Protein Engineering Protocols - Mycobacteriology research center

148 Denault and Pelletier3.2.1. The χ 2 TestThe statistic used in the χ 2 test is:( )Y − npq =np1 112( Y2 − np2)+np( Yk− npk)+ … +npThe hypothesis that we test, the so-called null hypothesis, is that the experimentwas run properly, i.e., in such a way as to yield the outcomes in their theoreticalproportions, on average. A large value of q suggests that there is significantdiscrepancy between the results of the actual experiment and the results expected,contradicting the null hypothesis.The statistic q is compared with the χ 2 distribution of parameter k – 1, at anychosen level α of significance, a value that is denoted χ 2 α. The parameter α is specifiedby the user (see Note 9 regarding the significance level). The test is as follows:For q < χ 2 α(k –1), the hypothesis is not rejected; there is no sufficient reasonto think there is a bias.For q > χ 2 α(k –1), the hypothesis is rejected; there is sufficient reason to thinkthere is a bias.Values for the χ 2 distribution can be found in tables in most statistics textbooksand many software programs, including Excel.Example 1. Two possible outcomes: an experiment is run 100 times; eachtime, a certain characteristic may or may not occur, thus, there are two possibleoutcomes and k = 2. The characteristic (call it “outcome 1”) should theoreticallyoccur in one sampling out of four; in the 100 items sampled, it occurred 29 times.The χ 2 test is based on the statistic:22k2( 29 −100× 0.25)( 71−100× 0.75)q =+100 × 0. 25 100 × 075 .2 2≅ 085 .Let us choose a level of significance α =0.05. Then the test is to compareq = 0.85 to:222χ ( k − 1) = χ ( 2− 1) = χ ( 1) = 3.84005 . 005 . 005 .Because:2q = 085 . < 384 . = χ ( 1)005 .the hypothesis (that the experiment was run properly, without bias) is notrejected (see Fig. 4). Figure 7 illustrates the number of outcomes with the characteristicfor which the hypothesis should be kept (17 to 33 outcomes with thecharacteristic) or rejected (all other results).