Volume 8 Supplement 1

## Genetic Analysis Workshop 18: Human sequence data in extended pedigrees

- Proceedings
- Open Access

# Practical investigation of the performance of robust logistic regression to predict the genetic risk of hypertension

- Miriam Kesselmeier
^{1, 4}, - Carine Legrand
^{1}, - Barbara Peil
^{1}, - Maria Kabisch
^{2}, - Christine Fischer
^{3}, - Ute Hamann
^{2}and - Justo Lorenzo Bermejo
^{1}Email author

**8 (Suppl 1)**:S65

https://doi.org/10.1186/1753-6561-8-S1-S65

© Kesselmeier et al.; licensee BioMed Central Ltd. 2014

**Published:**17 June 2014

## Abstract

Logistic regression is usually applied to investigate the association between inherited genetic variants and a binary disease phenotype. A limitation of standard methods used to estimate the parameters of logistic regression models is their strong dependence on a few observations deviating from the majority of the data.

We used data from the Genetic Analysis Workshop 18 to explore the possible benefit of robust logistic regression to estimate the genetic risk of hypertension. The comparison between standard and robust methods relied on the influence of departing hypertension profiles (outliers) on the estimated odds ratios, areas under the receiver operating characteristic curves, and clinical net benefit.

Our results confirmed that single outliers may substantially affect the estimated genotype relative risks. The ranking of variants by probability values was different in standard and in robust logistic regression. For cutoff probabilities between 0.2 and 0.6, the clinical net benefit estimated by leave-one-out cross-validation in the investigated sample was slightly larger under robust regression, but the overall area under the receiver operating characteristic curve was larger for standard logistic regression. The potential advantage of robust statistics in the context of genetic association studies should be investigated in future analyses based on real and simulated data.

## Keywords

- Robust Regression
- Genetic Analysis Workshop
- Standard Logistic Regression
- Cutoff Probability
- Robust Parameter Estimate

## Background

Hypertension is a common chronic medical condition characterized by elevated arterial blood pressure. High blood pressure is associated with an increased risk of stroke, heart attack, and other serious diseases. Age, gender, tobacco smoking, alcohol consumption, and high body mass index constitute established risk factors for hypertension [1]. A genetic component has also been postulated. It has been shown that individuals with a family history of hypertension have on average a higher blood pressure than individuals without a family history. Yanek et al found a 44% higher prevalence of hypertension in siblings of affected persons than in the general reference population [2]. In a Canadian study, standardized risk ratios of hypertension were higher for first-degree relatives than for spouses of probands with hypertension [3]. In genetic studies, a large number of polymorphisms has been associated with hypertension and validated in independent collectives; 14 loci have been identified (as of 2010) and many genetic studies are currently in progress [4–8].

The relationship between inherited genetic polymorphisms and a binary response variable (with/without hypertension) can be investigated using logistic regression models that simultaneously consider the effects of multiple risk factors. Standard methods used to estimate the parameters of logistic regression models--for example, iteratively reweighted least squares--are limited by their dependence on a few observations departing from the majority of the data. This contrasts with the purpose of genetic risk models that aim to predict a particular health outcome that holds for the bulk of individuals, and to identify persons with a deviating high risk of disease. We use data from the Genetic Analysis Workshop (GAW18) to explore the possible benefit of robust parameter estimates in logistic regression models for the genetic prediction of hypertension risk.

## Methods

The analysed data (real phenotypes) were derived from 142 unrelated individuals who participated in the San Antonio Family Heart or Family Diabetes/Gallbladder studies. Longitudinal information on hypertension, age, gender, and current tobacco smoking was measured up to 4 times per individual; the present analyses relied on the first available measurement. Further information is provided in several articles [9–12].

The original data was filtered according to the following criteria: (a) at least 1 measurement with complete information on hypertension and age, (b) monomorphisms were excluded and each polymorphism had to be represented by at least 2 individuals, (c) individuals with more than 5% missing genotypes were excluded, and, finally, (d) variants with missing data in any individual were removed.

