Abstract
The relationship between genetic variability and individual phenotypes is usually investigated by testing for association relying on called genotypes. Allele counts obtained from next-generation sequence data could be used for this purpose too. Genetic association can be examined by treating alternative allele counts (AACs) as the response variable in negative binomial regression. AACs from sequence data often contain an excess of zeros, thus motivating the use of Hurdle and zero-inflated models. Here we examine rough type I error rates and the ability to pick out variants with small probability values for 7 different testing approaches that incorporate AACs as an explanatory or as a response variable. Model comparisons relied on chromosome 3 DNA sequence data from 407 Hispanic participants in the Type 2 Diabetes Genetic Exploration by Next-generation sequencing in Ethnic Samples (T2D-GENES) project 1 with complete information on diastolic blood pressure and related medication. Our results suggest that in the investigation of the relationship between AAC as response variable and individual phenotypes as explanatory variable, Hurdle-negative binomial regression has some advantages. This model showed a good ability to discriminate strongly associated variants and controlled overall type I error rates. However, probability values from Hurdle-negative binomial regression were not obtained for approximately 25 % of the investigated variants because of convergence problems, and the mass of the probability value distribution was concentrated around 1.
Background
Diastolic blood pressure (DBP) measures the pressure during heart relaxation. High DBP increases the risk of heart attacks, stroke, and kidney failure. DBP is dependent on the age, gender, and medication of an individual [1]. A genetic component has been identified for high DBP [2].
The relationship between genetic variability and DBP is commonly investigated by testing for association between called genotypes and phenotypes. In sequence data, genotypes are called relying on sequence reads. Alternatively, association tests without calling genotypes have been suggested [3].
Depth in DNA sequence data refers to the number of reads at a position. In next-generation sequence data there are 2 counts of allelic depth, the number of reference alleles, and the number of alternative alleles. Here we use the number of alternative allele counts (AACs) and the total read depth. AACs are genotype measurements that are more informative than called genotypes in the sense that the 2 counts—“zero reference alleles out of 100” and “1 reference allele out of 100”—would translate into the same called genotype. In other words, after applying user-defined data quality filters, the uncertainty in genotype calling is rarely taken into account in statistical analyses.
Several statistical procedures can be used for testing association between allele counts and a phenotype of interest. When AAC is regarded as the response variable and the total read depth as an offset, standard models for count data can be used. The response variable can also be defined as the ratio of the AAC to the total read depth. In this case, linear regression can be used to investigate the genotype–phenotype relationship. It is also possible to model the phenotype as a response variable, and the ratio “AAC/total read depth” as an explanatory variable.
We used data from the Genetic Analysis Workshop (GAW) 19, which are provided for 1943 Hispanic samples that have been whole-exome sequenced as part of the Type 2 Diabetes Genetic Exploration by Next-generation sequencing in Ethnic Samples (T2D-GENES) Project 1 to compare different regression techniques.
Methods
Phenotype data were provided for 1943 Hispanics who participated in the T2D-GENES Project 1, and included 4 variables: DBP, age, gender, and medication [4, 5]. We excluded individuals with missing information on medication; that is, the investigated data set included 407 patients. To be able to treat DBP both as response and as explanatory variable, we regressed the measured DBP on age, sex, and medication, and used the residuals as a new phenotype variable that we denominated “adjusted-DBP.”
Regarding genotype measurements, exonic regions were isolated using Agilent TruSeq capture reagents, and individually barcoded samples sequenced in Illumina HiSeq 2000 instruments. Across the coding sequence of 18,281 genes, an average read depth of approximately 82-fold was reached. We used VCFtools (version 0.1.4a) to extract the data needed: counts of reference and alternative alleles (AD field in the FORMAT tag of the variant call format [vcf] file), genotype (GT field in the FORMAT tag), and average genotype quality (GQ field in the FORMAT tag). Total read depth was calculated as the sum of the counts for the reference and the alternative alleles. We also used VCFtools to calculate the minor allele frequency (MAF). Our analysis focused on biallelic variants in chromosome 3. Variants with a MAF of less than 0.003 were excluded, leaving a total of 8957 variants for analysis.
