Author: Mason Garrison

# No Database? No Problem: Using discord with Simple Family Structures

#### Mason Garrison

# Introduction

The {discord} package was originally developed for use with the National Longitudinal Survey of Youth (NLSY), but its functionality extends far beyond that. When paired with its sister package {BGmisc}, discord can be applied to any dataset containing basic family structure information, allowing researchers to explore genetic and environmental influences without requiring pre-constructed kinship links.

This vignette demonstrates how to:

- Construct kinship links from simple family data (e.g., individual ID, mother ID, father ID).
- Simulate phenotyipic data based on known genetic and environmental structures.
- Fit a discordant-kinship regression model using the simulated data.

We use tools from {BGmisc} and a toy dataset to illustrate the workflow.

# Loading Packages and Data

We begin by loading the required packages and a built-in dataset from {BGmisc}.

    library(BGmisc)
    library(ggpedigree)
    library(tidyverse)
    library(discord)

    data(potter)

We rename the family ID column to avoid naming conflicts and generate a pedigree-encoded data frame.

    df_potter <- potter

    names(df_potter)[names(df_potter) == "famID"] <- "oldfam"

    df_potter <- ped2fam(df_potter,
      famID = "famID",
      personID = "personID"
    )

We also verify and repair sex coding to ensure compatibility with downstream pedigree operations.

    df_potter <- checkSex(df_potter,
      code_male = 1,
      code_female = 0,
      verbose = FALSE, repair = TRUE
    )

    ggpedigree(potter, config = list(
      label_method = "geom_text",
      label_nudge_y = .25
    )) +
      labs(title = "Pedigree Plot of the Potter Dataset") +
      theme(legend.position = "bottom")

Pedigree plot of the Potter dataset

The pedigree plot provides a visual representation of the kinship structure in the dataset. Each node represents an individual, and the edges indicate familial relationships.

# Constructing Kinship Links

To extract the necessary kinship information, we need to compute two matrices: the additive genetic relatedness matrix and the shared environment matrix. These matrices are derived from the pedigree data and are essential for understanding the genetic and environmental relationships among individuals.

Using {BGmisc}, we compute:

- The additive genetic relatedness matrix (add).

- The shared environment matrix (cn), indicating whether kin were raised together (1) or apart (0).

    add <- ped2add(df_potter)
    cn <- ped2cn(df_potter)

The `ped2add()` function computes the additive genetic relatedness matrix, which quantifies the genetic similarity between individuals based on their pedigree information. The `ped2cn()` function computes the shared environment matrix, which indicates whether individuals were raised in the same environment (1) or different environments (0).

The resulting matrices are symmetric, with diagonal elements representing self-relatedness (1.0). The off-diagonal elements represent the relatedness between pairs of individuals, with values ranging from 0 (no relatedness) to 0.5 (full siblings) to 1 (themselves).

We convert the component matrices into a long-form data frame of kin pairs using `com2links()`. Self-pairs and duplicate entries are removed.

    df_links <- com2links(
      writetodisk = FALSE,
      ad_ped_matrix = add,
      cn_ped_matrix = cn,
      drop_upper_triangular = TRUE
    ) %>%
      filter(ID1 != ID2)

    df_links %>%
      slice(1:10) %>%
      knitr::kable()

ID1 | ID2 | addRel | cnuRel
--- | --- | --- | ---
1 | 2 | 0.500 | 1
3 | 4 | 0.500 | 1
1 | 6 | 0.500 | 0
2 | 6 | 0.250 | 0
3 | 6 | 0.500 | 0
4 | 6 | 0.250 | 0
3 | 7 | 0.250 | 0
4 | 7 | 0.500 | 0
5 | 7 | 0.500 | 0
6 | 7 | 0.125 | 0

As you can see, the `df_links` data frame contains pairs of individuals (ID1 and ID2) along with their additive genetic relatedness (`addRel`) and shared environment status (`cnuRel`). These data are in wide format, with each row representing a unique pair of individuals.

Further, we can tally the number of pairs by relatedness and shared environment to understand the composition of the dataset.

    df_links %>%
      group_by(addRel, cnuRel) %>%
      tally()
    #> # A tibble: 5 × 3
    #> # Groups:   addRel [4]
    #>   addRel cnuRel     n
    #>    <dbl>  <dbl> <int>
    #> 1 0.0625      0     3
    #> 2 0.125       0    47
    #> 3 0.25        0   104
    #> 4 0.5         0    50
    #> 5 0.5         1    32

As you can see, the dataset contains a variety of kinship pairs, including full siblings, parent-child, aunt-nephew, cousins, and unrelated individuals, with varying degrees of shared environment.

