The Weighted-Sum Definition of a Polygenic Score

From Genotypes to a Score

A polygenic score compresses genotype information across multiple genetic variants into one number for each individual. The basic operation is straightforward: represent each genotype as a dosage, multiply each dosage by its corresponding effect weight, and add the products. The meaning of the resulting number depends on how the genotypes were coded and scaled, so those choices belong in the definition of the score.

For this definition, suppose the variants and weights have already been selected. At variant $j$, the dosage and its weight must refer to the same allele: if $G_{ij}$ counts copies of one allele for person $i$, then $\beta_j$ must be the weight for that allele under the chosen coding. This tutorial assumes that correspondence is correct; deciding how to align alleles across data sources is a separate construction task.

Definition: For $n$ individuals and $m$ variants, let $G_{ij}$ be the dosage for individual $i$ at variant $j$, and let $\beta_j$ be the weight attached to that dosage. The raw polygenic score for individual $i$ is $$ S_i=\sum_{j=1}^{m}G_{ij}\beta_j. $$ For a biallelic variant, a dosage typically ranges from 0 to 2 and represents the number of copies of the counted allele; with imputed data, it may be a fractional expected dosage between 0 and 2.

The formula defines a weighted sum, not a particular procedure for choosing the variants or estimating their weights. A zero weight contributes nothing, a positive weight raises the score as dosage increases, and a negative weight lowers it. Those statements describe arithmetic in the defined coding; they do not by themselves establish a causal interpretation.

Raw Dosages: The Basic Calculation

A raw score uses the dosage as recorded, without subtracting a reference mean or dividing by a scale. Each product $G_{ij}\beta_j$ is the contribution of variant $j$ to person $i$'s score. The score is the sum of these contributions over the included variants.

Worked Example: A Raw Score From Three Variants

Suppose a hypothetical individual has dosages $G_{i1}=2$, $G_{i2}=1$, and $G_{i3}=0$. The corresponding weights are $\beta_1=0.30$, $\beta_2=-0.10$, and $\beta_3=0.25$. Assume each dosage counts the allele to which its weight applies.

The raw score is $$ S_i=(2)(0.30)+(1)(-0.10)+(0)(0.25) =0.60-0.10+0 =0.50. $$

The first variant contributes $0.60$, the second contributes $-0.10$, and the third contributes zero. Thus this individual’s raw score is $0.50$ in the score scale determined by these weights. The result is a numerical summary of the specified genotype-weight pairs, not a probability or a diagnosis.

The same formula applies to every individual in a data set. If genotype dosages are assembled in an $n$-by-$m$ matrix $G$, and the weights are collected in a vector $\boldsymbol{\beta}$, then the scores for all individuals can be written as $G\boldsymbol{\beta}$. This matrix expression is just a compact way to perform the same row-by-row weighted sums.

Centered Genotypes and Centered Scores

A raw dosage has a reference value of zero copies of the counted allele. Sometimes it is useful to express each person’s dosage as a deviation from a reference mean dosage. Let $f_j$ be the frequency of the counted allele at variant $j$ in a specified reference population. Under a biallelic coding, the expected dosage is $2f_j$. The centered dosage is $G_{ij}-2f_j$.

Definition: Given reference allele frequencies $f_j$, the centered-genotype score for individual $i$ is $$ S_i^{(c)}=\sum_{j=1}^{m}(G_{ij}-2f_j)\beta_j. $$ The centering reference must be stated: for example, it may be a specified reference sample or population. A score centered using a different reference can have a different numerical zero.

Centering subtracts a constant from each genotype at a variant. It does not change the weight assigned to a one-unit dosage difference. Expanding the sum gives an exact relation to the raw score:

\begin{aligned} S_i^{(c)} &=\sum_{j=1}^{m}G_{ij}\beta_j-\sum_{j=1}^{m}2f_j\beta_j\\ &=S_i-\sum_{j=1}^{m}2f_j\beta_j. \end{aligned}

The second term is the same for every individual when the frequencies and weights are fixed. Therefore, raw and centered scores differ by a constant across individuals. Their ordering is the same, although their zero points and numerical values differ. This fact can be useful when comparing score calculations, but only if the variant set, allele coding, weights, and reference frequencies are held fixed.

Worked Example: Centering Changes the Zero Point

Consider two variants with weights $\beta_1=0.40$ and $\beta_2=-0.20$. The reference frequencies of the counted alleles are $f_1=0.25$ and $f_2=0.10$. A hypothetical individual has dosages $G_{i1}=2$ and $G_{i2}=0$.

First calculate the raw score: $$ S_i=(2)(0.40)+(0)(-0.20)=0.80. $$ The reference mean dosages are $2f_1=0.50$ and $2f_2=0.20$, so the centered dosages are $2-0.50=1.50$ and $0-0.20=-0.20$. The centered score is $$ S_i^{(c)}=(1.50)(0.40)+(-0.20)(-0.20) =0.60+0.04=0.64. $$

The constant subtracted from the raw score is $$ (0.50)(0.40)+(0.20)(-0.20)=0.20-0.04=0.16, $$ and $0.80-0.16=0.64$, agreeing with the direct calculation. The centered score is lower here because of the chosen reference and weights; it does not mean the individual’s genotype data changed.

Standardized Genotypes and Standardized-Input Scores

Centering changes the reference point but leaves the genotype scale unchanged. Standardization also divides each centered dosage by a positive scale. One conventional scale follows from the biallelic allele frequency under Hardy–Weinberg equilibrium. If the counted allele has frequency $f_j$, the genotype dosage has mean $2f_j$ and variance $2f_j(1-f_j)$ under that model.

