Chi-Square test worksheet - pafaculty.net

1 Statistical Test of Significance: the Chi-Square Test The Chi-Square Test is generally used to evaluate differences between experimental or observed data and expected or hypothetical data. As a goodness of fit test, it tells us how well a set of observations fits the outcome predicted by the hypothesis being tested. It tells us whether there is a statistically significant difference between what we observed and what we expected. For example, if we tossed a balanced coin 100 times, we would expect it to come up heads about 50 times and tails about 50 times.

2 How far from this 50-50 prediction could the observations be and still fit our prediction? Assume the results of our 100 trials were as follows: observed expected Heads 55 50 Tails 45 50 Could we get this variation by chance alone? If so, what is the probability of getting these results by chance? Is it likely or extremely unlikely? Because it is impossible to prove the correctness of a good hypothesis, scientists (and statisticians) choose instead to work their hypotheses so they can reject a poor hypothesis.

3 We will follow this convention. (And if we must reject a poor hypothesis, when we analyze the observations, we can then consider what might have caused our unexpected results, what else might be going on. These ideas often appear in the discussion section of a scientific paper and form the basis for the next experiment.) With the coin example above, our null hypothesis is that there is no significant difference between our observed results and the ones we expected. To test this hypothesis, we use the Chi-Square test.

4 First we calculate a number called the Chi-Square value. We then use a probability table to tell us how likely it is that we would get these results by chance alone. If the probability of getting our observed results by chance is greater than or equal to 5% (p= in the table below), then we conclude there is no significant difference between the observed and expected results. We accept our hypothesis. If, however, the probability of getting these results is less than 5% (p< ), it is highly unlikely that this is due to chance.

5 (Something else is going on, even if we don t know what.) In this case, we reject our hypothesis. (Usually we would try to find and test a better one.) CALCULATING Chi-Square (Chi is the Greek letter ) The formula for calculating the Chi-Square value is Chi-Square (X2) = (observed expected)2 + (observed-expected)2 etc, etc expected expected For the example above: X2 = (55-50)2 + (45-50)2 = 25 + 25 = 50 50 50 50 To analyze the data, we then use a probability table to see how likely it is that we could get this X2 value (from these observations) purely by chance.

6 To do this, we also need to know the degrees of freedom (df), or how many of the numbers in our data are determined independently of the rest. (For example, if we had 2 numbers that had to add to 10, and one were 4, the other would have to be 6. We could pick anything for the first number, and determine it independently, but the second number would then have to have a particular value in order for the sum to be 10. If we had 3 numbers that had to add to 10, the first two could be independent, but the third would have to have a particular value to add to 10.)

7 Try it.) The degrees of freedom equal the number of categories (in our example, heads + tails = 2) minus one. In our example then, there are 2 categories and 1 degree of freedom (df=1). The table below shows the critical values of Chi-Square and the probability of getting each value as a function of the number of degrees of freedom. To use the table, use the line that corresponds to your degrees of freedom. Read left to right across the table until you find the interval that corresponds to the Chi-Square value you calculated.

8 Is the probability (the p= at the top of the table) greater than or less than If p= or > , will you accept or reject your hypothesis that there is no significant difference between your observed and expected results? What if p< (Most scientific experiments use p = or > as the standard for deciding whether or not to reject the hypothesis. Occasionally a scientist will choose a more (p< ) or less (p> ) rigorous value for p as the standard.) CRITICAL VALUES OF CHI square df p= p= p= p= p= p= 1 2 3 4

9 5 In our example from above, X2 = and df=1. In the table below, note that falls between p= and p= ; since p> in both cases, there is no significant difference between our observed results and those we expected. 55 heads/45 tails is not significantly different from the expected ratio of 50/50. If the results had been 65 heads and 35 tails, would there be a significant difference between observed and expected?

10 Calculate X2, show df = ? , and show the value of p. What does the value of p tell you? When we use the chi square test to evaluate the results of genetic crosses, the number of phenotypes becomes the number of categories used to determine the degrees of freedom. The expected values are derived from the phenotypic ratios one would expect from particular crosses if the parents are known, or from the common genetic ratio you think is closest to the actual counts. (For example, if you had a monohybrid cross and expected a 3:1 phenotypic ratio, then with 500 offspring, you would expect 375 of the dominant phenotype and 125 of the recessive phenotype.)