# A Versatile and Efficient Novel Approach for Mendelian Randomization Analysis with Application to Assess the Causal Effect of Fetal Hemoglobin on Anemia in Sickle Cell Anemia

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## Abstract

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## 1. Introduction

## 2. Method

#### 2.1. Model

#### 2.1.1. MREPS for Samples from the Extreme Tails of Outcome Distribution

^{Y}throughout the paper.

#### 2.1.2. MREPS for Samples from the Extreme Tails of Risk Factor Distribution

^{X}, let the n individuals whose risk factors fall into either $({x}_{0},{x}_{1})$ or $({x}_{2},{x}_{3})$, with ${x}_{1}\ne {x}_{2}$, are chosen as the EPS samples without imposing any restriction on the outcome values by choosing ${y}_{0}=-\infty ,{y}_{3}=\infty ,$ and ${y}_{1}={y}_{2}$. Note that under this setting, $\mathcal{X}=({x}_{0},{x}_{1})\cup ({x}_{2},{x}_{3})$, and ${S}_{i}$ reduces to ${S}_{i}=I({X}_{i}\in \mathcal{X})$ as $\mathcal{Y}=(-\infty ,\infty ).$

#### 2.1.3. MREPS for Random Samples

^{X}and EPS

^{Y}. With this adaptive capability, MREPS turns out to be a versatile, universal approach for MR, regardless of if the samples are random or EPS.

#### 2.2. Estimation and Hypothesis Testing

#### 2.2.1. Samples Drawn from the Extreme Tails of Outcome

#### 2.2.2. Samples Drawn from the Extreme Tails of Risk Factor

#### 2.2.3. Random Samples

## 3. Simulations

^{Y}and EPS

^{X}. For the EPS designs, the n subjects were drawn from the upper and lower q quantile from the distribution of primary trait values ${Y}_{i}$ for EPS

^{Y}and risk factor values ${X}_{i}$ for EPS

^{X}. For EPS

^{Y}, the support $\mathcal{X}$ was set to be $(-\infty ,\infty )$ and for EPS

^{X}, the support $\mathcal{Y}$ was set to be $(-\infty ,\infty )$. For these two designs, n satisfies the relation $n=2qN$, where N is the original cohort size. This selection sets the cutoffs ${x}_{1},{x}_{2}$ and ${y}_{1},{y}_{2}$ under the the corresponding EPS

^{X}and EPS

^{Y}, accordingly. For EPS

^{X}, the values ${x}_{0},{x}_{3}$ were chosen to be -∞ and ∞, respectively. The cutoffs ${y}_{0},{y}_{3}$ were, respectively, chosen to be -∞ and ∞ for EPS

^{Y}. We first considered $q=0.01$ for assessing the different effect sizes of ${\beta}_{2}$. We then considered a set of q ranging from 0.005 to 0.045 with increment of 0.01 for assessing the effect of different n given $N=20,000$ on the performance of MREPS and 2SLS.

