Jean-Baptiste Pingault, Tabea Schoeler & Frank Dudbridge 13 November 2020
The following script provides an example on how to run Gsens, a genetically informed sensitivity analysis, in R. Please cite the following papers where the concept was first proposed and then implemented:
-
Pingault, J.-B., O’Reilly, P. F., Schoeler, T., Ploubidis, G. B., Rijsdijk, F., & Dudbridge, F. (2018). Using genetic data to strengthen causal inference in observational research. Nature Reviews Genetics, 19(9), 566–580. https://doi.org/10.1038/s41576-018-0020-3
-
Pingault, J.-B., Rijsdijk, F., Schoeler, T., Choi, S. W., Selzam, S., Kraphol, E., O’Reilly, P. F., & Dudbridge, F. Genetic sensitivity analysis: adjusting for genetic confounding in epidemiological associations. BioRxiv.
In this tutorial, we illustrate the use of Gsens to estimate the role of genetic confounding in explaining the associations between maternal educational and three developmental outcomes in the offspring
- educational achievement
- BMI
- ADHD
The following examples are based on the correlation matrix between polygenic scores and variables, as shown below (also available in the supplementary material of article 2). Study variables are residualised for sex and PCAs before computing the correlations.
Correlations between study variables
| Maternal Education | GCSE | BMI | ADHD | EDU PS | BMI PS | ADHD PS | |
|---|---|---|---|---|---|---|---|
| Maternal Education | 1 | 0.3975 | -0.0894 | -0.1240 | 0.2909 | -0.0839 | -0.0630 |
| GCSE | 0.3975 | 1 | -0.0894 | -0.3398 | 0.3462 | -0.0885 | -0.1103 |
| BMI | -0.0894 | -0.0894 | 1 | -0.0088 | -0.0269 | 0.2524 | 0.0441 |
| ADHD | -0.1240 | -0.3398 | -0.0088 | 1 | -0.0902 | 0.0820 | 0.1188 |
| EDU PS | 0.2909 | 0.3462 | -0.0269 | -0.0902 | 1 | -0.1856 | -0.1865 |
| BMI PS | -0.0839 | -0.0885 | 0.2524 | 0.0820 | -0.1856 | 1 | 0.1413 |
| ADHD PS | -0.0630 | -0.1103 | 0.0441 | 0.1188 | -0.1865 | 0.1413 | 1 |
To begin, please install and load the Gsens package with the code below:
# Install devtools
install.packages("devtools")
library(devtools)
# Install Gsens
install_github("JBPG/Gsens")
library(Gsens)Three functions are available:
gsensY(): sensivity analysis based on one polygenic score for the outcome (Y)
gsensX(): sensivity analysis based on one polygenic score for the exposure (X)
gsensXY(): sensivity analysis based on two polygenic scores for X and Y
As noted in the manuscript, gsensY() should be preferred in almost all situations.
In the following example, we will test the association between maternal years of education (X) and child GCSE scores (Y) after controlling for the best fitting polygenic score for years of education estimated in the child.
To do so, we will use the "gsensY" function.
For this, we need to specify 5 parameters:
-
rxy = the observed phenotypic correlation between exposure X and outcome Y (here: correlation between maternal years of education and child GCSE);
-
rgx = the observed correlation between phenotype X (maternal years of education) and the observed polygenic score for education;
-
rgy = the observed correlation between phenotype Y (child GCSE) and the observed polygenic score for education (adjusted for sex, age, and principal components);
-
n = sample size;
-
h2 = is the variance explained in the outcome, here by the observed polygenic score (hence why h2 is rgy^2).
gsensY(rxy=0.3975,
rgx = 0.2909,
rgy = 0.3462,
n=3785,
h2=0.3462^2)## est se z pvalue ci.lower ci.upper
## Adjusted Bxy 0.324 0.017 19.396 8.4077e-84 0.291 0.357
## Genetic confounding 0.073 0.003 22.881 7.1055e-116 0.067 0.080
## Total effect 0.397 0.015 26.479 1.6783e-154 0.368 0.427
The output provides:
-
Adjusted Bxy This is the standardized estimate of the relationship between X and Y, adjusted for G (i.e. the residual association between maternal education and child GCSE scores adjusting for the offspring's polygenic score). Note that this estimate should be the same as a regression of Y on X adjusting for the polygenic score in the dataset from which the correlations were obtained.
