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 Reliability Checks on the Indo-US Stellar Spectral Library Using Artificial Neural Networks and Principal Component AnalysisThe Indo-US coudé feed stellar spectral library (CFLIB) madeavailable to the astronomical community recently by Valdes et al. (2004,ApJS, 152, 251) contains spectra of 1273 stars in the spectral region3460 to 9464Å at a high resolution of 1Å (FWHM) and a widerange of spectral types. Cross-checking the reliability of this databaseis an important and desirable exercise since a number of stars in thisdatabase have no known spectral types and a considerable fraction ofstars has not so complete coverage in the full wavelength region of3460-9464Å resulting in gaps ranging from a few Å to severaltens of Å. We use an automated classification scheme based onArtificial Neural Networks (ANN) to classify all 1273 stars in thedatabase. In addition, principal component analysis (PCA) is carried outto reduce the dimensionality of the data set before the spectra areclassified by the ANN. Most importantly, we have successfullydemonstrated employment of a variation of the PCA technique to restorethe missing data in a sample of 300 stars out of the CFLIB. The Vertical Stellar Kinematics in Face-On Barred Galaxies: Estimating the Ages of BarsIn order to perform a detailed study of the stellar kinematics in thevertical axis of bars, we obtained high signal-to-noise spectra alongthe major and minor axes of the bars in a sample of 14 face-on galaxiesand used them to determine the line-of-sight stellar velocitydistribution, parameterized as a Gauss-Hermite series. With these data,we developed a diagnostic tool that allows one to distinguish betweenrecently formed and evolved bars, as well as to estimate their ages,assuming that bars form in vertically thin disks that are recognizableby low values for the vertical velocity dispersion σz.Through N-body realizations of bar unstable disk galaxies we were alsoable to check the timescales involved in the processes that give bars animportant vertical structure. We show that σz inevolved bars is roughly 100 km s-1, which translates to aheight scale of about 1.4 kpc, giving support to scenarios in whichbulges form through disk material. Furthermore, the bars in ournumerical simulations have values for σz generallysmaller than 50 km s-1, even after evolving for 2 Gyr,suggesting that a slow process is responsible for making bars asvertically thick as we observe. We verify theoretically that theSpitzer-Schwarzschild mechanism is quantitatively able to explain theseobservations if we assume that giant molecular clouds are twice asconcentrated along the bar as in the rest of the disk. Chromospherically Active Stars. XXIV. The Giant, Single-lined Binaries HD 37824, HD 181809, and HD 217188We have obtained spectroscopy and photometry of three chromosphericallyactive, single-lined spectroscopic binaries, HD 37824 (V1149 Ori), HD181809 (V4138 Sgr), and HD 217188 (AZ Psc). HD 37824 has a circularorbit with a period of 53.57 days. Its primary is a K0 III star, whilethe secondary is likely a G or K dwarf. HD 181809 has an orbit with aperiod of 13.04667 days and a low eccentricity of 0.040. The primary hasa spectral type of K0 III-IV, and its secondary is probably an M dwarf.The orbit of HD 217188 has a period of 47.1209 days and a moderatelyhigh eccentricity of 0.470. The spectral type of the primary is K0 III,while the secondary is likely an M dwarf. All three systems areestimated to have near solar iron abundances. Photometric observationsspanning 15-16 years for all three stars yield mean photometric periodsof 53.12, 59.85, and 90.89 days for HD 37824, HD 181809, and HD 217188,respectively. Thus, HD 37824 is rotating synchronously with the orbitalperiod, while HD 181809 and HD 217188 are both rotating considerablyslower than synchronously. All three stars show long-term variations inmean brightness and photometric amplitude, but no correlations areobserved between the seasonal mean brightness, photometric amplitude,and seasonal photometric period in any of the stars. No clear evidencefor long-term periodic variations in any of these parameters is present.The circular orbit of HD 37824 and the synchronous rotation of its Kgiant argue that the star is in the core helium-burning phase of itsevolution. The giant components of HD 181809 and HD 217188 areasynchronous rotators, and both systems have eccentric orbits. Thus,those two stars are likely first-ascent giants. First results from the ESO VLTI calibrators programThe ESO Very Large Telescope Interferometer (VLTI) is one of the leadinginterferometric facilities. It is equipped with several 8.2 and 1.8 mtelescopes, a large number of baselines up to 200 m, and with severalsubsystems designed to enable high quality measurements and to improvesignificantly the limits of sensitivities currently available tolong-baseline interferometry. The full scientific potential of the VLTIcan be exploited only if a consistent set of good quality calibrators isavailable. For this, a large number of observations of potentialcalibrators have been obtained during the commissioning phase of theVLTI. These data are publicly available. We briefly describe theinterferometer, the VINCI instrument used for the observations, the dataflow from acquisition to processed results, and we present and commenton the volume of observations gathered and scrutinized. The result is alist of 191 calibrator candidates, for which a total of 12 066observations can be deemed of satisfactory quality. We present a generalstatistical analysis of this sample, using as a starting point theangular diameters previously available in the literature. We derive thegeneral characteristics of the VLTI transfer function, and its trendwith time in the period 2001 through mid-2004. A second paper will bedevoted to a detailed investigation of a selected sample, aimed atestablishing a VLTI-based homogeneous system of calibrators. CHARM2: An updated Catalog of High Angular Resolution MeasurementsWe present an update of the Catalog of High Angular ResolutionMeasurements (CHARM, Richichi & Percheron \cite{CHARM}, A&A,386, 492), which includes results available until July 2004. CHARM2 is acompilation of direct measurements by high angular resolution methods,as well as indirect estimates of stellar diameters. Its main goal is toprovide a reference list of sources which can be used for calibrationand verification observations with long-baseline optical and near-IRinterferometers. Single and binary stars are included, as are complexobjects from circumstellar shells to extragalactic sources. The presentupdate provides an increase of almost a factor of two over the previousedition. Additionally, it includes several corrections and improvements,as well as a cross-check with the valuable public release observationsof the ESO Very Large Telescope Interferometer (VLTI). A total of 8231entries for 3238 unique sources are now present in CHARM2. Thisrepresents an increase of a factor of 3.4 and 2.0, respectively, overthe contents of the previous version of CHARM.The catalog is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/431/773 Local kinematics of K and M giants from CORAVEL/Hipparcos/Tycho-2 data. Revisiting the concept of superclustersThe availability of the Hipparcos Catalogue has triggered many kinematicand dynamical studies of the solar neighbourhood. Nevertheless, thosestudies generally lacked the third component of the space velocities,i.e., the radial velocities. This work presents the kinematic analysisof 5952 K and 739 M giants in the solar neighbourhood which includes forthe first time radial velocity data from a large survey performed withthe CORAVEL spectrovelocimeter. It also uses proper motions from theTycho-2 catalogue, which are expected to be more accurate than theHipparcos ones. An important by-product of this study is the observedfraction of only 5.7% of spectroscopic binaries among M giants ascompared to 13.7% for K giants. After excluding the binaries for whichno center-of-mass velocity could be estimated, 5311 K and 719 M giantsremain in the final sample. The UV-plane constructed from these datafor the stars with precise parallaxes (σπ/π≤20%) reveals a rich small-scale structure, with several clumpscorresponding to the Hercules stream, the Sirius moving group, and theHyades and Pleiades superclusters. A maximum-likelihood method, based ona Bayesian approach, has been applied to the data, in order to make fulluse of all the available stars (not only those with precise parallaxes)and to derive the kinematic properties of these subgroups. Isochrones inthe Hertzsprung-Russell diagram reveal a very wide range of ages forstars belonging to these groups. These groups are most probably relatedto the dynamical perturbation by transient spiral waves (as recentlymodelled by De Simone et al. \cite{Simone2004}) rather than to clusterremnants. A possible explanation for the presence of younggroup/clusters in the same area of the UV-plane is that they have beenput there by the spiral wave associated with their formation, while thekinematics of the older stars of our sample has also been disturbed bythe same wave. The emerging picture is thus one of dynamical streamspervading the solar neighbourhood and travelling in the Galaxy withsimilar space velocities. The term dynamical stream is more appropriatethan the traditional term supercluster since it involves stars ofdifferent ages, not born at the same place nor at the same time. Theposition of those streams in the UV-plane is responsible for the vertexdeviation of 16.2o ± 5.6o for the wholesample. Our study suggests that the vertex deviation for youngerpopulations could have the same dynamical origin. The underlyingvelocity ellipsoid, extracted by the maximum-likelihood method afterremoval of the streams, is not centered on the value commonly acceptedfor the radial antisolar motion: it is centered on < U > =-2.78±1.07 km s-1. However, the full data set(including the various streams) does yield the usual value for theradial solar motion, when properly accounting for the biases inherent tothis kind of analysis (namely, < U > = -10.25±0.15 kms-1). This discrepancy clearly raises the essential questionof how to derive the solar motion in the presence of dynamicalperturbations altering the kinematics of the solar neighbourhood: doesthere exist in the solar neighbourhood a subset of stars having no netradial motion which can be used as a reference against which to measurethe solar motion?Based on observations performed at the Swiss 1m-telescope at OHP,France, and on data from the ESA Hipparcos astrometry satellite.Full Table \ref{taba1} is only available in electronic form at the CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/430/165} VINCI-VLTI measurements of HR 4049: The physical size of the circumbinary envelopeWe present the first detection of the envelope which surrounds thepost-AGB binary source HR 4049. VINCI-VLTI K-band interferometricobservations of this source imply the existence of a large structurewith a Gaussian angular FWHM 22.4 ± 1.4 mas or uniform diskdiameter of 34.9 ± 1.9 mas. With the Hipparcos parallax of 1.50± 0.64 mas these values correspond to a physical size of14.9+11.1-4.4 AU and23.3+17.3-7.0 AU, respectively. Our measurements,covering an azimuth range of 60 ° for the sky-projectedbaseline, provide information on the geometry of the emitting region andshow that there is only a slight variation of the measured angularvalues along the different directions sampled. Thus, our results areconsistent with a spherical geometry of the envelope. However, we cannotcompletely rule out the existence of an asymmetric envelope (like thecircumbinary disk envisaged by some recent models) because of thelimited spatial frequency and azimuth range covered by the observations.Based on public shared risk science program data released from the ESOVLTI: http://www.eso.org/projects/vlti/instru/vinci/vinci_data_sets.html The Indo-US Library of Coudé Feed Stellar SpectraWe have obtained spectra for 1273 stars using the 0.9 m coudéfeed telescope at Kitt Peak National Observatory. This telescope feedsthe coudé spectrograph of the 2.1 m telescope. The spectra havebeen obtained with the no. 5 camera of the coudé spectrograph anda Loral 3K×1K CCD. Two gratings have been used to provide spectralcoverage from 3460 to 9464 Å, at a