For this demonstration, we will focus on cousins. Cousins share some genetic relatedness but typically do not share the same environment. (In contrast, full siblings share both genetic relatedness and environment. Half-siblings share genetic relatedness but sometimes do not share the same environment, making them less ideal for this demonstration.)

We then extract two subsets:

- Full siblings: additive relatedness = 0.5 and shared environment = 1

- Cousins: additive relatedness = 0.125 and shared environment = 0

    df_siblings <- df_links %>%
      filter(addRel == .5) %>% # only full siblings %>%
      filter(cnuRel == 1) # only kin raised in the same home

    df_cousin <- df_links %>%
      filter(addRel == .125) %>% # only cousins %>%
      filter(cnuRel == 0) # only kin raised in separate homes

Now for the rest of the vignette, we will use the cousin subset (`df_cousin`) to illustrate the process of simulating phenotypic data and fitting a discordant-kinship regression model. However, given the small sample size of cousins in this dataset, we will simulate four datasets worth of cousins and combine them to increase the sample size.

    df_cousin <- rbind(
      df_cousin,
      df_cousin %>% mutate(ID1 = ID1 + 1000, ID2 = ID2 + 1000),
      df_cousin %>% mutate(ID1 = ID1 + 2000, ID2 = ID2 + 2000),
      df_cousin %>% mutate(ID1 = ID1 + 3000, ID2 = ID2 + 3000)
    )

# Simulate Phenotypic Data

To simulate phenotypic data, we need to create a data frame that includes the kinship information and the outcome variables. We will simulate two outcome variables (y1 and y2) for each kin pair in the dataset. The `kinsim()` function from {discord} is used to generate the simulated data based on the specified variance structure. For convenience, we will generate data for all the cousins in `df_links`. However, we could also generate data for the full siblings or any other kinship pairs.

    set.seed(1234)
    syn_df <- discord::kinsim(
      mu_all = c(2, 2),
      cov_a = .4,
      cov_e = .4,
      c_vector = df_cousin$cnuRel,
      r_vector = df_cousin$addRel
    ) %>%
      select(-c(
        A1_1, A1_2, A2_1, A2_2,
        C1_1, C1_2, C2_1, C2_2,
        E1_1, E1_2, E2_1, E2_2,
        r
      ))

The simulated data reflect a known variance structure: additive genetic covariance = .4, genetic relatedness of 0.125, no shared environment, and residual (unique environment) variance = 0.4. Latent component scores are dropped from the final dataset, but they can be useful for debugging and understanding the underlying structure of the data.

We bind the simulated outcome data to the links data to prepare it for modeling.

    data_demo <- cbind(df_cousin, syn_df) %>%
      arrange(ID1, ID2)

    summary(data_demo)
    #>       ID1              ID2             addRel          cnuRel
    #>  Min.   :   3.0   Min.   :   7.0   Min.   :0.125   Min.   :0
    #>  1st Qu.: 759.2   1st Qu.: 781.2   1st Qu.:0.125   1st Qu.:0
    #>  Median :1515.5   Median :1555.5   Median :0.125   Median :0
    #>  Mean   :1521.6   Mean   :1536.6   Mean   :0.125   Mean   :0
    #>  3rd Qu.:2271.8   3rd Qu.:2329.8   3rd Qu.:0.125   3rd Qu.:0
    #>  Max.   :3028.0   Max.   :3104.0   Max.   :0.125   Max.   :0
    #>       y1_1              y1_2              y2_1             y2_2
    #>  Min.   :-3.3672   Min.   :-1.5255   Min.   :-2.959   Min.   :-2.3276
    #>  1st Qu.: 0.6044   1st Qu.: 0.8779   1st Qu.: 0.787   1st Qu.: 0.9394
    #>  Median : 1.7633   Median : 2.1013   Median : 1.847   Median : 2.0066
    #>  Mean   : 1.7469   Mean   : 2.0855   Mean   : 2.039   Mean   : 2.1207
    #>  3rd Qu.: 2.9137   3rd Qu.: 3.1359   3rd Qu.: 3.208   3rd Qu.: 3.3210
    #>  Max.   : 6.0358   Max.   : 6.1543   Max.   : 6.783   Max.   : 6.7877
    #>        id
    #>  Min.   :  1.00
    #>  1st Qu.: 47.75
    #>  Median : 94.50
    #>  Mean   : 94.50
    #>  3rd Qu.:141.25
    #>  Max.   :188.00

    data_demo %>%
      slice(1:5) %>%
      knitr::kable()

