Many existing cohort studies initially designed to investigate disease risk as a function of environmental exposures have collected genomic data in recent years with the objective of testing for gene-environment interaction (

It is now clear from many lines of evidence that pure genetics or pure environmental factors play only a partial role in the etiology of most complex diseases. Instead, it is now accepted that the majority of chronic diseases likely stem from interactions between genetic traits, “

Established environmental health cohorts that have demonstrated modest health effects of the environment are now collecting genomic data to test

Latent variable (LV) models have been used in environmental health studies to extract features from a set of correlated biomarkers, thus reducing dimensionality of exposure data and multiple testing burden, and enhancing power (e.g.,

The Early Life Exposures in MExico City to Neuro-Toxicants (ELEMENT) study motivates our work. ELEMENT consists of four longitudinal birth cohorts in Mexico City, constituting over 2,000 mother infant pairs with prospectively collected exposure data and several anthropometric, cardiovascular, neuro-development, and behavioral outcomes. Genotyping for these cohorts is underway, with genotyping for the first cohort completed on a set of candidate genes (≈ 400 pairs).

In Section 2, we describe a model structure to summarize a group of biomarkers into latent variables, and the spectrum of

For the ^{th}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

The latent variable model is then specified in two stages: a health outcome model and an exposure model. In the _{i}_{i}_{i}_{i}_{gi}_{i}_{i} has variance σ^{2}. _{0,g}, _{U,g}, _{Z,g} are parameters specific to class _{0} : β_{U,0} = ⋯ = β_{U,g}, i.e., homogeneity of the environment’s effects across genetic groups, and is equivalent to testing the interaction parameters in _{0} : _{g×U} = _{U,g} − _{U,0} = 0 for all _{U,g}, are of primary interest.

The exposure model consists of a _{i}_{0g} and _{W} are _{g} = _{0g} − α_{00} being effect of genotype class on exposure _{i}_{i}_{i}, are assumed independent of ε_{i}, and have category-specific _{g}. Means vector _{g} and factor loading matrix _{g} are _{i} has zero mean and _{g}.

Although LV models are helpful in many respects, one well known problem is the potential for lack of identifiability. Standard identifiability constraints have been developed for linear latent variable models (_{g} and _{g} are fixed to 0 or 1, although sometimes algebraic proofs of identifiability are needed (_{g} = (0, ν_{g,2}, ν_{g,3}, ν_{g,4})^{⊤} and _{g} = (1, λ_{g,2}, λ_{g,3}, λ_{g,4})^{⊤} fix the mean and scale of the latent exposure to those of patella lead. Parameters in _{g} are typically unconstrained, while the off-diagonal elements of _{g} are typically, although not necessarily, restricted to be zero denoting conditional independence between _{i}_{i}

In many

Varying degrees of _{0} and

A first step at relaxing _{g} = α_{0g} − α_{00} be the gene effect on the latent exposure we have
_{i}_{i}_{g} to be equal across genotype subgroups; that is, a slightly modified assumption _{g}, _{g}, _{g}, _{g}) = (

Next, constraints of equal means _{4} may vary by genotype.

Lastly, all equality constraints on the parameters can be removed, namely,

Assumptions

Although various estimation procedures have been proposed for LV models (

Let _{i}, _{i} and _{i} are normally distributed, and integrating over the latent variable, the joint marginal distribution of the observed outcome and exposures, ^{th}_{θ} = −^{2}ℓ(^{⊤}), or by computing robust variances

Shrinkage estimators have been used in outcome-dependent sampling based studies as a way to balance bias and efficiency gains from assuming _{1}, θ_{2}, θ_{3}) as _{1})_{2})_{3})/_{1}, θ_{2}, θ_{3}) with θ_{1} being the outcome model parameters, θ_{2} and θ_{3} describing the _{2}). Because _{1}, θ_{2}, θ_{3}) = ∑_{G, E, Z}
_{1})_{2})_{3}) (in the denominator) depends on the specification of _{2}), the MLE of θ_{1} depends on the assumed model for the _{2}) = _{2}), in case-control studies leads to large efficiency gain for estimating the _{1}). However, under violation of the independence assumption, these estimators are biased. Shrinkage estimators then arise as a weighted average of two estimators: one obtained under dependence and the other obtained under independence; the weights are chosen in a data-adaptive fashion and reflect the uncertainty around the conditional