resolution of ~1 Å FWHMand at an original dispersion of 0.44 Å pixel-1. For885 stars we have complete spectra over the entire 3460 to 9464 Åwavelength region (neglecting small gaps of less than 50 Å), andpartial spectral coverage for the remaining stars. The 1273 stars havebeen selected to provide broad coverage of the atmospheric parametersTeff, logg, and [Fe/H], as well as spectral type. The goal ofthe project is to provide a comprehensive library of stellar spectra foruse in the automated classification of stellar and galaxy spectra and ingalaxy population synthesis. In this paper we discuss thecharacteristics of the spectral library, viz., details of theobservations, data reduction procedures, and selection of stars. We alsopresent a few illustrations of the quality and information available inthe spectra. The first version of the complete spectral library is nowpublicly available from the National Optical Astronomy Observatory(NOAO) via ftp and http. High-Precision Near-Infrared Photometry of a Large Sample of Bright Stars Visible from the Northern HemisphereWe present the results of 8 yr of infrared photometric monitoring of alarge sample of stars visible from Teide Observatory (Tenerife, CanaryIslands). The final archive is made up of 10,949 photometric measuresthrough a standard InSb single-channel photometer system, principally inJHK, although some stars have measures in L'. The core of this list ofstars is the standard-star list developed for the Carlos SánchezTelescope. A total of 298 stars have been observed on at least twooccasions on a system carefully linked to the zero point defined byVega. We present high-precision photometry for these stars. The medianuncertainty in magnitude for stars with a minimum of four observationsand thus reliable statistics ranges from 0.0038 mag in J to 0.0033 magin K. Many of these stars are faint enough to be observable with arraydetectors (42 are K>8) and thus to permit a linkage of the bright andfaint infrared photometric systems. We also present photometry of anadditional 25 stars for which the original measures are no longeravailable, plus photometry in L' and/or M of 36 stars from the mainlist. We calculate the mean infrared colors of main-sequence stars fromA0 V to K5 V and show that the locus of the H-K color is linearlycorrelated with J-H. The rms dispersion in the correlation between J-Hand H-K is 0.0073 mag. We use the relationship to interpolate colors forall subclasses from A0 V to K5 V. We find that K and M main-sequence andgiant stars can be separated on the color-color diagram withhigh-precision near-infrared photometry and thus that photometry canallow us to identify potential mistakes in luminosity classclassification. A catalogue of calibrator stars for long baseline stellar interferometryLong baseline stellar interferometry shares with other techniques theneed for calibrator stars in order to correct for instrumental andatmospheric effects. We present a catalogue of 374 stars carefullyselected to be used for that purpose in the near infrared. Owing toseveral convergent criteria with the work of Cohen et al.(\cite{cohen99}), this catalogue is in essence a subset of theirself-consistent all-sky network of spectro-photometric calibrator stars.For every star, we provide the angular limb-darkened diameter, uniformdisc angular diameters in the J, H and K bands, the Johnson photometryand other useful parameters. Most stars are type III giants withspectral types K or M0, magnitudes V=3-7 and K=0-3. Their angularlimb-darkened diameters range from 1 to 3 mas with a median uncertaintyas low as 1.2%. The median distance from a given point on the sky to theclosest reference is 5.2degr , whereas this distance never exceeds16.4degr for any celestial location. The catalogue is only available inelectronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/393/183 CHARM: A Catalog of High Angular Resolution MeasurementsThe Catalog of High Angular Resolution Measurements (CHARM) includesmost of the measurements obtained by the techniques of lunaroccultations and long-baseline interferometry at visual and infraredwavelengths, which have appeared in the literature or have otherwisebeen made public until mid-2001. A total of 2432 measurements of 1625sources are included, along with extensive auxiliary information. Inparticular, visual and infrared photometry is included for almost allthe sources. This has been partly extracted from currently availablecatalogs, and partly obtained specifically for CHARM. The main aim is toprovide a compilation of sources which could be used as calibrators orfor science verification purposes by the new generation of largeground-based facilities such as the ESO Very Large Interferometer andthe Keck Interferometer. The Catalog is available in electronic form atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/386/492, and from theauthors on CD-Rom. Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition - Comments and statisticsThe Catalogue, available at the Centre de Données Stellaires deStrasbourg, consists of 13 573 records concerning the results obtainedfrom different methods for 7778 stars, reported in the literature. Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521 Research Note Hipparcos photometry: The least variable starsThe data known as the Hipparcos Photometry obtained with the Hipparcossatellite have been investigated to find those stars which are leastvariable. Such stars are excellent candidates to serve as standards forphotometric systems. Their spectral types suggest in which parts of theHR diagrams stars are most constant. In some cases these values stronglyindicate that previous ground based studies claiming photometricvariability are incorrect or that the level of stellar activity haschanged. Table 2 is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/367/297 Sixth Catalogue of Fundamental Stars (FK6). Part III. Additional fundamental stars with direct solutionsThe FK6 is a suitable combination of the results of the HIPPARCOSastrometry satellite with ground-based data, measured over a longinterval of time and summarized mainly in the FK5. Part III of the FK6(abbreviated FK6(III)) contains additional fundamental stars with directsolutions. Such direct solutions are appropriate for single stars or forobjects which can be treated like single stars. Part III of the FK6contains in total 3272 stars. Their ground-based data stem from thebright extension of the FK5 (735 stars), from the catalogue of remainingSup stars (RSup, 732 stars), and from the faint extension of the FK5(1805 stars). From the 3272 stars in Part III, we have selected 1928objects as "astrometrically excellent stars", since their instantaneousproper motions and their mean (time-averaged) ones do not differsignificantly. Hence most of the astrometrically excellent stars arewell-behaving "single-star candidates" with good astrometric data. Thesestars are most suited for high-precision astrometry. On the other hand,354 of the stars in Part III are Δμ binaries in the sense ofWielen et al. (1999). Many of them are newly discovered probablebinaries with no other hitherto known indication of binarity. The FK6gives, besides the classical "single-star mode" solutions (SI mode),other solutions which take into account the fact that hidden astrometricbinaries among "apparently single-stars" introduce sizable "cosmicerrors" into the quasi-instantaneously measured HIPPARCOS proper motionsand positions. The FK6 gives, in addition to the SI mode, the "long-termprediction (LTP) mode" and the "short-term prediction (STP) mode". TheseLTP and STP modes are on average the most precise solutions forapparently single stars, depending on the epoch difference with respectto the HIPPARCOS epoch of about 1991. The typical mean error of anFK6(III) proper motion in the single-star mode is 0.59 mas/year. This isa factor of 1.34 better than the typical HIPPARCOS errors for thesestars of 0.79 mas/year. In the long-term prediction mode, in whichcosmic errors are taken into account, the FK6(III) proper motions have atypical mean error of 0.93 mas/year, which is by a factor of about 2better than the corresponding error for the HIPPARCOS values of 1.83mas/year (cosmic errors included). K-Band Calibration of the Red Clump LuminosityThe average near-infrared (K-band) luminosity of 238 Hipparcos red clumpgiants is derived and then used to measure the distance to the Galacticcenter. These Hipparcos red clump giants have been previously employedas I-band standard candles. The advantage of the K-band is a decreasedsensitivity to reddening and perhaps a reduced systematic dependence onmetallicity. In order to investigate the latter, and also to refer ourcalibration to a known metallicity zero point, we restrict our sample ofred clump calibrators to those with abundances derived fromhigh-resolution spectroscopic data. The mean metallicity of the sampleis [Fe/H]=-0.18 dex (σ=0.17 dex). The data are consistent with nocorrelation between MK and [Fe/H] and only weakly constrainthe slope of this relation. The luminosity function of the sample peaksat MK=-1.61+/-0.03 mag. Next, we assemble published opticaland near-infrared photometry for ~20 red clump giants in a Baade'swindow field with a mean metallicity of [Fe/H]=-0.17+/-0.09 dex, whichis nearly identical to that of the Hipparcos red clump. Assuming thatthe average (V-I)0 and (V-K)0 colors of these twored clumps are the same, the extinctions in the Baade's window field arefound to be AV=1.56, AI=0.87, andAK=0.15, in agreement with previous estimates. We derive thedistance to the Galactic center: (m-M)0=14.58+/-0.11 mag, orR=8.24+/-0.42 kpc. The uncertainty in this distance measurement isdominated by the small number of Baade's window red clump giantsexamined here. Revision and Calibration of MK Luminosity Classes for Cool Giants by HIPPARCOS ParallaxesThe Hipparcos parallaxes of cool giants are utilized in two ways in thispaper. First, a plot of reduced parallaxes of stars brighter than 6.5,as a function of spectral type, for the first time separates members ofthe clump from stars in the main giant ridge. A slight modification ofthe MK luminosity standards has been made so that luminosity class IIIbdefines members of the clump, and nearly all of the class III stars fallwithin the main giant ridge. Second, a new calibration of MK luminosityclasses III and IIIb in terms of visual absolute magnitudes has beenmade. Spectral Irradiance Calibration in the Infrared. X. A Self-Consistent Radiometric All-Sky Network of Absolutely Calibrated Stellar SpectraWe start from our six absolutely calibrated continuous stellar spectrafrom 1.2 to 35 μm for K0, K1.5, K3, K5, and M0 giants. These wereconstructed as far as possible from actual observed spectral fragmentstaken from the ground, the Kuiper Airborne Observatory, and the IRAS LowResolution Spectrometer, and all have a common calibration pedigree.From these we spawn 422 calibrated spectral templates'' for stars withspectral types in the ranges G9.5-K3.5 III and K4.5-M0.5 III. Wenormalize each template by photometry for the individual stars usingpublished and/or newly secured near- and mid-infrared photometryobtained through fully characterized, absolutely calibrated,combinations of filter passband, detector radiance response, and meanterrestrial atmospheric transmission. These templates continue ourongoing effort to provide an all-sky network of absolutely calibrated,spectrally continuous, stellar standards for general infrared usage, allwith a common, traceable calibration heritage. The wavelength coverageis ideal for calibration of many existing and proposed ground-based,airborne, and satellite sensors, particularly low- tomoderate-resolution spectrometers. We analyze the statistics of probableuncertainties, in the normalization of these templates to actualphotometry, that quantify the confidence with which we can assert thatthese templates truly represent the individual stars. Each calibratedtemplate provides an angular diameter for that star. These radiometricangular diameters compare very favorably with those directly observedacross the range from 1.6 to 21 mas. A catalog of rotational and radial velocities for evolved starsRotational and radial velocities have been measured for about 2000evolved stars of luminosity classes IV, III, II and Ib covering thespectral region F, G and K. The survey was carried out with the