Covariates in linkage analysis

Incorporation of covariates in simultaneous localization of two linked loci using affected relative pairs

Background

Many dichotomous traits for complex diseases are often involved more than one locus and/or associated with quantitative biomarkers or environmental factors. Incorporating these quantitative variables into linkage analysis as well as localizing two linked disease loci simultaneously could therefore improve the efficiency in mapping genes. We extended the robust multipoint Identity-by-Descent (IBD) approach with incorporation of covariates developed previously to simultaneously estimate two linked loci using different types of affected relative pairs (ARPs).

Results

We showed that the efficiency was enhanced by incorporating a quantitative covariate parametrically or non-parametrically while localizing two disease loci using ARPs. In addition to its help in identifying factors associated with the disease and in improving the efficiency in estimating disease loci, this extension also allows investigators to account for heterogeneity in risk-ratios for different ARPs. Data released from the collaborative study on the genetics of alcoholism (COGA) for Genetic Analysis Workshop 14 (GAW 14) were used to illustrate the application of this extended method.

Conclusions

The simulation studies and example illustrated that the efficiency in estimating disease loci was demonstratively enhanced by incorporating a quantitative covariate and by using all relative pairs while mapping two linked loci simultaneously.

Methods for combining multiple genome-wide linkage studies

Cardiovascular disease, metabolic syndrome, schizophrenia, diabetes, bipolar disorder, and autism are a few of the numerous complex diseases for which researchers are trying to decipher the genetic composition. One interest of geneticists is to determine the quantitative trait loci (QTLs) that underlie the genetic portion of these diseases and their risk factors. The difficulty for researchers is that the QTLs underlying these diseases are likely to have small to medium effects which will necessitate having large studies in order to have adequate power. Combining information across multiple studies provides a way for researchers to potentially increase power while making the most of existing studies.Here, we will explore some of the methods that are currently being used by geneticists to combine information across multiple genome-wide linkage studies. There are two main types of meta-analyses: (1) those that yield a measure of significance, such as Fisher's p-value method along with its extensions/modifications and the genome search meta-analysis (GSMA) method, and (2) those that yield a measure of a common effect size and the corresponding standard error, such as model-based methods and Bayesian methods. Some of these methods allow for the assessment of heterogeneity. This chapter will conclude with a recommendation for usage.

Interpreting analyses of continuous covariates in affected sibling pair linkage studies

Datasets collected for linkage analyses of complex human diseases often include a number of clinical or environmental covariates. In this study, we evaluated the performance of three linkage analysis methods when the relationship between continuous covariates and disease risk or linkage heterogeneity was modeled in three different ways: (1) The covariate distribution is determined by a quantitative trait locus (QTL), which contributes indirectly to the disease risk; (2) the covariate is not genetically determined, but influences the disease risk through statistical interaction with a disease susceptibility locus; (3) the covariate distribution differs in families linked or unlinked to a particular disease susceptibility locus. We analyzed simulated datasets with a regression-based QTL analysis, a nonparametric analysis of the binary affection status, and the ordered subset analysis (OSA). We found that a significant OSA result may be due to a gene that influences variability in the population distribution of a continuous disease risk factor. Conversely, a regression-based QTL analysis may detect the presence of gene-environment (GxE) interaction in a sample of primarily affected individuals. The contribution of unaffected siblings and the size of baseline lod scores may help distinguish between QTL and GxE models. As illustrated by a linkage study of multiplex families with age-related macular degeneration, our findings assist in the interpretation of analysis results in real datasets. They suggest that the side-by-side evaluation of OSA and QTL results may provide important information about the relationship of measured covariates with either disease risk or linkage heterogeneity.

Testing genetic linkage with relative pairs and covariates by quasi-likelihood score statistics

BACKGROUND/AIMS

Genetic linkage analysis of common diseases is complicated by the heterogeneity of genetic and environmental factors that increase disease risk, and possibly interactions among them. Most linkage methods that account for covariates are restricted to sib pairs, with the exception of the conditional logistic regression model [1] implemented in LODPAL in the S.A.G.E. software [2]. Although this model can be applied to arbitrary pedigrees, at times it can be difficult to maximize the likelihood due to model constraints, and it does not account for the dependence among the different types of relative pairs in a pedigree.

