Methods of Estimation and Precision of Estimates of Animal Data | Animal Husbandry & Veterinary Science Optional for UPSC PDF Download

Introduction

Understanding the precision of estimates is crucial in the assessment of heritability. Precision is indicated by the standard error, with the sampling variance playing a key role. Experimental planning involves choosing methods and designing experiments to maximize precision within given limitations.

Sampling Variance:

• Precision depends on sampling variance.
• Lower sampling variance leads to greater precision.
• Standard error is the square root of sampling variance.

Experimental Design Considerations

Method Selection

• Choosing the most suitable method for estimation is critical.
• Precision is a key factor in method selection.

Determining Sample Size:

• Balancing the number of individuals and families is essential.
• Compromises between large families and many families must be considered.

Limitations on Experiment Scale:

• The scale of the experiment is limited by factors such as labor, cost, and space.
• Identifying limitations is crucial for effective experimental design.

a. Labor Limitation:

• Total number of individuals measured is the limiting factor.
• Consideration of the scale based on labor constraints.

b. Space Limitation:

• Limitations in breeding and rearing space, common with larger animals.
• Choice between limiting the number of families or offspring.

Comparison of Methods and Designs:

• Assessing efficiencies requires comparing experiments on the same scale.
• Consideration of circumstances limiting the experiment scale.

a. Labor as the Limiting Factor:

• Total number of individuals, including parents, is limited by labor.
• Consideration of measurement labor as a crucial factor.

b. Space as the Limiting Factor:

• Number of families or total offspring may be limited by space.
• Inclusion of parents in measurements without additional cost.

Offspring-Parent Regression

Regression Analysis:

• Estimations based on regression of offspring on parents.
• Variables include X (independent) and Y (dependent).

Variables and Symbols:

• X and Y represent parent and offspring values.
• Variance of X and Y denoted by σ²x and σ²y.
• Regression coefficient represented by 'b'.
• N is the number of paired observations or families in the experiment.
• T is the total number of individuals measured.

Equations for Variance of Regression Coefficient:

• Formulas for variance of the estimate of the regression coefficient.
• Simplified and approximate forms for practical application.

Phenotypic Variance and Parental Measurement

Single Parent Measurement:

• Variance of parental values equals phenotypic variance (σ²).
• Phenotypic variance for both parents measured is half the total phenotypic variance (V/2).
• Offspring variance (σ²f) is influenced by the phenotypic correlation (r) within families.

Phenotypic Correlation Influence:

• Offspring variance is determined by the phenotypic correlation (r) among family members.
• Expression for offspring variance involves phenotypic correlation and variances.

a. Substituting Variables:

• Substitution of σ²f and σ² in the equation for sampling variance of regression on one parent.

b. Sampling Variance Expression:

• Derivation of an approximate expression for the sampling variance, enabling method and offspring number comparison.

One Parent Measurement Design

Denominator Consideration:

• Focus on the measurement of one parent.
• The denominator 'N' in the equation represents the total number of offspring measured.

Efficient Design under Limited Scale:

• When the scale is limited by the total offspring measured (N is fixed), efficiency is maximized by having as many families as possible with only one offspring measured per family.
• Standard error of heritability estimate provided as an approximate formula.

a. Standard Error Calculation:

• Formula for standard error of heritability estimate, considering one parent measurement.

b. Precision Requirements:

• Illustration: To achieve a standard error of 0.3, 400 parents and 400 offspring need to be measured.

c. Dependency on Offspring Number:

• Precision increase depends on phenotypic correlation (r); low correlation benefits from additional offspring.

Both Parents Measurement Design

Regression on Mid-Parent Values:

• Measurement of both parents for regression on mid-parent values.
• Evaluation of standard error for one offspring per family and two offspring per family scenarios.

Comparison of Precision:

• Comparison between regression on mid-parent and single-parent measurements.
• Illustration that, under most circumstances, regression on mid-parent values provides better precision.

