simulation strategy from simulation 4 to create 16 different scenarios with differing and . We examined the distribution of the -values (using unstructured covariance and compound symmetry) for the 16 scenarios, each having 36 combination of variance values (similar to simulation 2), each of them having 100 covariates, 10 outcome variable, and the entire process repeated 1000 times.

Simulation 1: FEMA-Long accurately models unstructured covariances

For the fixed effects, the average (over 50 iterations) RMSE across 84 simulations was (Figs 1a and S3). For the random effects parameters, the average RMSE for the family effect was (Figs 1b and S4), and for the subject effect was (Figs 1c and S5), indicating that the estimates were close to the ground truth.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. Comparison of estimated parameters from FEMA-Long with ground truth.

Each panel shows the scatterplot of the simulated ground truth and the estimated parameters from FEMA-Long across 50 iterations of 84 simulation settings with different amounts of variances explained by fixed, family, and subject effects, as well as noise; the amount of noise in the simulated phenotype is indicated by the color of the points; the orange line indicates the least square fit and the correlation between ground truth and estimated parameters are indicated in orange text; a) beta coefficients for the covariates; b) variance-covariance components for the family random effect; and c) variance-covariance components for the subject effect.

https://doi.org/10.1371/journal.pgen.1012184.g001

The average RMSE between the estimates from FEMA-Long and from glmmTMB toolbox for the fixed effects was (S6 Fig), for the family effect was (S7 Fig), and for the subject effect was (S8 Fig), showing that the FEMA-Long estimates were almost identical to glmmTMB.

Simulation 2: FEMA-Long controls for false positives

Across the 36 simulations, when using unstructured covariance, the -values were uniformly distributed under the null (-values following the diagonal; green points in Figs 2 and S9). However, for the compound symmetry covariance pattern, in almost all scenarios, the -values had a non-uniform distribution with several of the -values being smaller than expected (leftward deflection of the -values from the diagonal; orange points in Figs 2 and S9). These results show that the unstructured covariances -values are well-calibrated.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. Examining false positives under the null.

a) Distribution of -values across 1000 iterations of 36 simulation settings with differing amounts of variances explained by family effect, subject effect, and noise for unstructured covariance (green points) and compound symmetry (orange points); each simulation iteration consisted of 100 X variables and 10 outcome variables; the purple filled area indicates the 95% confidence interval based on inverse beta distribution; b) box-plots showing the average number of false positives (rounded off) at different alpha values across 100 X variables and 10 y variables for unstructured covariance (green bars) and compound symmetry (orange bars); the whiskers mark the values 1.5 times the interquartile range away from 75^th and 25^th percentile; under the null, the expected number of false positives are 50 (α = 0.05), 10 (α = 0.01), and 1 (α = 0.001).

https://doi.org/10.1371/journal.pgen.1012184.g002

Simulation 3: FEMA-Long is faster and greener

FEMA-Long was fast across a range of observations and outcomes (Fig 3). For a single outcome, for 1,000 observations, FEMA-Long took ~0.14 seconds; in comparison, glmmTMB took ~9.76 seconds and lmer took ~85.66 seconds (i.e., FEMA-Long was ~ 71 times faster than glmmTMB and ~622 times faster than lmer); for 100,000 observations, this time went up to ~34.8 seconds (FEMA-Long; Fig 3a), ~ 1013 seconds (glmmTMB; S1 Table), and ~3105 seconds (lmer; S5 Table); therefore, for a single phenotype, FEMA-Long was 29–106 times faster than glmmTMB and 89–622 times faster than lmer. The carbon footprint for glmmTMB for a single outcome varied between 1.78 gCO₂e and 185.01 gCO₂e; for lmer the carbon footprint varied between 8.52 gCO₂e and 567 gCO₂e; in comparison, for FEMA-Long, the carbon footprint was between 0.03 gCO₂e and 6.35 gCO₂e; therefore, similar to computational timing, FEMA-Long was between 29 and 106 times greener than glmmTMB (S7 Table) and between 89 and 622 times greener than lmer (S9 Table).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Time taken and carbon footprint for fitting linear mixed-effects models with unstructured covariances.