METHODS

To overcome these limitations, we developed a new approach based on score statistics for quasi- likelihoods, implemented as weighted least squares. Our methods can be used to test three different hypotheses: (1) a test for linkage without covariates; (2) a test for linkage with covariates, and (3) a test for effects of covariates on identity by descent sharing (i.e., heterogeneity). Furthermore, our methods are robust because they account for the dependence among different relative pairs within a pedigree.

RESULTS AND CONCLUSION

Although application of our methods to a prostate cancer linkage study did not find any critical covariates in our data, the results illustrate the utility and interpretation of our methods, and suggest, nonetheless, that our methods will be useful for a broad range of genetic linkage heterogeneity analyses.

Linkage analysis using co-phenotypes in the BRIGHT study reveals novel potential susceptibility loci for hypertension

Identification of the genetic influences on human essential hypertension and other complex diseases has proved difficult, partly because of genetic heterogeneity. In many complex-trait resources, additional phenotypic data have been collected, allowing comorbid intermediary phenotypes to be used to characterize more genetically homogeneous subsets. The traditional approach to analyzing covariate-defined subsets has typically depended on researchers' previous expectations for definition of a comorbid subset and leads to smaller data sets, with a concomitant attrition in power. An alternative is to test for dependence between genetic sharing and covariates across the entire data set. This approach offers the advantage of exploiting the full data set and could be widely applied to complex-trait genome scans. However, existing maximum-likelihood methods can be prohibitively computationally expensive, especially since permutation is often required to determine significance. We developed a less computationally intensive score test and applied it to biometric and biochemical covariate data, from 2,044 sibling pairs with severe hypertension, collected by the British Genetics of Hypertension (BRIGHT) study. We found genomewide-significant evidence for linkage with hypertension and several related covariates. The strongest signals were with leaner-body-mass measures on chromosome 20q (maximum LOD = 4.24) and with parameters of renal function on chromosome 5p (maximum LOD = 3.71). After correction for the multiple traits and genetic locations studied, our global genomewide P value was .046. This is the first identity-by-descent regression analysis of hypertension to our knowledge, and it demonstrates the value of this approach for the incorporation of additional phenotypic information in genetic studies of complex traits.

Covariate-based linkage analysis: application of a propensity score as the single covariate consistently improves power to detect linkage

Successful identification of genetic risk loci for complex diseases has relied on the ability to minimize disease and genetic heterogeneity to increase the power to detect linkage. One means to account for disease heterogeneity is by incorporating covariate data. However, the inclusion of each covariate will add one degree of freedom to the allele sharing based linkage test, which may in fact decrease power. We explore the application of a propensity score, which is typically used in causal inference to combine multiple covariates into a single variable, as a means of allowing for multiple covariates with the addition of only one degree of freedom. In this study, binary trait data, simulated under various models involving genetic and environmental effects, were analyzed using a nonparametric linkage statistic implemented in LODPAL. Power and type I error rates were evaluated. Results suggest that the use of the propensity score to combine multiple covariates as a single covariate consistently improves the power compared to an analysis including no covariates, each covariate individually, or all covariates simultaneously. Type I error rates were inflated for analyses with covariates and increased with increasing number of covariates, but reduced to nominal rates with sample sizes of 1000 families. Therefore, we recommend using the propensity score as a single covariate in the linkage analysis of a trait suspected to be influenced by multiple covariates because of its potential to increase the power to detect linkage, while controlling for the increase in the type I error.

Genome wide significant linkage in schizophrenia conditioning on occurrence of depressive episodes

BACKGROUND

Schizophrenia shows substantial clinical heterogeneity. One common important clinical variable in presentation is the occurrence of episodes of major depression.

METHODS

We undertook analyses in an attempt to detect loci that influence susceptibility to, or modify the clinical expression of, schizophrenia according to the occurrence of episodes of major depression. We used a logistic regression framework in which lifetime presence/absence of major depression was entered as a covariate in the linkage analysis of our UK schizophrenia affected sibling pair series (168 affected sibling pairs typed for a 10 cM map of microsatellite markers).

RESULTS

Inclusion of presence/absence of depression as a covariate detected a genome wide significant linkage signal on chromosome 4q28.3 at 130.7 cM (LOD = 4.59; p = 0.038; increase in maximum LOD over univariate analysis (ILOD) = 3.62). Inclusion of the depression covariate also showed suggestive evidence of linkage on 20q11.21 (LOD = 4.10; expected to occur by chance 0.093 times per genome scan, ILOD = 2.83).

