data<-read.csv("reg_herit_corr.csv", header=T)
attach(data)
head(data)
##   Pair Parent Offspring
## 1    1   21.6      22.3
## 2    2   20.3      21.0
## 3    3   20.6      19.4
## 4    4   31.2      23.6
## 5    5   20.4      28.2
## 6    6   25.4      23.8
str(data)
## 'data.frame':    20 obs. of  3 variables:
##  $ Pair     : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ Parent   : num  21.6 20.3 20.6 31.2 20.4 25.4 26.6 21.4 23.6 24.6 ...
##  $ Offspring: num  22.3 21 19.4 23.6 28.2 23.8 25.6 24.2 20.9 29.2 ...
### Fit a linear regression model
model <- lm(Offspring ~ Parent)
### Summary of the regression model
summary(model)
## 
## Call:
## lm(formula = Offspring ~ Parent)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.4515 -1.1235 -0.2818  0.5050  5.0817 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  22.4776     3.8638   5.817 1.64e-05 ***
## Parent        0.0667     0.1572   0.424    0.676    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.503 on 18 degrees of freedom
## Multiple R-squared:  0.009906,   Adjusted R-squared:  -0.0451 
## F-statistic: 0.1801 on 1 and 18 DF,  p-value: 0.6763
plot(Parent, Offspring, 
     main = "Offspring vs Parent", 
     xlab = "Parent", 
     ylab = "Offspring", 
     pch = 4, col = "blue")

###fitted values with line
# Same plot as above
plot(Parent, Offspring, 
     main = "Offspring vs Parent (with fitted line)", 
     xlab = "Parent", 
     ylab = "Offspring", 
     pch = 4, col = "blue")
lines(Parent, fitted(model), col = "red", lwd = 2)

### Heritability estimate
heri_reg <- 2*coef(model)[2]  # The slope coefficient
print(paste("Heritability estimate:", round(heri_reg, 2)))
## [1] "Heritability estimate: 0.13"
## Heritablility using the correlation method
Parent_num <- as.numeric(as.character(Parent))
Offspring_num <- as.numeric(as.character(Offspring))

heri_cor <- 2 * cor(Parent_num, Offspring_num)
print(heri_cor)
## [1] 0.1990625
#### heritability using the covariance method
covariance <- cov(Parent_num, Offspring_num)
variance_parent <- var(Parent_num)
variance_offspring <- var(Offspring_num)
heri_cov <- 2 * (covariance / sqrt(variance_parent * variance_offspring))
print(paste("Heritability estimate:", round(heri_cov, 2)))
## [1] "Heritability estimate: 0.2"

Interpretation

1. The plot

Heritability in the narrow sense (ℎ²) can often be estimated as the slope of the regression line of offspring on parent (or mid-parent). Here’s how to interpret it:

1.1 Positive but Shallow Slope

  • The red line has a slight positive slope , meaning there is some genetic resemblance between parents and offspring.

  • However, the slope is very shallow , suggesting low heritability .

  • In other words, while offspring tend to slightly resemble their parents, much of the variation may be due to environmental factors or non-additive genetics .

2.2 High Scatter Around the Line

  • The points are widely scattered around the line, which means the fit of the model is weak .

  • This reinforces the idea that parental traits explain only a small portion of offspring traits —another sign of low heritability.

2. Regression Method

heri_reg <- 2 * coef(model)[2]

  • This method estimates heritability as twice the slope of the regression line of offspring on one parent .
Estimate: **0.13**
  • Interpretation : Only 13% of the variation in the trait can be explained by additive genetic effects from the parent. This suggests low heritability.
### 3. **Correlation Method**

`heri_cor <- 2 * cor(Parent_num, Offspring_num)`

  • Estimates heritability as twice the Pearson correlation between parent and offspring traits.

    Estimate: 0.20

    Interpretation : About 20% of the variation in offspring traits is due to resemblance with parents. Again, this reflects low heritability.

    1. Covariance Method

    heri_cov <- 2 * (cov(Parent_num, Offspring_num) / sqrt(var_parent * var_offspring))

    This uses covariance standardized by variances (like correlation), then multiplies by 2.

    Estimate: 0.20

    Interpretation : Also suggests low additive genetic influence .

Summary :

  • All three methods aim to estimate narrow-sense heritability , and values ​​are fairly close (0.13 to 0.20).

  • The regression estimate is slightly lower because it is sensitive to the scaling and distribution of variables.

  • Correlation and covariance-based methods give identical results in this case because they are mathematica

End of regression, correlation and covariance estimation of heritability