Testing a 9:7 Epistasis Model with Chi-Square
In a plant species, purple flower colour requires at least one dominant allele at each of two independently assorting loci, A and B. Any plant with aa or bb has white flowers. A dihybrid cross AaBb × AaBb produces 160 offspring: 94 purple and 66 white. A student proposes a single-gene model with purple dominant and white recessive, expecting 120 purple and 40 white. A second student proposes the two-locus model, expecting a 9:7 ratio (90 purple and 70 white). The class uses a chi-square test at the 5% significance level; the critical value for df=1 is 3.84. The instructor asks the class to compare the models and to state what the result does and does not prove about the genetic pathway. No other loci are considered, and the offspring are assumed to be a random sample.
1. Calculate chi-square for both models and decide which model is supported at the 5% level.
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9:7 model: expected purple=160×9/16=90, white=160×7/16=70. χ²=(94−90)²/90+(66−70)²/70=16/90+16/70=0.178+0.229=0.407. df=1; 0.407<3.84, so fail to reject 9:7. 3:1 model: expected 120 and 40. χ²=(94−120)²/120+(66−40)²/40=676/120+676/40=5.63+16.9=22.5. df=1; 22.5>3.84, so reject 3:1. The two-locus 9:7 model is supported.
2. A classmate says the 9:7 fit proves that purple pigment requires both A and B and that the loci act in the same pathway. Evaluate this claim.
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The 9:7 fit is consistent with a model in which purple requires at least one dominant allele at each locus, but a chi-square failure to reject does not prove that mechanism. Other genetic models or sampling variation could produce a similar ratio, and the test only compares the specified expectations. Proving that A and B act in the same biochemical pathway would require additional evidence, such as complementation tests, molecular data, or reciprocal crosses. The correct conclusion is that the data are compatible with 9:7, not that the pathway is proven.
3. Suppose the observed numbers were 100 purple and 60 white. Recalculate chi-square for 9:7 and state the conclusion at the same significance level.
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Expected for 9:7: 90 and 70. χ²=(100−90)²/90+(60−70)²/70=100/90+100/70=1.11+1.43=2.54. df=1; 2.54<3.84, so still fail to reject 9:7 at the 5% level. The conclusion is unchanged; the data remain compatible with the two-locus model, though the fit is less close.