In a cohort or cross-sectional study, maximization of the joint likelihood with respect to outcome model parameters, even from modeling the joint distribution _{1}, θ_{2}, θ_{3}), will be independent of the specification of _{2}, θ_{3}). Because of the lack of outcome dependent sampling, and thus absence of conditioning on

We follow _{A0} and _{A*},
_{MV} = ^{⊤})^{−1}, with ψ̂ = _{A0} − _{A*} and _{CW} where the ^{th}_{k}^{th}_{k} the ^{th}_{CW} leads to ‘component-wise’ shrinkage, since the weights used for a given component of _{shrink} depend only on the variance and bias related to that component; we call these estimates _{CW}. In contrast, using _{MV} leads to so-called multivariate shrinkage (_{MV}. In both cases we use superscripts to denote which estimates were combined, e.g., _{A*} in _{A0}.

Additional considerations about shrinkage estimators are worth mentioning. First, CW shrinkage may be desirable in terms of efficiency gain, compared to multivariate shrinkage in small samples, because large sampling error in the off-diagonals of _{MV} estimates will be more prone to favoring the more flexible models. To see this, note that the component-wise shrinkage weights (the ^{th}_{CW}) can be re-written as ^{th}^{2}) where χ^{2} = ψ̂^{⊤}
^{−1}ψ̂. ^{2} can be interpreted as a bias-variance ratios; when they are smaller than one, the ^{2} is a weighted sum of all the bias-variance ratios for all model parameters. Hence, in MV shrinkage, a given parameter might be shrunk given not only its own bias-variance ratio ^{2} has expectation equal to the number of model parameters estimated with assumption ^{2}) will almost always be small, leading to EB estimates closer to those from more flexible models. Third, the weights summarize information about model fit in the sense of comparing differences in estimated parameters. If the models fit equally well, then corresponding parameters would likely be similar. Large differences in corresponding parameters indicate a poorer fitting model (e.g., constrained model). Hence, shrinkage estimates can help assess model adequacy, with smaller weights for the constrained model implying the more flexible model is preferred. Finally, both CW and MV shrinkage estimators will asymptotically converge to those from the more flexible model.

_{shrink}. Heuristically, the variance can be obtained by treating _{shrink} as a function of two random variables, _{A0} and _{A*}, with joint covariance matrix _{A0}, _{A*}) = _{A*} + ^{⊤})^{−1}
_{A0} − _{A*}, and employing the multivariate Delta theorem, then _{shrink}) ≈ ^{⊤}
^{−1}
^{−⊤} is constructed using the sandwich-variance formula where

Instead of positing a latent exposure model,

We conducted a small scale simulation study to examine the finite sample properties of estimators under various settings of the true data generating model using _{G}, β_{U}, β_{G×U}. We investigate the estimators’ properties under two scenarios of the _{U} = β_{G} = β_{G×U} = 0 or β_{U} = 1, β_{G} = β_{G×U} = 2 (i.e., standardized effects of 0.2, 0.4, 0.4, respectively, since outcome variance was σ^{2} = 5^{2}). See

When _{U} and β̂_{G}. While _{CW} retain inflated Type I error rates, _{MV} estimates do not.

When _{G} and β̂_{G×U} estimated from _{G×U} estimated under _{G×U} is 0.44 under _{U} and β̂_{G×U}.

When _{U} and β_{G×U}. Simply relaxing the assumption of different mean and variance for the latent variable, but not the measurement model, may not be sufficient to reduce bias, and could in fact increase it.

_{CW} estimates are approximately half way between _{U} and β_{G×U} than the bias in

As would be expected from the measurement error literature, parameter estimates using only _{U} and β̂_{G×U} are attenuated. However, note that β̂_{G} have large bias as well, due to the _{1}) can induce bias in regression coefficients of covariates measured without error (e.g., _{G} can be in either positive or negative under the alternative hypothesis (_{U} = β_{G×U} = 0,

_{CW} estimators achieve a better bias-variance compromise in small samples compared to _{MV}.