Assortative Mating

Increased Precision through Assortative Mating:

• Assertion that mating parents assortatively increases precision.
• Focus on measuring both parents for regression on mid-parent values.

a. Comparison with Other Designs:

Emphasis on the efficiency of regression on mid-parent values in terms of precision.

Assortative Mating and Sampling Variance

Effect of Assortative Mating:

• Increase in the variance of mid-parent values under assortative mating (Vp(1+r)).
• Impact on sampling variance in regression equations.

Sampling Variance Comparison:

• Substituting assortative mating variance into equations (2) and (3).
• Precision improvement: Sampling variance is approximately 1/(1+r) times that with random mating.

a. Complete Assortative Mating:

• Precision enhancement by a factor of 1 or √λ if assortative mating is complete (r=1).

Weighting Unequal Family Sizes

Variable Offspring Numbers:

• Measurement of varying offspring numbers per family introduces weighting challenges.
• Weighting based on the phenotypic correlation 't' between offspring in families.

Weighting Principle:

• Families of size 's' weighted in proportion to the reciprocal of the regression variance.
• Adjustment ensures families of size 1 always have a weight of 1.

a. Weighting Formula:

• Weight 'W' for the mean of an offspring based on family size and correlation.

b. Effect on Precision:

• Impact of weighting on precision, particularly with significant variations in family sizes.

Sibling Analysis and Intraclass Correlation

Intraclass Correlation Estimates:

• Estimation from intraclass correlation of full and half-sibling families.
• Initial assumption: No subdivision in half-sib families, measuring only one offspring per dam.

Optimal Family Size:

• Determining optimal family size for efficiency, considering the correlation and heritability.
• Formula for minimal sampling variance of intraclass correlation.

a. Correlation and Heritability Influence:

• Optimal family size depends on correlation and heritability assumptions.
• Importance of considering heritability in designing efficient half-sibling analyses.

b. Efficiency Trade-off:

• Balancing efficiency and uncertainty in heritability estimates.
• Preferably erring on the side of larger families due to greater efficiency losses with smaller sizes.

c. Guidelines for Half-Sib Analysis:

• Design considerations for half-sib analyses, suggesting family sizes between 20 and 30 in the absence of precise heritability knowledge.

Sampling Variance in Optimal Designs

Correlation Sampling Variance:

• Deriving sampling variance of the correlation in the optimal design by substituting values in equation (8).
• Approximate expression for improved precision.

Heritability Sampling Variance:

• Obtaining sampling variance of heritability by multiplying the variance of full-sib correlation by 4 and half-sib correlation by 16.
• Substituting values in equation (3) for full-sib and equation (9) for half-sib.

a. Heritability Sampling Variance Expressions:

• Approximate formulas for sampling variance of heritability estimates from full-sib and half-sib families.

b. Comparison of Precision:

• Illustration: Estimate from full-sib families is approximately twice as precise, considering variances, compared to half-sib families.

Combined Approach for Heritability Estimation

Dual Method Design:

• Designing experiments to estimate heritability using both offspring-parent regression and sib-correlation.
• Evaluation: Optimal design does not significantly differ from individual method designs.

Effect of Parental Selection:

• In experimental and farm animal populations, parents are often selected based on heritable traits.
• Selection reduces variance between parents, impacting covariance and heritability estimates.

Influence on Heritability Estimates:

• Intraclass correlation heritability estimates are affected by downward bias due to parental selection.
• Up to a 50 percent reduction in estimated heritability values.

Impact on Regression Estimates:

• Selection based on the trait being estimated does not affect offspring-parent regression but decreases precision due to reduced parental variance.

Potential Improvement through Dual Selection:

• Improved precision possible by selecting two groups of parents: one with high and one with low trait values.
• Precision gain from focusing on extreme families, providing more information about regression.

a. Optimal Proportion of Parents:

• When equal numbers of offspring to parents are measured, the optimal proportion of parents in each group is around 5 percent.