-
Genetic confounding This is the estimate of genetic confounding.
-
Total effect This is the total effect, which should add up to the observed initial association between X and Y when no constraints are added.
We will now implement the sensitivity analysis to examine genetic confounding under a scenario in which polygenic scores explain SNP-heritability in the outcome (here child GCSE scores).
We will do this by adding a "h2" option which provides the chosen heritability estimate.
Here, h2 = 0.31, which corresponds to the SNP-heritability of Y (child GCSE scores).
Heritability and genetic correlation under different scenarios
| Education | |
|---|---|
| Best-Fitting Polygenic score | 0.119 |
| SNP-based scenario | 0.31 |
| Twin scenario | 0.63 |
gsensY(rxy=0.3975,
rgx = 0.2909,
rgy = 0.3462,
n=3785,
h2=0.31)## est se z pvalue ci.lower ci.upper
## Adjusted Bxy 0.175 0.024 7.359 1.8564e-13 0.129 0.222
## Genetic confounding 0.222 0.014 15.914 5.0431e-57 0.195 0.249
## Total effect 0.397 0.016 25.212 2.9876e-140 0.367 0.428
The results show that under a SNP heritability scenario, the effect of maternal education on child educational achievement is attenuated (B=0.176) relative to when controlling for observed polygenic scores (B=0.325).
As noted in the manuscript, it is possible to fix the ratio k between rgy and rgx when a priori knowledge is available, for example the genetic relationship between the child and the mother.
We do this below by specifying that rgx (i.e., the correlation between the child's genetics with maternal education) is half of the correlation between the child's genetics with their own educational achievement (i.e., the SNP heritability).
We also specify that the rgy (i.e., the correlation between the child's genetics with their own educational achievement) reflects SNP heritability (by taking the square root the heritability estimate to get the corresponding path value).
gsensY(rxy=0.3975,
rgx = 0.5*sqrt(.31),
rgy = sqrt(.31),
n = 3785,
h2 = 0.31)## est se z pvalue ci.lower ci.upper
## Adjusted Bxy 0.263 0.016 16.941 2.2374e-64 0.232 0.293
## Genetic confounding 0.135 0.006 21.208 8.1051e-100 0.122 0.147
## Total effect 0.397 0.015 26.092 4.5077e-150 0.368 0.427
In this fixed solution, rgx and rgy and are therefore set to their value under the heritability scenario, rather than the observed value from the polygenic score.
When a polygenic score is available for each of X and Y, both can be modelled using gsensXY, for example, for maternal years of education and BMI. The gsensXY function takes a number of additional arguments:
-
rxy = as before, the observed phenotypic correlation between exposure X and outcome Y (here: correlation between maternal years of education and BMI);
-
rg1x = the correlation between the observed exposure X and the polygenic score for X (maternal education) and the observed polygenic score for education;
-
rg2x = the correlation between the observed exposure X (maternal education) and the observed polygenic score for BMI;
-
rg1y = the correlations between the observed outcome Y (BMI) and the observed polygenic score for education;
-
rg2y = the correlation between the observed outcome Y (BMI) and the observed polygenic score for BMI;
-
rg1g2 = the correlation between the observed polygenic scores for education and BMI;
-
h2.x = the additive genetic variance explained in the observed exposure X (maternal education) under the scenario of interest. When calculating genetic confounding for with the two observed polygenic score, then (h2.x = rg1x^2), otherwise h2.x is the chosen heritability estimate for exposure X.
-
h2.y = the additive genetic variance explained in the observed outcome Y (BMI) under the scenario of interest. When calculating genetic confounding for with the two observed polygenic score, then (h2.y =rg2y^2) otherwise h2.y is the chosen heritability estimate for outcome Y.