Definition: For a variant with $0<f_j<1$, define its allele-frequency-standardized genotype as $$ Z_{ij}=\frac{G_{ij}-2f_j}{\sqrt{2f_j(1-f_j)}}. $$ The score using these standardized genotypes is $$ S_i^{(z)}=\sum_{j=1}^{m}Z_{ij}\beta_j. $$ Here the denominator is the genotype standard deviation implied by the allele frequency under Hardy–Weinberg equilibrium. A different explicitly chosen scale, such as an empirical genotype standard deviation, defines a different standardized score.

This transformation gives each variant a centered input measured in units of its chosen genotype standard deviation. It does not make the weights identical, nor does it make the resulting score a standardized outcome. Also, the score is not generally equal to the raw score minus one constant: each variant’s genotype has been divided by its own scale as well as centered.

Worked Example: Calculating a Score With Standardized Genotypes

Use three hypothetical variants with counted-allele frequencies $f_1=0.25$, $f_2=0.50$, and $f_3=0.10$. Suppose one individual has dosages $G_{i1}=2$, $G_{i2}=1$, and $G_{i3}=0$, with weights $0.40$, $-0.20$, and $0.30$, respectively. Assume the stated frequencies are the reference frequencies and use the Hardy–Weinberg variance scale.

For variant 1, the centered dosage is $2-2(0.25)=1.50$, and the scale is $\sqrt{2(0.25)(0.75)}=\sqrt{0.375}$. For variant 2, the centered dosage is $1-2(0.50)=0$, and the scale is $\sqrt{0.50}$. For variant 3, the centered dosage is $0-2(0.10)=-0.20$, and the scale is $\sqrt{0.18}$. Thus $$ \begin{aligned} S_i^{(z)} &=\frac{1.50}{\sqrt{0.375}}(0.40) +\frac{0}{\sqrt{0.50}}(-0.20) +\frac{-0.20}{\sqrt{0.18}}(0.30)\\ &\approx (2.4494897)(0.40)+0+(-0.4714045)(0.30)\\ &\approx 0.9797959-0.1414214\\ &\approx 0.8384. \end{aligned} $$

The calculation uses more precision in the intermediate values than in the reported result, so rounding to four decimal places gives $0.8384$. Using rounded standardized genotypes too early could alter the final digit. This score is on the scale defined by the standardized inputs and the given weights; it is not numerically interchangeable with the raw score.

What the Different Scores Do and Do Not Say

The distinction among raw, centered, and standardized-input scores is a distinction in construction. Raw dosage counts copies relative to zero. Centered dosage counts deviations relative to a chosen mean. Standardized dosage measures those deviations relative to a chosen genotype scale. For the frequency-based standardization above, the scale uses the Hardy–Weinberg variance implied by the reference allele frequency.

Genotype input	Variant-level value	Effect on score construction
Raw dosage	$G_{ij}$	Uses the recorded copy count directly
Centered dosage	$G_{ij}-2f_j$	Changes the reference zero; with fixed weights, shifts all scores by a common constant
Standardized dosage	$(G_{ij}-2f_j)/\sqrt{2f_j(1-f_j)}$	Changes the reference zero and the scale for each variant

A score value alone does not say whether an individual has a particular condition, nor does it have a universal interpretation across score definitions. Its meaning depends on the included variants, allele coding, weights, and genotype transformation. Even when two methods use the same weights, changing from raw to standardized inputs changes the contributions and may change how individuals are ranked. Centering alone, with fixed frequencies and weights, changes the common zero point but preserves the ranking.

Common Mistakes

Adding dosages without applying weights. A polygenic score is not generally the total number of counted alleles. Each dosage is multiplied by its corresponding weight before summation.
Using a dosage and weight for different alleles. The product is meaningful only when the dosage’s counted allele matches the allele represented by the weight.
Calling a centered score raw. Subtracting $2f_j$ changes the score’s reference point, even though it does not change the one-dosage increment at a variant.
Treating standardization as centering alone. Standardization also divides by a scale, so it can change the relative contributions of variants.
Interpreting a score as a probability or causal effect. The weighted sum is a constructed predictor. Its value alone is neither a probability nor evidence that a genotype caused an outcome.

Key takeaway: State the dosage coding, weights, and any centering or scaling rule when defining a polygenic score. The compact formula is a weighted sum, but its numerical scale and reference point follow from those choices.

Check Your Understanding

Use the definitions and calculations above to answer the following questions.

For dosages $1$ and $2$ with weights $0.3$ and $-0.1$, calculate the raw score.
What reference value is subtracted from dosage $G_{ij}$ when centering by counted-allele frequency $f_j$?
With fixed frequencies and weights, explain why centering changes all individuals’ scores by the same amount.
Under what model-based assumption does the scale $\sqrt{2f_j(1-f_j)}$ represent the genotype standard deviation?
Why can standardized-input scores not generally be treated as raw scores shifted by one common constant?

Genotype input	Variant-level value	Effect on score construction
Raw dosage	\(G_{ij}\)	Uses the recorded copy count directly
Centered dosage	\(G_{ij}-2f_j\)	Changes the reference zero; with fixed weights, shifts all scores by a common constant
Standardized dosage	\((G_{ij}-2f_j)/\sqrt{2f_j(1-f_j)}\)	Changes the reference zero and the scale for each variant