CORAVELspectrometer. The precision for the radial velocities is better than0.30 km s-1, whereas for the rotational velocity measurementsthe uncertainties are typically 1.0 km s-1 for subgiants andgiants and 2.0 km s-1 for class II giants and Ib supergiants.These data will add constraints to studies of the rotational behaviourof evolved stars as well as solid informations concerning the presenceof external rotational brakes, tidal interactions in evolved binarysystems and on the link between rotation, chemical abundance and stellaractivity. In this paper we present the rotational velocity v sin i andthe mean radial velocity for the stars of luminosity classes IV, III andII. Based on observations collected at the Haute--Provence Observatory,Saint--Michel, France and at the European Southern Observatory, LaSilla, Chile. Table \ref{tab5} also available in electronic form at CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html Catalogs of temperatures and [Fe/H] averages for evolved G and K starsA catalog of mean values of [Fe/H] for evolved G and K stars isdescribed. The zero point for the catalog entries has been establishedby using differential analyses. Literature sources for those entries areincluded in the catalog. The mean values are given with rms errors andnumbers of degrees of freedom, and a simple example of the use of thesestatistical data is given. For a number of the stars with entries in thecatalog, temperatures have been determined. A separate catalogcontaining those data is briefly described. Catalog only available atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html Towards a fundamental calibration of stellar parameters of A, F, G, K dwarfs and giantsI report on the implementation of the empirical surface brightnesstechnique using the near-infrared Johnson broadband { (V-K)} colour assuitable sampling observable aimed at providing accurate effectivetemperatures of 537 dwarfs and giants of A-F-G-K spectral-type selectedfor a flux calibration of the Infrared Space Observatory (ISO). Thesurface brightness-colour correlation is carefully calibrated using aset of high-precision angular diameters measured by moderninterferometry techniques. The stellar sizes predicted by thiscorrelation are then combined with the bolometric flux measurementsavailable for a subset of 327 ISO standard stars in order to determineone-dimensional { (T, V-K)} temperature scales of dwarfs and giants. Theresulting very tight relationships show an intrinsic scatter induced byobservational photometry and bolometric flux measurements well below thetarget accuracy of +/- 1 % required for temperature determinations ofthe ISO standards. Major improvements related to the actual directcalibration are the high-precision broadband { K} magnitudes obtainedfor this purpose and the use of Hipparcos parallaxes for dereddeningphotometric data. The temperature scale of F-G-K dwarfs shows thesmallest random errors closely consistent with those affecting theobservational photometry alone, indicating a negligible contributionfrom the component due to the bolometric flux measurements despite thewide range in metallicity for these stars. A more detailed analysisusing a subset of selected dwarfs with large metallicity gradientsstrongly supports the actual bolometric fluxes as being practicallyunaffected by the metallicity of field stars, in contrast with recentresults claiming somewhat significant effects. The temperature scale ofF-G-K giants is affected by random errors much larger than those ofdwarfs, indicating that most of the relevant component of the scattercomes from the bolometric flux measurements. Since the giants have smallmetallicities, only gravity effects become likely responsible for theincreased level of scatter. The empirical stellar temperatures withsmall model-dependent corrections are compared with the semiempiricaldata by the Infrared Flux Method (IRFM) using the large sample of 327comparison stars. One major achievement is that all empirical andsemiempirical temperature estimates of F-G-K giants and dwarfs are foundto be closely consistent between each other to within +/- 1 %. However,there is also evidence for somewhat significant differential effects.These include an average systematic shift of (2.33 +/- 0.13) % affectingthe A-type stars, the semiempirical estimates being too low by thisamount, and an additional component of scatter as significant as +/- 1 %affecting all the comparison stars. The systematic effect confirms theresults from other investigations and indicates that previousdiscrepancies in applying the IRFM to A-type stars are not yet removedby using new LTE line-blanketed model atmospheres along with the updatedabsolute flux calibration, whereas the additional random component isfound to disappear in a broadband version of the IRFM using an infraredreference flux derived from wide rather than narrow band photometricdata. Table 1 and 2 are only available in the electronic form of thispaper A catalogue of [Fe/H] determinations: 1996 editionA fifth Edition of the Catalogue of [Fe/H] determinations is presentedherewith. It contains 5946 determinations for 3247 stars, including 751stars in 84 associations, clusters or galaxies. The literature iscomplete up to December 1995. The 700 bibliographical referencescorrespond to [Fe/H] determinations obtained from high resolutionspectroscopic observations and detailed analyses, most of them carriedout with the help of model-atmospheres. The Catalogue is made up ofthree formatted files: File 1: field stars, File 2: stars in galacticassociations and clusters, and stars in SMC, LMC, M33, File 3: numberedlist of bibliographical references The three files are only available inelectronic form at the Centre de Donnees Stellaires in Strasbourg, viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5), or viahttp://cdsweb.u-strasbg.fr/Abstract.html Transformations from Theoretical Hertzsprung-Russell Diagrams to Color-Magnitude Diagrams: Effective Temperatures, B-V Colors, and Bolometric CorrectionsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1996ApJ...469..355F&db_key=AST Vitesses radiales. Catalogue WEB: Wilson Evans Batten. Subtittle: Radial velocities: The Wilson-Evans-Batten catalogue.We give a common version of the two catalogues of Mean Radial Velocitiesby Wilson (1963) and Evans (1978) to which we have added the catalogueof spectroscopic binary systems (Batten et al. 1989). For each star,when possible, we give: 1) an acronym to enter SIMBAD (Set ofIdentifications Measurements and Bibliography for Astronomical Data) ofthe CDS (Centre de Donnees Astronomiques de Strasbourg). 2) the numberHIC of the HIPPARCOS catalogue (Turon 1992). 3) the CCDM number(Catalogue des Composantes des etoiles Doubles et Multiples) byDommanget & Nys (1994). For the cluster stars, a precise study hasbeen done, on the identificator numbers. Numerous remarks point out theproblems we have had to deal with. Stellar effective temperatures and angular diameters determined by the infrared flux method (IRFM): Revisions using improved Kurucz LTE stellar atmospheresInfrared flux method (IRFM) determinations of stellar effectivetemperatures and angular diameters are revised using new Kurucz localthermodynamic equilibrium (LTE) line-blanketed model atmospheres, whichmore accurately predict the emergent stellar radiation flux than modelsused previously. An improved method for deriving integrated stellarfluxes is described, together with polynomial coefficients forevaluating them from V and V-K. Tables were given for making smallcorrections appropiate to changes in log g, metallicity, fluxcalibration in J, K and L and interstellar extinction, to avoid the needfor additional tabular material. The determined temperatures areexpressed in terms of V and V-K, giving a mean absolute deviation of0.53%. A comparison of the derived angular diameters for three starswith determinations using Michelson interferometer shows an averageagreement to better than 1%. Evolved GK stars near the sun. I - The old disk populationA sample of nearly two thousand GK giants with intermediate band, (R,I),DDO and Geneva photometry has been assembled. Astrometric data is alsoavailable for most of the stars. The some 800 members of the old diskpopulation in the sample yield accurate luminosities (from two sources),reddening values and chemical abundances from calibrations of thephotometric parameters. Less than one percent of the objects arepeculiar in the sense that the flux distribution is abnormal. Thepeculiarity is signaled by strong CH (and Ba II) and weak CH. The CH+stars are all spectroscopic binaries, probably with white dwarfcompanions, whereas the CH- stars are not. A broad absorption band,centered near 3500 A, is found in the CH+ stars whereas the CH- objectshave a broad emission feature in the same region. The intensity of theseabsorptions and emissions are independent of the intensity of abnormalspectral features. Ten percent of the old disk sample have a heavyelement abundance from one and a half to three times the solar value.The distribution of the heavy element abundances is nearly a normal onewith a peak near solar abundance and ranges three times to one sixthsolar. The distribution of the (U, V) velocities is independent of theheavy element abundance and does not appear to be random. Ten percent ofthe old disk stars show a CN anomaly, equally divided between CN strongand CN weak. Several stars of individual astrometric or astrophysicalimportance are isolated. The truth about star colors.Not Available Velocity dispersions and mean abundances for Roman's G5-K1 spectroscopic groupsThe velocity dispersions and U-V distributions of Roman's (1950, 1952)four spectroscopic groups (weak CN, weak line, strong line, and 4150)are compared with those of groups based only on Fe/H ratio. It is shownthat the velocity-dispersion gradient for the Roman spectroscopic groupsis greater than for the comparison group for all three velocitycomponents and for the mean orbital eccentricity, with a clearly definedminimum for the strong-line stars and a sharp upturn for the 4150 stars.The results suggest that the Roman's assignment to distinctspectroscopic groups results in more homogeneous groups than the binningon the basis of metallicity. K Giant Magnitude Calibration from ParallaxesAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1992A&A...254..105G&db_key=AST Fifth fundamental catalogue. Part 2: The FK5 extension - new fundamental starsThe mean positions and proper motions for 3117 new fundamental starsessentially in the magnitude range about 4.5 to 9.5 are given in thisFK5 extension. Mean apparent visual magnitude is 7.2 and is on average2.5 magnitudes fainter then the basic FK5 which has a mean magnitude of4.7. (The basic FK5 gives the mean positions and proper motions for theclassical 1535 fundamental stars). The following are discussed: theobservational material, reduction of observations, star selection, andthe system for the FK5 extension. An explanation and description of thecatalog are given. The catalog of 3117 fundamental stars for the equinoxand epoch J2000.0 and B1950.0 is presented. The parallaxes and radialvelocities for 22 extension stars with large forecasting effects aregiven. Catalogs used in the compilation of the FK5 fundamental catalogare listed. A critical appraisal of published values of (Fe/H) for K II-IV stars'Primary' (Fe/H) averages are presented for 373 evolved K stars ofluminosity classes II-IV and (Fe/H) values beween -0.9 and +0.21 dex.The data define a 'consensus' zero point with a precision of + or -0.018 dex and have rms errors per datum which are typically 0.08-0.16dex. The primary data base makes recalibration possible for the large(Fe/H) catalogs of Hansen and Kjaergaard (1971) and Brown et al. (1989).A set of (Fe/H) standard stars and a new DDO calibration are given whichhave rms of 0.07 dex or less for the standard star data. For normal Kgiants, CN-based values of (Fe/H) turn out to be more precise than manyhigh-dispersion results. Some zero-point errors in the latter are alsofound and new examples of continuum-placement problems appear. Thushigh-dispersion results are not invariably superior to photometricmetallicities. A review of high-dispersion and related work onsupermetallicity in K III-IV star is also given.
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Article| Volume 91, ISSUE 6, P1011-1021, December 07, 2012