^{2}tests. Covariates significantly associated at the 5% confidence level entered the intercept-only model to build the baseline model. Subsequently, standard logistic regression (iteratively reweighted least squares) was used to identify possible hypertension-associated single-nucleotide polymorphisms (SNPs) with minimal deviance, taking into account associated covariates. The deviance is defined as minus twice the logarithm of the likelihood. Genotypes were coded according to an additive penetrance model; that is, 0, 1, and 2. Departing observations (outliers) according to standard logistic regression were identified based on the Cook's distance in the baseline model. The Cook's distance for observation $\text{i}$ is defined as

where ${\widehat{\text{y}}}_{\text{j}}$ denotes the full regression model prediction for observation $\text{j}$, ${\widehat{\text{y}}}_{\text{j}\left(\text{i}\right)}$ represents the regression model prediction for observation $\text{j}$ estimated omitting observation $\text{i}$, and MSE indicates the mean square error of the regression model with $q$ explanatory variables.

where $\text{v}\left({\text{y}}_{\text{i}};{\mu}_{\text{i}}\right)=\frac{{\psi}_{\text{c}}\left({\u03f5}_{\text{i}}\right)}{{\text{V}}^{1/2}\left({\mu}_{\text{i}}\right)}$ with the Pearson residuals ${\u03f5}_{\text{i}}$ and the Huber function

${\psi}_{\text{c}}\left({\text{r}}_{\text{i}}\right)=\left\{\begin{array}{cc}\hfill {\text{r}}_{\text{i}}\hfill & \hfill \text{for}\phantom{\rule{2.77695pt}{0ex}}\left|{\text{r}}_{\text{i}}\right|\le \text{c}\hfill \\ \hfill \text{c}\phantom{\rule{2.77695pt}{0ex}}\text{sign}\left({\text{r}}_{\text{i}}\right)\hfill & \hfill \text{for}\phantom{\rule{2.77695pt}{0ex}}\left|{\text{r}}_{\text{i}}\right|>c,\hfill \end{array}\right.$$w\left({x}_{i}\right)={\left(1-{h}_{ii}\right)}^{1/2}$ with ${h}_{ii}$ the i^{th} diagonal element of the matrix $H=X{\left({X}^{T}X\right)}^{-1}{X}^{T}$, ${\mu}_{\text{i}}^{\text{'}}=\frac{\partial {\mu}_{\text{i}}}{\partial \beta}$ and $\alpha \left(\beta \right)=\frac{1}{2}\sum _{\text{i}=1}^{\text{n}}\text{E}\left[\text{v}\left({\text{y}}_{\text{i}};\phantom{\rule{2.77695pt}{0ex}}{\mu}_{\text{i}}\right)\right]\text{w}\left({\text{x}}_{\text{i}}\right){\mu}_{\text{i}}^{\text{'}}$.

where ${Q}_{M}\left({y}_{i},{\mu}_{i}\right)={\int}_{\stackrel{\sim}{s}}^{{\mu}_{i}}v\left({y}_{i},t\right)w\left({x}_{i}\right)dt-\frac{1}{n}{\sum}_{j=1}^{n}{\int}_{\stackrel{\sim}{t}}^{{\mu}_{j}}E\left[v\left({y}_{j},t\right)w\left({x}_{j}\right)\right]dt$ with $\stackrel{\sim}{s}$ such that $v\left({y}_{i},\stackrel{\sim}{s}\right)=0$ and $\stackrel{\sim}{t}$ such that $E\left[v\left({y}_{i},\stackrel{\sim}{t}\right)\right]=0$ and the estimated linear predictor $\widehat{\mu}$ is associated to the estimate $\widehat{\beta}$ of $\beta $ and $\stackrel{.}{\mu}$ is associated to $\stackrel{.}{\beta}$ which is the estimate of $\left({\beta}_{\left(1\right)},0\right)$. Linkage disequilibrium was not accounted for during variant selection neither for standard logistic regression nor for robust logistic regression.

