Using Chi-squared analysis to confirm expected phenotypic ratios of offspring

In a recent question the original poster was faced with deciding which ratios to use in a Chi-squared analysis. They were supposed to decide on, "logically", what test-ratio to use.

Example question:

A supplied sample contains four types of seeds and the total number is 64. The types of seeds are large red 42, large white 8, small red 10 and small white 4. Calculate goodness of fit.

My initial assumption was that in order to solve the problem, you would need to think about the genotype of each parent.

Lets say: L = large seeds; s = small seeds; R = red seeds; w = white seeds

If both parents were LsRw, the phenotype-distribution of the offspring would follow a 9:3:3:1 ratio.

If one parent was LsRw, and one parent was ssww, you would have a 1:1:1:1 ratio.

And if one parent was LsRw, and the other parent was Lsww or ssRw, you would have a 2:2:1:1 ratio.

Obviously, out of all of these ratios, the example question follows most closely with the 9:3:3:1 ratio.

So, to solve this question statistically, would you make the null hypothesis There is no difference between the observed and expected ratios?

While the alternative hypothesis would be There is a different between the observation and the expected ratios?

And would a statistically significant result indicate that the null hypothesis must be rejected in favour of the alternative hypothesis, thus the parents are unlikely to be heterozygotic for both traits?

• I suspect that the actual question contained more information than what was posted. If not, presumably you could formulate a null and alternate hypothesis for each expected ratio to determine the phenotypes of the parents, though as I type this I can't help but think there would be some statistical reason not to do it this way. Perhaps you would get a better answer at Cross Validated. – canadianer Mar 16 '17 at 0:01
• You're correct, each time you run a statistical model you are introducing an error of ~5% to your conclusion - that would be the reason to not do it that way. But this question has had me thinking for a couple days so I thought I'd see if anyone has any other approaches to the problem with the limited information available. If indeed it is a matter of just thinking about inheritance, I'm not sure if the statisticians would be able to catch on. – Bob Mar 16 '17 at 0:08
• For any statistical test the definition of the null hypothesis is essential. You cannot guess the null hypothesis based on a given question. The null hypothesis is based on some prior knowledge. There is no way to guess it. Moreover, what the alternative hypothesis is does not really matter because the statistical test does not say which one of the infinite possible alternatives is correct. The test just accepts or rejects the null hypothesis and for any test the first step is to define the null hypothesis; everything else is after that. Plus, a test cannot be performed with just one sample! – WYSIWYG Mar 17 '17 at 8:23
• In the example you considered, the null hypothesis is basically that observed value is not different from an expected value for which the prerequisite is the knowledge of the expected value. Now what the expected value is, depends on your model. You assumed one possible inheritance model and proceeded towards calculation. However, this assumption has to be clearly stated and understood by the experimenter before performing the test. OP's professor did not talk about any underlying model and simply asked to check goodness of fit. It is ridiculous; I can assume anything as per my convenience. – WYSIWYG Mar 17 '17 at 8:29