# Improved Heritability Estimation from Genome-wide SNPs

Open Access
Estimation of narrow-sense heritability, h2, from genome-wide SNPs genotyped in unrelated individuals has recently attracted interest and offers several advantages over traditional pedigree-based methods. With the use of this approach, it has been estimated that over half the heritability of human height can be attributed to the ∼300,000 SNPs on a genome-wide genotyping array. In comparison, only 5%–10% can be explained by SNPs reaching genome-wide significance. We investigated via simulation the validity of several key assumptions underpinning the mixed-model analysis used in SNP-based h2 estimation. Although we found that the method is reasonably robust to violations of four key assumptions, it can be highly sensitive to uneven linkage disequilibrium (LD) between SNPs: contributions to h2 are overestimated from causal variants in regions of high LD and are underestimated in regions of low LD. The overall direction of the bias can be up or down depending on the genetic architecture of the trait, but it can be substantial in realistic scenarios. We propose a modified kinship matrix in which SNPs are weighted according to local LD. We show that this correction greatly reduces the bias and increases the precision of h2 estimates. We demonstrate the impact of our method on the first seven diseases studied by the Wellcome Trust Case Control Consortium. Our LD adjustment revises downward the h2 estimate for immune-related diseases, as expected because of high LD in the major-histocompatibility region, but increases it for some nonimmune diseases. To calculate our revised kinship matrix, we developed LDAK, software for computing LD-adjusted kinships.

## Introduction

The linear mixed model, long a mainstay of heritability estimation,
• Henderson C.
• Kempthorne O.
• Searle S.
• von Krosigk C.
The estimation of environmental and genetic trends from records subject to culling.
• Hartley H.O.
• Rao J.N.
Maximum-likelihood estimation for the mixed analysis of variance model.
• Robinson G.
That BLUP is a good thing: The estimation of random effects.
fits a covariance structure specified by a matrix of kinship coefficients to a vector of measured phenotypes. The term “mixed” refers to the presence of both an unobserved random effect, usually interpreted in terms of a polygenic contribution to the trait, and one or more fixed effects corresponding to individual SNPs or other covariates. For association analysis, the SNP effects are of interest and the random effect is used for overcoming confounding due to population structure and cryptic relatedness.
• Astle W.
• Balding D.
Population structure and cryptic relatedness in genetic association studies.
In plant and animal breeding, estimation of the random effects (“breeding values”) is of primary importance because these values reflect a “true” phenotype adjusted for environmental effects or measurement error. Narrow-sense-heritability estimates $(hˆ2)$ are based on the squared regression coefficient of the random effect.
Mixed-model analysis in quantitative genetics was developed by animal breeders decades ago, but the advent of dense genome-wide SNP data has radically enhanced its applicability, for example, to the use of apparently unrelated individuals. This might seem counterintuitive given that relatedness is central to heritability, but the key insight made by Yang et al.
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
is that dense genotype data permit the exploitation of small differences in the proportions of genome shared among apparently unrelated individuals, and this has advantages over the traditional pedigree-based approaches. Specifically, it is the short genomic regions passed down to unrelated individuals from their remote common ancestors that generate linkage disequilibrium (LD). So, h2 estimated from unrelated individuals corresponds only to the causal-variant heritability that is tagged by the genotyped SNPs (this is sometimes referred to as the “chip heritability”). Investigating the genetic architecture of complex traits thus becomes possible through the estimation of h2 components tagged by different SNP sets, which one can obtain, for example, by imposing minor-allele-frequency (MAF) thresholds or restricting attention to specific pathways or genomic regions (termed “genomic partitioning”).
• Yang J.
• Manolio T.A.
• Pasquale L.R.
• Boerwinkle E.
• Caporaso N.
• Cunningham J.M.
• de Andrade M.
• Feenstra B.
• Feingold E.
• Hayes M.G.
• et al.
Genome partitioning of genetic variation for complex traits using common SNPs.
The usual mixed models in quantitative genetics assume independent Gaussian effect sizes. The “thin tails” property of the Gaussian distribution makes it unrealistic for individual SNP effect sizes, but we illustrate by simulating from distributions with thicker tails that $hˆ2$ is reasonably robust to this assumption. Further, the standardization made by Yang et al.
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
and other authors when they computed SNP-based kinship coefficients implies a specific relationship between MAF and the variance of effect sizes such that rarer SNPs tend to have larger effect sizes. We show that $hˆ2$ is somewhat robust to the relationship between MAF and effect size. A polygenic assumption, one of many small phenotypic effects distributed genome-wide, is often employed for justifying a mixed-model analysis. This assumption proves to be unnecessary because $hˆ2$ remains unbiased (although precision is eroded) as the number of causal variants is reduced, even down to a monogenic model.
Although these results support the use of SNP-based mixed-model analysis for estimating h2 from unrelated individuals, we did uncover serious cause for concern: linkage disequilibrium (LD) can generate large biases. Contributions to h2 from causal variants tend to be overestimated in regions of strong LD and underestimated in regions of low LD. Of course, SNP-based $hˆ2$ cannot capture any causal-variation component that is not tagged by a genotyped SNP, but we show that even when all causal variants are genotyped and hence fully tagged, SNP-based $hˆ2$ reflects patterns of LD, in addition to causal variation. Likewise, the contributions to $hˆ2$ from untyped causal variants can be overestimated or underestimated according to their LD with genotyped SNPs.
Patterns of LD are strongly linked to MAF: on average, the signals from low-frequency variants are less replicated than those from high-frequency variants. Therefore, $hˆ2$ will be too low for traits with predominantly low-frequency causal variants and will be too high for those with predominantly high-frequency causal variants. This has consequences when one performs genomic partitioning to investigate the frequency spectrum of causal variants for complex traits. Yang et al.
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
suggested a transformation that involves uniformly scaling the usual SNP-based kinship coefficients; this transformation counteracts the average bias caused by LD but requires prior knowledge of the MAF spectrum of causal variants. We propose a different adjustment in which SNPs are weighted according to how well they are tagged by their neighbors. Using simulated data, we demonstrate that our adjustment both greatly reduces bias and increases the precision of SNP-based $hˆ2$.
When we reanalyzed the height data
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
with our LD-adjusted kinship matrix, $hˆ2$ changed only slightly, suggesting that any underestimation of contributions to h2 in low-LD regions is balanced by overestimation elsewhere. We also analyzed seven traits studied by the Wellcome Trust Case Control Consortium (WTCCC).
Wellcome Trust Case Control Consortium
Genome-wide association study of 14,000 cases of seven common diseases and 3,000 shared controls.
For both hypertension and type 2 diabetes, $hˆ2$ increased by nearly a quarter when we used our LD-adjusted kinships instead of a standard kinship matrix, suggesting that these traits’ causal variants tend to be poorly tagged and thus have a lower-than-average MAF. By contrast, for rheumatoid arthritis, $hˆ2$ was reduced by one-tenth when we used LD-adjusted kinships. This disease, along with type 1 diabetes, has major-histocompatibility-complex (MHC) risk variants that tend to be well tagged; the estimated heritability attributed to chromosome 6 was substantially reduced for both these diseases.

## Material and Methods

### The Linear Mixed Model

The essence of mixed-model analysis in quantitative genetics is partitioning the phenotypic variance-covariance matrix between two (or more) specified matrices. Given phenotypic values $Y=(Y1,…,Yn)$, a typical form is
$Var(Y)=σg2G+σe2I,$
(Equation 1)

where G is a matrix of kinship coefficients and I is the n × n identity matrix, which implicitly assumes independence across individuals of environmental effects and measurement error. Estimates $σˆg2$ and $σˆe2$ are typically obtained via restricted maximum likelihood (REML).
• Corbeil R.
• Searle S.
Restricted maximum likelihood (REML) estimation of variance components in the mixed model.
Then, h2, the $Var(Y)$ proportion explained by additive genetic effects, is estimated by
$hˆ2=σˆg2σˆg2+σˆe2.$
(Equation 2)

We can derive Equation 1 by assuming
$Yi=μ+∑j=1mZijuj+ei,$
(Equation 3)

in which $Zij$ is the genotype of individual i at the jth of m diallelic causal loci given an additive coding of genotypes standardized to have zero mean and unit variance for each j and in which $ei$ and $uj$ are independently Gaussian with zero mean and variances $σe2$ and $σg2/m$, respectively. Then, Y is multivariate Gaussian with $E(Yi)=μ$ and
$Var(Y)=σg2mZZT+σe2I,$

where Z denotes the matrix of $Zij$ values. We thus have Equation 1 with $ZZT/m$ in place of G. If Z were not column standardized, Equation 2 might not hold; a more general form is given in Appendix B.
With Z column standardized, $ZTZ/n$ is a sample correlation matrix of causal variants. It follows that each $ZijZi′j$ can be regarded as an estimator of the population correlation coefficient at locus j on the basis of just one pair of individuals, i and i′. Therefore, the ii′ entry of $ZZT/m$ is an average over loci of allelic correlation estimates for i and i′. This average can be interpreted as a measure of the excess sharing of causal alleles by i and i′ relative to independent allele assignments. Equation 3 specifies an additive model, and so $hˆ2$ in Equation 2 is an estimate of h2.
For binary outcomes, the same analysis can be applied, but the resulting $hˆ2$ depends on the case-control ratio, which can be chosen arbitrarily. One solution is to implement this analysis with a subsequent transformation of the estimate to an underlying liability scale adjusted for ascertainment effects.
• Dempster E.R.
• Lerner I.M.
Heritability of threshold characters.
• Lee S.H.
• Wray N.R.
• Goddard M.E.
• Visscher P.M.
Estimating missing heritability for disease from genome-wide association studies.