In each panel, the line(s) indicate the time taken (in seconds) on the left y-axis and the carbon footprint (in grams of carbon dioxide emission) on the right y-axis for fitting linear mixed-effects models with unstructured covariances for family and subject random effects and 50 covariates across a range of number of observations (shown on the x-axis). Panel a) shows the time taken by and the carbon footprint for FEMA-Long (green), glmmTMB (orange), and lmer (from the lme4 toolbox; purple) for a single outcome variable across a range of number of observations. Panel b) shows the time taken by and the carbon footprint of FEMA-Long with serial computing, and panel c) shows the time taken by and the carbon footprint of FEMA-Long with parallel computing (32 parallel processes) across a range of increasing number of outcome variables, indicated by the color of the line. In all the panels, both axes are log-spaced and the y-axes tick labels are rounded off.

https://doi.org/10.1371/journal.pgen.1012184.g003

For 100,000 observations and 10,000 outcome variables, FEMA-Long took ~47.5 minutes and had a carbon footprint of 520.51 gCO₂e (serial, Fig 3b and S2 and S7 Tables) and ~4.2 minutes with a carbon footprint of 46.46 gCO₂e (parallel, Fig 3c and S2 and S7 Tables); in contrast, extrapolating the timing for glmmTMB, the same analysis would have taken ~117.25 days and would have had a carbon footprint of 1.85 TCO₂e (S2 and S7 Tables); similarly, extrapolating the timing for lmer, the same analysis would have taken ~359.36 days and would have had a carbon footprint of ~5.67 TCO₂e (S6 and S9 Tables). Overall, for more than one phenotype, across a range of observations, FEMA-Long was between ~86–3,554 times faster and greener than glmmTMB (serial) and between ~84–42,096 times faster and greener than glmmTMB (parallel; S2 and S7 Tables); compared to lmer, FEMA-Long was between ~263–15,785 times faster and greener (serial) and between ~259–287,190 times faster and greener (parallel; S6 and S9 Tables). We encountered convergence issues with glmmTMB; increasing the maximum number of iterations and evaluations to 5000 and re-examining the time taken by glmmTMB, we found FEMA-Long to be up to 6,902 (serial) and 93,659 times faster and greener (parallel) than glmmTMB (S3, S4, and S8 Tables).

Simulation 4: FEMA-Long estimates are stable across sample sizes and missingness

When we compared the FEMA-Long estimates with the ground truth across a range of sample size and differing minimum number of observations between pairs of visits, we found the FEMA-Long estimates to be highly correlated to the ground truth (; S10–S25 Figs). The average (over 50 iterations) RMSE across 84 simulations for the fixed effects ranged between (, ) and (, ); for the family effect covariance parameters, the average RMSE ranged between (, ) and (, ); for the subject effect covariance parameters, the average RMSE ranged between (, ) and (, ); see S10 Table for details. For smaller such as 12,000 observations, increasing resulted in reduced RMSE with the ground truth; at larger such as 20,000 observations, we did not see any differences when increasing , likely indicating a stable performance for large samples.

On comparing the FEMA-Long estimates with glmmTMB across a range of sample sizes and minimum number of observations between pairs of visits, the FEMA-Long parameters were comparable to the glmmTMB estimated parameters (S11 Table). The average (over 50 iterations) RMSE across 84 simulations for the fixed effects ranged between (, ) and (, ); for the family effect covariance parameters, the average RMSE ranged between (, ) and (, ); for the subject effect covariance parameters, the average RMSE ranged between (, ) and (, ). Similar to the comparison with ground truth, we observed that for smaller such as 12,000 observations, increasing resulted in reduced RMSE, while at larger such as 20,000 observations, increasing did not make a difference to the average RMSE.

Simulation 5: FEMA-Long false positive calibration improves with sample size

We examined the distribution of -values under a range of and ; in each of these 16 scenarios, across the 36 simulations, unstructured covariance generally resulted in uniformly distributed -values (S26, S29, S32, and S35 Figs), except for two settings: and ; in these two settings, we observed that there were a few combinations of and , where the distribution of the unstructured covariance -values closely followed the null distribution but deviated away from the null towards the tail of the distribution: , (S27 Fig), , (S28 Fig), , (S30 Fig), , (S31 Fig), , (S33 Fig), and , (S34 Fig). On further examination of these cases, we observed that the miscalibration of the -values was caused by one or few iterations (out of the 1000) where the resulting distribution of visits was extremely imbalanced (S36 Fig), leading to a small number of common observations between the last pairs of visits, possibly leading to an unstable estimation of the covariance components. As the sample size increased, the false positive calibration for the unstructured covariance improved; in contrast, the compound symmetry covariance pattern resulted in inflated Q-Q plots.

