Chapter 14-7. Chi-squared Test (3 types)
Higher category : 【Statistics】 Chapter 14. Statistcal Test
1. Chi-square goodness-of-fit test
2. Chi-square test of independence
3. Chi-square test of homogeneity
4. Chi-Square distribution and central limit theorem
Figure 1. chi-squared distribution table
1. chi-square goodness-of-fit test
⑴ problem situation (contingency table): if the situation is genetic experiment, Oi and Ej are not proportions but numbers of population
Figure 2. problem situation of chi-square goodness-of-fit test
⑵ setting hypotheses
H0 : the sample Xi follows the given probability distribution
H1 : the sample Xi does not follow the given probability distribution
⑶ calculation of test statistic
⑷ critical region: significance level is α. it is well established when the sample size is big
⑸ Chi-square goodness-of-fit test for multiple variables is also possible.
| 1 | 2 | ⋯ | S | Total | |
|---|---|---|---|---|---|
| X(1) | N1(1) | N2(1) | ⋯ | NS(1) | n(1) |
| X(2) | N1(2) | N2(2) | ⋯ | NS(2) | n(2) |
| ⋮ | ⋮ | ⋮ | ⋱ | ⋮ | ⋮ |
| X(K) | N1(K) | N2(K) | ⋯ | NS(K) | n(K) |
| m1 | m2 | ⋯ | mS | n |
Table 1. Chi-square goodness-of-fit test for multiple variables
2. chi-square test of independence
⑴ problem situation (contingency table)
Figure 3. problem situation of chi-square test of independence
⑵ setting hypotheses
H0 : X and Y are independent
H1 : X and Y are not independent
⑶ calculation of test statistic: calculate the predicted value by using marginal probability distribution and the relationship of independence
⑷ critical region: significance level is α. it is well established when the sample size is big
3. chi-square test of homogeneity
⑴ problem situation (contingency table)
Figure 4. problem situation of chi-square test of homogeneity
⑵ setting hypotheses
H0 : Group1 and Group2 follow the same probability distribution
H1 : Group1 and Group2 don’t follow the same probability distribution
⑶ calculation of test statistic
① equal probability distribution is clearly different from independence conceptually
② however, chi-square test of homogeneity follows a very similar calculation to a chi-square test of independence
⑷ critical region: significance level is α. it is well established when the sample size is big
4. Chi-Square distribution and central limit theorem
⑴ Central limit theorem (CLT)
⑵ Derivation of the Chi-Square distribution: Using the fact that for Xi following a binomial distribution, the variance is Var(Xi) = npq = np - np2 ≈ np = 𝔼[Xi], we get…
Input : 2019.10.05 11:13