Additional simulation results for the case of null main effects and small interaction parameter: β_{U} = 0, β_{G} = 0, and β_{G×U} = 0.1 demonstrate that efficiency gains in _{G×E} persisted when incorrectly assuming

For hypothesis testing alone, PCA approaches may be just as good as using a full latent variable model because they maintain Type I error (

We use data from the first ELEMENT cohort, where the following prenatal lead exposure biomarkers were collected on the mother and child: maternal blood lead levels at delivery and umbilical cord blood lead as well as maternal bone lead levels (patella and tibia) (

Deleterious effects of prenatal lead exposure on birth weight have been demostrated (_{U} = −54.12, _{U}) = 25.1, _{U}

Increasing lead exposure among wild types is associated with decreased birth weight (negative β̂_{U} in

Estimates and standard errors for β̂_{G} are fairly constant across _{G} from MLR and PCA are much higher (more negative) than those from MLE. This can be due to bias arising due to exposure-gene correlation and exposure measurement error (

Being variant for iron metabolism genes is protective against reduced the birth weight due to lead exposure (β̂_{G×U} are positive), as hypothesized (_{U}_{G×U} are impacted by the assumed _{G×U} = 1.46 (

Differences in MLE estimates for the outcome parameters can be largely explained by a few key differences exposure model parameters by genotype (_{g}), the variance of the latent variable is twice as high among variants (Φ̂_{g=1} = 2.11) than among wild types (Φ̂_{g=0} = 1.05). Residual variances for _{1} and _{2}, Θ_{11} and Θ_{22}, also appear to differ between genotypes (51% and 72% difference, respectively), as does λ_{4} (17% difference). This deserves further study–e.g., differences in Θ_{11} and Θ_{22} might be due to maternal genotypes, which are inherently correlated to infant genotype. Such investigation is out of the scope of the current work, but this finding highlights the utility of LV models in elucidating potential biological pathways.

In this example, implementing multivariate shrinkage was possible only for combining estimates from

The presence of multiple correlated measures of exposure exacerbates existing challenges in _{i}_{i}_{i}_{i}_{i}

Because of the flexibility afforded by LV models, one challenge is the potential for model misspecification. In this particular application of LV models, we described various specifications of the

It is possible that one may use the proposed approach for screening

The availability of higher dimensional genomic data, and multiple continuous or categorical outcomes point to several extensions of our work. General LV models encompass latent class models (_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

The authors thank ELEMENT investigators for proving data for the example, as well as the following NIEHS grants that supported data collection: K23ES000381; P01 ES012874; P42 ES05947; R01 ES013744; R01 ES014930; R01 ES007821. The authors also acknowledge salary support from grants NSF DMS 1007494, NIEHS R01 ES016932 and R01 ES017022, and NIEHS/EPA 1-P20-SE018171-01.

Path diagram showing relationships between exposure biomarkers, latent prenatal lead exposure, iron metabolism genes, and birth weight.

Path diagrams showing gene-environment dependence assumptions.

Bias, variance ratios (Var.R), mean squared error (MSE), and rejection probabilities (P(R)) for outcome model parameter estimates under two scenarios of true exposure model parameters. Outcome model parameters set at β_{U} = β_{G} = β_{G×U} = 0, σ^{2} =5; sample size was N = 350, with 500

Data | Est. | β̂_{U} | Var.R | P(R) | β̂_{G} | Var.R | P(R) | β̂_{G×U} | Var.R | P(R) | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A0 | 0.00 | 1(Ref) | 0.063 | −0.05 | 1(Ref) | 0.048 | −0.04 | 1(Ref) | 0.036 | ||||

0.00 | (1.02) | 0.059 | −0.05 | (1.05) | 0.044 | −0.04 | (1.10) | 0.038 | |||||

0.00 | (1.01) | 0.059 | −0.04 | (1.11) | 0.042 | −0.05 | (1.15) | 0.036 | |||||

0.00 | (1.02) | 0.061 | −0.04 | (1.12) | 0.042 | −0.04 | (1.15) | 0.044 | |||||

| 0.00 | (1.02) | 0.061 | −0.05 | (0.99) | 0.046 | −0.04 | (0.99) | 0.050 | ||||

| 0.00 | (1.02) | 0.069 | −0.04 | (1.02) | 0.046 | −0.04 | (1.01) | 0.050 | ||||

| 0.00 | (1.02) | 0.069 | −0.05 | (1.03) | 0.044 | −0.04 | (1.04) | 0.052 | ||||

| 0.00 | (1.03) | 0.063 | −0.05 | (1.02) | 0.044 | −0.04 | (1.04) | 0.046 | ||||

| 0.00 | (1.03) | 0.063 | −0.04 | (1.09) | 0.044 | −0.05 | (1.08) | 0.044 | ||||

| 0.00 | (1.03) | 0.063 | −0.04 | (1.09) | 0.042 | −0.04 | (1.07) | 0.055 | ||||

E1 | 0.00 | (0.23) | 0.044 | −0.04 | (1.38) | 0.052 | 0.00 | (0.24) | 0.048 | ||||

PCA | 0.00 | (0.23) | 0.063 | −0.04 | (1.02) | 0.048 | 0.01 | (0.24) | 0.042 | ||||

A3 | 0.03 | 1(Ref) | 0.126 | −0.07 | 1(Ref) | 0.142 | 0.03 | 1(Ref) | 0.043 | ||||