We compared different regression models and treated AAC both as response and as explanatory variable. The relationship between AAC as response variable (total read depth was used as offset), and adjusted-DBP as explanatory variable was first investigated by negative binomial regression. Zero-inflated negative binomial (ZI-NB) and Hurdle-negative binomial (Hu-NB) models are more flexible than negative binomial regression in the presence of zero inflation [6, 7]. Comparisons between the complete model with adjusted-DBP, and a null model with only the intercept relied on the deviance (minus twice the difference of log null and complete model likelihoods) investigated with a chi-squared test.
We also investigated the genotype–phenotype relationship based on the ratio “AAC/total read depth” and adjusted-DBP, as response and explanatory variables in standard linear and robust linear regression [8].
Type I error rates were roughly estimated assuming that the large majority of the 8957 investigated variants were under the null hypothesis of no genetic association. Quantile–quantile (Q-Q) plots were used to explore possible disparities between small probability values (p values) from different regression models. These analyses do not amount to a systematic model comparison relying on extensive simulations, but they could be suggestive of the most promising modeling approaches.
R-packages stats, pscl, and robust were used to fit negative binomial/linear regression, ZI-NB/Hu-NB, and robust linear regression models, respectively.
Results
Figure 1 shows the distribution of AACs for all 8957 investigated variants. Panel A shows the distribution of mean AACs; panel B shows the distribution of median AACs; and panel C, which represents the AAC distribution variant in position Ch3:391100, illustrates the frequent large proportion of zero counts. Noteworthy is that panel B revealed 105 variants with a median AAC exactly equal to 254. This peak was not apparent when mean AACs were represented. To investigate the possible origin of this peak, we plotted a histogram of AACs for the variant in position Ch3:16249998, which presented a median (mean) AAC of 254 (40.03). This variant showed a MAF of 0.168 (panel D: 280 reference allele homozygotes, 117 heterozygotes, and 10 alternative allele homozygotes). The presence of 35 variants with a median AAC greater than 254 in panel B, combined with panel (D), suggest that downsampling, if present, was incomplete. A field with information on the possible downsampling was not available. Panel E compares the distribution of the “AAC/total read depth” ratio with the distribution of called genotypes.
Fig. 1.
Distribution of AAC. a Mean AACs per variant. b Median AACs per variant. c Exemplary zero-inflated AACs distribution grouped by genotype for the variant in position Chr3:391100 with MAF equal to 0.0258. d Exemplary skewed AACs distribution grouped by genotype for the variant in position Chr3:16249998 with MAF equal to 0.1683. e Exemplary comparison of the ratios (AAC/total read depth) and called genotypes for the variant in position Chr3:16249998
Table 1 shows rough type I error rates for the 8957 investigated variants in chromosome 3. For example, the use of negative binomial regression to investigate the relationship between AAC as response variable, and adjusted-DBP as explanatory variable, resulted in 8871 nonmissing p values, 39 % of which were smaller than 0.05. No convergence was the principal cause of missing p values and increasing the number of iterations did not alleviate the problem. Consideration of zero inflation in negative binomial regression increased the percentage of missing p values, which amounted to 26 % for both ZI-NB and Hu-NB. The proportion of nonmissing p values less than 0.05 was 6.4 % for ZI-NB and 3.5 % for Hu-NB regression models.
Table 1.
