Pitfalls of predicting age-related traits by polygenic risk scores.

Polygenic risk scores (PRS) are a method increasingly used to capture the combined effect of genome-wide significant variants and those which individually do not show genome-wide significant association but are likely to contribute to the risk of developing diseases. However, their practical use incurs complications and inconsistencies that so far limit their clinical applicability. The aims of the present review are to discuss the PRS for age-related diseases and to highlight pitfalls and limitations of PRS prediction accuracy due to ageing and mortality effects. We argue that the PRS is widely used but the individual's PRS values differ substantially depending on the number of genetic variants included, the discovery GWAS and the method employed to generate them. Moreover, for neurodegenerative disorders, although an individual's genetics do not change with age, the actual score depends on the age of the sample used in the discovery GWAS and is likely to reflect the individual's disease risk at this particular age. Improvement of PRS prediction accuracy for neurodegenerative disorders will come from two sides, both the precision of clinical diagnoses, and a careful attention to the age distribution in the underlying samples and validation of the prediction in longitudinal studies.

What does heritability of Alzheimer's disease represent?

INTRODUCTION: Both late-onset Alzheimer's disease (AD) and ageing have a strong genetic component. In each case, many associated variants have been discovered, but how much missing heritability remains to be discovered is debated. Variability in the estimation of SNP-based heritability could explain the differences in reported heritability. METHODS: We compute heritability in five large independent cohorts (N = 7,396, 1,566, 803, 12,528 and 3,963) to determine whether a consensus for the AD heritability estimate can be reached. These cohorts vary by sample size, age of cases and controls and phenotype definition. We compute heritability a) for all SNPs, b) excluding APOE region, c) excluding both APOE and genome-wide association study hit regions, and d) SNPs overlapping a microglia gene-set. RESULTS: SNP-based heritability of late onset Alzheimer's disease is between 38 and 66% when age and genetic disease architecture are correctly accounted for. The heritability estimates decrease by 12% [SD = 8%] on average when the APOE region is excluded and an additional 1% [SD = 3%] when genome-wide significant regions were removed. A microglia gene-set explains 69-84% of our estimates of SNP-based heritability using only 3% of total SNPs in all cohorts. CONCLUSION: The heritability of neurodegenerative disorders cannot be represented as a single number, because it is dependent on the ages of cases and controls. Genome-wide association studies pick up a large proportion of total AD heritability when age and genetic architecture are correctly accounted for. Around 13% of SNP-based heritability can be explained by known genetic loci and the remaining heritability likely resides around microglial related genes.

Modelling of Lemna minor L. growth as influenced by nutrient supply, supplemental light, CO2 and harvest intervals for a continuous indoor cultivation.

Given the proper conditions, Lemna spp. rapidly produce a high amount of valuable biomass which is considered as an alternative source for feed and food. For a continuous and long-term indoor production under controlled conditions, environmental and harvest parameters have to be optimized to suppress algal growth and constantly yield a high-quality product. Experimentally assessing the effect of a larger number of parameters on the growth rate ri is impossible due to the theoretically high number of parameter combinations. Thus, a SIMILE® - based model has been developed. This enables production parameters to be assessed individually for its effect on the growth rate r i by a differential equation. Start values for numerical integration were taken from measured data and analytical solutions of the differential growth equation. At 400 ppm CO2, the regrowth rate ri in an optimized laboratory set-up amounted to 216 g FM·m-2d-1, harvesting one third of the biomass at intervals of 5 days. In up-scaled set-ups, lower regrowth rates ri of about 173 g FM·m-2d-1 (Kalkar) and 190 g FM·m-2d-1 (Berlin) were obtained, because temperature and light conditions were below optimum. At 3,500 ppm CO2, the regrowth rate ri in laboratory set-up increased to 323 g FM·m-2d-1 by shortening the harvest interval to three days. Maximum growth rates ri were obtained with an NH4 +/NO3 - ratio of 1/9 at 1.14 mM total N concentration. The results indicate how to optimize culture conditions and harvest intervals. Model runs closely match the experimental data taken from the three different approaches and thus confirm the validity of the model.

Probability of Alzheimer's disease based on common and rare genetic variants.