Our comparison of the performance of standard and robust logistic regression was based on different statistics. First, standard and robust estimates of age effects were used to exemplify the potential influence of departing observations. Because of a different handling of outliers, it was expected that different age-genotype models were selected under standard and robust logistic regression. Consequently, the areas under the receiver operating characteristic curves (AUCs) were subsequently compared in order to investigate the discriminative performance of the selected models. Comparisons were conducted for the complete data set and after exclusion of potential outliers.

where ${\widehat{\text{p}}}_{\text{rob},\phantom{\rule{2.77695pt}{0ex}}\text{i}}$, ${\widehat{\text{p}}}_{\text{rob},\phantom{\rule{2.77695pt}{0ex}}\text{j}}$, ${\widehat{\text{p}}}_{\text{stand},\phantom{\rule{2.77695pt}{0ex}}\text{i}}$, and ${\widehat{\text{p}}}_{\text{stand},\phantom{\rule{2.77695pt}{0ex}}\text{j}}$ denote the probability estimates from the robust and standard logistic regression models for cases and controls [15]. This index represents the difference in the discrimination slopes of the 2 compared models. A positive IDI indicates that the robust model discriminates better between hypertensive and normotensive individuals than the standard model. Statistical analyses were carried out using the statistical language R, version 2.15.1 [16].

## Results

χ^{2} tests revealed no influence of gender (*p* = 0.95) and tobacco smoking (*p* = 1.00) on hypertension risk. Hence, only age was included in the logistic regression models as covariate. Filter criteria resulted in 130 individuals (43 cases and 87 controls) with complete genotype and phenotype information. The age of the individuals ranged between 20 and 95 years with a median age of 52 years. The total number of measured SNPs on chromosome 3 in the investigated GAW18 data set was 35,045.

Estimated odds ratios per year of age

Excluded individuals | HTN | Age | Standard logistic regression | Robust logistic regression | ||
---|---|---|---|---|---|---|

OR-Age (95% CI) | % Change | OR-Age (95% CI) | % Change | |||

None | 1.085 (1.050, 1.121) | ref. | 1.084 (1.048, 1.122) | ref. | ||

62 | 0 | 90.23 | 1.095 (1.057, 1.133) | +11.2% | 1.091 (1.052, 1.131) | +7.8% |

58 | 0 | 87.66 | 1.094 (1.056, 1.132) | +10.0% | 1.091 (1.052, 1.131) | +7.9% |

60 | 1 | 38.44 | 1.091 (1.054, 1.128) | +6.5% | 1.089 (1.051, 1.128) | +5.1% |

24 | 0 | 80.27 | 1.091 (1.054, 1.128) | +6.6% | 1.091 (1.052, 1.131) | +7.6% |

Overall odds of hypertension per age interval

Age interval (number of cases-to-controls) | |||
---|---|---|---|

<39.0 (1:22) | [39.0, 46.0) (2:20) | [46.0, 56.2) (9:23) | ≥56.2 (31:22) |

0.05 | 0.10 | 0.39 | 1.41 |

*ULK4*gene as the variant that most improved the model fit. Robust logistic regression identified SNP rs11918360 in

*RP11-408H1.3*as the variant with the strongest association signal. Under both standard and robust regression, model selection clearly favored the 2 identified SNPs as represented in Figure 2. The pairwise

*r*

^{ 2 }between SNP rs3934103 and SNP rs11918360 was 0.003.

Area under the receiver operating characteristic curve (AUC)

Excluded individuals | Standard logistic regression | Robust logistic regression | ||||||
---|---|---|---|---|---|---|---|---|

AUC-Age (% Change) | AUC-Age + SNP (% Change) | AUC-Age (% Change) | AUC-Age + SNP (% Change) | |||||

None | 0.811 | (ref.) | 0.852 | (ref.) | 0.811 | (ref.) | 0.843 | (ref.) |

62 | 0.820 | +1.1% | 0.861 | +1.1% | 0.820 | +1.1% | 0.852 | +1.0% |

58 | 0.820 | +1.1% | 0.861 | +1.1% | 0.820 | +1.1% | 0.853 | +1.2% |

60 | 0.825 | +1.7% | 0.859 | +0.9% | 0.825 | +1.7% | 0.851 | +0.9% |

24 | 0.819 | +1.0% | 0.859 | +0.9% | 0.819 | +1.0% | 0.844 | +0.0% |

Concordance, sensitivity, specificity, clinical net benefit, and overall AUCs.

