2-1 ANOVA.pptx materi statiska untuk perhitungan anova

anova perhitungan statistika

1. ANOVA

2. A Scenario for t-test Membandingkan tingkat stress akademik antara 2 sekolah

3. A Scenario for t-test Membandingkan stress akademik anatara siswa laki laki dan siswa perempuan .

4. Bagaimana jika kita ingin membandingkan tingkat stress akademik dari siswa di 3 sekolah berbeda ? ?

5. A Scenario for ANOVA

6. A scenario for ANOVA Kita ingin membandingkan nilai RATA-RATA UN dari 3 sekolah . School A School B School C Kita bisa melakukan beberapa kali uji t: School A versus School B School B versus School C School C versus School A  You will need more than one t-test for multiple comparison of pairs of means.

7. Several t-tests for the multiple comparison of pairs of means (1 ) Multiple t-tests will inflate the overall Type I error rate. By performing s everal t-tests for pairs of means, we capitalize on chance when we perform repeated tests. (2) These t-tests are not independent ! The multiple tests of contrasts are not independent.  ANOVA

8. ANOVA ANOVA avoids this type of Type I error inflation by conducting a single test whether the entire set of sample means suggests that the samples were drawn from the same general population. ANOAVA is used to determine the probability that differences in means (across several groups) are due solely to sampling error. p -value Like in t-tests i.e., standard error

9. ANOVA More flexible than t-test More than two groups More than one independent variable (i.e., Factor) Levels in factors can have many levels as desired. Remember: ANOVA still takes one dependent variable! What about in the previous NAPLAN example?

10. ANOVA Independent variables (IV): One or more categorical variables (e.g., gender, social class, school type, treatment, education program) D ependent variable (DV ): A continuous variable (e.g., test scores, etc . ). ANOVA (One-way, Two-way, Factorial)

11. The Purpose of ANOVA To determine whether the means are significantly different from each other . To determine whether the means of the dependent variable are significantly different from each other . To determine whether the means of the dependent variable for an independent variable are significantly different from each other. To determine whether the means of the dependent variable for each level of an independent variable are significantly different from each other. Same as t-tests Same as t-tests Same as t-tests Same as t-tests

12. Hypothesis Testing in ANOVA N ull hypothesis: There is no difference in group means Alternative hypothesis: There is difference in group means Ho : μ1 = μ2 = μ3 = … = μ k H A : μ1 ≠ μ2 ≠ μ3 ≠ … ≠ μ k

13. ANOVA - post hoc tests Examination of the group means enables you to assess the relative standing of each group on the dependent measure. T he F statistic test assesses the null hypothesis of equal means, and it does not address the question of which means are different . For example, in a three‐group situation, all three groups may differ significantly , or two may be equal but differ from the third. To assess these differences, the researcher can employ either planned comparisons or post hoc tests.

14. The Purpose of ANOVA ANOVA can tell you whether: G roups scores differ from one another. Additional analysis can be done to find out: One group has a higher mean score than all the other groups. One group has a higher mean score than another group.

15. ANOVA: F-statistic F-statistic is the ratio of MS B to MS W . It is a measure of: how much variance is attributable to the different groups versus the variance expected from random sampling . The ratio of MS B to MS W is similar in concept to the t statistic, but in this case gives us a value for the F statistic.

16. Essence of ANOVA An ANOVA is an examination of means (i.e., mean differences) based on means themselves and variations from the means. Concept: If everyone did have the same score, then the overall mean would have zero variance. When not everyone scores exactly the same, then, it is possible to express each person’s score as a deviation from the grand mean. ANOVA is based on a measure of variation, called sum of squares (SS) . Definition of The S um of Squares (SS) The Sum of Squares (SS ) refers to the sum of the squares of the deviations from a mean for an entire group. The Sum of Squares (SS) refers to the sum of the squared deviations about a mean . The Sum of Squares (SS) refers to the sum of the squared deviations about a mean , which is the primary component that is used to compute variance.

17. Essence of ANOVA Definition of The S um of Squares (SS) The Sum of Squares (SS ) refers to the sum of the squares of the deviations from a mean for an entire group. Student1 4 Student2 3 Student3 6 Student4 4 Student5 7 Student6 3 Student7 7 Student8 6 Student9 2 Student10 8 5 Mean -1 -2 1 -1 2 -2 2 1 -3 3 Deviation 1 4 1 1 4 4 4 1 9 9 38 SS

18. Essence of ANOVA The ANOVA seeks to determine how much total variation can be explained by the independent variable (or group identity), by dividing the total variance (i.e., sum of squares, SST ) into two parts : the variance attributable to the variables in the study (between sums of squares, SSB ) the variance attributable to variables not included in the study (within sums of squares, SSW )

19. Essence of ANOVA The elements of an ANOVA Differences between a group mean and the grand mean ( SSB ) Differences between an individual’s score and his or her group means ( SSW ) Differences between an individual’s score and the grand mean ( SST ) ANOVA tests whether the amount of variance explained by the independent variable ( SSB ) is a significant proportion relative to the variance that has not been explained ( SSW ).