0.04 | (2.59) | 0.038 | −0.10 | (2.46) | 0.063 | 0.02 | (1.21) | 0.050 | |||||

0.03 | (1.42) | 0.036 | −0.08 | (2.07) | 0.056 | 0.03 | (1.08) | 0.043 | |||||

0.03 | (1.27) | 0.038 | −0.09 | (2.26) | 0.047 | 0.04 | (1.19) | 0.050 | |||||

| 0.03 | (1.38) | 0.020 | −0.07 | (1.18) | 0.101 | 0.02 | (1.81) | 0.007 | ||||

| 0.03 | (0.83) | 0.171 | −0.07 | (1.10) | 0.119 | 0.03 | (1.65) | 0.011 | ||||

| 0.03 | (0.84) | 0.153 | −0.08 | (1.16) | 0.106 | 0.03 | (1.61) | 0.018 | ||||

| 0.04 | (2.59) | 0.036 | −0.10 | (2.37) | 0.074 | 0.02 | (1.16) | 0.050 | ||||

| 0.03 | (1.39) | 0.050 | −0.08 | (1.98) | 0.056 | 0.03 | (1.04) | 0.054 | ||||

| 0.03 | (1.25) | 0.054 | −0.09 | (2.14) | 0.059 | 0.04 | (1.13) | 0.061 | ||||

0.01 | (0.28) | 0.070 | −0.00 | (1.97) | 0.052 | −0.01 | (0.23) | 0.047 | |||||

PCA | −0.01 | (0.36) | 0.038 | −0.05 | (1.57) | 0.056 | −0.01 | (0.24) | 0.061 |

Ratios of empirical variances, comparing to variance of

Multiple regression using one exposure marker, _{1}

Multiple regression using 1st principal component of (_{1}, _{2}, _{3}, _{4}) as exposure marker

4.6% (

Percent bias, variance ratios (Var.R), mean squared error (MSE), and rejection probabilities (P(R)) for outcome model parameter estimates under two scenarios of true exposure model parameters. Outcome model parameters set at β_{U} = 1, β_{G} = β_{G×U} = 2, σ^{2} = 5; sample size was N = 350, with 500

Data | Est. | β̂_{U} | Var.R | P(R) | β̂_{G} | Var.R | P(R) | β̂_{G×U} | Var.R | P(R) | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A0 | −1.3% | 1(Ref) | 0.74 | 3.4% | 1(Ref) | 0.85 | −1.4% | 1(Ref) | 0.66 | ||||

−1.0% | (1.02) | 0.74 | 3.0% | (1.14) | 0.78 | 1.3% | (1.18) | 0.62 | |||||

−1.0% | (1.03) | 0.74 | 3.0% | (1.73) | 0.63 | 3.1% | (1.74) | 0.47 | |||||

−0.7% | (1.04) | 0.74 | 2.9% | (1.76) | 0.63 | 3.1% | (1.90) | 0.44 | |||||

| −1.2% | (1.01) | 0.73 | 3.2% | (1.08) | 0.82 | −0.3% | (1.08) | 0.64 | ||||

| −1.2% | (1.02) | 0.74 | 3.4% | (1.45) | 0.71 | −0.5% | (1.36) | 0.56 | ||||

| −1.0% | (1.02) | 0.75 | 3.3% | (1.47) | 0.72 | −1.1% | (1.46) | 0.53 | ||||

| −1.0% | (1.02) | 0.73 | 3.0% | (1.12) | 0.79 | 1.3% | (1.13) | 0.64 | ||||

| −1.0% | (1.03) | 0.73 | 3.0% | (1.66) | 0.64 | 3.1% | (1.64) | 0.50 | ||||

| −0.7% | (1.04) | 0.73 | 2.9% | (1.67) | 0.65 | 2.8% | (1.74) | 0.49 | ||||

E1 | −65.4% | (0.22) | 0.46 | 3.2% | (1.33) | 0.83 | −63.6% | (0.23) | 0.41 | ||||

PCA | −53.0% | (0.22) | 0.74 | 3.3% | (0.94) | 0.86 | −51.5% | (0.23) | 0.65 | ||||

A3 | 12.1% | 1(Ref) | 0.86 | 35.7% | 1(Ref) | 0.94 | −27.6% | 1(Ref) | 0.48 | ||||