- (a)
- Equal-sized combined effects of 9 and 25 independent SNPs: two separate cases were considered by using different numbers, 9 and 25, of SNPs. The genotype data of each SNP was independently generated under HWE as mentioned above. Then, the equal sized combined effect was considered by using ${\gamma}_{2}{G}_{i}={\gamma}_{2}{\sum}_{j=1}^{9}{G}_{ij}$, where ${\gamma}_{2}=0.17$ and ${\sum}_{j=1}^{9}{G}_{ij}$ is the summation of the 9 independent SNP genotype values for the ith individual. Similarly, for the 25 SNPs, ${\gamma}_{2}{G}_{i}={\gamma}_{2}{\sum}_{j=1}^{25}{G}_{ij}$, where ${\gamma}_{2}=0.07$ and ${\sum}_{j=1}^{25}{G}_{ij}$ is the summation of the 25 independent SNP genotype values for the ith individual. This case represents the utilization of the summary of multiple SNPs with equal-sized association with the risk factor as an IV.
- (b)
- Different-sized combined effects of 9 and 25 SNPs: with this setting, ${\gamma}_{2}{G}_{i}$ turned out to be ${\sum}_{j=1}^{J}{\gamma}_{{2}_{j}}{G}_{ij}$ for $J=9,25$. Each independently generated SNP under HWE was multiplied by randomly generated ${\gamma}_{{2}_{j}}$ from $N(0.18,0.018)$ for 9 SNPs and from $N(0.06,0.018)$ for 25 SNPs separately. Then, the summation of these products considered as ${\gamma}_{2}{G}_{i}$ under 9 and 25 SNPs separately for each i. This scenario represents the utilization of the summary of multiple SNPs with different-sized association with the risk factor as an IV.
- (c)
- A few large and many small effects of 9 and 25 SNPs: two SNPs generated under H-WE were multiplied by 0.46 and the rest SNPs multiplied by 0.092 separately for 9 and 25 SNPs. Next, for each i, the summations of these products separately considered as ${\gamma}_{2}{G}_{i}$ for 9 and 25 SNPs. This represents the utilization of the summary of multiple SNPs with very few large associations and a large number of week associations with the risk factor as an IV.
- (d)
- A combination of valid and invalid SNPs as an IV, 9 and 25 SNPs: Among 9 SNPs g-enerated under HWE, 4 SNPs were multiplied by 0.37 while multiplying the rest by 0, and the summation of the resulting values was considered as ${\gamma}_{2}{G}_{i}$ for the ith individual. Similarly, among 25 SNPs, 12 SNPs were multiplied by 0.14 while multiplying the rest by 0, and the resulting summation was considered as ${\gamma}_{2}{G}_{i}$ for the ith individual. This case represents the utilization of the summary of few valid and many invalid SNPs as an IV.

#### 3.1. Comparison: Bias, Rejection Proportion, Standard Error, and Coverage Percentage

^{X}and EPS

^{Y}. Under the random sampling, both MREPS and 2SLS produced almost the same results. The ${\beta}_{2}$ vs. ${\hat{\beta}}_{2}$ curves overlapped with the true ${\beta}_{2}$ vs. ${\beta}_{2}$ curve. The rejection proportion plots of MREPS and 2SLS overlapped with each other, becoming comparable approaches under the random sampling, by producing almost same unbiased estimates and the rejection proportions. However, under EPS

^{Y}, 2SLS became a highly biased approach, and the bias gradually increased as ${\beta}_{2}$ increases. MREPS still overlapped with the true curve, becoming an unbiased estimator. Although the rejection proportion curves overlap with each other, the estimate of the true causal effect by 2SLS was uninterpretable due to highly biased estimator. Under EPS

^{X}, MREPS was still the unbiased estimator, overlapping with the true curve. 2SLS consistently maintained the bias by consistently overestimating the true ${\beta}_{2}$. 2SLS produced unreliable rejection proportion, which is almost always 1, even when ${\beta}_{2}=0$. Therefore, under EPS

^{X}and EPS

^{Y}, we cannot use 2SLS for MR analysis because of biased and uninterpretable parameter estimates and uncontrolled type I error rate.

**(a)**–

**(d)**, we accessed MREPS by comparison with 2SLS quantitatively in terms of average bias, average SE, bias percentages, rejection proportion of the null hypothesis and the coverage percentage of the 95% CI under 9 and 25 SNPs, separately. We kept the true values fixed at the same vectors above and considered the values $\{0,0.2,0.4\}$ for ${\beta}_{2}$. Table 1 summarizes the results under EPS

^{Y}. The corresponding results for EPS

^{X}and random sampling are given in Supplemental Table S1. From Table 1, we observed that MREPS controlled the bias percentages well below 10% for all none zero ${\beta}_{2}$ settings, while 2SLS failed to control the bias percentages. The resulting bias percentages were above 100% for all nonzero ${\beta}_{2}$ settings, becoming a highly biased approach. MREPS produced the smallest average SEs in all settings. The coverage percentages of 95% CIs of MREPS were all close to 95 for all settings. However, the coverage percentages of 2SLS CI became close to 95 only when ${\beta}_{2}=0$. For all nonzero ${\beta}_{2}$ coverage percentages decreased significantly as ${\beta}_{2}$ increased under each setting. The only acceptable statistics under 2SLS was the rejection rate of the null hypothesis. However, reliability of these rejection rates was questionable due to the highly biased estimates and the relatively large SEs. From Supplement Table S1, under the random sampling, comparable results were observed from both 2SLS and MREPS. However, for EPS

^{X}, the worst results were observed from 2SLS, while observing overall acceptable better results from MREPS. High bias percentages, poor coverage percentages (below 2.2 for all cases) and unreliable (almost 100% rejection rates for the null hypothesis, regardless of the change of the magnitude of the effect size) were observed from 2SLS. These overall observations are consistent with the observations displayed in Figure 1.