-
print = T = enables the examination of model parameters
gsensXY(rxy=-0.0894,
rg1x=0.2909,
rg2x=-0.0839,
rg1y=-0.0269,
rg2y=0.2524,
rg1g2=-0.1856,
n=3663,
h2.x=0.2909^2,
h2.y=0.2524^2,
print=T)## lavaan 0.6-6 ended normally after 70 iterations
##
## Estimator GLS
## Optimization method NLMINB
## Number of free parameters 12
##
## Number of observations 3663
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## GG1 =~
## G1 (lg1) 1.000 0.059 16.903 0.000
## GG2 =~
## G2 (lg2) 1.000 0.068 14.809 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## Y ~
## X (bxy) -0.081 0.017 -4.820 0.000
## GG1 (bg1y) 0.044 0.018 2.404 0.016
## GG2 (bg2y) 0.254 0.004 67.390 0.000
## X ~
## GG1 (bg1x) 0.285 0.003 89.869 0.000
## GG2 (bg2x) -0.031 0.017 -1.805 0.071
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## GG1 ~~
## GG2 (bg12) -0.186 0.022 -8.408 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## GG1 1.000
## GG2 1.000
## .Y (vy) 0.930 0.023 39.974 0.000
## .X (vx) 0.914 0.023 39.407 0.000
## .G1 (vg1) 0.000 0.111 0.000 1.000
## .G2 (vg2) 0.000 0.129 0.000 1.000
##
## Defined Parameters:
## Estimate Std.Err z-value P(>|z|)
## conf -0.009 0.006 -1.343 0.179
## total -0.089 0.016 -5.431 0.000
##
## Constraints:
## |Slack|
## bg1x+bg1g2*bg2x - (sqrt(0.08462281)) 0.000
## bg2y+(bg2x+bg1g2*b1)*+12*1-((0.06370576)) 0.000
## est se z pvalue ci.lower ci.upper
## Adjusted Bxy -0.081 0.017 -4.820 1.4336e-06 -0.114 -0.048
## Genetic confounding -0.009 0.006 -1.343 0.17943 -0.021 0.004
## Total effect -0.089 0.016 -5.431 5.5933e-08 -0.122 -0.057
Similar to the one polygenic score case, the model is specified with paths between variables X and Y and polygenic scores for X (g1) and Y (g2). In some cases, parameters may take unlikely values. For example, the cross path bg1y flips from a negative to a positive value, which would imply that a higher polygenic score for education is linked to higher BMI. In such cases, constraints can be imposed on the the model in the following way.
gsensXY(rxy=-0.0894,
rg1x=0.2909,
rg2x=-0.0839,
rg1y=-0.0269,
rg2y=0.2524,
rg1g2=-0.1856,
n=3663,
h2.x=0.2909^2,
h2.y=0.2524^2,
print=T,
constrain='lg1 < 1 \n lg2 < 1 \n vg1 > 0 \n vg2 > 0 \n bg1y < 0 \n bg2x < 0 \n bxy < 0')## lavaan 0.6-6 ended normally after 183 iterations
##
## Estimator GLS
## Optimization method NLMINB
## Number of free parameters 12
## Number of inequality constraints 7
##
## Number of observations 3663
##
## Model Test User Model:
##
## Test statistic 6.665
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## GG1 =~
## G1 (lg1) 0.998 0.011 87.425 0.000
## GG2 =~
## G2 (lg2) 1.000 NA
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## Y ~
## X (bxy) -0.069 0.016 -4.297 0.000
## GG1 (bg1y) -0.000 NA
## GG2 (bg2y) 0.247 0.002 141.325 0.000
## X ~
## GG1 (bg1x) 0.285 0.003 94.417 0.000
## GG2 (bg2x) -0.031 0.016 -1.921 0.055
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## GG1 ~~
## GG2 (bg12) -0.186 0.016 -11.753 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## GG1 1.000
## GG2 1.000
## .Y (vy) 0.928 0.022 42.712 0.000
## .X (vx) 0.914 0.021 42.778 0.000
## .G1 (vg1) 0.000 NA
## .G2 (vg2) 0.000
##
## Defined Parameters:
## Estimate Std.Err z-value P(>|z|)
## conf -0.021 0.004 -5.328 0.000
## total -0.089 0.016 -5.458 0.000
##
## Constraints:
## |Slack|
## bg1x+bg1g2*bg2x - (sqrt(0.08462281)) 0.000
## bg2y+(bg2x+bg1g2*b1)*+12*1-((0.06370576)) 0.000
## 1 - (lg1) 0.002
## 1 - (lg2) 0.000
## vg1 - 0 0.000
## vg2 - 0 0.000
## 0 - (bg1y) 0.000
## 0 - (bg2x) 0.031
## 0 - (bxy) 0.069
## est se z pvalue ci.lower ci.upper
## Adjusted Bxy -0.069 0.016 -4.297 1.7339e-05 -0.100 -0.037
## Genetic confounding -0.021 0.004 -5.328 9.9505e-08 -0.028 -0.013
## Total effect -0.089 0.016 -5.458 4.8156e-08 -0.122 -0.057
Loadings for the latent part of the model are constrained to be less than 1 (lg1 < 1 & lg2 < 1) and residual variances to be positive (vg1 > 0 & vg2). Cross paths and the residual association are constrained to be negative, ie not flip sign (e.g. bg1y < 0).