45.6% | (2.57) | 0.78 | −18.7% | (2.57) | 0.35 | −42.0% | (1.24) | 0.27 | |||||

8.9% | (1.43) | 0.79 | 10.3% | (3.79) | 0.51 | −13.5% | (1.86) | 0.40 | |||||

3.7% | (1.26) | 0.80 | −1.4% | (5.18) | 0.41 | 0.9% | (2.65) | 0.39 | |||||

| 32.8% | (2.51) | 0.72 | 0.0% | (2.83) | 0.44 | −37.5% | (1.94) | 0.19 | ||||

| 10.4% | (0.94) | 0.88 | 28.4% | (3.47) | 0.67 | −21.3% | (1.98) | 0.31 | ||||

| 7.4% | (0.95) | 0.86 | 21.8% | (4.45) | 0.57 | −14.6% | (2.48) | 0.30 | ||||

| 45.5% | (2.54) | 0.77 | −18.6% | (2.55) | 0.37 | −42.0% | (1.22) | 0.27 | ||||

| 9.0% | (1.38) | 0.79 | 10.7% | (4.03) | 0.50 | −13.7% | (1.97) | 0.39 | ||||

| 3.8% | (1.23) | 0.79 | −0.7% | (4.89) | 0.44 | 0.4% | (2.50) | 0.41 | ||||

−63.5% | (0.27) | 0.51 | 110.7% | (1.97) | 0.99 | −78.0% | (0.23) | 0.23 | |||||

PCA | −44.2% | (0.33) | 0.78 | 37.3% | (1.52) | 0.88 | −63.7% | (0.23) | 0.48 |

Ratios of empirical variances, comparing to variance of

Multiple regression using one exposure marker, _{1}

Multiple regression using 1st principal component of (_{1}, _{2}, _{3}, _{4}) as exposure marker

3.6% (

Outcome model parameter estimates, robust standard errors, and t-statistics obtained using MLE under assumptions A0–A3 and shrinkage-based estimates combining assumptions. Coefficients β̂_{U} and β̂_{G×U} have been re-scaled to represent changes in birth weight(g) associated with an increase of 10 µg Pb/g in patella bone mass. Models are adjusted for maternal age, parity, education and marital status

Est. Method^{a}, | β̂_{U} | _{U}) | _{U} | β̂_{G} | _{G}) | _{G} | β̂_{G×U} | _{G×U}) | _{G×U} |
---|---|---|---|---|---|---|---|---|---|

−80.59 | 29.97 | −2.69 | −97.75 | 51.30 | −1.91 | 92.42 | 63.50 | 1.46 | |

−79.51 | 29.79 | −2.67 | −99.09 | 51.45 | −1.93 | 91.92 | 62.56 | 1.47 | |

−98.44 | 39.54 | −2.49 | −98.47 | 51.30 | −1.92 | 103.09 | 52.71 | 1.96 | |

−85.14 | 33.75 | −2.52 | −98.83 | 51.42 | −1.92 | 90.66 | 53.57 | 1.69 | |

−86.62 | 35.01 | −2.47 | −100.74 | 51.41 | −1.96 | 105.00 | 49.24 | 2.13 | |

| −80.05 | 29.76 | −2.69 | −98.02 | 51.31 | −1.91 | 91.90 | 63.25 | 1.45 |

| −91.12 | 40.15 | −2.27 | −97.89 | 51.31 | −1.91 | 101.95 | 61.06 | 1.67 |

| −81.73 | 32.13 | −2.54 | −97.90 | 51.24 | −1.91 | 92.58 | 60.07 | 1.54 |

| −82.40 | 33.04 | −2.49 | −98.93 | 51.37 | −1.93 | 94.63 | 62.10 | 1.52 |

| −85.02 | 33.64 | −2.53 | −98.80 | 51.40 | −1.92 | 90.70 | 53.76 | 1.69 |

| −86.28 | 34.66 | −2.49 | −100.58 | 51.39 | −1.96 | 104.30 | 49.81 | 2.09 |

_{1} | −41.58 | 15.06 | −2.76 | −191.50 | 71.07 | −2.69 | 60.59 | 31.19 | 1.94 |

PCA | −46.58 | 15.34 | −3.04 | −187.92 | 69.27 | −2.71 | 58.52 | 30.04 | 1.95 |

Estimates from models

Multiple regression using patella lead, _{1}

Multiple regression using 1st principal component of _{1}, _{2}, _{3}, _{4} as exposure marker