^{X}and EPS

^{Y}, becoming a versatile approach that can be utilized under the random as well as EPS samples in MR analysis.

#### 3.2. Comparison: The Optimum Cost-Effective Subsample Size

^{X}, EPS

^{Y}or random subsample drawing. To explore this capability further numerically, we carried out simulations to explore the optimal subsample size n that required to chosen under EPS

^{X}, EPS

^{Y}and random sampling. We considered a cohort of size N = 20,000. For parameter ${\beta}_{2}$, the values 0.1 annd 0.4 were considered. For subsample size n the sequence of values from 200 to 2000 were considered with the increment of 200, which corresponds to quantile, q, ranging from 0.005 to 0.045 with the increment of 0.01 under EPS

^{X}and EPS

^{Y}. For IV, a single SNP was considered.

^{X}and EPS

^{Y}. Under the considered simulation setting, for the random subsample, $n=600$ is an ideal sample size, and there is not much gain for other large sample sizes, indicating the consistency property is achieved. Both 2SLS and MREPS procedures produced roughly the same estimators. For both EPS

^{X}and EPS

^{Y}, the ideal optimal sample size can be as small as 200 (100 in both tails) under MREPS, and there is not any substantial gain for the larger sample sizes. However, regardless of the increment of the sample size n, 2SLS consistently produced nearly the same bias, indicating that 2SLS fails to recover the tradeoff loss due to the subsample selection under EPS

^{X}and EPS

^{Y}.

## 4. Unravelling the Effect of HbF on Anemia in Sickle Cell Anemia

^{HbF}) as the summation of the number of high HbF alleles across these 11 SNPs, which is opposite to the PGS

^{HbF}defined by [22] that equals to the summation of the number of low HbF alleles. The first five principal components (PCs) were calculated in the original cohort [22]. The covariates we used here were the same as those used in [22], but we further added blood transfusion history as an additional covariate because it was associated with Hb and HbF [48].

^{HbF}and each individual SNP identified in the discovery cohort of SCCRIP/BCM, we analyzed the 724 patients in SIT Trial. The cohort description and GWAS data have been reported by [23]. The missing genotypes for this study were imputed using the Michigan Imputation Server for chromosomes 2, 6, and 11 using the 1000 genomes Phase 3 data as African reference population samples [49]. After imputation, one SNP ($rs66650371$) from $HBS1L-MYB$ remained unavailable. The dose data for the remaining ten SNPs were extracted and analyzed as continuous variables. PGS

^{HbF}was calculated as the unweighted sum of the dosage for the high HbF from ten SNPs using dosage data and was analyzed as a continuous variable. The first five principal components (PCs) were calculated in the original cohort [23] and included in the model to control for potential population stratification. The covariates for which we adjusted here are age, sex, diagnosis (HbSS or S${\beta}_{0}$), and 5 PCs.

^{Y}and EPS

^{X}, we then combined the two cohorts to increase the cohort size and mimicked the two EPS designs. For choosing the two extreme tail regions for EPS sample selection under EPS