CONCLUSIONS

Our findings identify loci that may harbour genes that play a role in susceptibility to, or modify the risk of, episodes of major depression in people with schizophrenia.

Stage 2 of the Wellcome Trust UK-Irish bipolar affective disorder sibling-pair genome screen: evidence for linkage on chromosomes 6q16-q21, 4q12-q21, 9p21, 10p14-p12 and 18q22

Bipolar affective disorder (BPAD) is a common psychiatric disorder with complex genetic aetiology. We have undertaken a genome-wide scan in one of the largest samples of bipolar affected sibling pairs (ASPs) using a two-stage approach combining sample splitting and marker grid tightening. In this second stage analysis, we have examined 17 regions that achieved a nominally significant maximum likelihood LOD score (MLS) threshold of 0.74 (or 1.18 for the X-chromosome) in stage one. The second stage has added 135 ASP families to bring the total stage 2 sample to 395 ASPs. In total, 494 microsatellite markers have been used to screen the human genome at a density of 10 cM in the first stage sample (260 ASPs) and 5 cM in the second stage. Under the broad diagnostic model, two markers gave LOD scores exceeding 3 with two-point analysis: D4S392 (LOD=3.30) and D10S197 (LOD=3.18). Multipoint analysis demonstrated suggestive evidence of linkage between BPAD and chromosomal regions 6q16-q21 (MLS=2.61) and 4q12-q21 (MLS=2.38). 6q16-q21 is of particular interest because our data, together with those from two recent genome scans, make this the best supported linkage region in BPAD. Further, our data show evidence of a gender effect at this locus with increased sharing predominantly within the male-male pairs. Our scan also provides support for linkage (MLS> or =1.5) at several other regions that have been implicated in meta-analyses of bipolar disorder and/or schizophrenia including 9p21, 10p14-p12 and 18q22.

Genome screen for loci influencing age at onset and rate of decline in late onset Alzheimer's disease

We performed an affected sib-pair (ASP) linkage analysis to test for the effects of age at onset (AAO), rate of decline (ROD), and Apolipoprotein E (APOE) genotype on linkage to late-onset Alzheimer's disease (AD) in a sample comprising 428 sib-pairs. We observed linkage of mean AAO to chromosome 21 in the whole sample (max LOD = 2.57). This came entirely from the NIMH sample (max LOD = 3.62), and was strongest in pairs with high mean AAO (>80). A similar effect was observed on chromosome 2q in the NIMH sample (max LOD = 2.73); this region was not typed in the IADC/UK sample. Suggestive evidence was observed in the combined sample of linkage of AAO difference to chromosome 19q (max LOD = 2.33) in the vicinity of APOE and 12p (max LOD = 2.22), with linkage strongest in sib-pairs with similar AAO. Mean ROD showed suggestive evidence of linkage to chromosome 9q in the whole sample (max LOD = 2.29), with the effect strongest in the NIMH sample (max LOD = 3.58), and in pairs with high mean ROD. Additional suggestive evidence was also observed in the NIMH sample with AAO difference on chromosome 6p (max LOD = 2.44) and 15p (max LOD = 1.87), with linkage strongest in pairs with similar AAO, and in the UK sample with mean ROD on chromosome 1p (max LOD = 2.73, linkage strongest in pairs with high mean ROD). We also observed suggestive evidence of increased identical by descent (IBD) in APOE epsilon4 homozygotes on chromosome 1 (max LOD = 3.08) and chromosome 9 (max LOD = 3.34). The previously reported genome-wide linkage of AD to chromosome 10 was not influenced by any of the covariates studied.

The Familial Intracranial Aneurysm (FIA) study protocol

Background

Subarachnoid hemorrhage (SAH) due to ruptured intracranial aneurysms (IAs) occurs in about 20,000 people per year in the U.S. annually and nearly half of the affected persons are dead within the first 30 days. Survivors of ruptured IAs are often left with substantial disability. Thus, primary prevention of aneurysm formation and rupture is of paramount importance. Prior studies indicate that genetic factors are important in the formation and rupture of IAs. The long-term goal of the Familial Intracranial Aneurysm (FIA) Study is to identify genes that underlie the development and rupture of intracranial aneurysms (IA).