### Pedigree- and SNP-Based Kinship Matrices

Because $ZZT/m$ is unknown, it has traditionally been replaced by the kinship matrix G, whose ii′ entry is the probability that homologous alleles from i and i′ are identical by descent from common ancestors within a specified pedigree. Assuming that pedigree founders are completely unrelated, then G reflects genome-wide average allelic correlation, which might be a reasonable proxy for allelic correlation at the causal loci, for example, if little is known about the genetic architecture of the phenotype. However, the identity-by-descent (IBD) approach is unsatisfactory because IBD values depend on the available pedigree, which is always incomplete. In any case, the phenomenon of interest is genome sharing, for which pedigrees only give expected and not realized values.
Nowadays, the proxy for G can be kinship coefficients computed from genome-wide SNP genotypes rather than from a known pedigree. There are at least two popular coefficients for measuring the relatedness of two individuals from genome-wide SNPs:
• Astle W.
• Balding D.
Population structure and cryptic relatedness in genetic association studies.
their average allelic correlation and their proportion of shared alleles. These are sometimes labeled as IBD and IBS (identity by state) methods, but we avoid this terminology; in the absence of an explicit pedigree, meaningfully defining IBD is difficult. If we assume Equation 3, a natural choice is the average allelic correlation matrix $A=XXT/m′$, where X is defined in the same way as Z except that SNP genotypes replace causal-locus genotypes and m′ is the number of genotyped SNPs. A slight modification of A was adopted by Yang et al.
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
Although A is computed from data, in practice it is treated as known in Equation 1, as was the case for pedigree-based G.

### Simulation Study

We implemented a simulation study to check four key assumptions underlying $hˆ2$ obtained via mixed-model analysis in which G was replaced by $A=XXT/m′$, as well as to check its sensitivity to LD. We merged the 2,699 and 2,501 individuals who passed quality control (QC) from the UK 1958 Birth Cohort samples and National Blood Service samples, respectively.
Wellcome Trust Case Control Consortium
Genome-wide association study of 14,000 cases of seven common diseases and 3,000 shared controls.
Following Yang et al.,
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
we removed individuals so that $|Aii′|$ < 0.024 for all pairs ii′, leaving n = 5,127. Unless otherwise stated, results were obtained with only the m′ = 81,327 SNPs (on chromosomes 1 and 2) that passed QC, which included requiring a MAF > 0.01.
We generated 50 replicate phenotype vectors Y for each scenario. Except where specified otherwise, we chose for each scenario 100 causal variants and simulated Gaussian effect sizes to achieve h2 equal to either 0.5 or 0.8. Because SNP-based $hˆ2$ only measures the h2 proportion that is tagged by the SNPs, the true value of h2 is usually unknown. To allow us to assess the bias and precision of $hˆ2$, for the simulations reported here, we chose causal variants from among the genotyped SNPs so that h2 was known. Given Y and the kinship matrix, estimates $σˆg2$ and $σˆe2$, as well as estimates of their SDs, were obtained with the REML algorithm incorporated in the software GCTA (Genome-wide Complex Trait Analysis).
• Yang J.
• Lee S.H.
• Goddard M.E.
• Visscher P.M.
GCTA: A tool for genome-wide complex trait analysis.

### SNP-Based $hˆ2$: Implicit Assumptions

First, using $A=XXT/m′$ in place of G in Equation 1 supposes a polygenic model in which all SNPs contribute to Y. We investigated this assumption by varying the numbers of causal SNPs for fixed h2.
Second, assigning a common variance to the standardized effect sizes uj in Equation 3 implies an equal contribution to h2 from each causal locus. Under the assumption of Hardy-Weinberg equilibrium, this implies that vj, the per-allele effect size at the jth causal locus, has variance
$Var(vj)∝1pj(1−pj),$
(Equation 4)

where pj is the population MAF. A corresponding relationship applies to the SNP effect sizes when $ZZT/m$ is replaced with A. A similar assumption is implicit in many tests of genetic association,
• Wakefield J.
Bayes factors for genome-wide association studies: Comparison with P-values.
and a tendency for alleles of greater effect to have lower MAF is expected under a range of evolutionary models.
• Pritchard J.K.
• Cox N.J.
The allelic architecture of human disease genes: Common disease-common variant...or not?.
Using known susceptibility loci, Park et al.
• Park J.H.
• Wacholder S.
• Gail M.H.
• Peters U.
• Jacobs K.B.
• Chanock S.J.
• Chatterjee N.
Estimation of effect size distribution from genome-wide association studies and implications for future discoveries.
• Park J.H.
• Gail M.H.
• Weinberg C.R.
• Carroll R.J.
• Chung C.C.
• Wang Z.
• Chanock S.J.
• Fraumeni Jr., J.F.
• Chatterjee N.
Distribution of allele frequencies and effect sizes and their interrelationships for common genetic susceptibility variants.
found empirical evidence for this trend in all eight traits that they studied, even after adjusting for power to detect, and this relationship was significant for three of the traits. To investigate the impact of assuming Equation 4, we considered four relationships of the form
$Var(vj)∝[pj(1−pj)]α,$
(Equation 5)

which can be implemented in the computation of A by appropriate scaling of the column-centered genotypes (the standard scaling, dividing each column by the square root of its variance, corresponds to α = −1). When α < 0, per-allele effect sizes increase as the MAF decreases, whereas they decrease when α > 0. We simulated Y by assuming α = −2, −1, 0, and 1, and for each Y, we calculated $hˆ2$ by assuming each of the four effect-size models when calculating A. Here, it was important to use the full equation for calculating $hˆ2$ (Appendix B) because Equation 2 only applies when A is computed with α2 = −1.
The third implicit assumption that we checked is that of a Gaussian distribution for genetic effect sizes. This can be criticized because the “thin tails” of the Gaussian means that larger-than-usual effect sizes are strongly penalized. The normal exponential gamma (NEG) has been proposed as more realistic for SNP effects
• Hoggart C.J.
• Whittaker J.C.
• De Iorio M.
• Balding D.J.
Simultaneous analysis of all SNPs in genome-wide and re-sequencing association studies.
in that it has thicker tails than both the Gaussian and (when the shape parameter is small) the Laplace (double exponential) distributions. It can be modeled as a Laplace distribution with a gamma-distributed rate. Decreasing the gamma shape parameter leads to thicker tails, whereas increasing it recovers the Laplace distribution. We simulated Y by assuming NEG effect sizes with shape parameters of 10, 2, and 1, and we adjusted the scale parameter in each case to obtain the desired h2.
In a similar vein, our fourth test was to investigate the Gaussian assumption for the noise term ei in Equation 3. We did this by estimating h2 when ei was simulated from five other distributions, including both heavier-tailed and skewed distributions (see Figure S4A, available online).

### The Effect of LD on $hˆ2$

If a causal variant is tagged by multiple genotyped SNPs, then some or all of its signal can be replicated, and this can lead to overestimation of its contribution to h2. For the dense SNP sets used in current genome-wide association studies, LD induces strong correlations between SNPs, and SNP-based $hˆ2$ reflects patterns of LD, in addition to the architecture of causal variants. This problem does not depend on the total amount of tagging: if all SNPs were duplicated exactly once, the amount of tagging would double, yet A, and hence $hˆ2$, would remain unchanged. Instead, the problem concerns relative amounts of tagging: some of the causal variants are tagged more than others, distorting their estimated contributions to h2.
Pruning (or “thinning”) the columns of X is a common approach to reducing correlation among SNPs.
• Purcell S.
• Neale B.
• Todd-Brown K.
• Thomas L.
• Ferreira M.A.
• Bender D.
• Maller J.
• Sklar P.
• de Bakker P.I.
• Daly M.J.
• Sham P.C.
PLINK: A tool set for whole-genome association and population-based linkage analyses.
Pruning has limited consequences for identifying relatedness because genomic segments shared from a recent common ancestor will usually extend well beyond the usual range of LD. However, it is not a satisfactory solution for the problem of uneven tagging in obtaining $hˆ2$ from unrelated individuals because these share short genomic segments and so tagging of causal variation will be lost. Instead, we propose to overcome the problem by scaling SNP genotypes according to local patterns of LD. The goal of the scaling is that the signal from each SNP is downweighted so that replication of its signal by neighboring SNPs can be compensated for. We denote by A the weighted allelic correlation matrix, which is defined the same as A except that each column Xj of X is replaced by $wjXj$ and the denominator m′ is replaced by $∑jwj$. The wj’s are chosen so that
$wj+∑j′wj′rjj′2e−λdjj′$
(Equation 6)

is constant over j. Here,$rjj′2$ denotes the squared correlation between SNPs j and j′, a standard measure of LD, and the summation is over SNPs j′ such that $e−λdjj′$ > 0.125, where λ is a constant and djj′ denotes the base-pair distance between SNPs j and j′. The motivation for requiring constancy of Equation 6 is that it (almost) represents the total replication of the signal from SNP j (after weighting). Identifying wj’s satisfying this requirement is a linear programming problem (see Appendix A, where we also discuss appropriate values for λ). For example, a set of k SNPs all in perfect LD but in linkage equilibrium (LE) with all other SNPs would each be assigned (approximately) wj = 1/k, so the signal from a causal variant tagged by these SNPs would be counted once overall. Zou et al.,
• Zou F.
• Lee S.
• Knowles M.R.
• Wright F.A.
Quantification of population structure using correlated SNPs by shrinkage principal components.
when considering a similar problem, suggested weights satisfying
$1wj=1+∑rjj′2,$
(Equation7)

which also reduces to 1/k in this simple setting but provides a less satisfactory solution for incomplete LD (see Figure S1D). The value of $∑j′rjj′2e−λdjj′$ can be viewed as a measure of the tagging of SNP j; for the simulations of Figure 3, weakly and very weakly tagged causal variants were those in the bottom 40% and 20%, respectively, of values for this sum, whereas strongly and very strongly tagged causal variants were those in the top 40% and 20%, respectively.

## Results

### Polygenic Assumption

Figure 1A shows that $hˆ2$ remained approximately unbiased in our simulations as the number of causal variants m decreased down to one. A polygenic assumption is thus not required for $hˆ2$ to be useful, and even a large $hˆ2$ does not, taken alone, indicate a polygenic component to the genetic architecture of a trait. Intuitively, it seems reasonable that A remains a good proxy for $ZZT/m$ even for a single, randomly placed locus. As expected, the precision of $hˆ2$ reduced as m declined; this effect was largely eliminated when A was replaced with A.
Figure 1B compares the REML estimates of SD provided by GCTA with the empirical SD based on the 50 replicates for each scenario. GCTA never reported SD > 0.04, which greatly exaggerated the precision of $hˆ2$ when m was small. In contrast, when using our LD-adjusted kinship matrix, GCTA always gave an SD within 0.01 of the empirical estimate, even for small m.
When m is small, many SNPs do not tag any causal variant, so estimation could be improved by the removal of redundant SNPs from X.
• Listgarten J.
• Lippert C.
• Davidson R.I.
• Eskin E.
• Heckerman D.
Improved linear mixed models for genome-wide association studies.
As an illustration of the potential gain, in a further simulation (Figure S2), we found that when m′ included five times as many redundant SNPs as SNPs tagging causal variants, the empirical standard deviation of $hˆ2$ was approximately twice as much as when only the nonredundant SNPs were used. In practice, it would be challenging to identify SNPs that do not tag any causal variation. However, these results would suggest that inclusion of the redundant SNPs has limited consequence.

