Question

Problem 3: Understanding the chi-squared test of independence of categorical random variables The problem has two sub-problems that should help understand the chi-squared test of independence. a) For the chi-squared test of independence between two categorical variables, we use the fact that the MLE for the probability that a multinomial trial is of category i is ni /n, where ni is the number of trials that fall into category i, and n is the total number of trials. Prove the previously stated fact. Recall that the multinomial distribution is Definition 5.12 of WMS:

b) Explain how two independent multinomial experiments, denoted as E1 and E2, can actually be thought of as a single multinomial experiment, denoted as E. In particular: i) How does the number of categories of E relate to the number of categories in E1 and E2? ii) Additionally, how do the category probabilities of E relate to the category probabilities of E1 and E2?

Answer 1

Let us consider that there are k calegones for a categorical variable (say 2) hat s en var be the number of observations that fall into each Categones respectively & n-Eri Then the probability distributions of mine is feismas nu) - a on! p pn -pthi Peel-EP: mi melmel where Pi is the probability of th category To find M.LE Let us considee a sample of seze ses, Thens La fost so where she is the results of the sample aus 2 Pi nk n-En a los 2 alos L - - Pipi Sanu 0 Puller ар k- &nk=nE ne 1-Epi ni k-1 *-) n-Emi 1-Epi р I- - From ① we can say Pi - MLE of pumi

Consider two multinorial experiments E, & Ez to home atmost a categories, Thens the expenment E which consists of these two experiment is thought to have toue categories in Ella Eiz, E21 E22 where the first subscript alonotes the for category of E, & second subscript denotes the category of & cè Eu will have the entries which satisfies. the seategory of E, & ' category of Ea. Vrly the rest of the categories of E. can be found. Pictonically the representing the above situations using a table EX E12 Ev & stappening Non happening Happening Nos happening En Eze E can also be considered to have a multinomial distribution with tous calegories. het A be the event of happening of El 62 & B then pa- Probability of happening of E, & E = P(ADB) = P(A). P(B) Since E, & te are independente lllsly Prz = P(ANB) = P(A). P(5) P21 = PDB) = PA P (B) P2z = PAB) = P(A) - P(B). where Puspiz, P21, P22 are the probabilities of categories of E. E & E are considered to have benomial be probability since we consider two classificat for each experiment [happening & Non happening). Bad & bars fore categories & E, & Ez bave multiple categories say t, has k categories & E has heategories Then E must have ke categories. ...