ID1 | ID2 | addRel | cnuRel | y1_1 | y1_2 | y2_1 | y2_2 | id
--- | --- | --- | --- | --- | --- | --- | --- | ---
3 | 21 | 0.125 | 0 | 0.6984681 | -0.9463635 | 0.9203754 | 0.8795634 | 2
3 | 22 | 0.125 | 0 | 3.4082317 | 0.1747639 | 4.5457972 | 0.2776425 | 3
3 | 23 | 0.125 | 0 | -1.9437220 | -1.4571830 | 2.8360088 | 2.4356576 | 4
6 | 7 | 0.125 | 0 | 3.0188293 | 5.1542516 | 0.7041064 | 0.9333481 | 1
21 | 24 | 0.125 | 0 | 0.5056531 | 0.0700135 | 1.1275003 | 4.2852528 | 5

The `data_demo` data frame now contains the kinship information along with the simulated outcome variables y1 and y2. Each row represents a pair of cousins, and the columns include the IDs of the individuals, their relatedness, and the simulated phenotypic data.

# Fitting a Discordant-Kinship Regression Model

We then use `discord_regression()` to fit a discordant-kinship model, predicting y1 from y2. Based on the structure of the data, we expect that there will be a significant association between the two outcome variables, as there is a known overlapping non-shared environment covariance.

The model is fit using the `discord_regression()` function, which takes the following arguments:

    model_output <- discord_regression(
      data = data_demo,
      outcome = "y1",
      predictors = "y2",
      id = "id",
      sex = NULL,
      race = NULL,
      pair_identifiers = c("_1", "_2")
    )
    summary(model_output)
    #>
    #> Call:
    #> stats::lm(formula = y1_diff ~ y1_mean + y2_diff + y2_mean, data = preppedData)
    #>
    #> Residuals:
    #>     Min      1Q  Median      3Q     Max
    #> -1.8511 -0.8476 -0.1705  0.6317  3.6083
    #>
    #> Coefficients:
    #>             Estimate Std. Error t value Pr(>|t|)
    #> (Intercept)  1.38903    0.17337   8.012 1.25e-13 ***
    #> y1_mean      0.06308    0.06038   1.045  0.29749
    #> y2_diff      0.13111    0.04081   3.213  0.00155 **
    #> y2_mean     -0.04300    0.05654  -0.761  0.44789
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Residual standard error: 1.108 on 184 degrees of freedom
    #> Multiple R-squared:  0.06089,    Adjusted R-squared:  0.04557
    #> F-statistic: 3.976 on 3 and 184 DF,  p-value: 0.00893

The output of the model includes estimates of the regression coefficients, standard errors, and p-values for the association between the two outcome variables.

# Alternative Specifications

If desired, one can manually prepare the data using `discord_data()` and then fit separate models for the individual-level, between-pair, and within-pair effects. This approach provides more flexibility in specifying the models and allows for a deeper understanding of the different components of the association.

    data_df <- discord_data(
      data = data_demo,
      outcome = "y1",
      predictors = "y2",
      id = "id",
      sex = NULL,
      race = NULL,
      demographics = "none",
      pair_identifiers = c("_1", "_2")
    )
    summary(data_df)
    #>        id              y1_1             y1_2            y1_diff
    #>  Min.   :  1.00   Min.   :-1.457   Min.   :-3.3672   Min.   :0.003215
    #>  1st Qu.: 47.75   1st Qu.: 1.643   1st Qu.: 0.1777   1st Qu.:0.707853
    #>  Median : 94.50   Median : 2.723   Median : 1.1478   Median :1.311807
    #>  Mean   : 94.50   Mean   : 2.679   Mean   : 1.1538   Mean   :1.524696
    #>  3rd Qu.:141.25   3rd Qu.: 3.882   3rd Qu.: 2.1351   3rd Qu.:2.207785
    #>  Max.   :188.00   Max.   : 6.154   Max.   : 4.9615   Max.   :5.157103
    #>     y1_mean            y2_1             y2_2           y2_diff
    #>  Min.   :-1.700   Min.   :-2.109   Min.   :-2.959   Min.   :-3.9241
    #>  1st Qu.: 1.006   1st Qu.: 1.291   1st Qu.: 0.574   1st Qu.:-0.4878
    #>  Median : 1.912   Median : 2.364   Median : 1.505   Median : 0.8402
    #>  Mean   : 1.916   Mean   : 2.478   Mean   : 1.682   Mean   : 0.7951
    #>  3rd Qu.: 2.884   3rd Qu.: 3.602   3rd Qu.: 2.775   3rd Qu.: 2.0682
    #>  Max.   : 5.053   Max.   : 6.788   Max.   : 6.783   Max.   : 6.9832
    #>     y2_mean
    #>  Min.   :-1.154
    #>  1st Qu.: 1.077
    #>  Median : 1.936
    #>  Mean   : 2.080
    #>  3rd Qu.: 2.911
    #>  Max.   : 6.119