BACKGROUND: Alzheimer's disease, among other neurodegenerative disorders, spans decades in individuals' life and exhibits complex progression, symptoms and pathophysiology. Early diagnosis is essential for disease prevention and therapeutic intervention. Genetics may help identify individuals at high risk. As thousands of genetic variants may contribute to the genetic risk of Alzheimer's disease, the polygenic risk score (PRS) approach has been shown to be useful for disease risk prediction. The APOE-ε4 allele is a known common variant associated with high risk to AD, but also associated with earlier onset. Rare variants usually have higher effect sizes than common ones; their impact may not be well captured by the PRS. Instead of standardised PRS, we propose to calculate the disease probability as a measure of disease risk that allows comparison between individuals. METHODS: We estimate AD risk as a probability based on PRS and separately accounting for APOE, AD rare variants and the disease prevalence in age groups. The mathematical framework makes use of genetic variants effect sizes from summary statistics and AD disease prevalence in age groups. RESULTS: The AD probability varies with respect to age, APOE status and presence of rare variants. In age group 65+, the probability of AD grows from 0.03 to 0.18 (without APOE) and 0.07 to 0.7 (APOE e4e4 carriers) as PRS increases. In 85+, these values are 0.08-0.6 and 0.3-0.85. Presence of rare mutations, e.g. in TREM2, may increase the probability (in 65+) from 0.02 at the negative tail of the PRS to 0.3. CONCLUSIONS: Our approach accounts for the varying disease prevalence in different genotype and age groups when modelling the APOE and rare genetic variants risk in addition to PRS. This approach has potential for use in a clinical setting and can easily be updated for novel rare variants and for other populations or confounding factors when appropriate genome-wide association data become available.

Challenges of Adjusting Single-Nucleotide Polymorphism Effect Sizes for Linkage Disequilibrium.

BACKGROUND: Genome-wide association studies (GWAS) were successful in identifying SNPs showing association with disease, but their individual effect sizes are small and require large sample sizes to achieve statistical significance. Methods of post-GWAS analysis, including gene-based, gene-set and polygenic risk scores, combine the SNP effect sizes in an attempt to boost the power of the analyses. To avoid giving undue weight to SNPs in linkage disequilibrium (LD), the LD needs to be taken into account in these analyses. OBJECTIVES: We review methods that attempt to adjust the effect sizes (ß-coefficients) of summary statistics, instead of simple LD pruning. METHODS: We subject LD adjustment approaches to a mathematical analysis, recognising Tikhonov regularisation as a framework for comparison. RESULTS: Observing the similarity of the processes involved with the more straightforward Tikhonov-regularised ordinary least squares estimate for multivariate regression coefficients, we note that current methods based on a Bayesian model for the effect sizes effectively provide an implicit choice of the regularisation parameter, which is convenient, but at the price of reduced transparency and, especially in smaller LD blocks, a risk of incomplete LD correction. CONCLUSIONS: There is no simple answer to the question which method is best, but where interpretability of the LD adjustment is essential, as in research aiming at identifying the genomic aetiology of disorders, our study suggests that a more direct choice of mild regularisation in the correction of effect sizes may be preferable.

POLARIS: Polygenic LD-adjusted risk score approach for set-based analysis of GWAS data.

Polygenic risk scores (PRSs) are a method to summarize the additive trait variance captured by a set of SNPs, and can increase the power of set-based analyses by leveraging public genome-wide association study (GWAS) datasets. PRS aims to assess the genetic liability to some phenotype on the basis of polygenic risk for the same or different phenotype estimated from independent data. We propose the application of PRSs as a set-based method with an additional component of adjustment for linkage disequilibrium (LD), with potential extension of the PRS approach to analyze biologically meaningful SNP sets. We call this method POLARIS: POlygenic Ld-Adjusted RIsk Score. POLARIS identifies the LD structure of SNPs using spectral decomposition of the SNP correlation matrix and replaces the individuals' SNP allele counts with LD-adjusted dosages. Using a raw genotype dataset together with SNP effect sizes from a second independent dataset, POLARIS can be used for set-based analysis. MAGMA is an alternative set-based approach employing principal component analysis to account for LD between markers in a raw genotype dataset. We used simulations, both with simple constructed and real LD-structure, to compare the power of these methods. POLARIS shows more power than MAGMA applied to the raw genotype dataset only, but less or comparable power to combined analysis of both datasets. POLARIS has the advantages that it produces a risk score per person per set using all available SNPs, and aims to increase power by leveraging the effect sizes from the discovery set in a self-contained test of association in the test dataset.

