Jednoczynnikowa analiza wariancji (ANOVA)

Transcription

1 Wydział Matematyki Jednoczynnikowa analiza wariancji (ANOVA) Wykład 07

2 Example 1 An accounting firm has developed three methods to guide its seasonal employees in preparing individual income tax returns. In comparing the effectiveness of these methods, a test is set up in which each of 10 seasonal employees is randomly assigned to use one of the three methods in preparing a hypothetical income tax return. The preparation times (in minutes) are shown in below table. At the α = level, can we conclude that the three methods could be equally effective?

3 Some terminology for ANOVA

4 Example 1 Factors/Treatment Observations of dependent variable

5 What is ANOVA? Analysis of variance (ANOVA) is a statistical methodology for comparing several means in an experiment in which we try to determine whether various levels of a given factor might be having different effects on something we re observing or measuring. We consider two ANOVA techniques: One-way ANOVA one factor used in analysis Two-way ANOVA more than one factor used

6 The basis of ANOVA is comparing two kinds of variation: a) variation between the groups, reflecting the effect of the factor levels b) variation within the groups, which represents random error from the sampling process Variation Between and Within the Groups

7 One-way analysis of variance The one-way analysis of variance examines two or more independent samples to determine whether their population means could be equal.

8 One-way analysis of variance In one-way analysis of variance each individual observation is considered to be the sum of three separate components:

9 One-way analysis of variance The null and alternative hypotheses shown previously can now be presented in an equivalent form that is relevant to the model described here:

10 Assumptions for one-way ANOVA 1. The samples have been independently selected, 2. The population variances are equal 3. The population distributions are normal.

11 Procedure for one-way ANOVA Part A: The Null and Alternative Hypotheses The null and alternative hypotheses are expressed in terms of the equality of the population means for all of the treatment groups. Part B: The Format of the Data to Be Analyzed The data can be listed in tabular form, as shown, with a separate column for each of the t treatments. The number of observations in the columns (n1, n2, n3,..., nt) need not be equal.

12 Procedure for one-way ANOVA Part C: The Calculations for One-Way ANOVA The Sum of Squares Terms: Quantifying the Two Sources of Variation Treatments, TR SSTR is the sum of squares value reflecting variation between individual treatment means and the overall mean for all treatments Weighted according to the sample sizes for the respective treatment groups, SSTR expresses the amount of variation that is attributable to the treatments. Sampling error, E SSE is the sum of the squared differences between observed values and the means for their respective treatment groups; SSE expresses the amount of variation due to sampling error. Total variation, T SST is the total amount of variation, or SST SSTR SSE.

13 Procedure for one-way ANOVA Part D: The Test Statistic, the Critical Value, and the Decision Rule

14 Example 1 An accounting firm has developed three methods to guide its seasonal employees in preparing individual income tax returns. In comparing the effectiveness of these methods, a test is set up in which each of 10 seasonal employees is randomly assigned to use one of the three methods in preparing a hypothetical income tax return. The preparation times (in minutes) are shown in below table. At the α = level, can we conclude that the three methods could be equally effective?

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22 Example 1 - solution We need to find critical value of F from the α = F distribution table.

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25 Example 1 1 2

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28 Example 1 1

29 Example 2 From each of four suppliers, a quality-control technician collects a random sample of 10 rivets, then measures the number of pounds each will withstand before it fails. Perform the one-way ANOVA analysis at the 0.01 level of significance

30 Example 2 The null and alternative hypotheses are: The quality of rivets offered by the four suppliers are comparable At least one of the suppliers offers significantly different quality

31 Example 2 Checking the assumptions for ANOVA 1. The samples have been independently selected, 2. The population variances are equal 3. The population distributions are normal.

32 Example 2 Checking the normality 2 1 3

33 Example 2 Checking the normality 1

34 Example 2 Checking the normality

35 Example 2 Checking the normality 2 1

36 Example 2 Checking the normality Normality assumption is satisfied

37 Example 2 Checking the homoscedasticity 2 1 3

38 Example 2 Checking the homoscedasticity variances are equal

39 Example

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42 Example 2 observed value of F statistic p value 1 2

43 Example 2 1 Treatment Sum of Squares SSTR = Error Sum of Squares SSE = Treatment Mean Square MSTR = 5709/3 = 1903 Error Mean Square MSE = 36780/36 = 1022 F = 1903/1022 = 1,863

44 Example 2 numerator denominator F critical value for d2 = 36 lies between 4.31 and 4.51 Observed value = 1,863 < F critical value

45 Example F critical value = Observed value = 1,863 < F critical value

46 Example 2 - conclusion We fail to reject the null hypothesis. It means that the quality of rivets offered by the four suppliers are comparable