Using R for Introductory Statistics : John Verzani

Goodness of fit 241This gives the total discrepancy between the observed and the expected. We use thesquare as ∑Y i− np i =0. This sum gets larger when a category is larger or smaller thanexpected. So a larger-than-expected value contributes, and any correlated smaller-thanexpectedvalues do, too. As usual, we scale this by the right amount to yield a test statisticwith a known distribution. In this case, each term is divided by the expected amount,producing Pearson’s chi-squared statistic (written using the Greek letter chi):(9.1)Figure 9.1 Simulation of χ 2 statisticwith n=20 and probabilities 3/12,4/12, and 5/12. The chi-squareddensity with 2 degrees of freedom isadded.If the multinomial model is correct, then the asymptotic distribution of Y i is known to bethe chi-squared distribution with k−1 degrees of freedom. The number of degrees offreedom coincides with the number of free ways we can specify the values for p i in thenull hypothesis. We are free to choose k−1 of the values but not k, as the values must sumto 1.The chi-squared distribution is a good fit if the expected cell counts are all five ormore. Figure 9.1 shows a simulation and a histogram of the corresponding χ 2 statistic,along with a theoretical density.Using this statistic as a test statistic allows us to construct a significance test. Largervalues are now considered more extreme, as they imply more discrepancy from thepredicted amount.