= 1 i. normal approximation to χ 2 df > df

Transcription

1 χ tests 1) 1 categorical variable χ test for goodness-of-fit ) categorical variables χ test for independence (association, contingency) 3) categorical variables McNemar's test for change χ df k (O i 1 i E i) E normal approximation to χ df > 30

2 χ goodness-of-fit test In a random sample of 1000 housewives, 55% state a preference for Brand A and 45% for Brand B. Is this result consistent with the hypothesis that 50% of all housewives prefer Brand A (i.e., no brand preference)? category #observed #expected O i E i O i -E i (O i -E i ) ( Oi Ei ) Ei A B n H o : O 1 O O 3 O K H 1 : not H o k (O i Ei ) χ df 1 Ei df k-1 where k #categories χ 1 10 critical χ 1, , reject H o

3 χ df O E O E k (Oi 1 i OE E Ei ) E E + E O + E O OE + E E O E O E OE E n + n E + E O E n O i E i O i Oi Ei A B n χ

4 Example: A die is rolled 10 times and the following distribution is obtained. Is the die honest? H o : f 1 f f 3 f 4 f 5 f 6 H 1 : not H o O i E i O i Oi Ei n χ critical χ 5, reject Ho

5 χ test for independence (association, contingency)

6 A college professor distributes a teacher-effectiveness questionnaire to 00 students registered in a statistics class. One question is: How do you rate the instructor in her ability to explain difficult concepts (above average, average, below average). The results are summarized by rating and Level. Do ratings depend on level? I (C1) II (C) III (C3) above (R1) average (R) below (R3) H o : the variables are independent H 1 : the variables are dependent expected frequencies are determined assuming H o is true

7 P (A) n(a ) n P(A and B) P(A)P(B) if A and B are independent P(R1) n(r1) n fr1 n n(c1) fc1 50 P(C1) n n 00 P(R1 and C1) P(R1)P(C1) if R1 and C1 are independent P(R1andC1) f n r1 f n fr1 n(r1andc1) np(r1andc1) n n c1 f n c1 f f n r1 c1 (50)(100) 00

8 I (C1) II (C) III (C3) above (R1) 34 (5) 36 (40) 30 (35) 100 average (R) 1(15) 4(4) 4(1) 60 below (R3) 4(10) 0(16) 16(14) O i E i O i -E i (O i -E i ) Oi Ei Ei n df (r-1)(c-1) k (O i Ei ) χ df 1 Ei χ O i χ df n Ei χ critical χ 4, reject Ho ( Oi Ei ) O i

9

10 McNemar Change Test This test studies the change in a group of 75 respondents measured twice on a dichotomous variable. One group of voters is asked twice about their voting intention, before and after a television debate. 13 respondents changed their preference from Carter to Reagan while 7 respondents changed their preference from Reagan to Carter. Is the number of respondents changing similar in the direction from Reagan to Carter as in the other direction. a c b d before Reagan Carter after Reagan 7 7 (10) 34 Carter 13 (10) H 0 : the two changes are the same H 0 : the two changes are different ( O E) (7 10) (13 10) χ E 10 10

11 after before Reagan Carter Reagan Carter χ 1 (b c) b + c (7 13) after before Carter Reagan Reagan Carter χ 1 (a d) a + d (7 13)

12 after before yes no yes 6 0 (15) 6 no 10 (15) χ 1 ( O E) E (0 15) 15 + (10 15) χ 1 (b c) b + c (0 10)

13 Example 1 Example Mann-Whitney U Test Rank T1 T T1 T R 1 8 R Rank T1 T T1 T R 1 40 R Ho: identical population distributions H 1 : different population distributions rank all n 1 + n n scores n(n + 1) Σ ( R1 + R) Σ ( R1 + R ) 55 n(n + 1) (10)(11) 55

14 Example 1 n1(n1 + 1) (5)(6) U1 n1n + R1 (5)(5) n (n + 1) (5)(6) U n1n + R (5)(5) U 1 + U n 1 n 5 U obs is the smaller of U 1 and U U obs 1 Example n1(n1 + 1) (5)(6) U1 n1n + R1 (5)(5) n (n + 1) (5)(6) U n1n + R (5)(5) U 1 + U n 1 n 5 U obs is the smaller of U 1 and U U obs 0 5 0