^{Y}, we carried out a pilot simulation by using the simulation setting of Figure 2 as a foundation. The size of the combined cohort of SIT and SCCRIP is 1309, which was considered as the size of the original cohort. The percentiles 10%, 20%, 30%, 40%, and 50% were chosen as the candidate cutoff values for the extreme regions. The corresponding sample sizes determined by these cutoffs are 261, 523, 785, 1047, and 1309, respectively. Note that 50% percentile represents the entire cohort of size 1309. The parameters ${\beta}_{2}$ and ${\gamma}_{2}$ were set at 0.11 and 1.4 based on the literature [50] and this study. MAF was set as 0.3. The corresponding results are given by Figure S3 and Table S3 of Supplementary Materials. Based on these results, the 20% percentile for the extreme tail sample selection gave a simulated 80% power to detect a causal effect of HbF on Hb in this study at a significance level of 0.05. Without financial budget information, we chose 20% percentile only based on the power. In the real practice, the extreme percentile might be determined based on the available resources and the statistical power. We analyzed 492 participants with Hb in the lowest and highest 20% percentile, which was used to mimic EPS

^{Y}. We also analyzed 512 participants with HbF in the lowest and highest 20% percentile, which was used to mimic EPS

^{X}. For comparison, we also randomly sampled 518 individuals for these MR analyses. Because in SIT, PGS

^{HbF}was defined based on 10 SNPs, we then removed SNP $rs6650371$ from PGS

^{HbF}in SCCRIP/BCM. In the combined data analysis, we adjusted for the common covariates described above with additional covariate cohort (SCCRIP/BCM or SIT). These combined cohort and mimicked EPS analyses were only used to demonstrate the performance of the proposed new MREPS approach.

^{HbF}was not associated with any potential confounding variables, including age, sex, blood transfusion history, site, SS/S${\beta}_{0}$, and 5 PCs based on the Spearman correlation analysis and the two-sample Wilcoxon rank sum test. In SCCRIP/BCM, the Sargan test provided evidence of pleiotropic effects in HbF ($p=0.0005$), which was not supported by the MR Egger intercept test ($p=0.36$). In contrast, in SIT, both Sargan test ($p=0.15$) and MR Egger intercept test ($p=0.72$) did not provide any evidence of pleiotropic effects in HbF.

^{HbF}was also associated with HbF (est = 1.64, SE = 0.1, $p<0.0001$) and Hb (est = 0.17, SE = 0.02, $p<0.0001$) (Supplementary Figure S1). After including both HbF and PGS

^{HbF}in the same model, PGS

^{HbF}was not independently significantly associated with Hb (est = 0.04, SE = 0.03, $p=0.13$), although HbF was still independently significantly associated with Hb (est = 0.08, SE = 0.008, $p<0.0001$). This implies that PGS

^{HbF}does not have a direct effect on Hb except its effect on Hb via HbF, which supports the third MR analysis assumption [53].

^{HbF}as the IV showed that there was a significant relationship between HbF and Hb (est = 0.10, SE = 0.013, p = 1 $\times {10}^{-16}$). All 10 SNPs were significantly associated with HbF. Using each of 10 individual SNP as an IV in the model, there was a significant relationship between HbF and Hb (min p = 6.7 $\times {10}^{-10}$ for $BCL11A$ SNP $rs1427407$ and max p$=0.029$ for $HBB$ SNP $rs28440105$) at a level of 0.05 (Table 2, Supplemental Table S2). The strongest effect for testing the causal effect of HbF on Hb induced by the $BCL11A$ SNP $rs1427407$ is because this SNP explains more than 20% of the variance of HbF, which is almost 2/3 of the variance explained by PGS

^{HbF}that includes the contributions from all 11 SNPs from three genes [22,54]. In contrast, 2SLS only rejected PGS

^{HbF}and 9 out of 10 analyzed SNPs (90%). p-values for each test obtained by 2SLS were larger than those obtained by MREPS. Similarly, even though Benjamini–Hochberg approach [55] was used to control false discovery rate (FDR) to adjust for multiple comparisons, MREPS still rejected all 11 tests (PGS

^{HbF}+ 10 SNPs) at FDR controlled at 0.05. However, 2SLS only rejected 9 (82%) of them: $rs968857$ with a raw p value of 0.04 was not significant after multiple comparison correction. This was consistent with our previous simulation result.

^{HbF}(see Supplementary Figure S1). There were no relations between PGS

^{HbF}and any potential confounding factor, including age, sex, SS/S${\beta}_{0}$, and 5 PCs based on the Spearman correlation analysis and the two-sample Wilcoxon rank sum test. When we included both HbF and PGS

^{HbF}in the same model, both PGS

^{HbF}and HbF were significantly associated with Hb although the variance in Hb explained by HbF was 3.5 times higher than that by PGS

^{HbF}(14% vs. 4%).

