genetika

The Goodness-of-Fit Chi-Square Test

If you expected a 1:1 ratio of brown and yellow cockroaches

but the cross produced 22 brown and 18 yellow, you probably

wouldn’t be too surprised even though it wasn’t a perfect

1:1 ratio. In this case, it seems reasonable to assume that

chance produced the deviation between the expected and

the observed results. But, if you observed 25 brown and 15

yellow, would the ratio still be 1:1? Something other than

chance might have caused the deviation. Perhaps the inheritance of this character is more complicated than was

assumed or perhaps some of the yellow progeny died before

they were counted. Clearly, we need some means of evaluating

how likely it is that chance is responsible for the deviation

between the observed and the expected numbers.

To evaluate the role of chance in producing deviations

between observed and expected values, a statistical test

called the goodness-of-fit chi-square test is used. This test

provides information about how well observed values fit

expected values. Before we learn how to calculate the chi

square, it is important to understand what this test does and

does not indicate about a genetic cross.

The chi-square test cannot tell us whether a genetic

cross has been correctly carried out, whether the results are

correct, or whether we have chosen the correct genetic

explanation for the results.What it does indicate is the probability

that the difference between the observed and the

expected values is due to chance. In other words, it indicates

the likelihood that chance alone could produce the deviation

between the expected and the observed values.

If we expected 20 brown and 20 yellow progeny from a

genetic cross, the chi-square test gives the probability that

we might observe 25 brown and 15 yellow progeny simply

owing to chance deviations from the expected 20:20 ratio.

When the probability calculated from the chi-square test is

high, we assume that chance alone produced the difference.

When the probability is low, we assume that some factor

other than chance—some significant factor—produced

the deviation.

To use the goodness-of-fit chi-square test, we first determine

the expected results. The chi-square test must always

be applied to numbers of progeny, not to proportions

or percentages. Let’s consider a locus for coat color in domestic

cats, for which black color (B) is dominant over gray

(b). If we crossed two heterozygous black cats (Bb _ Bb), we

would expect a 3:1 ratio of black and gray kittens. A series

of such crosses yields a total of 50 kittens—30 black and 20

gray. These numbers are our observed values.We can obtain

the expected numbers by multiplying the expected proportions

by the total number of observed progeny. In this case,

the expected number of black kittens is _ 50 _ 37.5 and

the expected number of gray kittens is _ 50 _ 12.5. The

chi-square (_2) value is calculated by using the following

formula:

_2

where means the sum of all the squared differences

between observed and expected divided by the expected values.

To calculate the chi-square value for our black and gray

kittens, we would first subtract the number of expected

black kittens from the number of observed black kittens

(30 _ 37.5 _ _7.5) and square this value: _7.52 _ 56.25.

We then divide this result by the expected number of black

kittens, 56.25/37.5, _ 1.5.We repeat the calculations on the

number of expected gray kittens: (20 _ 12.5)2/12.5 _ 4.5.

To obtain the overall chi-square value, we sum the (observed The next step is to determine the probability associated

with this calculated chi-square value, which is the probability

that the deviation between the observed and the

expected results could be due to chance. This step requires

us to compare the calculated chi-square value (6.0) with

theoretical values that have the same degrees of freedom in

a chi-square table. The degrees of freedom represent the

number of ways in which the observed classes are free to

vary. For a goodness-of-fit chi-square test, the degrees of

freedom are equal to n _ 1, where n is the number of different

expected phenotypes. In our example, there are two

expected phenotypes (black and gray); so n _ 2 and the

degree of freedom equals 2 _ 1 _ 1.

Now that we have our calculated chi-square value and

have figured out the associated degrees of freedom, we are

ready to obtain the probability from a chi-square table

(Table 3.4). The degrees of freedom are given in the lefthand

column of the table and the probabilities are given at

the top; within the body of the table are chi-square values

associated with these probabilities. First, find the row for

the appropriate degrees of freedom; for our example with 1

degree of freedom, it is the first row of the table. Find

where our calculated chi-square value (6.0) lies among the

theoretical values in this row. The theoretical chi-square

values increase from left to right and the probabilities decrease

from left to right. Our chi-square value of 6.0 falls

between the value of 5.024, associated with a probability of

.025, and the value of 6.635, associated with a probability

of .01.

Thus, the probability associated with our chi-square

value is less than .025 and greater than .01. So, there is less

than a 2.5% probability that the deviation that we observed

between the expected and the observed numbers of black

and gray kittens could be due to chance.

Most scientists use the .05 probability level as their cutoff

value: if the probability of chance being responsible for the

deviation is greater than or equal to .05, they accept

that chance may be responsible for the deviation between the

observed and the expected values. When the probability is

less than .05, scientists assume that chance is not responsible

and a significant difference exists. The expression significant

difference means that some factor other than chance is

responsible for the observed values being different from the

expected values. In regard to the kittens, perhaps one of the

genotypes experienced increased mortality before the progeny

were counted or perhaps other genetic factors skewed the

observed ratios.

In choosing .05 as the cutoff value, scientists have

agreed to assume that chance is responsible for the deviations

between observed and expected values unless there is

strong evidence to the contrary. It is important to bear in

mind that even if we obtain a probability of, say, .01, there is

still a 1% probability that the deviation between the

observed and expected numbers is due to nothing more

than chance. Calculation of the chi-square value is illustrated

in ( FIGURE 3.15).