# Goodness of fit: How to decide which ratio to deal with?

While solving problems of Goodness of fit, I'm faced with an issue, how to decide which ratio to consider to find the test statistic from given set of observations.

E.g. 1

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.

Problem: As df=3, Ratio= 9:3:3:1 / 1:1:1:1?

E.g.2

You are supplied with two different varieties of plant samples;tall-76 and short-24. Determine the observed number, apply Chi square test to state whether it is in agreement with expected ratio.

Problem: As df=1, Ratio= 3:1 /1:1?

I was vaguely told by my propessor that which ever ratio seems to apply(by logical guess) we should choose that one to determine the expected values and thus the statistic.

I couldn't find any good read in this regard, except those that are full of mistakes. I've been reading Statistics blogs to understand the concepts but they don't cover these typical biological problems and those that did had the ratios mentioned.

I've another question in mind, from experimental result it is also likely that we won't get one of the four types of seeds (given that mating is random and the progenies appeared by dihybrid crosses) so determining the actual ratio behind becomes more difficult as df = 2 =/= 3!

• Without more information, as in a specific hypothesis to test (which should come from your professor), you can't be expected to come up with the hypothesis yourself. If your professor won't be clearer with the questions, then I say you expect 42:8:10:4 for the first and 76:24 ratios for the second. Chi^2 = 0 and you are done.
– kmm
Commented Mar 13, 2017 at 22:38
• For Pearson-χ² test, you can use any one frequency (for instance the frequency of heterozygotes). However, both your example questions lack important information regarding what is the expected frequency. Why should you even expect 3:1? Any expected model has to have some basis. These examples do not indicate any. The first step of a statistical test is to define the null hypothesis. Commented Mar 14, 2017 at 7:45
• @SanjuktaGhosh You don't have to find out what the H₀ is. The null hypothesis is your basic question. You will, of course, know that before performing the experiment. In your case of 3:1 you will have the assumptions such as: there is dominance, the cross is between two heterozygotes etc. This appears to be the case but I can invent arbitrary cases in which you may see 3:1 ratios. I can also argue that 76:24 is not 3:1 but actually different from it. Unless you have multiple samples, there is no way to prove it. Commented Mar 15, 2017 at 9:11
• @SanjuktaGhosh, you don't always need to disprove the null hypothesis. In some cases it makes more sense to run a model to check to see if the null hypothesis is correct - particularly when you are not entirely certain about what the alternative hypothesis could be. This is what Chi-square analysis is for. You have determine whatever ratio would make the most sense, logically. Then you run the observed values against the null hypothesis's ratio - using the same number of samples. So, run a Chi-squared test with a ratio of 75:25 against the observed ratio of 76:24.
– Bob
Commented Mar 15, 2017 at 22:41
• Statistical test cannot be performed if you do not have a prior hypothesis. Your hypothesis can be as simple as "both samples are the same". When you do a t-test you have one additional clause that both samples are normally distributed. P-value is the probability that the null hypothesis is correct. The smaller that value is the higher will be your confidence. In general, we define our confidence interval before performing the test. The choice of confidence interval is arbitrary; usual choice is 95% (p < 0.05 is considered significant) but you can be more stringent depending on the situation. Commented Apr 2, 2017 at 9:46

To solve this question you need to have a basic understanding of Mendelian genetics. Contrary to peoples' comments, this problem does provide enough information.

Genes are inherited from two parents. That is, the offspring will inherit one allele from each parent. Genes also interact with each other in a dominant/recessive nature. A dominant trait will be expressed, no matter what if this is carried by an organism. A recessive trait will only be expressed if the organism is carrying two copies of the same allele - the organism must receive the recessive gene from both parents.

In the first question, there are four genes. One gene encoding large seeds. One gene encoding small seeds. On gene encoding red seeds. One gene encoding white seeds.

The dominant genes are those that code for: large & red The recessive genes are those that code for: small & white

For this question, you assume that both parents are carrying a single copy of each gene. This means, each parent has one copy of a gene coding for large seeds, one copy of a gene coding for small seeds, one copy of a gene coding for red seeds, and one copy of a gene coding for white seeds.

As a side note, this would mean that each parent came from a large red seed. There is also a chance for any of the four genes to be passed to the offspring.

Each parent could pass the genes for: Large red seeds, Large white seeds, small red seeds or small white seeds.

This means that the following gene-combinations are possible:

-large red, large red

-large red, large white

-large red, small red

-large red, small white

-large white, large red

-large white, large white

...etc...

And, I'm sorry, I've been trying for twenty minutes to upload a picture to show you what I mean. But, you can do this yourself also to see what I mean:

Draw a 4x4 grid, and in each box fill in each possible combination of genes that can be inherited by an offspring. They call this a Punnett square, and it is used to predict the phenotype of an offspring bred by two parents. If you search Punett square you will understand how this works.

But the thing to remember is this: If the offspring carries one copy of a dominate gene (large seed, red seed) then these phenotypes will be expressed. In order for a recessive trait to be expressed, the offspring must carry two copies of the genes (small seed, white seed), ie. it must have received the same recesssive allele from both parents.

Now, to solve the problem statistically, you will use the ratio 9:3:3:1. The null hypothesis is that there is no difference between the observation and this ratio. The alternative hypothesis is that there is a statistically significant difference observed from the sample in comparison to the test-ratio.

This means, a p-value < 0.05 indicates that the 9:3:3:1 ratio is false. And the gene-interaction must be explained by another means - not by Mendelian inheritance. Or, I suppose it could also indicate that the parents did not in fact carry one copy of each allele and some other combination, that maybe you need to find. Eg. One parent could have had two copies of each recessive trait, while the other parent could have had in had one copy of each trait. (Play with the Punnett square, and maybe you can figure out if this is true)

• Oh yeah, and obviously they aren't a 1:1:1:1 ratio. So isn't this question basically "free marks"?
– Bob
Commented Mar 13, 2017 at 23:07
• @Sanjukta Ghosh, my apologies - when I first answered your question I was under the assumption that you had an understanding of Mendelian genetics and inheritance. I have edited my answer, and provided you with some background information required to solve this question. I was also trying to upload pictures, but I had no luck with this. I'm sorry. I hope my new answer clears things up for you. If you have any more questions, please feel free to ask.
– Bob
Commented Mar 14, 2017 at 23:54
• And this is a link to dihybrid crosses. It will explain why 9:3:3:1 ratios are expected. en.wikipedia.org/wiki/Dihybrid_cross
– Bob
Commented Mar 15, 2017 at 0:33
• I'm aware of Mendelian genetics. Thanks for your efforts and this comment. I agree with them that the information(data) provided is not enough to establish a scientific hypothesis but questions like this are mentioned in books followed by my university. I prefer your previous answer though. I would like to ask can you support your answer with some authoritative reference. Commented Mar 16, 2017 at 20:12
• By your answer I meant the previous version where you talked about the technique. Commented Mar 16, 2017 at 20:29