^{HbF}or 6 SNPs as the IV (Figure 3, Table 2). In contrast, 2SLS only showed a significant relationship between HbF and Hb when PGS

^{HbF}and 5 SNPs used as IVs at a level of 0.05 (Table 2, Supplemental Table S2). With FDR controlled at 0.05, the same conclusions held. Using SIT cohort, we replicated the finding that there is a causal effect of HbF on Hb predicted from genetic variants from these three genes. This is the first time that the causal effect of HbF on Hb was shown in SCA patients using the MR approach.

^{HbF}or 8 SNPs (90%) as the IV (Figure 4, Table 2, Supplemental Table S2). However, 2SLS missed two SNPs. The analysis results under the mimicked EPS and random sampling design are displayed in Figure 4, Supplemental Table S2 and Figure S2. Using the samples selected under EPS

^{Y}(Figure 4), MREPS identified 8 out of 10 tests (80%), while 2SLS identified 6 (60%) at a level of 0.05. Similarly, using the samples selected under EPS

^{X}(Supplemental Figure S2), both MREPS and 2SLS only identified the relationship via IVs of PGS

^{HbF}and two $BCL11A$ SNPs ($rs1427407$ and $rs7606173$) at a level of 0.05. In contrast, using the randomly selected samples (Supplemental Figure S2), MREPS identified 7 out of 10 (70%) tests with correct effect direction at a level of 0.05, but 2SLS only identified 6 (60%). With FDR controlled at 0.05, the same conclusions held.

## 5. Discussion

^{HbF}or some individual SNPs protects patients from anemia using the observation SCCRIP/BCM cohort data, which was validated in another independent SIT study. This is the first study in the literature to show the causal effect of HbF on Hb in SCA patients using MR analysis of observation data. In the data analyses using the combined SCCRIP/BCM and the SIT data, the corresponding EPS

^{Y}, EPS

^{X}and randomly sampling from the combined cohort demonstrated the same conclusions as those obtained from the simulation studies. An important implication of the mimicked EPS

^{Y}, EPS

^{X}, and random sampling compared to whole cohort analysis is that only a 40% of samples was used from the large combined cohort for these designs to get close conclusions to the conclusions derived from the combined large cohort. A 60% percent of the data from the large cohort was not used, which means that about 60% of associated cost, time, and resources under the utilization of EPS was saved compared to collecting samples/data on the whole samples, no matter EPS design is for prospective or retrospective studies.

^{HbF}as an IV, allows us to examine whether there is potential causal effect of HbF on these SCA-related clinical complications using observation data. Our work demonstrating the causal effect of HbF on anemia provides a theoretical foundation for assessing the causal effect of HbF on the other clinical complications listed above because of the strong biological evidence showing the causal effect of three genes of $BCL11A,HBS1L-MYB$ and the extended $\beta $-globin locus on HbF. We will investigate these research topics in the future.

## Supplementary Materials

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Code Availability

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MR | Mendalian randomization |

