. Author manuscript; available in PMC: 2020 Jan 29.

Published in final edited form as: Nat Genet. 2019 Jul 29;51(8):1244–1251. doi: 10.1038/s41588-019-0465-0

Table 1.

Existing methods to estimate SNP-heritability impose additional assumptions on top of the generalized random effects (GRE) model. Under the GRE model, the causal effects at any two SNPs are assumed to be independent ( $E [β_{i} β_{j}] = 0$ for all $i \neq j$ ) and genome-wide SNP-heritability is defined as $h_{g}^{2} \equiv \sum_{i = 1}^{M} σ_{i}^{2}$ , where each $σ_{i}^{2}$ can be an arbitrary nonnegative real number as long as $0 \leq h_{g}^{2} \leq 1$ (Methods). All existing methods make assumptions on the distribution of $β_{i}$ and/or the form of $σ_{i}^{2}$ that can be subsumed under the GRE model. To simplify notation, we assume for each model that phenotypes are standardized in the population (i.e. $Var [y_{n}] = 1$ for every individual $n$ ).

Model	Assumptions on $β_{i}$	Description
Generalized random effects	$E [β_{i}] = 0$ , $Var [β_{i}] = σ_{i}^{2}$ , $σ_{i}^{2} \geq 0$	Each SNP $i$ has a nonnegative SNP-specific variance $σ_{i}^{2}$ . Total SNP-heritability is $h_{g}^{2} \equiv \sum_{i = 1}^{M} σ_{i}^{2}$ .
GREML-SC ^3,8,16	$β_{i} \sim N (0, h_{g}^{2} / M)$	Each SNP explains an equal portion of $h_{g}^{2}$ . In other words, $σ_{i}^{2} = h_{g}^{2} / M$ for all $i = 1, \dots, M$ .
GREML-MC ^7,8,18,42,43	$β_{i} \sim N (0, \sum_{c \in C} [{SNP}_{i} \in c] h_{c}^{2} / m_{c})$	$h_{g}^{2}$ is partitioned by a set of disjoint SNP partitions $C$ that span all $M$ SNPs. Partition $c \in C$ contains $m_{c}$ SNPs that have per-SNP variances $h_{c}^{2} / m_{c}$ . Total SNP-heritability is $h_{g}^{2} = \sum_{c \in C} h_{c}^{2}$ .
LDAK^6,9	$β_{i} \sim N (0, σ_{i}^{2})$ , $σ_{i}^{2} \propto w_{i} {[f_{i} (1 - f_{i})]}^{1 + α}$	Each SNP-specific variance is proportional to a function of $f_{i}$ (the MAF of SNP $i$ ) and to $w_{i}$ (a SNP-specific weight that is a function of the inverse of the LD score of SNP $i$ ). $α$ controls the relationship between $σ_{i}^{2}$ and $f_{i}$ . The most recent recommendation by ref.⁹ is to assume $α = - 0.25$ .
LDSC¹¹	$E [β_{i}] = 0$ , $Var [β_{i}] = h_{g}^{2} / M$	Each SNP explains an equal portion of $h_{g}^{2}$ (similar to the GREML-SC model when $h_{g}^{2}$ is defined with respect to the same set of $M$ SNPs).
S-LDSC^12,13,30	$E [β_{i}] = 0$ , $Var [β_{i}] = \sum_{a \in A} τ_{a} a (i)$	Each SNP-specific variance is a linear function of a set of annotations $A$ where each $a \in A$ represents a binary or continuous-valued annotation. $a (i)$ is the value of annotation $a$ at SNP $i$ . $τ_{a}$ is the expected contribution of a one-unit increase in annotation $a$ to each SNP-specific variance.
SumHer¹⁴	$E [β_{i}] = 0$ , $Var [β_{i}] \propto w_{i} {[f_{i} (1 - f_{i})]}^{1 + α}$	An extension of the LDAK model to operate on summary-level data; can also efficiently partition $h_{g}^{2}$ by multiple annotations. The most recent recommendations by refs.^9,14 is to set $α = - 0.25$ .