Study sample

To demonstrate the features of FEMA-Long (unstructured covariance, non-linear modeling, scalability to large datasets, performing GWAS, and discovery of time-dependent genetic effects), we performed a GWAS on the length, weight, and BMI of infants in the first year of life using the large-scale Norwegian Mother, Father, and Child Cohort Study (MoBa) [30–32]. The MoBa Study is a population-based pregnancy cohort study conducted by the Norwegian Institute of Public Health. Participants were recruited from all over Norway from 1999-2008. The women consented to participation in 41% of the pregnancies. The cohort includes approximately 114,500 children, 95,200 mothers and 75,200 fathers. The current study is based on version 12 of the quality-assured data files released for research in 2019. The establishment of MoBa and initial data collection was based on a license from the Norwegian Data Protection Agency and approval from The Regional Committees for Medical and Health Research Ethics. The MoBa cohort is currently regulated by the Norwegian Health Registry Act. The current study was approved by The Regional Committees for Medical and Health Research Ethics (2016/1226/REK Sør-Øst C).

Genetic data

Blood samples were obtained from both parents during pregnancy and from mothers and children (umbilical cord) at birth [33], which were used for genotyping. We used the quality controlled genetic data from the MoBaPsychGen pipeline v1 [34]. The release consists of data from 76,577 children, 77,634 mothers, and 53,358 fathers; we restricted ourselves to the data on children who had not withdrawn their consent from MoBa as of November 1, 2024.

Phenotypes

The phenotypes of interest were the length, weight, and BMI of infants at six timepoints during the first year of life: birth, six weeks, three months, six months, eight months, and twelve months. Phenotypic information was retrieved from questionnaire data, and from the Medical Birth Registry of Norway (MBRN); the MBRN is a national health registry containing information about all births in Norway (see S1 Text for details on data curation and quality checks prior to GWAS). The final sample size for GWAS was 68,273 infants with 299,447 observations having complete data on length, weight, and BMI (i.e., each of the 68,273 subjects had all three measurements); see S10 Fig for distribution of age at each timepoint, and S11–S13 Figs for plots showing the trajectories of length, weight, and BMI.

Genome-wide association study

For the 68,273 infants, with up to six repeated measurements totaling 299,447 observations on length, weight, and BMI during the first year of life, we performed a longitudinal GWAS using FEMA-Long on the autosomes. We standardized the phenotypes to have zero mean and unit standard deviation. The covariates included the intercept, six spline basis functions of age (after the transformations described in the Methods section), dummy-coded sex, twenty genetic principal components (PC), and 23 dummy-coded genotyping batch variables, resulting in a total of 51 covariates. We created the spline basis functions by placing a knot at the median age for each timepoint: 44, 94, 181, 246, and 369 days, and boundary knots at the extremes (0 and 425 days; see, S14 and S15 Figs).

The random effects included the effect of family or shared environment, the genetic relationship matrix (GRM), and the subject effect (or repeated measurements). We defined families using the mother ID of the participants; in addition, for a subset of participants where the mother ID was different, but the father ID was the same, we assigned those individuals to the same family; this resulted in 46,453 unique families. To calculate GRM, we used a subset of 500,243 SNPs which were directly genotyped in at least one imputation batch or had an imputation INFO score > 0.99785 in all imputation batches; we mean imputed any missing genotyping data and standardized the SNPs (zero mean, unit standard deviation); then, the GRM was calculated as the correlation coefficient between the standardized SNPs. Therefore, the stage-1 model (without GWAS) for each phenotype was:

(25)

We performed GWAS on 6,981,748 SNPs [34] after mean imputing missing genotyping data, and examined both the main effect of the SNPs as well as the interaction of each SNP with the six basis functions, resulting in seven coefficients per SNP per phenotype. Specifically, after fitting the stage-1 model and residualizing each genotype for the covariates in , the model for each residual SNP, , was:

(26)

where is the element-wise multiplication of with every column of the basis function matrix. Then, for each phenotype, and each SNP, we performed a Wald test to determine if any of these coefficients had a non-zero effect. To accelerate computing, we split the genetic data into chunks of 5,000 SNPs, processing eighteen chunks in parallel, with each parallel worker having seven threads, run on a compute cluster with two AMD EPYC 7702 64-core processers; the job specification specified 128 CPUs per task and 30 GB RAM per task. Additionally, we also ran a standard longitudinal GWAS, without including and terms (run on the same hardware with the same specification, but twenty parallel chunks with six threads each).

To quantify genomic inflation factor (, we used the Wald test values, converted them to chi-squared statistics (using values as input to the inverse of the chi-square cumulative distribution function with 1 degree of freedom); then, was calculated as:

(27)

where were the observed chi-squared statistics, while were the expected chi-squared statistics under the null (this simplifies to using the chi-squared statistics for a value of 0.5, since the under the null, the values are uniformly distributed). We quantified for both the scenarios: when allowing SNP effects to vary as smooth function of age, and when performing a standard longitudinal GWAS without any interactions or smooth terms.