Note that constraints need to be imposed cautiously on the model for two reasons:
-
To avoid nonsensical constraints. In another example, where the polygenic scores are expected to be positively associated with both X and Y keeping the < 0 constraint on the cross path above would be nonsensical. Instead > 0 should be used to keep the cross path positive.
-
Constraining parameters can decrease standard error even when the value of the parameter is not changed. That is, if lg1 is estimated to 1, constraining it to 1 will not change the estimates but will prevent the model from estimating a standard error for lg1 and can reduce standard errors of other parameters including the estimates of interest.
All model parameters should be systematically checked after fitting a model, and before and after constraints are imposed. In addition to implausible parameters, so-called 'heywood cases' should be checked. Here, as the model is standardized, no parameter should be above 1 or below -1. For example, a heritability parameter above 1 would correspond to a heritability above 100%.
Findings for the constrained vs the unconstrained model diverge. We can use gsensY to adjust for the outcome-related polygenic score only, here BMI. Note that in theory, if the polygenic score for the outcome was perfect, adjusting for that polygenic score would be sufficient as it would capture shared genetic influences between X and Y. Findings from adjusting for the BMI polygenic score only converge with the constrained version of the two polygenic score approach.
gsensY(rxy=-0.0894,
rgx = -0.0839,
rgy = 0.2524,
n=3663,
h2=0.2524^2)## est se z pvalue ci.lower ci.upper
## Adjusted Bxy -0.069 0.016 -4.233 2.3022e-05 -0.101 -0.037
## Genetic confounding -0.021 0.004 -5.063 4.1229e-07 -0.029 -0.013
## Total effect -0.089 0.016 -5.429 5.6544e-08 -0.122 -0.057
gsensXY(rxy=-0.0894,
rg1x=0.2909,
rg2x=-0.0839,
rg1y=-0.0269,
rg2y=0.2524,
rg1g2=-0.1856,
n=3785,
h2.x=0.25*0.31,
h2.y=0.186,
constrain='lg1 < 1 \n lg2 < 1 \n vg1 > 0 \n vg2 > 0 \n bg1y < 0 \n bg2x < 0 \n bxy < 0',
print=T)## Warning in lav_object_post_check(object): lavaan WARNING: some estimated ov
## variances are negative
## lavaan 0.6-6 ended normally after 163 iterations
##
## Estimator GLS
## Optimization method NLMINB
## Number of free parameters 12
## Number of inequality constraints 7
##
## Number of observations 3785
##
## Model Test User Model:
##
## Test statistic 41.768
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## GG1 =~
## G1 (lg1) 0.989 0.011 87.840 0.000
## GG2 =~
## G2 (lg2) 0.681 0.036 19.002 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## Y ~
## X (bxy) -0.043 0.017 -2.464 0.014
## GG1 (bg1y) 0.000
## GG2 (bg2y) 0.426 0.002 213.726 0.000
## X ~
## GG1 (bg1x) 0.264 0.006 46.959 0.000
## GG2 (bg2x) -0.063 0.024 -2.686 0.007
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## GG1 ~~
