Genotype Coding

Genotype Coding#

Genotype coding converts DNA nucleotide pairs into numerical values for statistical analysis, with the additive model being particularly valuable because genetic effects often accumulate proportionally with each additional copy of a variant.

Graphical Summary#

Fig

Key Formula#

We use \(\mathbf{X}_\text{raw}\) to denote the raw genotype matrix (an \(N \times M\) matrix for \(N\) individuals at \(M\) variants) under the additive coding model with values \(\{0, 1, 2\}\) representing alternative allele counts.

\(\mathbf{X}\) is the standardized genotype matrix. Why standardize? Variants have different allele frequencies and variances. Without standardization, a \(\beta = 0.1\) effect means different things for rare versus common variants. Standardization makes effect sizes comparable across variants, which is essential for multi-variant analyses (see proportion of variance explained, random effect, Bayesian normal mean model).

For each column \(j\) (the \(j\)-th variant):

\[ X_{ij} = \frac{X_{\text{raw},ij} - \mu_j}{\sigma_j} \]

Where:

  • \(\mu_j = \frac{1}{N}\sum_{i=1}^{N} X_{\text{raw},ij}\) (mean of the \(j\)-th variant)

  • \(\sigma_j = \sqrt{\frac{1}{N-1}\sum_{i=1}^{N} (X_{\text{raw},ij} - \mu_j)^2}\) (standard deviation of the \(j\)-th variant)

Technical Details#

Coding Models#

For diploid organisms (with two copies of each chromosome, e.g., human), we can use different coding models:

Genetic Models#

Model

Values

Genotype Encoding

Description

Additive (default)

{0, 1, 2}

0 = AA (homozygous reference)
1 = Aa (heterozygous)
2 = aa (homozygous alternative)

Represents the count of alternative alleles

Dominant

{0, 1}

0 = AA (homozygous reference)
1 = Aa or aa (carries alternative allele)

Represents the presence of at least one copy of the alternative allele

Recessive

{0, 1}

0 = AA or Aa (carries reference allele)
1 = aa (homozygous alternative)

Represents when both copies of the alternative allele are present

Standardized Genotype Matrix#

\(\mathbf{X}\) is a normalized version of \(\mathbf{X}_\text{raw}\) where each variant is scaled to have mean 0 and variance 1.

For each column \(j\) (the \(j\)-th variant):

  • \(\mu_j = \frac{1}{N}\sum_{i=1}^{N} \mathbf{X}_{\text{raw},ij}\) (mean of the \(j\)-th variant)

    • Represents the average number of alternative alleles at the \(j\)-th variant in the sample under the additive model

  • \(\sigma_j = \sqrt{\frac{1}{N-1}\sum_{i=1}^{N} (\mathbf{X}_{\text{raw},ij} - \mu_j)^2}\) (standard deviation of the \(j\)-th variant)

    • Measures the variability in the number of alternative alleles at the \(j\)-th variant

This standardization ensures that:

  • Each column of \(\mathbf{X}\) has mean 0 and variance 1

  • The standardized values reflect the deviation from the population mean in units of standard deviation

  • All variants contribute equally to downstream statistical analyses regardless of their allele frequencies

Example#

Let’s say we have genetic data from 5 individuals tested at 3 different variants. How do we go from the actual genotype calls (like “CC” or “CT”) to the numerical matrices used in statistical genetics? What happens when we apply different coding models to the same data? And how do we standardize these matrices in R?

rm(list = ls())

# Define genotypes for 5 individuals at 3 variants
# These represent actual alleles at each position
# For example, Individual 1 the following nucleotides on three variants: CC, CT, AT
genotypes <- c(
 "CC", "CT", "AT",  # Individual 1
 "TT", "TT", "AA",  # Individual 2
 "CT", "CT", "AA",  # Individual 3
 "CC", "TT", "AA",  # Individual 4
 "CC", "CC", "TT"   # Individual 5
)
# Reshape into a matrix
N = 5
M = 3
geno_matrix <- matrix(genotypes, nrow = N, ncol = M, byrow = TRUE)
rownames(geno_matrix) <- paste("Individual", 1:N)
colnames(geno_matrix) <- paste("Variant", 1:M)

The raw genotype matrix is:

geno_matrix
A matrix: 5 × 3 of type chr
Variant 1Variant 2Variant 3
Individual 1CCCTAT
Individual 2TTTTAA
Individual 3CTCTAA
Individual 4CCTTAA
Individual 5CCCCTT

Then we assign the alternative allele for each variant:

# Define alternative alleles for each variant
alt_alleles <- c("T", "C", "T")
names(alt_alleles) <- colnames(geno_matrix)