Probability cutoff | Standard logistic regression | Robust logistic regression | ||||||
---|---|---|---|---|---|---|---|---|

Concordance N (%) | Sensitivity | Specificity | Clinical net benefit | Concordance N (%) | Sensitivity | Specificity | Clinical net benefit | |

0.0 | 43 (33.1) | 1.00 | 0.00 | 0.33 | 43 (33.1) | 1.00 | 0.00 | 0.33 |

0.1 | 79 (60.8) | 0.95 | 0.44 | 0.27 | 82 (63.1) | 0.88 | 0.51 | 0.26 |

0.2 | 90 (69.2) | 0.86 | 0.61 | 0.22 | 97 (74.6) | 0.86 | 0.69 | 0.23 |

0.3 | 98 (75.4) | 0.81 | 0.72 | 0.19 | 99 (76.2) | 0.81 | 0.74 | 0.19 |

0.4 | 98 (75.4) | 0.70 | 0.78 | 0.13 | 102 (78.5) | 0.72 | 0.82 | 0.16 |

0.5 | 101 (77.7) | 0.60 | 0.86 | 0.11 | 107 (82.3) | 0.67 | 0.90 | 0.15 |

0.6 | 97 (74.6) | 0.40 | 0.92 | 0.05 | 102 (78.5) | 0.51 | 0.92 | 0.09 |

0.7 | 99 (76.2) | 0.35 | 0.97 | 0.06 | 100 (76.9) | 0.42 | 0.94 | 0.05 |

0.8 | 93 (71.5) | 0.19 | 0.98 | 0.00 | 97 (74.6) | 0.30 | 0.97 | 0.01 |

0.9 | 91 (70.0) | 0.12 | 0.99 | −0.03 | 93 (71.5) | 0.19 | 0.98 | −0.08 |

1.0 | 87 (66.9) | 0.00 | 1.00 | - | 87 (66.9) | 0.00 | 1.00 | - |

AUC | 0.835 | 0.830 |

## Discussion

Present results confirmed that single individuals (1/130 = 0.8% of the observations) with a departing risk of hypertension may substantially affect the overall risk estimates in the baseline model, causing up to an 11.2% change in the estimated excess risk of hypertension per year according to standard logistic regression in the present exercise.

The identification of outliers is relatively straightforward using routine diagnostic plots, but outlier management is extremely challenging. For example, the specification of thresholds for outlier definition is often arbitrary. Robust statistics aim to generate estimates that hold for the majority of the population using complete data. The unequal weighting of outliers by standard and robust regression resulted in prediction models that included different genetic variants.

Although robust estimates of age effects and AUCs for age-genotype models were less sensitive to outliers than standard estimates in the investigated sample, cross-validation AUCs based on standard and robust logistic regression, as well as IDI, were almost identical. The other investigated performance characteristics (concordance, sensitivity, specificity, and clinical net benefit) were equal or better for robust logistic regression around the probability that reflects the case-control ratio.

The standard logistic regression model selected 1 variant in the *ULK4* gene. It was previously shown that variants in this gene are associated with hypertension [4, 17]. Among others, 4 variants (rs2272007, rs3774372, rs1716975, rs1052501) mentioned in the 2 publications were also genotyped in the GAW18 collective, and we found them to be in linkage disequilibrium (*r*^{
2
} values 0.83, 0.73, 0.83, and 0.83, respectively) with the associated SNP rs3934103.

## Conclusions

Preliminary findings suggest some advantage of robust statistics in the context of genetic association studies. However, present results were limited to a given sample size, as well as to particular genetic effect sizes and proportions of outliers. Additional analyses based on both real data and more general simulated scenarios should be conducted to validate initial findings.

## Declarations

### Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) grant SFB/TRR77 (Project Z2).

The GAW18 whole genome sequence data were provided by the T2D-GENES Consortium, which is supported by NIH grants U01 DK085524, U01 DK085584, U01 DK085501, U01 DK085526, and U01 DK085545. The other genetic and phenotypic data for GAW18 were provided by the San Antonio Family Heart Study and San Antonio Family Diabetes/Gallbladder Study, which are supported by NIH grants P01 HL045222, R01 DK047482, and R01 DK053889. The Genetic Analysis Workshop is supported by NIH grant R01 GM031575.

This article has been published as part of *BMC Proceedings* Volume 8 Supplement 1, 2014: Genetic Analysis Workshop 18. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcproc/supplements/8/S1. Publication charges for this supplement were funded by the Texas Biomedical Research Institute.

## Authors’ Affiliations

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