Overall type I error rates
| Response variable | Explanatory variable | Regression model | # p values | # Nonmissing p values | % Nonmissing p values | % Nonmissing p values under 0.05 |
|---|---|---|---|---|---|---|
| AAC | Adjusted DBP | Negative binomial (NB) | 8957 | 8871 | 99.0 | 0.388 |
| Zero-inflated NB | 8957 | 6608 | 73.8 | 0.064 | ||
| Hurdle NB | 8957 | 6585 | 73.5 | 0.035 | ||
| AAC/total read depth | Adjusted DBP | Standard linear | 8957 | 8947 | 99.9 | 0.062 |
| Robust linear | 8957 | 1509 | 16.8 | 0.068 | ||
| Adjusted DBP | AAC/total read depth | Standard linear | 8957 | 8947 | 99.9 | 0.062 |
| Robust linear | 8957 | 8917 | 99.6 | 0.127 |
Among the 7 investigated models, standard linear regression generated the lowest proportion of missing p values (approximately 1 per 1000 tested variants). Note that treating the ratio “AAC/total read depth” as a response or explanatory variable in standard linear regression resulted in identical p values. In robust linear regression, treating the ratio “AAC/total read depth” as a response variable resulted in a high proportion of missing p values (83 %). The reason was a perfect model fit after downweighting outliers (observations with a large “AAC/total read depth” ratio) by robust regression. Figure 2 illustrates this problem.
Fig. 2.
Scatter graph and fitted model for the variant in position Chr3:361487
Motivated by this result, the rest of the manuscript focuses on the 4 regression models with the best calibrated overall rough type I error rates: ZI-NB and Hu-NB, standard linear regression, and robust linear regression with the ratio “AAC/total read depth” as a response variable. Table 2 shows rough type I error rates stratified by 3 characteristics of the variants: median AAC, MAF, and the average genotype quality. Only 1.6 % of the variants presented a median AAC equal to or larger than 254, but this cutoff was chosen based on the observed peak in Fig. 1, panel b. The proportion of p values of less than 0.05 from ZI-NB and Hu-NB regression models was lower for variants with a median AAC of less than 60 than for variants with higher median AACs. By contrast, this proportion did not depend on the median AAC for standard and robust linear regression.
Table 2.
Type I error rates stratified by variant-specific AAC, MAF, and average genotype quality
| Response variable | Regression model | Median AAC | # P values | # Nonmissing p values | % Nonmissing p values | % Nonmissing p values under 0.05 |
| AAC | ZI-NB | [0,60] | 4728 | 3598 | 76.1 | 0.050 |
| (60,254) | 4089 | 2917 | 71.3 | 0.080 | ||
| 254 | 105 | 84 | 80.0 | 0.083 | ||
| (254,426] | 35 | 9 | 25.7 | 0.000 | ||
| Hu-NB | [0,60] | 4728 | 3580 | 75.7 | 0.029 | |
| (60,254) | 4089 | 2912 | 71.2 | 0.043 | ||
| 254 | 105 | 84 | 80.0 | 0.036 | ||
| (254,426] | 35 | 9 | 25.7 | 0.111 | ||
| AAC/total read depth | Standard linear (SL) | [0,60] | 4728 | 4719 | 99.8 | 0.062 |
| (60,254) | 4089 | 4088 | 100.0 | 0.062 | ||
| 254 | 105 | 105 | 100.0 | 0.029 | ||
| (254,426] | 35 | 35 | 100.0 | 0.114 | ||
| Robust linear (RL) | [0,60] | 4728 | 767 | 16.2 | 0.068 | |
| (60,254) | 4089 | 708 | 17.3 | 0.068 | ||
| 254 | 105 | 10 | 9.5 | 0.100 | ||
| (254,426] | 35 | 24 | 68.6 | 0.083 | ||
| Response variable | Regression model | MAF | # P values | # Nonmissing p values | % Nonmissing p values | % Nonmissing p values under 0.05 |
| AAC | ZI-NB | [0.003,0.01] | 3707 | 3694 | 99.6 | 0.036 |
| (0.01,0.05] | 1927 | 1895 | 98.3 | 0.144 | ||
| >0.05 | 3323 | 1019 | 30.7 | 0.016 | ||
| Hu-NB | [0.003,0.01] | 3707 | 3695 | 99.7 | 0.044 | |
| (0.01,0.05] | 1927 | 1886 | 97.9 | 0.031 | ||
| >0.05 | 3323 | 1004 | 30.2 | 0.011 | ||
| AAC/total read depth | SL | [0.003,0.01] | 3707 | 3698 | 99.8 | 0.058 |
| (0.01,0.05] | 1927 | 1927 | 100.0 | 0.053 | ||
| >0.05 | 3323 | 3322 | 100.0 | 0.072 | ||
| RL | [0.003,0.01] | 3707 | 2 | 0.1 | 0.000 | |