Exposure model parameter estimates and robust standard errors obtained under assumptions A0–A3 described by

Est(SE) | Est(SE) | Est(SE) | Est(SE) | Est(SE) | |

α_{0} | 1.536 (0.078) | 1.519 (0.086) | 1.526 (0.081) | 1.506 (0.087) | 1.506 (0.086) |

γ_{g} | 0.084 (0.178) | 0.050 (0.182) | 0.148 (0.207) | 0.148 (0.208) | |

Φ_{g=0} | 1.230 (0.312) | 1.253 (0.317) | 0.835 (0.218) | 1.090 (0.306) | 1.056 (0.333) |

Φ_{g=1} | 1.645 (0.489) | 1.440 (0.460) | 2.112 (0.916) | ||

Estimates for wild types | |||||

ν_{2} | 1.031 (0.053) | 1.021 (0.057) | 1.024 (0.056) | 1.040 (0.057) | 1.040 (0.056) |

ν_{3} | 1.759 (0.024) | 1.757 (0.025) | 1.758 (0.025) | 1.756 (0.027) | 1.756 (0.027) |

ν_{4} | 2.043 (0.023) | 2.041 (0.023) | 2.042 (0.023) | 2.034 (0.026) | 2.034 (0.026) |

λ_{2} | 0.553 (0.126) | 0.542 (0.124) | 0.680 (0.136) | 0.527 (0.136) | 0.558 (0.157) |

λ_{3} | 0.140 (0.036) | 0.138 (0.035) | 0.156 (0.036) | 0.129 (0.041) | 0.131 (0.043) |

λ_{4} | 0.125 (0.032) | 0.124 (0.032) | 0.140 (0.033) | 0.114 (0.037) | 0.116 (0.039) |

Θ_{11} | 1.172 (0.283) | 1.148 (0.289) | 1.401 (0.219) | 1.236 (0.262) | 1.258 (0.311) |

Θ_{22} | 0.710 (0.099) | 0.717 (0.097) | 0.624 (0.103) | 0.671 (0.096) | 0.627 (0.104) |

Θ_{33} | 0.218 (0.017) | 0.218 (0.017) | 0.218 (0.017) | 0.218 (0.017) | 0.217 (0.019) |

Θ_{44} | 0.194 (0.015) | 0.194 (0.015) | 0.194 (0.015) | 0.195 (0.015) | 0.197 (0.017) |

Θ_{34} | 0.168 (0.014) | 0.168 (0.014) | 0.168 (0.014) | 0.168 (0.014) | 0.168 (0.016) |

Estimates for variants | |||||

ν_{2} | 0.880 (0.156) | 0.920 (0.139) | |||

ν_{3} | 1.746 (0.059) | 1.751 (0.056) | |||

ν_{4} | 2.053 (0.055) | 2.056 (0.051) | |||

λ_{2} | 0.768 (0.162) | 0.496 (0.205) | |||

λ_{3} | 0.180 (0.058) | 0.142 (0.063) | |||

λ_{4} | 0.161 (0.053) | 0.136 (0.058) | |||

Θ_{11} | 0.612 (0.809) | ||||

Θ_{22} | 1.077 (0.259) | ||||

Θ_{33} | 0.226 (0.040) | ||||

Θ_{44} | 0.183 (0.032) | ||||

Θ_{34} | 0.171 (0.033) | ||||

Model fit criteria | |||||

Num Param | 23 | 24 | 25 | 31 | 36 |

−2LL (smaller is better) | 12302.8 | 12302.6 | 12296.6 | 12291.4 | |

AIC (smaller is better) | 12348.8 | 12350.6 | 12353.4 | 12359.0 | |

BIC(smaller is better) | 12446.7 | 12446.6 | 12477.6 | 12503.2 | |

CFI | 0.963 | 0.962 | 0.968 | 0.967 | |

TLI | 0.961 | 0.959 | 0.962 | 0.958 | |

RMSEA | 0.041 | 0.042 | 0.041 | 0.043 |

Parameters estimates for variants are the same for as for wild types unless shown here

For the exposure and outcome model combined

Including outcome model parameters

See