Rheumatoid arthritis, item response theory, Blom transformation, and mixed models
- Aldi T Kraja^{1}Email author,
- Jon Corbett^{1},
- An Ping^{1},
- Rosa S Lin^{1},
- Petra A Jacobsen^{1},
- Michael Crosswhite^{1},
- Ingrid B Borecki^{1} and
- Michael A Province^{1}
https://doi.org/10.1186/1753-6561-1-S1-S116
© Kraja et al; licensee BioMed Central Ltd. 2007
Published: 18 December 2007
Abstract
We studied rheumatoid arthritis (RA) in the North American Rheumatoid Arthritis Consortium (NARAC) data (1499 subjects; 757 families). Identical methods were applied for studying RA in the Genetic Analysis Workshop 15 (GAW15) simulated data (with a prior knowledge of the simulation answers). Fifty replications of GAW15 simulated data had 3497 ± 20 subjects in 1500 nuclear families. Two new statistical methods were applied to transform the original phenotypes on these data, the item response theory (IRT) to create a latent variable from nine classifying predictors and a Blom transformation of the anti-CCP (anti-cyclic citrinullated protein) variable. We performed linear mixed-effects (LME) models to study the additive associations of 404 Illumina-genotyped single-nucleotide polymorphisms (SNPs) on the NARAC data, and of 17,820 SNPs of the GAW15 simulated data. In the GAW15 simulated data, the association with anti-CCP Blom transformation showed a 100% sensitivity for SNP1 located in the major histocompatibility complex gene. In contrast, the association of SNP1 with the IRT latent variable showed only 24% sensitivity. From the simulated data, we conclude that the Blom transformation of the anti-CCP variable produced more reliable results than the latent variable from the qualitative combination of a group of RA risk factors. In the NARAC data, the significant RA-SNPs associations found with both phenotype-transformation methods provided a trend that may point toward dynein and energy control genes. Finer genotyping in the NARAC data would grant more exact evidence for the contributions of chromosome 6 to RA.
Background
Rheumatoid arthritis (RA) is a complex disease with a heritability of about 60% as evaluated in twin studies. Univariate qualitative assessments of RA did not reveal a large variation, and quantitative assessment of anti-cyclic citrullinated protein antibodies (anti-CCP), showed a non-normal distribution, although anti-CCP has been considered a better predictor of erosive outcome compared to the rheumatoid factor (RF) IgM [1–4]. In our study we propose the application of item response theory (IRT) and the Blom transformation, respectively, on the qualitative and quantitative variables. Utilizing the data provided by the North American Rheumatoid Arthritis Consortium (NARAC) and the simulated data provided by the Genetic Analysis Workshop 15 (GAW15), we explore the above two statistical methods that transform the phenotypes. The association of the new RA phenotypes in combination with the linear mixed effects (LME) models summarizes the genetic effect of individual SNPs on RA. The significant single-nucleotide polymorphisms (SNPs) in the NARAC real data and from the GAW15 dense SNPs simulated ones on chromosome 6 setting are reported.
Methods
Item response theory (multivariate qualitative to quantitative latent variable transformation)
In the application of the IRT method in handling multivariate phenotypic data, it was assumed that each individual classified with RA had a latent RA variable. Such a latent feature for RA was the result of a statistical summary of information from nine observed risk factors. These factors were qualitative anti-CCP, qualitative IgM, gender, smoking status, and five severity classes of left and right hand erosions for RA, which were transformed to individual severity dummy variables as 1/0 for each corresponding combined class (see Data Input). Applying IRT, each member of the sample was assigned a latent (z_{ i }) score. Our analysis is constraint to one latent variable z_{1i}. The probability density function of y_{ i }given the latent variable z can be written as $p({y}_{i}|z,x,{\theta}_{i},{\varnothing}_{i})=e\left\{\frac{{y}_{i}{\theta}_{i}-{\beta}_{i}({\theta}_{i})}{{\alpha}_{i}({\varnothing}_{i})}+{d}_{i}({y}_{i},{\varnothing}_{i})\right\}$, where θ_{ i }and ∅_{ i }are the location and dispersion parameters, respectively. In the IRT, the item characteristic curves represent the positive response probability of each risk factor j relative to the latent RA score. Formally, this relationship is expressed as $p\left({y}_{ij}=1|{z}_{i}\right)=\frac{1}{1+e\left\{-{\alpha}_{i}({z}_{i}-{\beta}_{i})\right\}}$, and graphically it represents the importance of each risk factor toward the latent RA. The discrimination parameter α_{ i }is viewed as how well a risk factor discriminates among subjects with opposite extreme latent RA. The difficulty parameter β_{ i }characterizes the difficulty level of individual risk variables. In this study we applied the two-parameter IRT model (α_{ i }, β_{ i }), which was constrained to one latent variable (z_{ i }). Software for IRT analysis in R language is developed by Rizopoulos (ltm package version 0.5–1 for Linux OS and R version 2.3.0) [5, 6].