    lm_ind <- lm(y1_1 ~ y2_1, data = data_df)
    summary(lm_ind)
    #>
    #> Call:
    #> lm(formula = y1_1 ~ y2_1, data = data_df)
    #>
    #> Residuals:
    #>     Min      1Q  Median      3Q     Max
    #> -4.1295 -0.9876  0.0817  1.1612  3.7977
    #>
    #> Coefficients:
    #>             Estimate Std. Error t value Pr(>|t|)
    #> (Intercept)  2.30725    0.18995  12.146   <2e-16 ***
    #> y2_1         0.14987    0.06282   2.386    0.018 *
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Residual standard error: 1.493 on 186 degrees of freedom
    #> Multiple R-squared:  0.0297, Adjusted R-squared:  0.02448
    #> F-statistic: 5.692 on 1 and 186 DF,  p-value: 0.01804

    lm_ind2 <- lm(y1_2 ~ y2_2, data = data_df)

    summary(lm_ind2)
    #>
    #> Call:
    #> lm(formula = y1_2 ~ y2_2, data = data_df)
    #>
    #> Residuals:
    #>     Min      1Q  Median      3Q     Max
    #> -4.4598 -0.9950 -0.0377  0.9567  3.3692
    #>
    #> Coefficients:
    #>             Estimate Std. Error t value Pr(>|t|)
    #> (Intercept)  0.87682    0.14193   6.178    4e-09 ***
    #> y2_2         0.16466    0.05788   2.845  0.00494 **
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Residual standard error: 1.416 on 186 degrees of freedom
    #> Multiple R-squared:  0.0417, Adjusted R-squared:  0.03655
    #> F-statistic: 8.094 on 1 and 186 DF,  p-value: 0.004939

    lm_between <- lm(y1_mean ~ y2_mean, data = data_df)
    summary(lm_between)
    #>
    #> Call:
    #> lm(formula = y1_mean ~ y2_mean, data = data_df)
    #>
    #> Residuals:
    #>     Min      1Q  Median      3Q     Max
    #> -3.7107 -0.8579  0.0237  0.9435  3.3296
    #>
    #> Coefficients:
    #>             Estimate Std. Error t value Pr(>|t|)
    #> (Intercept)   1.5642     0.1713   9.131   <2e-16 ***
    #> y2_mean       0.1692     0.0675   2.507    0.013 *
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Residual standard error: 1.346 on 186 degrees of freedom
    #> Multiple R-squared:  0.03268,    Adjusted R-squared:  0.02748
    #> F-statistic: 6.284 on 1 and 186 DF,  p-value: 0.01304

    lm_within <- lm(y1_diff ~ y1_mean + y2_diff + y2_mean, data = data_df)
    summary(lm_within)
    #>
    #> Call:
    #> lm(formula = y1_diff ~ y1_mean + y2_diff + y2_mean, data = data_df)
    #>
    #> Residuals:
    #>     Min      1Q  Median      3Q     Max
    #> -1.8511 -0.8476 -0.1705  0.6317  3.6083
    #>
    #> Coefficients:
    #>             Estimate Std. Error t value Pr(>|t|)
    #> (Intercept)  1.38903    0.17337   8.012 1.25e-13 ***
    #> y1_mean      0.06308    0.06038   1.045  0.29749
    #> y2_diff      0.13111    0.04081   3.213  0.00155 **
    #> y2_mean     -0.04300    0.05654  -0.761  0.44789
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Residual standard error: 1.108 on 184 degrees of freedom
    #> Multiple R-squared:  0.06089,    Adjusted R-squared:  0.04557
    #> F-statistic: 3.976 on 3 and 184 DF,  p-value: 0.00893

# Conclusion

This vignette demonstrates how {BGmisc} and discord enable researchers to perform discordant-kinship analyses starting from simple family data. There’s no need for pre-constructed kinship links or specialized datasets like the NLSY—just basic family identifiers are sufficient to generate kinship structures and apply powerful behavior genetic models.

# References