## GG2 (bg12) -0.224 0.023 -9.634 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## GG1 1.000
## GG2 1.000
## .Y (vy) 0.826 0.023 36.388 0.000
## .X (vx) 0.911 0.021 43.139 0.000
## .G1 (vg1) -0.000
## .G2 (vg2) 0.537 0.046 11.780 0.000
##
## Defined Parameters:
## Estimate Std.Err z-value P(>|z|)
## conf -0.052 0.010 -5.193 0.000
## total -0.095 0.016 -5.832 0.000
##
## Constraints:
## |Slack|
## bg1x+bg1g2*bg2x - (sqrt(0.0775)) 0.000
## bg2y+(bg2x+bg1g2*bg1x)*bxy+12*1-((0.186)) 0.000
## 1 - (lg1) 0.011
## 1 - (lg2) 0.319
## vg1 - 0 0.000
## vg2 - 0 0.537
## 0 - (bg1y) 0.000
## 0 - (bg2x) 0.063
## 0 - (bxy) 0.043
## est se z pvalue ci.lower ci.upper
## Adjusted Bxy -0.043 0.017 -2.464 0.013755 -0.077 -0.009
## Genetic confounding -0.052 0.010 -5.193 2.0673e-07 -0.072 -0.033
## Total effect -0.095 0.016 -5.832 5.4841e-09 -0.127 -0.063
Heritability scenarios of interest can be modelled with two polygenic scores by replacing h2.x and h2.y by the chosen values, here SNP-heritability estimates. Not that in h2.x=0.25x0.31, the factor 0.25 corresponds to the genetic relatedness between child and mother (i.e. sqrt(h2.x) is computed in the model leading to the value of the path equal to 0.5xsqrt(0.31).)
Note that in the model above corresponding to SNP-heritability, the genetic correlation between the latent genetic factor is estimated based on the observed polygenic scores and the heritability estimates to be -0.224. We have an external estimate of -0.279 based on LD score regression (See Table 1). We can use a fixed solution to use this value instead of the estimated value:
gsensXY(rxy=-0.0894,
rg1x=sqrt(0.25*0.31),
rg2y=sqrt(0.186),
rg1y=sqrt(0.25*0.31)*-0.0269/0.2909,
rg2x=sqrt(0.186)*-0.0839/0.2524,
rg1g2=-0.279,
n=3663,
h2.x=0.25*0.31,
h2.y=0.186,
print=T)## Warning in lav_object_post_check(object): lavaan WARNING: some estimated ov
## variances are negative
## lavaan 0.6-6 ended normally after 87 iterations
##
## Estimator GLS
## Optimization method NLMINB
## Number of free parameters 12
##
## Number of observations 3663
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## GG1 =~
## G1 (lg1) 1.000 0.062 16.229 0.000
## GG2 =~
## G2 (lg2) 1.000 0.042 23.965 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## Y ~
## X (bxy) -0.057 0.016 -3.518 0.000
## GG1 (bg1y) 0.117 0.021 5.707 0.000
## GG2 (bg2y) 0.456 0.007 61.612 0.000
## X ~
## GG1 (bg1x) 0.259 0.005 50.452 0.000
## GG2 (bg2x) -0.071 0.019 -3.769 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## GG1 ~~
## GG2 (bg12) -0.279 0.024 -11.759 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## GG1 1.000
## GG2 1.000
## .Y (vy) 0.801 0.024 33.160 0.000
## .X (vx) 0.918 0.023 40.218 0.000
## .G1 (vg1) -0.000 0.116 -0.000 1.000
## .G2 (vg2) 0.000 0.073 0.000 1.000
##
## Defined Parameters:
## Estimate Std.Err z-value P(>|z|)
## conf -0.033 0.009 -3.615 0.000
## total -0.089 0.016 -5.426 0.000
##
## Constraints:
## |Slack|
## bg1x+bg1g2*bg2x - (sqrt(0.0775)) 0.000
## bg2y+(bg2x+bg1g2*bg1x)*bxy+12*1-((0.186)) 0.000
## est se z pvalue ci.lower ci.upper
## Adjusted Bxy -0.057 0.016 -3.518 0.0004345 -0.088 -0.025
## Genetic confounding -0.033 0.009 -3.615 0.00030046 -0.050 -0.015
## Total effect -0.089 0.016 -5.426 5.7666e-08 -0.122 -0.057
Note that rg1y and rg2x are computed by multiplying the heritabilities under the chosen scenario by a ratio comparable to k in the one polygenic score case.