2SLS | Two-stage least-squares |

GWAS | Genome-wide association study |

EPS | Extreme phenotype sequencing |

MREPS | MR analysis under extreme or random phenotype sampling |

IV | Instrumental variables |

NGS | Next generation sequencing |

WBC | White blood cell count |

CI | Confidence intervals |

HbF | Total fetal hemoglobin |

Hb | Hemoglobin |

SCA | Sickle cell anemia |

SCCRIP | Sickle cell clinical research intervention program |

BCM | Baylor college of medicine |

SIT | Silent Cerebral Infarct Transfusion |

LIML | Limited information maximum likelihood |

MLE | Maximum likelihood estimator |

SE | Standard errors |

${\mathrm{EPS}}^{\mathrm{X}}$ | EPS design constructed upon the risk factor |

${\mathrm{EPS}}^{\mathrm{Y}}$ | EPS design constructed upon the outcome |

HWE | Hardy–Weinberg equilibrium |

SNP | Single nucleotide polymorphisms |

HbS | Sickle hemoglobin |

FDR | False discovery rate |

CP | Coverage percentage |

${\mathrm{PGS}}^{\mathrm{HbF}}$ | Polygenic Score of HbF |

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**Figure 1.**(

**Top**) panel: comparison of estimators obtained via MREPS and 2SLS under the random (

**left**), EPS

^{Y}(

**middle**) and EPS

^{X}(

**right**) designs. (

**Bottom**) panel: rejection proportion of ${H}_{0}$ comparison for MREPS and 2SLS under the random (

**left**), EPS

^{Y}(

**middle**) and EPS

^{X}(

**right**) designs.

**Figure 2.**(

**Left**): the behavior of the estimators due to the change of sample size, obtained through MREPS and 2SLS under the random design. (

**Middle**and

**right**): the behavior of the estimators, due to the change of sample size or the proportion of the extreme tails, obtained through MREPS and 2SLS under the EPS

^{Y}and EPS

^{X}, respectively.

**Figure 3.**HbF induced by $BCL11A$, $HBS1L-MYB$, and the extended $\beta $-globin locus causally influences Hb in SCCRIP/BCM (

**left**) and SIT (

**right**) patients. (

**Top**) panels: X-axis: Genetic association with HbF (%). Y-axis: Genetic association with Hb (g/dL). The genetic associations were done using linear regression models, adjusting for the covariates age, sex, sickle diagnosis (Yes/No), and top 5 PCs. (

**Bottom**) panels: X-axis: Genetic association with HbF (%); Y-axis: (Causal) Effect of HbF on Hb (g/dL/%). The Mendelian randomization analysis was done using MREPS. SCCRIP/BMC = Sickle Cell Clinical Research Intervention Program study and Baylor College of Medicine; SIT = Silent Cerebral Infarct Transfusion trial.

**Figure 4.**Top left and right panels: comparison between the effects of PGS

^{HbF}on HbF and Hb and correlation between HbF and Hb in SCCRIP/BCM and SIT combined cohort and EPS

^{Y}derived from SCCRIP/BCM and SIT combined cohort, respectively. The p value of the top right plot was calculated using STEPS [16], and the p values of the other plots were calculated using linear regression model, adjusting for covariates of cubic-spline–defined age, sex, sickle diagnosis (Yes/No), and top 5 PCs. Bottom panel: HbF induced by $BCL11A$ and $HBS1L-MYB$ causally influences Hb in SCCRIP/BCM and SIT combined (

**left**) and EPS

^{Y}(

**right**) patients. X-axis: Genetic association with HbF (%). Y-axis: (Causal) Effect of HbF on Hb (g/dL/%). SCCRIP/BMC = Sickle Cell Clinical Research Intervention Program study and Baylor College of Medicine; SIT = Silent Cerebral Infarct Transfusion trial.

**Table 1.**Average estimators, average standard errors, bias percentages, rejection proportions, and coverage percentages obtained through MREPS and 2SLS under EPS

^{Y}design.

Mean ${\hat{\mathit{\beta}}}_{2}$ | Mean SE | Bias % | Rejection Proportion | CP | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Number of SNPs | ${\mathit{\beta}}_{2}$ | MREPS | 2SLS | MREPS | 2SLS | MREPS | 2SLS | MREPS | 2SLS | MREPS | 2SLS | |||||

(a) Equal-sized IV—risk factor associations | ||||||||||||||||

9 | 0 | −0.004 | −0.02 | 0.07 | 0.3 | - | - | 0.06 | 0.05 | 94.6 | 95.1 | |||||

0.2 | 0.21 | 0.54 | 0.08 | 0.17 | −3.19 | −169.66 | 0.77 | 0.86 | 93.5 | 42.1 | ||||||

0.4 | 0.41 | 0.82 | 0.09 | 0.1 | −2.02 | −105.3 | 0.99 | 1.0 | 94.5 | 5.9 | ||||||