How allele frequency and study design affect association test statistics with misrepresentation errors.

We evaluate the effect of genotyping errors on the type-I error of a general association test based on genotypes, showing that, in the presence of errors in the case and control samples, the test statistic asymptotically follows a scaled non-central $\chi ^2$ distribution. We give explicit formulae for the scaling factor and non-centrality parameter for the symmetric allele-based genotyping error model and for additive and recessive disease models. They show how genotyping errors can lead to a significantly higher false-positive rate, growing with sample size, compared with the nominal significance levels. The strength of this effect depends very strongly on the population distribution of the genotype, with a pronounced effect in the case of rare alleles, and a great robustness against error in the case of large minor allele frequency. We also show how these results can be used to correct $p$-values.

On multiple-testing correction in genome-wide association studies.

The interpretation of the results of large association studies encompassing much or all of the human genome faces the fundamental statistical problem that a correspondingly large number of single nucleotide polymorphisms markers will be spuriously flagged as significant. A common method of dealing with these false positives is to raise the significance level for the individual tests for association of each marker. Any such adjustment for multiple testing is ultimately based on a more or less precise estimate for the actual overall type I error probability. We estimate this probability for association tests for correlated markers and show that it depends in a nonlinear way on the significance level for the individual tests. This dependence of the effective number of tests is not taken into account by existing multiple-testing corrections, leading to widely overestimated results. We demonstrate a simple correction for multiple testing, which can easily be calculated from the pairwise correlation and gives far more realistic estimates for the effective number of tests than previous formulae. The calculation is considerably faster than with other methods and hence applicable on a genome-wide scale. The efficacy of our method is shown on a constructed example with highly correlated markers as well as on real data sets, including a full genome scan where a conservative estimate only 8% above the permutation estimate is obtained in about 1% of computation time. As the calculation is based on pairwise correlations between markers, it can be performed at the stage of study design using public databases.

Susceptibility of biallelic haplotype and genotype frequencies to genotyping error.

With the availability of fast genotyping methods and genomic databases, the search for statistical association of single nucleotide polymorphisms with a complex trait has become an important methodology in medical genetics. However, even fairly rare errors occurring during the genotyping process can lead to spurious association results and decrease in statistical power. We develop a systematic approach to study how genotyping errors change the genotype distribution in a sample. The general M-marker case is reduced to that of a single-marker locus by recognizing the underlying tensor-product structure of the error matrix. Both method and general conclusions apply to the general error model; we give detailed results for allele-based errors of size depending both on the marker locus and the allele present. Multiple errors are treated in terms of the associated diffusion process on the space of genotype distributions. We find that certain genotype and haplotype distributions remain unchanged under genotyping errors, and that genotyping errors generally render the distribution more similar to the stable one. In case-control association studies, this will lead to loss of statistical power for nondifferential genotyping errors and increase in type I error for differential genotyping errors. Moreover, we show that allele-based genotyping errors do not disturb Hardy-Weinberg equilibrium in the genotype distribution. In this setting we also identify maximally affected distributions. As they correspond to situations with rare alleles and marker loci in high linkage disequilibrium, careful checking for genotyping errors is advisable when significant association based on such alleles/haplotypes is observed in association studies.

Individual SNP allele reconstruction from informative markers selected by a non-linear Gauss-type algorithm.

OBJECTIVES: In view of the linkage disequilibrium structure of the genome, the selection of maximally informative SNP markers is a fundamental issue in the design of association studies. Currently used selection methods rely on pairwise marker correlation or informativity measures for subsets of markers. Nevertheless, the selected markers do not provide a completely satisfactory description of the individual remaining markers. The number of tag markers can be further reduced by using haplotypic information, but then the results of association analysis are difficult to interpret. METHODS AND RESULTS: We propose a non-linear Gauss-type algorithm selecting a subset of markers which is optimal with respect to the informativity measures and allows an explicit reconstruction of all other known markers, thus permitting direct inference of allelic association. The selection is based on the haplotype distribution in the population, but can be adapted to work with unphased genotypes directly. CONCLUSIONS: The proposed algorithm provides a rational methodology of informative marker selection, allowing for control and optimisation of information content and full marker reconstruction. Moreover, the reconstruction step can also be applied to tag markers selected using a different method at the stage of study design, identifying those markers which cannot be uniquely recovered from the chosen tags.