Methods/Design

The FIA Study includes 26 clinical centers which have extensive experience in the clinical management and imaging of intracerebral aneurysms. 475 families with affected sib pairs or with multiple affected relatives will be enrolled through retrospective and prospective screening of potential subjects with an IA. After giving informed consent, the proband or their spokesperson invites other family members to participate. Each participant is interviewed using a standardized questionnaire which covers medical history, social history and demographic information. In addition blood is drawn from each participant for DNA isolation and immortalization of lymphocytes. High- risk family members without a previously diagnosed IA undergo magnetic resonance angiography (MRA) to identify asymptomatic unruptured aneurysms. A 10 cM genome screen will be performed to identify FIA susceptibility loci. Due to the significant mortality of affected individuals, novel approaches are employed to reconstruct the genotype of critical deceased individuals. These include the intensive recruitment of the spouse and children of deceased, affected individuals.

Discussion

A successful, adequately-powered genetic linkage study of IA is challenging given the very high, early mortality of ruptured IA. Design features in the FIA Study that address this challenge include recruitment at a large number of highly active clinical centers, comprehensive screening and recruitment techniques, non-invasive vascular imaging of high-risk subjects, genome reconstruction of dead affected individuals using marker data from closely related family members, and inclusion of environmental covariates in the statistical analysis.

Defining the Phenotype in Human Genetic Studies: Forward Genetics and Reverse Phenotyping

The definition of phenotypes for genetic study is a challenging endeavor. Just as we apply strict quality standards to genotype data, we should expect that phenotypes meet consistently high standards of reproducibility and validity. The methods for achieving accurate phenotype assignment in the research setting--the 'research diagnosis'--are different from the methods used in clinical diagnosis in the patient care setting. We evaluate some of the main challenges of phenotype definition in human genetics, and begin to outline a set of standards to which phenotypes used in genetics studies may aspire with the goal of increasing the quality and reproducibility of linkage and association studies. Revisiting the traditional phenotype definitions through a focus on familial components and heritable endophenotypes is a time-honored approach. Reverse phenotyping, where phenotypes are refined based on genetic marker data, may be a promising new approach. The stakes are high, since the success of gene mapping in genetically complex disorders hinges on the ability to delineate the target phenotype with accuracy and precision.

Comparison of methods incorporating quantitative covariates into affected sib pair linkage analysis

For complex traits, it may be possible to increase the power to detect linkage if one takes advantage of covariate information. Several statistics have been proposed that incorporate quantitative covariate information into affected sib pair (ASP) linkage analysis. However, it is not clear how these statistics perform under different gene-environment (G x E) interactions. We compare representative statistics to each other on simulated data under three biologically-plausible G x E models. We also compared their performance with a model-free method and with quantitative trait locus (QTL) linkage approaches. The statistics considered here are: (1) mixture model; (2) general conditional-logistic model (LODPAL); (3) multinomial logistic regression models (MLRM); (4) extension of the maximum-likelihood-binomial approach (MLB); (5) ordered-subset analysis (OSA); and (6) logistic regression modeling (COVLINK). In all three G x E models, most of these six statistics perform better when using the covariate C1 associated with a G x E interaction effect than when using the environmental risk factor C2 or the random noise covariate C3. Compared with a model-free method without covariates (S(all)), the mixture model performs the best when using C1, with the high-to-low OSA method also performing quite well. Generally, MLB is the least sensitive to covariate choice. However, most of these statistics do not provide better power than S(all). Thus, while inclusion of the "correct" covariate can lead to increased power, careful selection of appropriate covariates is vital for success.

Linkage analysis of susceptibility to leprosy type using an IBD regression method

Leprosy is a chronic disease caused by infection with Mycobacterium leprae, which is manifested across a wide clinical spectrum. There is evidence that susceptibility both to leprosy per se and to the clinical type of leprosy is influenced by host genetic factors. This paper describes the application of an identity by descent regression search for genetic determinants of leprosy type among families from Karonga District, Northern Malawi. Suggestive evidence was found for linkage to leprosy type on chr 21q22 (P<0.001). The methodological implications of the approach and the findings are discussed.