25 | 0 | −0.01 | −0.02 | 0.11 | 1.13 | - | - | 0.05 | 0.05 | 95.3 | 94.7 | |||||

0.2 | 0.19 | 0.51 | 0.12 | 0.28 | 4.84 | −155.91 | 0.48 | 0.64 | 95 | 62.7 | ||||||

0.4 | 0.41 | 0.82 | 0.13 | 0.16 | −2.6 | −104.03 | 0.89 | 0.97 | 94.6 | 25.7 | ||||||

(b) Different-sized IV—risk factor associations | ||||||||||||||||

9 | 0 | −0.005 | −0.04 | 0.07 | 0.28 | - | - | 0.04 | 0.03 | 96.2 | 96.8 | |||||

0.2 | 0.2 | 0.53 | 0.07 | 0.16 | −1.02 | −166.89 | 0.79 | 0.88 | 94.3 | 39.1 | ||||||

0.4 | 0.4 | 0.82 | 0.08 | 0.1 | −0.99 | −104.72 | 1.0 | 1.0 | 94.2 | 4.2 | ||||||

25 | 0 | −0.01 | −0.11 | 0.12 | 0.58 | - | - | 0.04 | 0.04 | 95.7 | 95.7 | |||||

0.2 | 0.18 | 0.51 | 0.12 | 0.28 | 8.28 | −153.52 | 0.46 | 0.61 | 95.4 | 61.4 | ||||||

0.4 | 0.41 | 0.81 | 0.13 | 0.17 | −1.96 | −102.81 | 0.86 | 0.96 | 94.8 | 27.8 | ||||||

(c) A combination of few large and many small IV - risk factor associations | ||||||||||||||||

9 | 0 | −0.0006 | −0.01 | 0.05 | 0.21 | - | - | 0.06 | 0.05 | 94.1 | 95.1 | |||||

0.2 | 0.2 | 0.54 | 0.06 | 0.12 | −1.45 | −170.85 | 0.94 | 0.97 | 93.2 | 20.1 | ||||||

0.4 | 0.41 | 0.82 | 0.06 | 0.07 | −1.34 | −105.3 | 1.0 | 1.0 | 93 | 0.5 | ||||||

25 | 0 | −0.0002 | −0.01 | 0.05 | 0.18 | - | - | 0.05 | 0.06 | 94.7 | 94.5 | |||||

0.2 | 0.2 | 0.54 | 0.05 | 0.1 | −1.15 | −171.2 | 0.97 | 0.99 | 93.3 | 12.4 | ||||||

0.4 | 0.4 | 0.82 | 0.06 | 0.06 | −1.1 | −104.93 | 1.0 | 1.0 | 93.1 | 0.1 | ||||||

(d) A combination of valid and invalid IV - risk factor associations | ||||||||||||||||

9 | 0 | −0.002 | −0.02 | 0.05 | 0.19 | - | - | 0.06 | 0.05 | 93.9 | 95.2 | |||||

0.2 | 0.2 | 0.54 | 0.05 | 0.11 | −1.42 | −170.38 | 0.95 | 0.97 | 91.5 | 18.7 | ||||||

0.4 | 0.4 | 0.82 | 0.06 | 0.07 | −1.04 | −104.89 | 1.0 | 1.0 | 93 | 0.2 | ||||||

25 | 0 | −0.005 | −0.03 | 0.08 | 0.31 | - | - | 0.05 | 0.05 | 94.9 | 95.5 | |||||

0.2 | 0.21 | 0.54 | 0.08 | 0.17 | −4.16 | −170.31 | 0.75 | 0.84 | 93.4 | 43.8 | ||||||

0.4 | 0.41 | 0.82 | 0.09 | 0.11 | −2.91 | −106.12 | 0.99 | 1.0 | 93.9 | 6.7 |

**Table 2.**MR analyses to assessing the causal effect of HbF on Hb of patients from the two cohorts of SCCRIP/BCM and SIT. Instrumental variables are PGS

^{HbF}and $rs1427407$.