### Relationship between Effect-Size Variance and MAF

Considering relationships between MAF and causal effect-size variance of the form of Equation 5, Figure 2 shows that, as expected, $hˆ2$ was most accurate when the analysis assumption (α2) matched that of the simulation (α1). However, the standard choice of α2 = −1 (the second block of eight boxes) appeared to give the most stable $hˆ2$ among the models considered here and would therefore seem the most prudent choice when the correct relationship is uncertain. A similar pattern was observed when A was used instead of A (results not shown).

### The Assumption of Gaussian Effect Sizes and Noise Terms

The four panels of Figure S3A show the densities of the Gaussian and the three NEG distributions. Despite the important differences in the tails, the distribution of $hˆ2$ was only modestly affected by the choice of effect-size distribution (Figure S3B): some precision was lost by a wrong Gaussian assumption, but little bias appeared to be introduced. These findings are consistent with the robustness to violations of the polygeneity assumption discussed above because when effect sizes are sampled from the NEG with low shape parameter, h2 is dominated by a few large effects. Once again, A outperformed A by showing lower variance and little, if any, bias.
Similarly, when we generated phenotypes by using five alternative distributions for $ei$ in Equation 3, we observed little effect on $hˆ2$ (Figure S4).

### The Effect of LD on $hˆ2$

The effects of uneven tagging are visible in the left half of Figure 3. In the underlying simulation, the causal variants were chosen only from areas of either high LD or low LD (see Material and Methods for definitions). When the causal variants were in regions of high LD and were therefore tagged better than average by neighboring SNPs, their contributions to the phenotype tended to be overestimated: $hˆ2$ > h2. Conversely, in regions of low LD, we typically found $hˆ2$ < h2. The degree of tagging is strongly correlated with MAF, and so we observed that the contribution to h2 of low-MAF causal variants tended to be underestimated, whereas that of high-MAF causal variants was overestimated (Figure S5).
The right half of Figure 3 illustrates the reduction in bias and increase in precision of $hˆ2$ when it is based on A rather than A. When A was used, the median $hˆ2$ in each scenario was within 2% of h2, whereas for A, it ranged from 50% to 133% of h2. Using the weights defined by Equation 7 instead gave results intermediate between those based on A and A (results not shown, but follow from Figure S1D). Even if causal variants are drawn at random so that $hˆ2$ is approximately unbiased for h2 (as for the gray boxes in Figure 3), individual causal variants can still be affected by uneven tagging, and this can reduce the precision of $h2ˆ$. This effect is particularly evident in Figure 1 when there are few causal variants; in this case, $h2ˆ$ based on A is much more precise than when it is based on A.
The wj’s are constructed to equalize the tagging of genotyped SNPs, but the tagging of untyped variants will not necessarily be equalized. However, in a simulation study in which only alternate SNPs were considered to be genotyped, we found that wj’s also greatly reduced the variation in the tagging of untyped SNPs (Figure S1C).

### Comparison with the Approach of Yang et al.

The impact of LD and MAF on $hˆ2$ was considered by Yang et al.
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
They proposed a transformation that can be used for negating the average bias caused by variable LD and also for estimating $hC2$, the total heritability due to common variation. They examined the accuracy of using the sharing of genotyped SNPs to estimate the sharing of causal variants by regressing $ZZT/m−I$ on AI. A slope coefficient β < 1 indicates that kinships are being overestimated by A (which typically leads to $hˆ2), and the opposite occurs when β > 1. By observing the results of simulations, they obtained for β an empirical formula that takes into account the MAF spectrum of the causal variants and the total number of genotyped SNPs, and they proposed replacing A in the mixed-model analysis with β (AI) + I. Although this approach can reduce the bias due to variable LD in $hˆ2$, the fact that it requires knowledge of the MAF spectrum of causal variants is not usually feasible. In any case, the bias is caused by levels of LD at the causal variants, and MAF is an imperfect indicator of such levels.
On can also use the proposal of Yang et al.
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
to estimate $hˆ2/hC2$ by taking into account the expected overestimation of kinship values caused by incomplete tagging of causal alleles. When our weighted kinship matrix A is used, this fraction can be estimated more directly by the calculation of the average proportion of untyped variation captured by the genotyped SNPs (Figure S6).

### A Test for Inflation Due to Population Structure

A criticism of SNP-based $hˆ2$ is its potential sensitivity to population structure. The danger was highlighted by Browning and Browning,
• Browning S.R.
• Browning B.L.
Population structure can inflate SNP-based heritability estimates.
who constructed a pseudo case-control association study by using the UK controls of the original WTCCC;
Wellcome Trust Case Control Consortium
Genome-wide association study of 14,000 cases of seven common diseases and 3,000 shared controls.
when treating 90% of English individuals as controls and 90% of Scottish and Welsh individuals as cases, they obtained a highly significant value of $hˆ2$ (7 SDs above 0). The inflation due to population structure for $hˆ2$ can be estimated by the difference between the sum of $hˆ2$ obtained from disjoint halves of the genome (say, chromosomes 1–8 and 9–22) and $hˆ2$ calculated from the whole genome.
• Yang J.
• Manolio T.A.
• Pasquale L.R.
• Boerwinkle E.
• Caporaso N.
• Cunningham J.M.
• de Andrade M.
• Feenstra B.
• Feingold E.
• Hayes M.G.
• et al.
Genome partitioning of genetic variation for complex traits using common SNPs.
In Figure S7, we apply this approach to a replicate of the study of Browning and Browning and show that it is effective in overcoming the problem they highlighted.

### Guarding against Genotyping Errors

Our simulation study avoided any problem arising from genotyping errors by using called SNP genotypes as causal variants. For real case-control studies, $hˆ2$ can be sensitive to different genotyping error rates in cases and controls; careful QC can reduce, but not eliminate, the problem. Replacing A with A could exacerbate the problem because poorly genotyped SNPs tend to have lower LD with neighboring SNPs, and this leads them to receive a relatively high weighting. To overcome this danger, we suggest calculating $rjj′2$ separately in cases and controls and then using the larger of the two values during computation of wj. Then, if a SNP is, say, poorly genotyped in cases, its wj will not artificially be increased provided that it is correctly genotyped in controls. We show in Figure S8 that $hˆ2$ remains approximately unbiased under this approach. An even more cautious approach would be to compute weightings from a completely independent (but ethnically matched) data set, which would not be affected by genotyping anomalies in the analysis data set.

### Application to Height and WTCCC Data Sets

We repeated the analysis of human height (MIM 606255) by using the post-QC data of Yang et al.
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
Using A as the kinship matrix, we obtained $hˆ2$ = 0.44 (0.09 SD), which was almost unchanged ($hˆ2$ = 0.45, 0.10 SD) when we replaced A with the LD-adjusted A. The lack of change in $hˆ2$ suggests that, when A is used, any underestimation of heritability from poorly tagged variants is compensated for by an overestimation of heritability from well-tagged variants. Yang et al.
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
suggested that the difference between their $hˆ2$ and the accepted value of h2 ≈ 0.8 could be explained by a model in which all causal variants have a MAF in the range [0.01, 0.1]. However, given that our LD-adjusted kinships better appreciate the contribution of low-MAF variants, explaining the gap between $hˆ2$ and h2 in terms of common (MAF > 0.01) causal variants would require them to have an even narrower range of MAF.
Next, we applied our method to the seven traits of the original WTCCC study:
Wellcome Trust Case Control Consortium
Genome-wide association study of 14,000 cases of seven common diseases and 3,000 shared controls.
bipolar disorder (BD [MIM 125480]), coronary artery disease (CAD [MIM 608320]), Crohn disease (CD [MIM 266600]), hypertension (HT [MIM 145500]), rheumatoid arthritis (RA [MIM 180300]), type 1 diabetes (T1D [MIM 222100]), and type 2 diabetes (T2D [MIM 125853]). For each study, approximately 2,000 patients were combined with a set of 3,004 common controls sourced from the 1958 Birth Cohort (58BC; 1,504 individuals) and the National Blood Service (NBS; 1,500 individuals). Three of these traits (BD, CD, and T1D) have been previously studied with the same WTCCC data by Lee et al.,
• Lee S.H.
• Wray N.R.
• Goddard M.E.
• Visscher P.M.
Estimating missing heritability for disease from genome-wide association studies.
whereas RA and T1D have been examined with different data.
• Stahl E.A.
• Wegmann D.
• Trynka G.
• Gutierrez-Achury J.
• Do R.
• Voight B.F.
• Kraft P.
• Chen R.
• Kallberg H.J.
• Kurreeman F.A.
• et al.
Diabetes Genetics Replication and Meta-analysis ConsortiumMyocardial Infarction Genetics Consortium
Bayesian inference analyses of the polygenic architecture of rheumatoid arthritis.
We also followed Lee et al.
• Lee S.H.
• Wray N.R.
• Goddard M.E.
• Visscher P.M.
Estimating missing heritability for disease from genome-wide association studies.
in contrasting the 58BC and NBS samples in a pseudo case-control study, for which we expect h2 = 0. Although these studies involve binary outcomes, for the purpose of comparing the standard and weighted kinship matrices, it suffices to estimate h2 on the observed scale rather than convert to the liability scale.
Because of the potential inflation caused by population structure and genotyping errors for $hˆ2$, Lee et al.
• Lee S.H.
• Wray N.R.
• Goddard M.E.
• Visscher P.M.
Estimating missing heritability for disease from genome-wide association studies.
employed stricter QC criteria than did the WTCCC,
Wellcome Trust Case Control Consortium
Genome-wide association study of 14,000 cases of seven common diseases and 3,000 shared controls.
and we applied similarly strict QC. Specifically, we first filtered out individuals with >2.5% missing genotypes or with heterozygosity (calculated across a pruned set of high-quality SNPs) not within [0.295, 0.345]. Next, for each of the seven case and two control data sets, we removed SNPs that had either a MAF < 0.01 or call rate < 0.99 or that were significant at 5% for Hardy-Weinberg equilibrium. Then, for each study, we rejected SNPs significant at 5% for differential missingness between cases and controls. Finally, for each study, we computed (unweighted) allelic-correlation kinship coefficients by using a set of about 50,000 SNPs in approximate LE. Apparent population outliers were removed on the basis of the first two principal components. Following Yang et al.,
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
we selectively removed individuals so that no pair remained with kinship greater than the absolute value of the smallest observed value (between 2.5% and 3.6%) for that data set. For the pseudo case-control study, 2,834 individuals and 297,894 SNPs remained; for each of the other studies, between 4,415 and 4,690 individuals and between 278,772 and 285,989 SNPs remained. For each study, we included the top 20 axes from principal-component analysis of the kinship matrix used above for identifying related individuals. We also assessed the inflation caused by population structure, and because cases and controls were genotyped separately, we implemented the procedure described above in “Guarding against Genotyping Errors.”
For the seven traits and the pseudo case-control study, Table 1 presents $hˆ2$ (calculated with both the standard and the LD-adjusted kinship matrices) for each case, evaluated either across all SNPs or across just those on chromosome 6 (Table S1 provides additional details, including implementation of the above test for inflation due to population structure and an assessment of the impact of genotyping errors). The largest differences in $hˆ2$ when A was used instead of A occurred for BD (+1.6 SDs), HT (+1.6 SDs), RA (−0.9 SD), and T2D (+1.5 SDs). Note that these are differences between statistics computed with the same data and which estimate the same quantity, so it does not make sense to test for significance of these differences. An increased $hˆ2$ provides evidence that BD, HT, and T2D are predominantly affected by variants of lower MAF, and this causes the standard analysis to underestimate their SNP-based heritability. RA and T1D both have their strongest (marginal) associations in a region of high LD in HLA-DRB1 (MIM 142857) at 6p21 in the MHC region and have estimated odds ratios of 2.36 and 5.49, respectively (see Figures S9–S16 for the results of marginal analyses). Unsurprisingly, the contribution to $hˆ2$ from chromosome 6 is reduced for these two conditions when A is used in place of A.
Table 1Analysis of WTCCC Traits
Trait or StudyTotal Heritability $hˆ2$ (SD)Chromosome 6 Heritability $hˆ62$ (SD)
StandardWeightedDifferenceStandardWeightedDifference
Bipolar disease59 (6)69 (8)+104 (2)5 (2)0
Coronary artery disease39 (6)41 (8)+32 (2)1 (2)−1
Crohn disease54 (6)58 (8)+55 (2)6 (2)+1
Hypertension42 (6)52 (8)+105 (2)8 (2)+4
Rheumatoid arthritis57 (6)52 (8)−619 (2)17 (2)−2
Type 1 diabetes73 (6)74 (8)037 (2)35 (2)−2
Type 2 diabetes35 (6)44 (8)+94 (2)5 (2)+1
Pseudo case-control study11 (10)7 (14)−40 (2)1 (3)+1
The variance explained by all SNPs and by just those on chromosome 6 with the use of both the standard and the weighted kinship matrix for the seven traits and the pseudo case-control study. The SD is the value given by the GCTA software.

