The chi square test is designed to test the statistical significance of an experimental outcome. We often use the coin flipping example, but probability can be applied to numerous other situations. For example, let’s say you are interested in the inheritance of small vs. large antlers in a population of Canadian Moose. You notice that large antlers appear to be dominant to small antlers (let’s assume antler size is encoded by a single gene). You’d like to test whether or not this gene is inherited in a Mendelian fashion. In a cross between two heterozygotes, you assume a 3:1 inheritance pattern in the offspring. You are setting out to test this expectation. Before you even count your actual numbers, you state for your experiment a null hypothesis. The null hypothesis is a statistical term that represents the standard against which your model/idea will be compared. In a genetics test, this means the gene(s) under scrutiny is/are Mendelian and the numbers you saw in your cross differ from expectations only by amounts that could occur by chance. Therefore, you should ALWAYS draw your cross before doing an experiment (make sure you know the parental genotypes) and write down what you would expect. Then and only then, count a predetermined number and make comparisons.
Ok, you have somehow engineered matings between some Canadian Moose that are heterozygous for antler size. When you count up the offspring in the F1, this is what you find: Based on these numbers, it looks pretty close to Mendelian inheritance,
right? Well, it may look that way, but we can’t jump to conclusions.
We must first determine how significant these numbers are. Significance
can be a very subjective
thing, so statisticians came up with the chi square (c2) test. The chi square
test allows the experimenter to determine whether or not he/she can say there
was anything other than chance acting on this scenario. At the end of the
chi square, we will retain or reject the null hypothesis. If we
retain the null hypothesis,
we can say these results could have been caused by chance (but most scientists
prefer the phrase “inconclusive result”). If we reject the null
hypothesis, we can say that it is unlikely that chance alone could not have
resulted in these numbers and, in
our moose scenario, there is probably some non-Mendelian force in action.
Once you have rejected the null hypothesis, you are allowed to entertain
(though have not
So without getting really involved in the derivation of the chi square test, here is what statisticians came up with: We can apply this equation to our values to obtain a chi square value: (134-150)2/150 + (66-50)2/50 = 1.7 + 5.1 = 6.8 WOW! A chi square value of 6.8!!!! Ummm, what does that mean?
Well, we need to compare this value before we draw any great conclusions.
We will
be looking
to
a chi square table to give us the groundbreaking answer. Before we do this,
you’ll
need to understand what the numbers on a chi square table mean. In the column
of the table, we have the The chi square table lists another value on the top row, it is
the a-value. You may know you need to look at the 0.05 a-value,
but why? The a-value
leads you
to your P-value. P stands for probability, and the P-value is the probability
that the test outcome would take a value as extreme or more extreme than
that observed. The smaller the P-value is, the stronger the evidence
against your
null hypothesis. We choose to always look at the 0.05 a-value. By doing
this, we are requiring that the data give evidence against the null hypothesis
so
strong that it would happen no more than 5% of the time when the null
hypothesis is
true. I think the best way to understand the a-value and P-value is with
a bell curve. When you measure a population of outcomes, you often end
up with
a bell
curve: In case you haven’t figured it out by now, we can reject the null hypothesis in our experiment regarding the Mendelian inheritance of large vs. small antlers. The P-value for 1 degree of freedom at an a-value of 0.05 is 3.84. When we compare our chi square value, 6.8 is greater than 3.84…so we can say that it is unlikely that Mendelian inheritance is responsible for the numbers we have seen. In other words, the null hypothesis has done a poor job of generating the data we see, and this allows us to entertain an alternate hypothesis. HOWEVER, this test alone does not allow us to prove that alternate hypothesis! There are several caveats of the chi square that we should acknowledge. First of all, the P-value indicates probability, so it can only state that it is XXX unlikely that the null hypothesis generated this data. So even though we could say, “there is a 99% chance that such and such…” there is always the chance of the alternative happening; so as good little scientists, we cannot be 100% sure of an outcome. Secondly, this test does not measure the plausibility of your alternate hypothesis generating this data, it only indicates there is “room” to entertain alternatives. Lastly, the chi square test in no way indicates how many alternate hypotheses there may be that can explain the data-yours is only one interpretation, there could be thousands! So now you have a lovely new hammer. As is so often the case, this does |