Blom score (quantitative univariate to quantitative rank score transformation)
In the second method we used the Blom transformation. Blom scores represent rank approximations of the exact order of a normal distribution. In the group of rank transformed quantitative variables, one extracts a Blom score by applying the following formula on the anti-CCP.
$Blo{m}_{i}=\frac{{\varphi}_{i}^{-1}({r}_{i}-\frac{3}{8})}{n+\frac{1}{4}}$, where ϕ_{ i }^{-1} is the inverse of cumulative normal function, r_{ i }is the rank of observation i, and n is the number of non-missing observations. The Blom scores of anti-CCP represented a better normal distribution than the original values. Software for the Blom transformation is available within the PROC RANK of SAS, v 9.1.3 for Linux OS.
Linear mixed effects models (association tests)
The LME model used in our study to test the association between an SNP and changes in the response variable (IRT RA latent variables/anti-CCP Blom transformed variables) follows in a matrix form: Y= XB+ ZU+ ε, where Yis an m × 1 vector of responses; Xis an m × p design matrix of the fixed effects; Bis the parameter p × 1 vector of fixed effects; Zis an m × q incidence matrix of random effects, and Uis a q × 1 vector of random effects with E(U) = 0, and covariance matrix G; 0is an m × 1 vector of random effects with E(0) = 0 and covariance matrix R. In the fixed effects we included SNP genotypes recoded as additive effects (-1 for one homozygote, 0 for the heterozygote, and 1 for the other homozygote genotype), and gender. In the random effects we included the family identification number. We tested whether the SNPs additive effects are different from zero, and especially we identified the highest significances, considering that multiple comparisons as well as correlated SNPs tests because of linkage disequilibrium are present. The LME in SAS (v. 9.1.3 for Linux OS) was applied with PROC MIXED using the option for EMPIRICAL variance, which computes the estimated variance-covariance matrix of the fixed-effects parameters by using the following asymptotically consistent estimator ${\left(X{{}^{\u2035}V}^{-1}X\right)}^{-}\left({\displaystyle {\sum}_{i=1}^{s}{X}_{i}{{}^{\u2035}V}_{i}^{-1}{\in}_{i}{\in}_{i}`{{}^{\u2035}V}_{i}^{-1}{X}_{i}}\right)\left(X{{}^{\u2035}V}^{-1}X\right)`$, where var(Y) = V= ZGZ' + R[7].
Results of association tests in 50 replications of the simulated data were analyzed with two definitions: sensitivity, defined as the proportion of the true causative SNPs that had a positive association result (p-value < 0.05); and specificity, defined as the proportion of true non-causative SNPs that had non-significant associations (p-value ≥ 0.05).
Data input
NARAC phenotypes in our analysis included anti-CCP, rheumatoid factor (RF), gender, severity of left and right hand side erosions on a scale of 1 to 5 (severityLH and severityRH, respectively), whether or not the subject smoked cigarettes (HxCigN). Qualitative dummy variables were created for each of the following variables, where the subject with anti-CCP > 49 [8], RF > 170 (to capture approximately the RF upper quartile in a very skewed RF distribution); smokers (HxCigN = 2); gender (if female); and whether severityLH or severityRH was 1, 2, 3, 4, or 5 were considered as affected. In the corresponding nine dummy variables (accp, igm, smok, sex, sev1, sev2, sev3, sev4, and sev5) affection was codes as 1, unaffected as 0, otherwise as missing. The same nine dummy variables were created also for the RA GAW15 simulated data. These dummy variables were analyzed individually for the two sources of data with the ltm package to extract the corresponding RA latent factors. Also, the anti-CCP variables were independently Blom-transformed within groups defined by gender and each age decade. LME model was performed to evaluate association of these traits, with 404 Illumina SNPs on chromosome 6 for NARAC data, and with 17,820 dense simulated SNPs on chromosome 6 for 50 replications on the simulated GAW15 data. In the NARAC data the sample size was 1499 subjects and 757 families, and 1340 subjects and 757 families for anti-CCP Blom transformation and IRT latent variable, respectively. In the simulated data the sample contained a mean of 3497 ± 20 subjects and 1500 nuclear families for both traits.