Genome-wide mapping of human loci for essential hypertension

BACKGROUND

Blood pressure may contribute to 50% of the global cardiovascular disease epidemic. By understanding the genes predisposing to common disorders such as human essential hypertension we may gain insights into novel pathophysiological mechanisms and potential therapeutic targets. In the Medical Research Council BRItish Genetics of HyperTension (BRIGHT) study, we aim to identify these genetic factors by scanning the human genome for susceptibility genes for essential hypertension. We describe the results of a genome scan for hypertension in a large white European population.

METHODS

We phenotyped 2010 affected sibling pairs drawn from 1599 severely hypertensive families, and completed a 10 centimorgan genome-wide scan. After rigorous quality control, we analysed the genotypic data by non-parametric linkage, which tests whether genes are shared in excess among the affected sibling pairs. Lod scores, calculated at regular points along each chromosome, were used to assess the support for linkage.

FINDINGS

Linkage analysis identified a principle locus on chromosome 6q, with a lod score of 3.21 that attained genome-wide significance (p=0.042). The inclusion of three further loci with lod scores higher than 1.57 (2q, 5q, and 9q) also show genome-wide significance (p=0.017) when assessed under a locus-counting analysis.

INTERPRETATION

These findings imply that human essential hypertension has an oligogenic element (a few genes may be involved in determination of the trait) possibly superimposed on more minor genetic effects, and that several genes may be tractable to a positional cloning strategy.

Multipoint affected sibpair linkage methods for localizing susceptibility genes of complex diseases

Recently, Liang et al. ([2001] Hum. Hered. 51:64-78) proposed a general multipoint linkage method for estimating the chromosomal position of a putative susceptibility locus. Their technique is computationally simple and does not require specification of penetrance or a mode of inheritance. In complex genetic diseases, covariate data may be available which reflect etiologic or locus heterogeneity. We developed approaches to incorporating covariates into the method of Liang et al. ([2001] Hum. Hered. 51:64-78) with particular attention to exploiting age-at-onset information. The results of simulation studies, and a worked data example using a family data set ascertained through probands with schizophrenia, suggest that utilizing covariate information can yield substantial efficiency gains in localizing susceptibility genes.

Multi-locus interactions predict risk for post-PTCA restenosis: an approach to the genetic analysis of common complex disease

The complexity of recognizing the potential contribution of a number of possible predictors of complex disorders is increasingly challenging with the application of large-scale single nucleotide polymorphism (SNP) typing. In the search for putative genetic factors predisposing to coronary artery restenosis following balloon angioplasty, we determined genotypes for 94 SNPs representing 62 candidate genes, in a prospectively assembled cohort of 342 cases and 437 controls. Using a customized coupled-logistic regression procedure accounting for both additive and interactive effects, we identified seven SNPs in seven genes that, together, showed a statistically significant association with restenosis incidence (P <0.0001), accounting for 11.6% of overall variance observed. Among them are candidate genes for cardiovascular pathophysiology (apolipoprotein-species and NOS), inflammatory response (TNF receptor and CD14), and cell-cycle control (p53 and p53-associated protein). Our results emphasize the need to account for complex multi-gene influences and interactions when assessing the molecular pathology of multifactorial medical entities.

Covariates in linkage analysis using sibling and cousin pairs

Using simulated data from GAW 12, problem 2, we further develop a novel technique to detect and use significant covariates in linkage analysis. The method, first introduced by Rice et al. [Genet Epidemiol 17(Suppl. 1):S691-5, 1999], uses logistic regression to model perturbation in sharing as a function of covariate levels. The original method allows use of all sib pairs (concordant affected, concordant unaffected, and discordant). Here we extend this method to include cousin pairs in analysis.

Marker selection by Akaike information criterion and Bayesian information criterion

We carried out a discriminant analysis with identity by descent (IBD) at each marker as inputs, and the sib pair type (affected-affected versus affected-unaffected) as the output. Using simple logistic regression for this discriminant analysis, we illustrate the importance of comparing models with different number of parameters. Such model comparisons are best carried out using either the Akaike information criterion (AIC) or the Bayesian information criterion (BIC). When AIC (or BIC) stepwise variable selection was applied to the German Asthma data set, a group of markers were selected which provide the best fit to the data (assuming an additive effect). Interestingly, these 25-26 markers were not identical to those with the highest (in magnitude) single-locus lod scores.