| (0.01,0.05] | 1927 | 0 | 0.0 | 0.000 | ||
| >0.05 | 3323 | 1507 | 45.4 | 0.068 | ||
| Response variable | Regression model | Average genotype quality | # P values | # Nonmissing p values | % Nonmissing p values | % Nonmissing p values under 0.05 |
| AAC | ZI-NB | Low | 2985 | 2973 | 99.6 | 0.067 |
| Medium | 2986 | 2810 | 94.1 | 0.060 | ||
| High | 2986 | 825 | 27.6 | 0.058 | ||
| Hu-NB | Low | 2985 | 2974 | 99.6 | 0.044 | |
| Medium | 2986 | 2812 | 94.1 | 0.031 | ||
| High | 2986 | 799 | 26.7 | 0.016 | ||
| AAC/total read depth | SL | Low | 2985 | 2975 | 99.7 | 0.056 |
| Medium | 2986 | 2986 | 100.0 | 0.062 | ||
| High | 2986 | 2986 | 100.0 | 0.068 | ||
| RL | Low | 2985 | 9 | 0.3 | 0.000 | |
| Medium | 2986 | 96 | 3.2 | 0.031 | ||
| High | 2986 | 1404 | 47.0 | 0.071 |
Interestingly, convergence problems and missing p values from ZI-NB and Hu-NB regression were particularly relevant for variants with a MAF greater than 0.05. The relationship between rough type I error rates and the MAF showed no clear pattern. For example, the proportion of p values less than 0.05 from Hu-NB regression decreased with increasing the MAF, and showed a U-pattern for standard linear regression. As expected, p values from robust linear regression were practically not available for variants with a MAF of less than 0.05.
Variants were also grouped into 3 balanced categories according to their average genotype qualities (low: GQ field in the vcf file from 30 to 60,000; medium: GQ from 60,000 to 210,000; and high: GQ >210,000). Rough type I error rates slightly decreased with increasing average genotype quality for ZI-NB and Hu-NB regression models. By contrast, approximate type I error rates increased with increasing average genotype quality for standard and for robust linear regression. P values from robust linear regression were practically not available for variants with an average genotype quality of less than 60,000.
Figure 3 investigates the distributions of p values for the 4 most promising regression models. To interpret results, the proportion of missing p values and the properties of each model discussed above should be kept in mind. Although 4 % of p values were smaller than 0.05 for Hu-NB and 6 % for ZI-NB regression, the mass of the p value distributions was concentrated around 1 (87 % of the p values from the Hu-NB and 80 % of the p values from the ZI-NB model were higher than 0.95). Hu-NB regression picked out variants with small p values more clearly than the ZI-NB version (A) with similar proportions of missing p values, and similar dependence patterns of the rough type I error rates on the median AACs and on the average genotype quality. However, test statistics from ZI-NB regression were likely not properly calibrated for variants with MAFs in the interval (0.01, 0.05] (see Table 2 in which 14 % of the p values were less than 0.05).
Fig. 3.
QQ plots for ZI-NB and Hu-NB models treating the AACs as a response variable (a), and for standard and robust linear regression models treating the ratio (AAC/total read depth) as a response variable (b). Gray shape represents the 95 % confidence bands regions
Figure 3b compares the standard and robust versions of linear regression. The robust model, somewhat more clearly, picked out variants with small p values, but p values were within 95 % confidence bands. It should be kept in mind that robust linear regression only generated p values for variants with a MAF greater than 0.05, and for variants with an average genotype quality greater than 60,000. Note the differential scales of the y-axis in Fig. 3a and b. Based on these preliminary results, which should be complemented with simulations in the future, Hu-NB regression seems to be a good candidate to investigate the relationship between AAC as a response variable, and individual phenotypes as an explanatory variable.