${\hat{\mathit{\beta}}}_{2}$ | SE (${\hat{\mathit{\beta}}}_{2}$) | p-Value (${\mathit{\beta}}_{2}$) | ${\hat{\mathit{\gamma}}}_{2}$ | SE (${\hat{\mathit{\gamma}}}_{2}$) | p-Value (${\mathit{\gamma}}_{2}$) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cohort/Design | IV | MREPS | 2SLS | MREPS | 2SLS | MREPS | 2SLS | MREPS | 2SLS | MREPS | 2SLS | MREPS | 2SLS |

SCCRIP/BCM | PGS^{HbF} | 0.1 | 0.1 | 0.013 | 0.013 | $1\times {10}^{-16}$ | $1\times {10}^{-14}$ | 1.6 | - | 0.083 | - | 0 | - |

$rs1427407$ | 0.096 | 0.095 | 0.016 | 0.016 | $7\times {10}^{-10}$ | $1\times {10}^{-8}$ | 4.5 | - | 0.3 | - | 0 | - | |

SIT | PGS^{HbF} | 0.15 | 0.15 | 0.014 | 0.023 | 0 | $3\times {10}^{-10}$ | 1.3 | - | 0.14 | - | 0 | - |

$rs1427407$ | 0.15 | 0.15 | 0.024 | 0.027 | $7\times {10}^{-10}$ | $1\times {10}^{-7}$ | 3.3 | - | 0.52 | - | $2\times {10}^{-10}$ | - | |

Combined | PGS^{HbF} | 0.11 | 0.11 | 0.01 | 0.01 | 0 | 0 | 1.4 | - | 0.076 | - | 0 | - |

$rs1427407$ | 0.11 | 0.11 | 0.011 | 0.012 | 0 | 0 | 4 | - | 0.27 | - | 0 | - | |

EPS^{Y} | PGS^{HbF} | 0.11 | 0.11 | 0.015 | 0.016 | $2\times {10}^{-13}$ | $3\times {10}^{-12}$ | 1.4 | - | 0.13 | - | 0 | - |

$rs1427407$ | 0.12 | 0.12 | 0.017 | 0.019 | $2\times {10}^{-12}$ | $1\times {10}^{-9}$ | 3.8 | - | 0.44 | - | 0 | - | |

EPS^{X} | PGS^{HbF} | 0.098 | 0.11 | 0.013 | 0.018 | $1\times {10}^{-14}$ | $5\times {10}^{-10}$ | 1.7 | - | 0.18 | - | 0 | - |

$rs1427407$ | 0.095 | 0.11 | 0.013 | 0.019 | $4\times {10}^{-13}$ | $3\times {10}^{-8}$ | 5.4 | - | 0.67 | - | $7\times {10}^{-16}$ | - |

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Liyanage, J.S.S.; Estepp, J.H.; Srivastava, K.; Rashkin, S.R.; Sheehan, V.A.; Hankins, J.S.; Takemoto, C.M.; Li, Y.; Cui, Y.; Mori, M.;
et al. A Versatile and Efficient Novel Approach for Mendelian Randomization Analysis with Application to Assess the Causal Effect of Fetal Hemoglobin on Anemia in Sickle Cell Anemia. *Mathematics* **2022**, *10*, 3743.
https://doi.org/10.3390/math10203743

**AMA Style**

Liyanage JSS, Estepp JH, Srivastava K, Rashkin SR, Sheehan VA, Hankins JS, Takemoto CM, Li Y, Cui Y, Mori M,
et al. A Versatile and Efficient Novel Approach for Mendelian Randomization Analysis with Application to Assess the Causal Effect of Fetal Hemoglobin on Anemia in Sickle Cell Anemia. *Mathematics*. 2022; 10(20):3743.
https://doi.org/10.3390/math10203743

**Chicago/Turabian Style**

Liyanage, Janaka S. S., Jeremie H. Estepp, Kumar Srivastava, Sara R. Rashkin, Vivien A. Sheehan, Jane S. Hankins, Clifford M. Takemoto, Yun Li, Yuehua Cui, Motomi Mori,
and et al. 2022. "A Versatile and Efficient Novel Approach for Mendelian Randomization Analysis with Application to Assess the Causal Effect of Fetal Hemoglobin on Anemia in Sickle Cell Anemia" *Mathematics* 10, no. 20: 3743.
https://doi.org/10.3390/math10203743