## Discussion

The two leading explanations for the “missing heritability” of complex phenotypes have been (1) many rare variants of small or modest effect size and (2) many common variants of weak effect.
• Manolio T.A.
• Collins F.S.
• Cox N.J.
• Goldstein D.B.
• Hindorff L.A.
• Hunter D.J.
• McCarthy M.I.
• Ramos E.M.
• Cardon L.R.
• Chakravarti A.
• et al.
Finding the missing heritability of complex diseases.
SNP-based mixed-model analysis of unrelated individuals has been used for providing support for the latter explanation,
• Yang J.
• Benyamin B.
• McEvoy B.P.
• Gordon S.
• Henders A.K.
• Nyholt D.R.
• Heath A.C.
• Martin N.G.
• Montgomery G.W.
• et al.
Common SNPs explain a large proportion of the heritability for human height.
• Yang J.
• Manolio T.A.
• Pasquale L.R.
• Boerwinkle E.
• Caporaso N.
• Cunningham J.M.
• de Andrade M.
• Feenstra B.
• Feingold E.
• Hayes M.G.
• et al.
Genome partitioning of genetic variation for complex traits using common SNPs.
• Stahl E.A.
• Wegmann D.
• Trynka G.
• Gutierrez-Achury J.
• Do R.
• Voight B.F.
• Kraft P.
• Chen R.
• Kallberg H.J.
• Kurreeman F.A.
• et al.
Diabetes Genetics Replication and Meta-analysis ConsortiumMyocardial Infarction Genetics Consortium
Bayesian inference analyses of the polygenic architecture of rheumatoid arthritis.
• Davies G.
• Tenesa A.
• Payton A.
• Yang J.
• Harris S.E.
• Liewald D.
• Ke X.
• Le Hellard S.
• Christoforou A.
• Luciano M.
• et al.
Genome-wide association studies establish that human intelligence is highly heritable and polygenic.
• Lee S.H.
• DeCandia T.R.
• Ripke S.
• Yang J.
• Sullivan P.F.
• Goddard M.E.
• Keller M.C.
• Visscher P.M.
• Wray N.R.
Schizophrenia Psychiatric Genome-Wide Association Study Consortium (PGC-SCZ)International Schizophrenia Consortium (ISC)Molecular Genetics of Schizophrenia Collaboration (MGS)
Estimating the proportion of variation in susceptibility to schizophrenia captured by common SNPs.
• Benjamin D.J.
• Cesarini D.
• van der Loos M.J.
• Dawes C.T.
• Koellinger P.D.
• Magnusson P.K.
• Chabris C.F.
• Conley D.
• Laibson D.
• Johannesson M.
• Visscher P.M.
The genetic architecture of economic and political preferences.
thus pointing researchers in the direction of, for example, pathway analyses for assisting in the identification of common variants with small effect sizes. SNP-based mixed-model analyses can help further by estimating h2 components attributable to pathways, genomic regions, or MAF classes.
• Yang J.
• Manolio T.A.
• Pasquale L.R.
• Boerwinkle E.
• Caporaso N.
• Cunningham J.M.
• de Andrade M.
• Feenstra B.
• Feingold E.
• Hayes M.G.
• et al.
Genome partitioning of genetic variation for complex traits using common SNPs.
We have examined the assumptions that underlie the use of SNP-based mixed-model analysis for estimating h2, and we have illustrated through simulation that the resulting $hˆ2$ is reasonably robust to four underpinning assumptions but is vulnerable to uneven LD. Use of the standard allelic-correlation kinship coefficients overestimates the contribution to h2 from causal variants that are well tagged and underestimates that from poorly tagged variants. This observation is particularly pertinent for traits where many of the causal variants have an intermediate MAF (say, between 0.01 and 0.1). For example, in a simulation with causal variants restricted to SNPs with a MAF in this range, (unadjusted) $hˆ2$ was typically 25% smaller than h2. When variants were selected from the 20% of SNPs with the lowest LD with neighboring SNPs, we observed $hˆ2$/ h2 as small as 0.5.
We have proposed estimating h2 instead with the use of A, a kinship matrix adjusted for local patterns of LD. This largely eliminates the biases based on the standard kinship matrix A in $hˆ2$ and increases its precision. This adjustment improves current methods for investigating the architecture of complex traits with the use of SNP-based genomic partitioning, and we expect that using A will provide benefits for other applications of the mixed model in quantitative genetics, such as association analysis and prediction of phenotype.

### Computational demands

Solving Equation 6 is computationally demanding for reasonably sized problems, but in Appendix A, we describe a good approximation obtained by a partitioning of the genome into regions. With this approximation, the time needed for computing the weightings is similar to that required for computing the standard kinship matrix A. Once the wj’s are available, calculating the weighted kinship matrix A can be significantly faster than calculating A because in a dense SNP set, many SNPs are assigned wj = 0 and can be ignored. Our LDAK software, which calculates both the wj’s and A, is freely available (see Web Resources below).

## Acknowledgments

We thank Will Astle and Vincent Plagnol for fruitful discussions, as well as the two anonymous reviewers for helpful comments. Access to Wellcome Trust data was authorized as work related to the project “Genome-wide association study of susceptibility and clinical phenotypes in epilepsy.” This work is funded by the UK Medical Research Council under grant G0901388.

## Appendix A: Calculating Weights for the LD-Adjusted Kinship Matrix

To explain the problems caused by LD, suppose that SNP 1 and SNP 2 are perfectly correlated. Use of the standard kinship matrix A in mixed-model analysis implicitly assumes the same effect-size distributions for all SNPs, so the analysis would be unchanged by the removal of SNP 1 but would double the prior effect-size variance for SNP 2. A similar effect arises if the two SNPs are correlated less than perfectly. As such, the effect of any causal allele, and its contribution to $hˆ2$, can be exaggerated by multiple tagging SNPs. Conversely, the contribution will be understated if the causal allele is tagged by relatively few SNPs.
To offset this effect of LD, we propose introduction of wj’s > 0 and the replacement of $A=XXT/m′$ with a weighted kinship matrix A, computed in the same way except that Xj (the allele counts for SNP j, scaled and centered) is replaced with
$Xj∗=wj∗Xjwherewj∗=wjm′∑jwj.$

Therefore, a $N(0,σg2/m′)$ prior distribution for the effect size for Xj is equivalent to a $N(0,wj∗σg2/m′)$ prior for the effect size of $Xj∗$. We seek wj’s that equalize the implied prior effect-size distributions of causal variants, irrespective of their tagging by SNPs. With genotypes standardized to have variance 1, the amount of a causal signal tagged by both X1 and X2 is r12, the correlation between SNPs 1 and 2. In a similar fashion, r13 will represent the extent to which a causal signal tagged by X1 is replicated by X3. Under the linear model, the combined phenotypic effect of X1, X2, and X3 is X1u1 + X2u2 + X3u3. Therefore, the contribution in the direction of X1 is u1 + r12u2 + r13u3, which has prior variance
$(w1∗+r122w2∗+r132w3∗)σg2m′.$