20. Essence of ANOVA Experimental Group Control Group Test score of Experimental Group Test score of Experimental Group What you want to see

21. Essence of ANOVA Experimental Group Control Group Test score of Experimental Group Test score of Control Group Within-group variation in the scores Within-group variation in the scores Error Error What you want to see

22. Essence of ANOVA Without the membership of the groups!

23. Essence of ANOVA Experimental Group Control Group Test score of Experimental Group Test score of Experimental Group

24. Within-groups Variation W ithin-groups variation : The extent to which individuals within a group differ from each other within their own group . It is expressed by the sum of squared deviations of their individual scores from their own group mean . The group differences are kept constant The mean of these within-groups variations gives an idea of the differences between the individuals, without this having anything to do with the group identity.

25. Between-groups Variation Between-groups variation: The extent to which groups differ from each other . It is expressed by the sum of squared deviations of the group means from the grand mean . (You need to calculate the group means, first, of course, to calculate the deviation of the group means from the grand mean.) A between-groups variation gives an idea of the differences between the groups, without this having anything to do with the individuals . If this variation is multiplied by the group size, we obtain the between-groups variation.

26. Student1 4 -1 1 Student2 3 -2 4 Student3 6 1 1 Student4 4 -1 1 Student5 7 2 4 Student6 3 -2 4 Student7 7 2 4 Student8 6 1 1 Student9 2 -3 9 Student10 8 3 9 Control Control Control Control Control Experiment Experiment Experiment Experiment Experiment 4.8 5.2 (4.8) - (5) = -0.2 (5.2) - (5) = 0.2 0.04 0.04 0.08 Between-groups Variation 5 5 38 Mean Deviation SS SSB (between sums of squares) 0.08 5 0.4 x =

27. Student1 4 Student2 3 Student3 6 Student4 4 Student5 7 Student6 3 Student7 7 Student8 6 Student9 2 Student10 8 Control Control Control Control Control Experiment Experiment Experiment Experiment Experiment 4.8 5.2 Within-groups Variation SSW (Within sums of squares) -0.8 -1.8 1.2 -0.8 2.2 -2.2 1.8 0.8 -3.2 2.8 0.64 3.24 1.44 0.64 4.84 4.84 3.24 0.64 10.24 7.84 37.6 5 38 Mean Deviation SS

28. SST = SSB + SSW 0.4 SSB (between sums of squares) 37.6 SSW (Within sums of squares) 38 = + SST (Total sums of squares)

29. Total Variation Total variation is the sum of squared deviations of individual scores from the grand mean. This gives an idea of the differences that have to do with both individuality as well as the group identity. Total variation = the between-groups variation + the within-groups variation

30. Student1 4 -1 1 Student2 3 -2 4 Student3 6 1 1 Student4 4 -1 1 Student5 7 2 4 Student6 3 -2 4 Student7 7 2 4 Student8 6 1 1 Student9 2 -3 9 Student10 8 3 9 Control Control Control Control Control Experiment Experiment Experiment Experiment Experiment Total Variation 5 38 Mean Deviation SS SST

31. Variation & Variance Variation divided by N (technically, degrees of freedom)  Variance

32. Degree of Freedom The between-groups variation indicates that part of the differences that is exclusively due to the groups. In dividing the between-groups variation by the number of degrees of freedom, we obtain the between-groups variance. The within-groups variation indicates that part of the differences that is exclusively due to the individuals. In dividing the within-groups variation by the number of degrees of freedom, we obtain the within groups variance.

33. Degree of Freedom Degree of freedom for the between-groups variance : d B = number of groups – 1 d B = number of groups – Number of Grand mean d B = ( k – 1 ) degrees of freedom when the number of groups is k. If you have 3 groups, d B = 3-1  3 groups and the grand mean is fixed to 1. Degree of freedom for the within-groups variance : d W = k(n -1) = kn – k = N - k d W = number of groups multiplied by (number of individuals within the group - 1). If you have 3 groups, and there are 8 individuals within the group, d W = 3(8 - 1) = 21, because for each of the three groups there are eight individuals, for which the group mean is fixed. F-tests are based on ( k – 1) and ( N – k ) degrees of freedom.

34. ANOVA: F-statistic ANOVA F-test is based on: F-statistic, F-ratio, F-score, F-value… SSW / df W SSB / df B Variation = Sum of Squares Variation = Sum of Squares

35. ANOVA ANOVA = AN alysis O f VA riance Variances (estimated) of the dependent variable of the groups are compared. Estimates of variances are divided into: Between-groups estimate of variance Within-groups estimate of variance The total variance is partitioned into the within-groups variation and the between-groups variation. TOTAL = WITHIN + BETWEEN

36. Between‐groups of variance Between‐groups estimate of variance ( MS B: mean square between groups): The estimate of variance is the variability of the group means on the dependent variable. It is based on deviations of group means from the overall grand mean of all scores. Under the null hypothesis of no group difference (i.e ., μ1 = μ2 = μ3 = … = μ k ), this variance estimate reflects any group effects that exist ; that is, differences in group means increase the expected value of MS B. Note that any number of groups can be accommodated.