Now let’s convert the variants information for the individuals into a raw genotype matrix, under three different models:

# Convert to raw genotype matrix using the additive / dominant / recessive model
Xraw_additive <- matrix(0, nrow=nrow(geno_matrix), ncol=ncol(geno_matrix)) # count number of non-reference alleles
Xraw_dominant <- matrix(0, nrow=nrow(geno_matrix), ncol=ncol(geno_matrix)) # presence of alternative allele
Xraw_recessive <- matrix(0, nrow=nrow(geno_matrix), ncol=ncol(geno_matrix)) # two copies of alternative allele required

rownames(Xraw_additive) <- rownames(Xraw_dominant) <- rownames(Xraw_recessive) <- rownames(geno_matrix)
colnames(Xraw_additive) <- colnames(Xraw_dominant) <- colnames(Xraw_recessive) <- colnames(geno_matrix)

for (i in 1:nrow(geno_matrix)) {
  for (j in 1:ncol(geno_matrix)) {
    alleles <- strsplit(geno_matrix[i,j], "")[[1]]
    Xraw_additive[i,j] <- sum(alleles == alt_alleles[j])
    Xraw_dominant[i,j] <- as.integer(any(alleles == alt_alleles[j]))
    Xraw_recessive[i,j] <- as.integer(all(alleles == alt_alleles[j]))
  }
}

Raw genotype matrix under additive model:

Xraw_additive
A matrix: 5 × 3 of type dbl
Variant 1Variant 2Variant 3
Individual 1011
Individual 2200
Individual 3110
Individual 4000
Individual 5022
 
print("Raw genotype matrix (dominant model):")
Xraw_dominant

[1] "Raw genotype matrix (dominant model):"
A matrix: 5 × 3 of type dbl
Variant 1Variant 2Variant 3
Individual 1011
Individual 2100
Individual 3110
Individual 4000
Individual 5011
 
print("Raw genotype matrix (recessive model):")
Xraw_recessive

[1] "Raw genotype matrix (recessive model):"
A matrix: 5 × 3 of type dbl
Variant 1Variant 2Variant 3
Individual 1000
Individual 2100
Individual 3000
Individual 4000
Individual 5011

Then we standardize the raw genotype matrix under each model:

# Standardize each raw genotype matrix
X_additive <- scale(Xraw_additive, center=TRUE, scale=TRUE)
X_dominant <- scale(Xraw_dominant, center=TRUE, scale=TRUE)
X_recessive <- scale(Xraw_recessive, center=TRUE, scale=TRUE)

Here are the standardized matrix \(\mathbf{X}\) for additive, dominant, and recessive models, respectively:

X_additive
A matrix: 5 × 3 of type dbl
Variant 1Variant 2Variant 3
Individual 1-0.6708204 0.2390457 0.4472136
Individual 2 1.5652476-0.9561829-0.6708204
Individual 3 0.4472136 0.2390457-0.6708204
Individual 4-0.6708204-0.9561829-0.6708204
Individual 5-0.6708204 1.4342743 1.5652476
 
X_dominant
A matrix: 5 × 3 of type dbl
Variant 1Variant 2Variant 3
Individual 1-0.7302967 0.7302967 1.0954451
Individual 2 1.0954451-1.0954451-0.7302967
Individual 3 1.0954451 0.7302967-0.7302967
Individual 4-0.7302967-1.0954451-0.7302967
Individual 5-0.7302967 0.7302967 1.0954451
 
X_recessive
A matrix: 5 × 3 of type dbl
Variant 1Variant 2Variant 3
Individual 1-0.4472136-0.4472136-0.4472136
Individual 2 1.7888544-0.4472136-0.4472136
Individual 3-0.4472136-0.4472136-0.4472136
Individual 4-0.4472136-0.4472136-0.4472136
Individual 5-0.4472136 1.7888544 1.7888544

Supplementary#

Beyond SNPs: Encoding Different Variant Types#

While we commonly encode SNPs, there are other types of genetic variations where the same principles also apply:

  1. Insertions and Deletions (Indels)

  • Small insertions or deletions in DNA sequence

  • Typically encoded using the same 0/1/2 scheme as SNPs

  1. Copy Number Variations (CNVs)

  • Deletions or duplications of larger DNA segments

  • Can be encoded as actual copy number (0, 1, 2, 3, etc.)

  1. Structural Variants

  • Inversions, translocations, and complex rearrangements

  • Often encoded as binary (presence/absence)

The choice of encoding scheme should reflect the biological hypothesis about how the variant affects the phenotype. For more information, refer to Figure 1 in Cardoso et al., 2015.

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