Results
In the simulated data, out of 50 replications, the association of SNP1 (locus DR) with the IRT latent variable based on LME was 24% sensitive and 94.79% specific. SNP2 and SNP3 together (Locus D) had a lower sensitivity (7%), but the same specificity as SNP1. When a larger region (~500 kb) was assumed as the corresponding genes, for GENE1 (corresponding to SNP1) the sensitivity was 7.3%, for GENE2 (corresponding to SNP2 and 3) the sensitivity was only 4.8%. In contrast, the sensitivity of the association of the Blom transformed anti-CCP with SNP1 was 100%. For SNP2 and SNP3 the sensitivity was 6%. Also for GENE1 the association sensitivity with anti-CPP Blom transformed was 46.2%.
NARAC data, additive model p-values and effects for the anti-CCP Blom-transformed and the IRT latent variable data^{a}
Anti-CCP Blom-transformed | IRT latent variable | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
No. | SNP | p-value | Additive effect | STD | bp | SNP | p-value | Additive effect | STD | bp |
1 | rs189512 | 0.00067 | -0.14680 | 0.04294 | 6923632 | rs2876143 | 0.00816 | 0.07944 | 0.02993 | 8928623 |
2 | rs214527 | 0.01890 | -0.11800 | 0.05014 | 18399329 | rs1863995 | 0.04430 | -0.05370 | 0.02665 | 13565596 |
3 | rs1402405 | 0.02528 | 0.08417 | 0.03755 | 24198476 | rs736794 | 0.04758 | -0.04813 | 0.02425 | 41044688 |
4 | rs1224485 | 0.02145 | 0.08928 | 0.03874 | 37043628 | rs1009130 | 0.00301 | 0.08090 | 0.02716 | 58068726 |
5 | rs1738240 | 0.01339 | 0.10040 | 0.04051 | 38810588 | rs977382 | 0.02981 | 0.06241 | 0.02866 | 62057200 |
6 | rs533393 | 0.00897 | -0.10350 | 0.03951 | 81645240 | rs1040802 | 0.01273 | -0.06005 | 0.02403 | 63469278 |
7 | rs1180237 | 0.01758 | 0.09720 | 0.04085 | 83939524 | rs2076874 | 0.00397 | 0.06580 | 0.02276 | 64150527 |
8 | rs910563 ^{b} | 0.01025 | 0.10250 | 0.03983 | 88054481 | rs1630182 | 0.02261 | -0.05252 | 0.02298 | 64205048 |
9 | rs1051131 | 0.04853 | 0.07951 | 0.04024 | 88217142 | rs1370436 | 0.03343 | 0.05014 | 0.02352 | 74520012 |
10 | rs2144363 | 0.02680 | 0.19480 | 0.08778 | 92037502 | rs910563 | 0.03853 | 0.04794 | 0.02312 | 88054481 |
11 | rs6454855 | 0.04535 | 0.07830 | 0.03906 | 92071536 | rs2610715 | 0.01464 | -0.05937 | 0.02425 | 89260502 |
12 | rs716192 | 0.04829 | -0.08171 | 0.04131 | 96557287 | rs240153 | 0.04093 | -0.04727 | 0.02307 | 101112733 |
13 | rs1321807 | 0.02545 | 0.08678 | 0.03876 | 117730562 | rs239229 | 0.02728 | -0.05129 | 0.02318 | 101151573 |
14 | rs941815 | 0.04541 | -0.08273 | 0.04128 | 123901148 | rs239189 | 0.03231 | 0.04970 | 0.02317 | 101174385 |
15 | rs839556 | 0.04100 | 0.08086 | 0.03950 | 142852448 | rs1158747 | 0.03314 | -0.05161 | 0.02417 | 112546220 |
16 | rs923459 | 0.00991 | 0.10150 | 0.03927 | 159471138 | rs763075 | 0.00505 | 0.06554 | 0.02330 | 122192096 |
17 | rs294882 | 0.02322 | -0.09232 | 0.04059 | 159512287 | rs1033540 | 0.04603 | -0.04924 | 0.02463 | 131717872 |
18 | rs225604 | 0.00643 | 0.06488 | 0.02373 | 142433136 | |||||
19 | rs1931992 | 0.00209 | 0.07516 | 0.02433 | 142581777 | |||||
20 | rs2151913 | 0.03291 | -0.05082 | 0.02377 | 150161633 | |||||
21 | rs718527 | 0.01903 | -0.05770 | 0.02454 | 152501127 | |||||
22 | rs231958 | 0.01449 | 0.06105 | 0.02490 | 155786394 | |||||
23 | rs923459 | 0.01838 | 0.05568 | 0.02356 | 159471138 | |||||
24 | rs409359 | 0.04983 | -0.04526 | 0.02303 | 159854284 | |||||
25 | rs916331 | 0.01358 | 0.05892 | 0.02380 | 167379847 |
