Self Assessment Answer # 5
for Lesson 11

by Dr Jamie Love Creative Commons Licence 2002 - 2005

Did you get the following chi-square value?

Phenotypes
Observed
Expected
O-E
(O-E)2
(O-E)2
E
Normal males
8
11
3
9
0.16
Colorblind males
8
5
3
9
1.8
Normal females
14
11
3
9
0.16
Colorblind females
2
5
3
9
1.8
Total
32
32
3.92

There are four classes here so there are three degrees of freedom.
That means we are "allowed" a chi-square as large as 7.81 before we get suspicious.
So, to a 5% significance level, there is nothing linked or strange about the colorblindness between the sexes in this population.

HOWEVER ...

Degrees of Freedom
5 % Significance Levels
1
3.84
2
5.99
3
7.81
4
9.49

It should be argued that this chi-square (above) is testing not only the distribution of colorblindness but also determining if the 1: 1 sex ratio is also correct! That's two tests in one and is not really fair. We already know there are equal numbers of females to males (16 to 16) and we expect that. This chi-square (above) is measuring obvious bits of data - things we already know to be a fact.
Let's see what happens when we ask ourselves, "Are there a reasonable number of colorblind males and females?".

We can do this chi-square - focusing only on the colorblindness - this way.

Phenotypes
Observed
Expected
O-E
(O-E)2
(O-E)2
E
Colorblind males
8
5
3
9
1.8
Colorblind females
2
5
3
9
1.8
Total
32
32
3.9

This chi-square is 3.9 so it is actually a little smaller than the first one (which was 3.92) BUT what about the degrees of freedom?

I hope you agree that there are now two classes (males and females) so there is now only ONE degree of freedom.
The significance table gives us a cut off of 3.84 with one degree of freedom so we should REJECT the idea that colorblindness is distributed equally between the males and females!

Clearly the males are more prone to colorblindness and we now have the correct chi-square to support our hunch!

Degrees of Freedom
5 % Significance Levels
1
3.84
2
5.99
3
7.81
4
9.49

I hope this problem has not confused or upset you. Frankly, I lead you down the wrong path because I wanted you to see how important it is to get your QUESTION right!
The first chi-square, with four classes, was asking, "Is the ratio of males to females 1 : 1 AND is colorblindness distributed among them in a ratio of 1 : 1?" That was the wrong question. This population has 16 males and 16 females so there is no doubt that the ratio of males to females is 1 : 1. Indeed, the sex ratio (of males to females) is so good that it overshadows the real data we want to use.
The correct question was, "Is colorblindness distributed between males and females in a ratio of 1 : 1?". The answer to that question gave you a slightly different value for the chi-square and (importantly) greatly restricted the number of classes under examination.

In point of fact, colorblindness is much more common among males than females for reasons that will be explained in "Part Three - Advanced Genetics".

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