Step 1: Understanding the Concept:
In plant breeding, genotypes often perform differently when grown across various locations, seasons, or soil types.
This interaction is called Genotype $\times$ Environment (G $\times$ E) interaction.
To identify crop varieties that perform consistently across different conditions, breeders use specialized statistical tools.
Step 2: Detailed Explanation:
Let us evaluate each of the listed statistical methods:
1. Correlation analysis:
This measures the linear relationship between two variables, but it cannot assess how a genotype adapts to different environments.
Thus, Option (A) is incorrect.
2. Line $\times$ Tester analysis:
This mating design is used to evaluate the combining ability of parents (general and specific combining ability) and estimate genetic components of variance.
Thus, Option (B) is incorrect.
3. $\text{D^2$ Mahalanobis:}
This multivariate statistical tool is used to measure genetic divergence among different populations or genotypes to help select parents for hybridization.
Thus, Option (C) is incorrect.
4. Stability analysis:
This method (such as the models proposed by Finlay and Wilkinson, or Eberhart and Russell) is designed to evaluate G $\times$ E interactions.
It measures a genotype's performance across multiple environments to determine its stability and adaptability.
A stable genotype is one that shows minimal variation in yield or performance across different environments.
Step 3: Final Answer:
The statistical analysis used to determine the adaptability of genotypes in different environments is Stability analysis.
This matches Option (D).