Chi-Square and Linked Genes

Testing whether a deviation matters

Real data never match a $3:1$ ratio exactly. The chi-square test asks one question: is the gap between observed and expected small enough to be chance?

The null hypothesis is always that the deviation is due to chance alone. Degrees of freedom $=$ (number of phenotype classes) $-1$. Compare $\chi^2$ with the critical value at $p=0.05$; if $\chi^2$ is larger, reject the null — something other than chance is at work.

Rejecting the null does not name the cause. Linkage, lethal alleles, unequal survival and a wrong model of inheritance all push $\chi^2$ up.

When genes travel together

Genes on the same chromosome do not assort independently. A dihybrid test cross that should give $1:1:1:1$ instead gives mostly parental combinations plus a few recombinants produced by crossing over.

That percentage is read directly as map units (centimorgans). The further apart two loci sit, the more often a crossover falls between them, so the higher the recombination frequency. At $50\%$ the genes behave as if unlinked — either they are on different chromosomes or so far apart on one chromosome that crossing over happens almost every time.