15 Rank A B A B R 1 46 R Ho: identical population distributions H 1 : different population distributions n(n + 1) Σ ( R1 + R) 66 U n1(n1 + 1) (5)(6) n1n + R1 (5)(6) n n n (n + + 1) R (6)(7) (5)(6) U 1 U 1 + U n 1 n 30 U obs 5 5 5

16 U distribution - Table A6 reject if U obs U CV U CV,.05 3 U obs 5 do not reject Ho

17 Rank A B A B R 1 94 R 77

18 Ho: identical population distributions H 1 : different population distributions Σ(R 1 + R ) n(n + 1) (18)(19) 171 n1(n1 + 1) (9)(10) U n1n + R1 (9)(9) n (n + 1) (9)(10) U n1n + R (9)(9) U 1 + U n 1 n 81 U obs 31 critical value of U do not reject Ho

19 normal approximation to U distribution U 00 n 1 n 5 n1n µ U if n1 and n are > 0 n n (5)(5) µ 1 U 31.5 σ U n1n (n1 + n 1 + 1) n1n (n1 + n + 1) (5)(5)( ) σ U Uobs µ U Z.18 σu critical Z reject Ho 51.54

20 Wilcoxon Matched-Pairs Signed-Ranks Test Example 1 X Y d X-Y rank of d signed R 1 Example X Y d X-Y rank of d signed R 1 Ho: identical population distributions H 1 : different population distributions n(n + 1) R where n # of non - zero differences R 1 n(n + 1) (6)(7) 1

21 Example 1 T(+) 10 T(-) 11 T obs smaller of T(+) and T(-) 10 Example T(+) 0 T(-) 1 T obs smaller of T(+) and T(-) 0

22 T(+) 3 T(-) 18 X Y d X-Y rank of d signed R 1 T obs smaller of T(+) and T(-) 3 critical T.05 0 T distribution - Table A7 reject if T obs T CV do not reject Ho

23 normal approximation to the T distribution n 60 T 615 µ n(n + 1) 4 (60)(61) 4 T 915 n(n + 1)(n + 1) (60)(61)(11) S T Z T µ σ T T critical Z reject Ho

24 Kruskal-Wallis H Test for k independent samples Ranks G1 G G3 G1 G G Σ Ho: identical populations H 1 : at least 1 is different rank all n 1 + n + + n k n scores n(n + 1) R 78 H 1 R n(n + 1) n j j 3(n + 1) H + + 1(1 + 1) H χ (k-1) where k # groups critical χ, reject Ho where n total # of subjects 3(1 + 1) (.0769)(585.5)

25 H critical χ, reject Ho T1 T T

26 Friedman Test for k dependent samples Ranks A1 A A3 A1 A A Σ Ho: identical populations H 1 : at least 1 is different 1 Fr R j 3n(k + 1) where n is the number of subjects nk(k + 1) 1 Fr [ ] (3)(6)(3 + 1) 7 (6)(3)(3 + 1) F r χ (k-1) where k # groups critical χ, reject Ho

27 Summary Design parametric nonparametric - ranks independent samples t-test Mann-Whitney U dependent samples t-test Wilcoxon k independent samples ANOVA between Kruskal-Wallis k dependent samples ANOVA within Friedman POWER

28 1. Four treatments were randomly assigned to four independent groups of six subjects each. a) How would you test the null hypothesis of equal population means? b) How would you test the null hypothesis of identical population distributions? a) between-subject ANOVA b) H test. The use of music to induce relaxation in dental patients is becoming popular. Suppose a dentist tested the effectiveness of music in relaxing patients by randomly assigning patients to two groups: music or no music. Patients were asked to rank their anxiety. How would you evaluate whether the groups differed in their anxiety estimates? Mann Whitney U test 3. An office manager randomly selects 10 smokers and 10 nonsmokers. He records the number of minutes wasted per working hour. Is there a difference in the amount of time wasted? -sample independent t, Mann Whitney U test 4. Fifty women followed a diet for 1 month. Their weight was recorded immediately before and at one week periods after the diet began. How would you evaluate whether the diet was effective in producing a weight change? within-subject ANOVA, Friedman