We can obtain similar expressions for the signals represented by X2 and X3 and similarly extend this to consider all m′ genotyped signals. Our aim is that the variance terms for all these prior distributions are equal, which we can achieve by solving a matrix equation of the form Cw = 1:
$[1C(1,2)2C(1,3)2⋯C(1,m'−1)2C(1,m')2C(2,1)21C(2,3)2⋯C(2,m'−1)2C(2,m')2C(3,1)2C(3,2)21⋯C(3,m′−1)2C(3,m′)2⋮⋮⋮⋱⋮⋮C(m′−1,1)2C(m′−1,2)2C(m′−1,3)2⋯1C(m′−1,m′)2C(m′,1)2C(m′,2)2C(m′,3)2⋯C(m′,m′−1)21][w1w2w3⋮wm′−1wm′]=[111⋮11],$

where for the moment, $C(j,j′)2$ represents the correlation squared between variants j and j′. Requiring wj > 0 means that there is usually no (exact) solution to this equation, and we instead seek the best achievable approximate solution in the sense of least absolute error. That is, we seek to minimize
$∑j|1−Cjw|,$

where Cj denotes the jth row of C. Equivalently, we minimize $∑jaj$, subject to
$−aj≤1−Cjw≤ajandaj≥0for allj,$

which is a linear programming problem that could, in theory, be solved with the simplex algorithm.
The approach above considers global correlations and could, if solvable, be used for addressing the effects of both short- and long-range LD. However, if we assume that long-range LD due, for example, to population structure has been addressed with other measures, we can restrict attention to local LD. Let djj′ represent the distance in base pairs between variants j and j′ and be set to $∞$ if the two SNPs are on different chromosomes. We define the elements of matrix C as
$C(j,j′)2={e−λdjj′rjj′2ife−λdjj′>0.125andrjj′2>0.010otherwise.$

The exponential term attempts to model the scope of LD, and λ reflects its rate of decay. We fix λ so that $exp(−λdjj′)$ = 0.125 when djj equals 3 Mbp. The requirement that $rjj′2$ > 0.01 corresponds loosely to requiring that $rjj′2$ be significantly above zero for typical sample sizes. We experimented with varying λ to achieve window sizes between 1 Mbp and 10 Mbp, as well as with setting the $rjj′2$ threshold to 0 and 0.05, and observed little change in $hˆ2$, except when we used the minimum window size of 1 Mbp, which seemed insufficient to account for the full effects of LD.
Now that matrix C is sparse (typically with only tens of nonzero elements in each row), solving the linear-programming problem becomes more manageable. In particular, solving for wj decomposes into independent solutions for each chromosome. Even with this shortcut, for our simulation study data, the largest C matrix, which contained about 40,000 rows for chromosome 2, and the corresponding optimization problem could still require a few days to be solved on a standard processor. Therefore, we further subdivided the C matrices to include at most 3,000 SNPs at a time and allowed a buffer of 500 SNPs at each boundary to minimize the effect of the subdivision. We checked the weights for the 1,000 SNPs in common between a pair of adjacent windows and found that the concordance between the central 500 of these SNPs was almost perfect. Typically, disruption was noticeable only for the last few tens of SNPs of each window, indicating that a buffer of 500 SNPs was sufficient.
When we used windows of size 3,000 SNPs (plus buffers of 500 SNPs), calculating the weights for the entire genome took approximately 40 computer hours for the simulation sample size of 5,127 (this process can be spread across multiple processors if desired). A potential future improvement of the method could allow for data-dependent λ that reduces window sizes in regions of low LD to reduce computing time but that expands them for high LD; this might improve h2 estimation, for example, in the MHC.

## Appendix B: Calculating Heritability from Estimates of $σg2$ and $σe2$

Let VT denote the total phenotypic variance, and let VR denote the residual variance (which equals the phenotypic variance when all genetic effects are correctly included in the model). Assuming that the phenotype has been centered to have mean of zero, we can write
$VT=∑i(yi2)n−(∑iyin)2andVR=∑i(ei)2n−(∑iein)2.$

To calculate the expected values of VT and VR, consider that
$E(ujuj′)=0forj≠j′,E(uj2)=σg2forj=1,2,…,m,E(eiei′)=0fori≠i′,E(ei2)=σe2fori=1,2,…,n,andE(ujei)=0,∀j,i.$

Considering the expectation of the terms in the first summation in VT, we obtain
$E(yi2)=E[(zi1u1+zi2u2+…+zimum+ei)2]=E[zi12u12+zi22u22+…+zim2um2+ei2]=(zi12+zi22+…+zim2)σg2+σe2,$

whereas the expected value required for the second summation is
$E[(y1+y2+…+yn)2]=E[(Z11u1+…+Z1mum+e1+…+Zn1u1+…+Znmum+en)2]=E[∑ii′Zi1Zi′1u1+…+∑ii′ZimZi′mum+e12+e22+…+en2]=(∑j∑ii′ZijZi′j)σg2+nσe2.∴E(VT)=trace(ZZT)nσg2+σe2−sum(ZZT)n2σg2−1nσe2.$

For the residual variance, calculation of the expectation is more straightforward:
$E(VR)=E[e12+e22+…+en2]n−E[(e1+e2+…+en)2]n2=σe2−1nσe2.$

Although the expected heritability is 1 − $E(VR/VT)$, because the fraction is bottom heavy, it should be reasonable to approximate this value with 1 − $E(VR)/E(VT)$, from which we obtain
$E(h2)≈1−E(VR)E(VT)=1−(1−1n)σe2(trace(ZZT)nσg2−sum(ZZT)n2σg2+(1−1n)σe2).$
(Equation B1)

With the usual column standardization, the genotype matrix has trace n and sum 0. Therefore, the heritability estimate will take the simpler form
$hˆ2=1−(1−1n)σˆe2(σˆg+(1−1n)σˆe2)=σˆg2σˆg2+(1−1n)σˆe2,$

which, for large n, tends to the form provided in the main text. However, for all other transformations, it is important to use the complete form, Equation B1. In particular, dividing the kinship matrix by a constant will lead to an estimate of $σg2$ multiplied by that constant. When the full form is used, $hˆ2$ will be invariant to any such scaling (or, indeed, shifting) of the kinship matrix; however, with the simple form, the heritability estimate would be unjustifiably affected.

## Supplemental Data

• Document S1. Figures S1–S16 and Table S1

## Web Resources

The URLs for data presented herein are as follows:

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Genome partitioning of genetic variation for complex traits using common SNPs.
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# Difference between revisions of "1983 IMO Problems/Problem 1"

## Problem

Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy the conditions: (i) $f(xf(y))=yf(x)$ for all $x,y$; (ii) $f(x)\to0$ as $x\to \infty$.

## Solution 1

Let $x=y=1$ and we have $f(f(1))=f(1)$. Now, let $x=1,y=f(1)$ and we have $f(f(f(1)))=f(1)f(1)\Rightarrow f(1)=[f(1)]^2$ since $f(1)>0$ we have $f(1)=1$.

Plug in $y=x$ and we have $f(xf(x))=xf(x)$. If $a=1$ is the only solution to $f(a)=a$ then we have $xf(x)=1\Rightarrow f(x)=\frac{1}{x}$. We prove that this is the only function by showing that there does not exist any other $a$:

Suppose there did exist such an $a\ne1$. Then, letting $y=a$ in the functional equation yields $f(xa)=af(x)$. Then, letting $x=\frac{1}{a}$ yields $f(\frac{1}{a})=\frac{1}{a}$. Notice that since $a\ne1$, one of $a,\frac{1}{a}$ is greater than $1$. Let $b$ equal the one that is greater than $1$. Then, we find similarly (since $f(b)=b$) that $f(xb)=bf(x)$. Putting $x=b$ into the equation, yields $f(b^2)=b^2$. Repeating this process we find that $f(b^{2^k})=b^{2^k}$ for all natural $k$. But, since $b>1$, as $k\to \infty$, we have that $b^{2^k}\to\infty$ which contradicts the fact that $f(x)\to0$ as $x\to \infty$.

## Solution 2

Let $x=1$ so $$f(f(y))=yf(1).$$ If $f(a)=f(b),$ then $af(1)=f(f(a))=f(f(b))=bf(1)\implies a=b$ because $f(1)$ goes to the real positive integers, not $0.$ Hence, $f$ is injective. Let $x=y$ so $$f(xf(x))=xf(x)$$ so $xf(x)$ is a fixed point of $f.$ Then, let $y=1$ so $f(xf(1))=f(x)\implies f(1)=1$ as $x$ can't be $0$ so $1$ is a fixed point of $f.$ We claim $1$ is the only fixed point of $f.$ Suppose for the sake of contradiction that $a,b$ be fixed points of $f$ so $f(a)=a$ and $f(b)=b.$ Then, setting $x=a,y=b$ in (i) gives $$f(ab)=f(af(b))=bf(a)=ab$$ so $ab$ is also a fixed point of $f.$ Also, let $x=\frac{1}{a},y=a$ so $$1=f(1)=f(\frac{1}{a}\cdot a)=f(\frac{1}{a}\cdot f(a))=af(\frac{1}{a})\implies f(\frac{1}{a})=\frac{1}{a}$$ so $\frac{1}{a}$ is a fixed point of $f.$ If $f(a)=a$ with $a>1,$ then $f(a^n)$ is a fixed point of $f$, contradicting (ii). If $f(a)=a$ with $0 then $f(\frac{1}{a^n})=\frac{1}{a^n}$ so $\frac{1}{a^n}$ is a fixed point, contradicting (ii). Hence, the only fixed point is $1$ so $xf(x)=1$ so $f(x)=\boxed{\frac{1}{x}}$ and we can easily check that this solution works.

 1983 IMO (Problems) • Resources Preceded byFirst question 1 • 2 • 3 • 4 • 5 • 6 Followed byProblem 2 All IMO Problems and Solutions


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# Sum and binomials

I have this sum $$\left(N+1\right)^{2}\underset{j=1}{\overset{N}{\sum}}\frac{\left(-1\right)^{j}}{2j+1}\dbinom{N}{j}\dbinom{N+j}{j-1}\underset{i=1}{\overset{N}{\sum}}\frac{\left(-1\right)^{i}}{\left(2i+1\right)\left(i+j\right)}\dbinom{N}{i}\dbinom{N+i}{i-1}$$ and numerical test indicates that is equal to $$\frac{\left(N+1\right)N}{\left(2N+1\right)^{2}}$$ but I'm not able to prove it. Thank you

• Where did this come from? It may help someone else answer it. – snulty Jul 17 '14 at 18:01


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# statsmodels.regression.mixed_linear_model.MixedLM.initialize¶

MixedLM.initialize()

Initialize (possibly re-initialize) a Model instance.

For example, if the the design matrix of a linear model changes then initialized can be used to recompute values using the modified design matrix.