Discussion
The direct inspection of raw counts is an established approach to investigate differential expression based on RNA sequence data (see, eg, the R-package DESeq2), but the analysis of DNA sequence data almost always focuses on called genotypes. Satten et al [3] proposed using the proportion of calls for the minor allele instead of called genotypes. This proposal motivated the present investigation, where we used the ratio “AAC/total read depth.” A large proportion of false-positive associations based on sequence data was reported in earlier versions of the GAW [9]. Recommendations to address this problem usually rely on called genotypes; for example, accounting for population substructure and conducting gene-based or collapsing association tests.
The possibility of using AAC instead of called genotypes motivated the present research. The underlying distribution of AAC reflects the reliability of called genotypes, and a direct analysis of AAC would permit integration of this uncertainty into association results. However, next-generation sequence data is noisy and the quality of the called genotypes depends on the chromosomal position. Lots of variants are artifactual and rely on small counts.
The exclusive use of real data without additional simulations is a limitation of the present study. Nonparametric and quintile regression could be potential alternatives to robust linear regression in the presence of outlying AACs. We investigated the dependency of results on AAC, but data was not available for variants with low-quality genotype calls. It would be interesting to examine the behavior of Hu-NB regression for these variants.
Conclusions
We took advantage of real data to compare different models that investigate the relationship between sequence allele counts and the adjusted-DBP. The methods with the best control of rough type I error rates were ZI-NB and Hu-NB regression for the relationship between AAC and adjusted-DBP, and standard and robust linear regression for the relationship between the ratio “AAC/total read depth” as a response variable and adjusted-DBP as an explanatory variable.
The simultaneous consideration of discriminative ability of variants with small p values, occurrence of convergence problems, and robustness of p values against departing DBP observations indicated that Hu-NB regression constitutes a promising approach to assess the association between AAC and individual phenotypes.
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Declarations
Travel funds for ÖK was provided by NIH R01 GM031575, which supports the Genetic Analysis Workshop. We thank 2 anonymous reviewers for helpful comments. The GAW19 whole genome sequence data were provided by the T2D-GENES Consortium, which is supported by National Institutes of Health (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 GAW is supported by NIH grant R01 GM031575.
This article has been published as part of BMC Proceedings Volume 10 Supplement 7, 2016: Genetic Analysis Workshop 19: Sequence, Blood Pressure and Expression Data. Summary articles. The full contents of the supplement are available online at http://bmcproc.biomedcentral.com/articles/supplements/volume-10-supplement-7. Publication of the proceedings of Genetic Analysis Workshop 19 was supported by National Institutes of Health grant R01 GM031575.
Authors’ contributions
RGS and ÖK analyzed and interpreted the data, created figures and tables, searched literature, and drafted the manuscript. BP, CF, MK, and CL supported data analysis and interpretation and reviewed the manuscript. JLB formulated study goals, supported data analysis and interpretation, and reviewed the manuscript. All authors read and approved the final version. RGS and ÖK contributed equally to this work.
Competing interests
The authors declare that they have no competing interests.
Contributor Information
Rosa González Silos, Email: gonzalez@imbi.uni-heidelberg.de.
Özge Karadag, Email: ozgekaradag@hacettepe.edu.tr.
Barbara Peil, Email: peil@imbi.uni-heidelberg.de.
Christine Fischer, Email: christine.fischer@med.uni-heidelberg.de.
Maria Kabisch, Email: m.kabisch@dkfz-heidelberg.de.
Carine Legrand, Email: legrand@imbi.uni-heidelberg.de.
Justo Lorenzo Bermejo, Email: lorenzo@imbi.uni-heidelberg.de.
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