1. 381 Introduction to ANOVA QSCI 381 – Lecture 43 (Larson and Farber, Sect 10.4)

2. 381 Introduction So far we have seen methods to test whether two means are different. Today we address the question whether the means of three or more populations are the same. The null and alternative hypotheses for this test are: H 0 : H a : At least one of the means is different from the others. Note that this test does not say which mean is different from the rest – that requires additional analysis.

3. 381 Examples We conduct a survey during each month of the year and wish to test the hypothesis that the survey catch-rate does not change over the year. We measure time to death for 5 types of poison and wish to determine whether the methods are equivalent. We collect data on the time it takes to find a salmon in various types of stream and wish to determine whether the mean times are the same across streams. General properties of the data: There are three (or more) populations. We have taken a sample (from which a mean and standard deviation can be calculated) from each population.

4. 381 Conditions To Use This Approach Each sample must be selected from a normal, or approximately normal, population. The samples must be independent of each other. Each population must have the same variance.

5. 381 One-way Analysis of Variance-I is a hypothesis-testing technique that is used to compare means from three or more populations. Analysis of variance is usually abbreviated ANOVA. There are many extensions to one-way ANOVA – this particular flavor is “one-way” because only one independent variable is being considered.

6. 381 One-way Analysis of Variance-II The test statistic for a one-way analysis of variance is the ratio of two variances: the variance between the samples and the variance within the samples, i.e.: MS B measures the differences related to the treatment given to each sample (often called the mean square between) MS W measures the differences related to the entries within each sample – sampling error - (often called the mean square within)

7. 381 One-way Analysis of Variance-III The test statistic is a ratio of variances so its sampling distribution (if the null hypothesis is correct) is the F-distribution. The degrees of freedom are: D.f. N =k-1; D.f. D =N-k where The idea is that if there is little (or no) difference among the means, the variability between the means should be the same as among the samples.

8. 381 Calculation Details-I Let x i,j be the j th data point from the i th population (m populations, n i samples per population). 1. Find the mean and variance of each sample, i.e.: 2. Find the “grand mean”: 3. Find the sum of squares between the samples:

9. 381 Calculation Details-II 4. Find the sum of squares within the samples: 5. Find the variance between the samples: 6. Find the variance within the samples: 7. Calculate the test statistic:

10. 381 An ANOVA Summary Table VariationSum of squares Degrees of freedom Mean Squares F BetweenSS B d.f. N SS B /d.f. N MS B /MS W WithinSS W d.f. D SS W /d.f. D The following is a convenient way to summarize an ANOVA

11. 381 Example-I Quarter 1234 12106 1498 1011819 514910 9 5455 118.212.6 11.5004.6672.20015.800 10.421 Quarter 1234 0.8861.34124.66523.739 SS B / d.f. N 50.632 / (4-1) = 16.877 46148.863.2 SS W / d.f. D 132 / (19-4) = 8.8 F16.877/8.8 = 1.918 Data SummaryDeveloping the Table

12. 381 Example-II VariationSum of squares Degrees of freedom Mean Squares F Between50.632316.8771.918 Within132158.8

13. 381 Example-III  =0.05; F crit =3.287 We fail to reject the null hypothesis.

14. 381 Using EXCEL To Conduct a ONE-WAY ANOVA-I

15. 381 Using EXCEL To Conduct a ONE-WAY ANOVA-II