Discussion
Of 404 Illumina SNPs typed on chromosome 6, none of the significant SNPs passed the 0.0001 Bonferroni threshold. Because such threshold is considered to be conservative, in Table 1 we show the original significant p-values of the SNPs associated with the two explored phenotypic transformations, Blom and IRT. Two SNPs, namely rs910563 and rs923459, overlapped as significant in both analyses. rs910563 is in an intron in the C6orf163 gene. Its function is unknown. Significant SNPs were also located in other interesting genes. Of them, rs533393 is in an intron in the DNAH8 gene, which is one of the dynein heavy chains responsible for force production and ATPase activity. It contains a highly conserved catalytic domain. The rs1180237 is in an intron in the ME1 gene, which encodes a cytosolic, NADP-dependent enzyme that generates NADPH for fatty acid biosynthesis. Another SNP, rs1051131, although located on the exon 8 region of gene SLC35A1, which is a solute carrier family 35 (CMP-sialic acid transporter), member A1, is untranslated. SNP rs716192 is located in the FUT9 (fucosyltransferase 9) gene. rs1321807 is part of the gene ROS1. Its protein may function as a growth or differentiation factor receptor. SNP rs941815 is part of the gene TRDN. Experimentally it is proposed that one of the Triadin genes overexpression blocks excitation-contraction coupling in rat skeletal myotubes. The SNP rs294882 is located in the gene FNDC1 (fibronectin type III). Fibronectin is a key extracellular matrix protein that not only provides a substrate for cell anchorage but also serves as a regulatory protein in processes such as cell adhesion, motility, differentiation, and proliferation.
Sensitivity and specificity of three SNPs associations with two different phenotypes (50 replications)
Sensitivity (%) | Specificity (%) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Traits(s) transformation | SNP1^{a} | SNP2&3^{b} | SNP1-3 | GENE1^{c} | GENE2 | SNP1 | SNP2&3 | SNP1-3 | GENE1 | GENE2 |
IRT latent variable | 24.00 | 7.00 | 12.70 | 7.30 | 4.80 | 94.79 | 94.79 | 94.79 | 94.80 | 94.78 |
Anti-CCP Blom transformation | 100.00 | 6.00 | 37.30 | 46.20 | 10.10 | 93.21 | 93.20 | 93.21 | 93.42 | 93.21 |
Conclusion
From our findings with the simulated data, we assume that the Blom-transformed NARAC anti-CCP data can be a more reliable phenotype in the analysis to identify true significant associations. Amos et al. documented the highest chromosome linkage peak with two SNPs, rs169679 and rs11908. Although these SNPs were present in our analysis, neither were significant. It is possible that differences stand in the facts that two analyses are different (linkage vs. association) and different traits were considered. In conclusion, our results provide a compelling trend that may point toward dynein and energy control genes involvement in the RA. In the future, denser genotyping may provide more exact evidence for chromosome 6 and its contribution to RA.
Declarations
Acknowledgements
This article has been published as part of BMC Proceedings Volume 1 Supplement 1, 2007: Genetic Analysis Workshop 15: Gene Expression Analysis and Approaches to Detecting Multiple Functional Loci. The full contents of the supplement are available online at http://www.biomedcentral.com/1753-6561/1?issue=S1.
Authors’ Affiliations
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