Meta-analysis of sib pair linkage studies of asthma and the interleukin-9 gene (IL9)

Asthma is a complex disease, with an etiology that includes both genetic and environmental influences that may interact. The moderate heritability of asthma has led researchers to investigate its molecular genetic basis using both exploratory investigations of linkage via genome scans, as well as targeted studies of specific candidate genes. Promising candidate genes include the cytokine genes on chromosome 5q23-31. Both genome scans and association studies of these candidate genes/genomic regions have yielded mixed findings, which raise the possibilities of a true relation that emerges more strongly in certain samples simply due to sampling variability, as well as of genetic heterogeneity. Meta-analytic approaches that combine data across samples and examine how findings may vary as a function of effect modifiers can address both of these possibilities. In this study, we used a meta-analytic approach to examine linkage between the interleukin-9 gene (IL9), one of the cytokine genes located on chromosome 5q31, and asthma diagnoses and serum IgE levels in four samples. We analyzed IBD allele sharing for affected, unaffected, and discordant sib pairs, and as a function of sibling differences in IgE levels. We used a recently developed logistic regression-based method that allows for the inclusion of covariates and/or effect modifiers in the analysis of allele sharing in sib-pairs [Rice et al., Genet Epidemiol 17(Suppl. 1):S691-5, 1999]. Sex of the siblings and transmitting parent were considered both as covariates and effect modifiers in analyses. The results provided little evidence for linkage, or for heterogeneity therein due to sex or transmitting parent, either within or across samples.

Mixture models for linkage analysis of affected sibling pairs and covariates

To determine the genetic etiology of complex diseases, a common study design is to recruit affected sib/relative pairs (ASP/ARP) and evaluate their genome-wide distribution of identical by descent (IBD) sharing using a set of highly polymorphic markers. Other attributes or environmental exposures of the ASP/ARP, which are thought to affect liability to disease, are sometimes collected. Conceivably, these covariates could refine the linkage analysis. Most published methods for ASP/ARP linkage with covariates can be conceptualized as logistic models in which IBD status of the ASP is predicted by pair-specific covariates. We develop a different approach to the problem of ASP analysis in the presence of covariates, one that extends naturally to ARP under certain conditions. For ASP linkage analysis, we formulate a mixture model in which a disease mutation is segregating in only a fraction alpha of the sibships, with 1 - alpha sibships being unlinked. Covariate information is used to predict membership within groups; in this report, the two groups correspond to the linked and unlinked sibships. For an ASP with covariate(s) Z = z and multilocus genotype X = x, the mixture model is alpha(z)g(x; lambda) + [1 - alpha(z)]g(0)(x), in which g(0)(x) follows the distribution of genotypes under the null IBD distribution and g(x; lambda) allows for increased IBD sharing. Two mixture models are developed. The pre-clustering model uses covariate information to form probabilistic clusters and then tests for excess IBD sharing independent of the covariates. The Cov-IBD model determines probabilistic group membership by joint consideration of covariate and IBD values. Simulations show that incorporating covariates into linkage analysis can enhance power substantially. A feature of our conceptualization of ASP linkage analysis, with covariates, is that it is apparent how data analysis might evaluate covariates prior to the linkage analysis, thus avoiding the loss of power described by Leal and Ott [2000] when data are stratified.

Genome scans for susceptibility genes in bipolar affective disorder

A genome-wide scan for genetic linkage can suggest fresh insights into disease aetiology. However, in the case of complex disorders such as bipolar affective disorder (BPAD), the results of genome-wide scans must be interpreted with caution. We review 10 published and 10 in-progress genome scans of BPAD, encompassing 3536 affected individuals in 1119 pedigrees. We find that ascertainment methods vary widely, with no two studies using identical methods. Sample sizes and marker densities have generally been well below what is now considered adequate, but several in-progress studies are using larger samples and more closely spaced markers. Few findings reach the 'suggestive' threshold, and fewer still reach the 'significant' threshold at genome-wide levels of significance. Strategies for pooling samples or subjecting findings in different samples to meta-analysis are being developed, but differences in ascertainment methods may have a large impact on the uniformity of different samples and hamper efforts at combining data or findings. There is also a need for methods that help define more genetically homogeneous phenotypes, take into account interactions between multiple susceptibility loci, and accommodate additional complexity (eg parent-of-origin effects) in the search for linkage.