István Nagy

Kaposvar University

In animal breeding repeatability has high importance. It can only be calculated when the trait can be measured on the same individual (eg. litter size at the successive parities) several times. Such trait is for example the litter size at the successive parities. The total variance can be partitioned into variance within individuals and variance between individuals. The within-individual component is caused by temporary differences of environment between successive performances (σ^{2}_{Es}). The differences between individuals are both genetic and (permanent) environmental. The general environmental variance (σ^{2}_{Eg}) refers to environmental variance arising from permanent or non-localized circumstances. The repeatability is the correlation between repeated measurements of the same individual (r). The repeatability expresses the proportion of the variance of single measurements that is due to permanent differences between individuals, both genetic and environmental. Expressing the repeatability by an equation:

r = σ^{2}_{A} + σ^{2}_{D} + σ^{2}_{Eg} / σ^{2}_{P}

In order to calculate repeatability the variance between individuals (σ^{2}_{A} + σ^{2}_{D} + σ^{2}_{Eg}) and within individuals (σ^{2}_{Es}), has to be separated then the former has to be divided by the total variance (σ^{2}_{A} + σ^{2}_{D} + σ^{2}_{Eg} + σ^{2}_{Es}) (sum of the between and within individual variance components).

Example:

Example: litter size of 5 sows at the first three parities was the following: (table 11.)

The repeatability has to be calculated.

Solution:

First the variance components between and within individuals have to be determined.

The within individuals variance component can be calculated as the average of the variances calculated from the individual measurements.

In the present example the first sow has three measurements: 10, 11, 11. Based on our earlier studies the variance of these measurements is:

(1 / N – 1) ⨯ (∑X2 – (∑X)2 / N),

where N = number of measurements per sow (3 in this example).

The variance of the first sow’s measurements is:

(1 / 3 - 1) ⨯ (102 + 112 + 112 - (10 + 11 +11)2 / 3) = 0.33

Similarly, the variance of the second sow’s measurements is:

(1 / 3 - 1) ⨯ (92 + 102 + 112 - (9 + 10 +11)2 / 3) = 1.0

For the third sow:

(1 / 3 - 1) ⨯ (112 + 112 + 122 - (11 + 11 +12)2 / 3) = 0.33

For the fourth sow:

(1 / 3 - 1) ⨯ (102 + 92 + 122 - (10 + 9 +12)2 / 3) = 0.83

For the fifth sow each measurement was 10 so the variance of these numbers is zero.

The within sow variance component is: (0.33 + 1.0 + 0.33 + 0.83 + 0.0) / 5 = 0.498

To calculate the between individuals variance component first the individual measurements have to be averaged. Then using these averages as "measurements" their variance has to be determined. This variance is the between individuals variance component + the within individuals variance component / n, where n = number of measurements per individual.

In the present example the litter size averages for the 5 sows were the following: 10.66; 10; 11.33; 10.3; 10. Their variance is: (1 / 5 – 1) ⨯ (10.66^{2} + 10^{2} + 11.33^{2} + 10.33^{2} + 10^{2}) = 0.31. From this variance the between individuals variance component can be obtained: 0.31 – (within individuals variance component / n). Thus the between individuals variance component is: 0.31 – (0.498 / 3) = 0.151. The repeatability is: 0.151 / (0.151 + 0.498) = 0.232. Litter size can be characterized as a reproductive trait which generally has low heritability. This is justified by the present result as the repeatability sets the upper limit of the heritability where the latter can only be smaller or equal than the former. The advantage of the repeated measurement is that with the increasing number of measurements the accuracy of the variance components also increases (proportionally to the total variance). The variance component (σ^{2}Es) is reduced by repeated measurements (n). The total variance can be calculated using the following equation:

σ^{2}_{P(N)} = σ^{2}_{A} +σ^{2}_{D} + σ^{2}_{Eg} + (1 / N) ⨯ σ^{2}_{Es}

where n = number of measurements.

From the equation it follows that when the repeatability is low and there is a large special environmental variance (σ^{2}_{Es}) multiple measurements may lead to a worthwhile gain in accuracy. In this case this variance component decreases substantially with the increasing number of measurements and repeatability also increases. However, when the special environmental variance (σ^{2}_{Es}) is small then the repeatability is large and in this case the increased number of measurements does not change accuracy substantially.

Repeatability can also be used for the prediction of future performance (when some parameters of the population are known). It is based on the fact that the repeatability is the correlation between the successive measurements. Assume that each individual has two measurements: X and Y. In this case repeatability is the correlation coefficient between X and Y. Correlation coefficient can be obtained by dividing the covariance between X and Y (σ_{XY}) by the product of the standard deviations of X and Y (σ_{X} ⨯ σ_{Y}):

r = σ_{XY} / (σ_{X} ⨯ σ_{Y})

In order to predict future performance the regression coefficient of Y on X has to be calculated. The regression coefficient is the covariance between X and Y (σ_{XY}) divided by the variance of the independent (X) variable:

b = σ_{XY} / (σ_{X}^{2}),

alternatively:

r ⨯ (σ_{Y} / σ_{X}).

For the prediction of future performance at population level the correlation coefficients between the successive measurements and their means and standard deviations have to be known.

Example:

The correlation (repeatability) between the milk yields of the first and second lactations is 0.4. The mean and standard deviation of the first and second lactations are: 4096 kg, and 696 kg; 4232 kg and 934 kg, respectively.

The second lactation milk yield of an individual has to be predicted when it is known that at the first lactation the milk yield of this cow was 5000 kg.

Solution:

Y - ̅Y = b ⨯ (X - ̅X),

where ̅Y = 4232;

b = 0.4 ⨯ (934 / 696);

X = 5000 kg; ̅X = 4096 kg

Using the equation: Y = 4716.5 kg.