37. Within‐groups of variance Within‐groups estimate of variance ( MS W: mean square within groups): The estimate of the average respondent variability on the dependent variable within the groups. It is based on deviations of individual scores from their respective group means. MS W is comparable to the standard error between two means calculated in the t test as it represents variability within groups. The value MS W is sometimes referred to as the error variance .

38. ANOVA Grand mean

39. ANOVA Within groups mean Within groups mean

40. ANOVA Between‐groups of variance

41. ANOVA Within‐groups of variance

42. ANOVA F-statistic in ANOVA: The differences between groups are compared with the differences between individuals within their own groups.

43. ANOVA: F-statistic The ratio of MS B to MS W is a measure of how much variance is attributable to the different groups versus the variance expected from random sampling .

44. ANOVA: Interpreting F-statistic F-test is based on F-statistic, F-ratio, F-score, F-value… F = 1 : the groups differ as much as the individuals already differ from each other. The groups have no effect on the dependent variable. F > 1 : the variance of groups rises above the variance of individuals within each group. Differences between groups Differences between individuals within each group

45. ANOVA: F-statistic Differences between the groups increase MS B  large values of the F statistic  lead to rejection of the null hypothesis of no difference in means across groups. If the analysis has several different factors/treatments (independent variables), then estimates of MS B are calculated for each factor/treatment and F statistics are calculated for each treatment. This approach allows for the separate assessment of each treatment.

46. ANOVA: Interpreting F-statistic Hypothesis testing for F-test follows a process similar to the t test. To determine whether the F statistic is sufficiently large to support rejection of the null hypothesis.

47. F-distribution

48. ANOVA: F-tests Step 1. Calculate/obtain the F statistic. Step 2. If the p-value attached to the calculated F statistic exceeds the significance level (acceptable level, error rate, often 0.05 level) , you can reject the null hypothesis that the means across all groups are equal.

49. F-tests

50. Results of F-tests The probability of finding an F-value of xxxx (or an even more extreme value) under Ho is xxxx . If this probability (p-value) is very low (e.g., less than a alpha level = 0.05), then we can reject the null hypothesis. If this probability (p-value) is not very small (e.g ., greater than a alpha level = 0.05), then we cannot reject the null hypothesis.

51. p-values The probability of observing an F-ratio of 1.638 or more given the null hypothesis is true is 0.203. In the SPSS computer output, we do not find a critical F* value, but rather the expression of the associated p-value (“Sig”). This is the empirical significance level, which represents a probability of finding an F-ratio (or an even more extreme value) under Ho.

52. Results of F-tests You set a significance level, 0.05  The probability of 0.05. When the p-value is less than 0.05 (i.e., significance level of 0.05), you reject the null hypothesis and concludes that the groups scores differ. When the p-value is greater than 0.05 (i.e., significance level of 0.05), you cannot reject the null hypothesis and concludes that the groups scores do not differ.

53. ANOVA & Sample size

54. ANOVA & Sample size Sample size requirements from increasing either the number of levels or the number of factors For each group, a sample size should be at least about 20 or so observations. Thus, increasing the number of levels in any factor requires an increase in sample size. Also, analyzing multiple factors can create a situation of large sample size requirements rather quickly. When two or more factors are included in the analysis, the number of groups formed is the product of the number of levels, not their sum.

55. ANOVA & Sample size School 1 School 2 25 25 Gender School 1 School 2 Male 25 25 Female 25 25 SES School 1 School 2 Low 25 25 Medium 25 25 High 25 25 50 100 150 By adding one factor with two levels and three levels

56. Sample size in Experiments Researchers should be careful when determining both the number of levels for a factor as well as the number of factors to be included, especially in controlled experimental settings where the necessary sample size per cell is much more difficult to achieve!

57. Assumption in ANOVA: Independence Critical assumption of ANOVA (or MANOVA) requires that the dependent measures for each respondent be totally uncorrelated with the responses from other respondents in the sample. A lack of independence severely affects the statistical validity of the analysis unless corrective action is taken.

58. A Final Note on the General F-test This F-test only confirms the existence of significant differences among the group means. It does not follow necessarily that every subset of two means shows a significant difference. The latter is investigated by means of separate tests (multiple comparison tests), in which the testing is more conservative (i.e., smaller than the usual alpha).

59. ANOVA: F-tests using Critical Values

60. ANOVA: F-tests As with the t test, a researcher can use certain F values as general guidelines when sample sizes are relatively large. These values are just the t crit value squared.

61. Critical Values in t-statistic & F-statistic