Design and Analysis of Single Factor Experiments: The Analysis of Variance (ANOVA)

1. Design and Analysis of Single Factor Experiments: The Analysis of Variance (ANOVA)

2. Terminology • Response variable • Measured output value • E.g. tensile strength • Factors • Input variables that can be changed • E.g. temperature, baking time, speed, workers • Levels • Specific values of factors (inputs) • 2 hours 4 hours 6 hours (baking time)

3. Terminology • Replication • Completely re-run experiment with same input levels • Used to determine impact of measurement error • Interaction • Effect of one input factor depends on level of another input factor

4. Example A product engineer is interested in maximizing the tensile strength of a new synthetic fiber that will be used to make cloth for men’s shirts. He knows from previous experience that the strength is affected by the percentage of cotton in the fiber. He also knows that cotton content should range between about 10% and 40%. He decides to test specimens at five levelsof cotton: 15%, 20%, 25%, 30%, and 35%. He also decides to test five specimens at each level of cotton content. The test results are shown in Table 1.

5. The Completely Randomized Single-Factor Experiment (One Way ANOVA) An Example Table 1

6. The Completely Randomized Single-Factor Experiment An Example • The levels of the factor are sometimes called treatments (15%, 20%, 25%, 30% and 35%). • Each treatment has five observations or replicates. • The runs are run in random order.

7. The Completely Randomized Single-Factor Experiment An Example Figure 1Box plots of tensile strength data.

8. The Completely Randomized Single-Factor Experiment The Analysis of Variance Suppose there are adifferent levels of a single factor that we wish to compare. The levels are sometimes called treatments. 2

9. We may describe the observations in Table 2 by the linear statistical model: = the j th observation taken under treatment i. = the overall mean = the i th treatment effect = A random error component The model could be written as = the mean of i th treatment

10. For hypothesis testing, the model errors are assumed to be normally and independently distributed random variables with mean zero and variance . Fixed Effects Model -The levels (treatments) of the factorare specifically chosen by the experimenter. -The experimenter wish to test hypotheses about the treatment means. -Conclusion will apply only to the factor levels considered in the analysis.

11. Random Effects Model -The levels (treatments) of the factorare randomly chosen by the experimenter from a large population of treatments . -The experimenter wish to test hypotheses about the variability of . -Conclusion can be extended to all the factor levels in the population.

12. The Completely Randomized Single-Factor Experiment The Analysis of Variance Fixed-effectsModel The treatment effects are usually defined as deviations from the overall mean so that: Also,

13. The Completely Randomized Single-Factor Experiment The Analysis of Variance We wish to test the hypotheses: or for at least one pair The analysis of variance partitions the total variability into two parts.

14. The Completely Randomized Single-Factor Experiment The Analysis of Variance The difference between the observed treatment averages and the grand average is a measure of the differences between treatment means. The difference of observations within a treatment from the treatment average can be due only to random error.

15. The Completely Randomized Single-Factor Experiment Definition = The sum of squares due to treatments = The sum of squares due to error

16. The Completely Randomized Single-Factor Experiment The Analysis of Variance The ratio MSTreatments = SSTreatments/(a – 1) is called the mean square for treatments.

17. The Completely Randomized Single-Factor Experiment If there are no differences in treatment means ( = 0 ) , then MSTreatments will estimates .

18. The Completely Randomized Single-Factor Experiment The Analysis of Variance The appropriate test statistic is We would reject H0 if F0 > F,a-1,a(n-1)

19. The Completely Randomized Single-Factor Experiment The Analysis of Variance Definition .

20. The Completely Randomized Single-Factor Experiment The Analysis of Variance Analysis of Variance Table

21. Estimation of Model Parameters The mean of the i th treatment is = A point estimator of would be

22. The Completely Randomized Single-Factor Experiment An Unbalanced Experiment

23. Comparison of Individual Treatment Means For the hypothesis a linear combination of treatment totals for this hypothesis is: For testing the hypothesis whether the average of cotton percentages 1 and 3 differs from the average of cotton percentages 2 and 4, a linear combination of treatment totals for this hypothesis is:

24. In general, the comparison of treatment means will imply a linear combination of treatment totals such as With the restriction that Such linear combinations are called contrasts. Contrasts -Contrast is linear combination of treatment totals.

25. The sum of squares for any contrast is and has a single degree of freedom. For the unbalanced design, the sum of squares for the contrast is and it is required that

26. A contrast is tested by comparing its sum of squares to the error mean square. The test statistic is F with 1 and N-a degrees of freedom. Orthogonal contrasts Two contrasts with coefficients and are orthogonal if or if for unbalanced design. For a treatments, the set of a-1 orthogonal contrasts partition the treatment sum of squares into a-1independent single-degree-of-freedom components.

27. For 4 treatments of the factor, the orthogonal contrasts might be as follows: For 5 treatments of the factor, the orthogonal contrasts might be as follows:

28. The Least Significant Difference (LSD) Method Suppose we wish to test for all . The t statistic could be employed: From therefore, The means and would be declared significantly different if

29. The least significant difference (LSD) is If the sample sizes are equal in each treatment: It is concluded that the population means and differ if > LSD

30. Duncan’s Multiple Range Test A procedure for comparing all pairs of means. • Arrange thetreatment averages in ascending order. • Determine the standard error of each average from: where For equal sample sizes, = n .

31. 3. From Duncan’s table of significant ranges, obtain the value for = 2, 3, …, a where is the significance level and is the number of degrees of freedom for error. 4. Convert the significant ranges into a set of a-1 least significant ranges for = 2, 3,…, a . for = 2, 3,…, a . 5. Test the differences between means, beginning with largest versus smallest, which is compared with .

32. 6. Compute the difference between the largest and second smallest means and compared with .The comparisons are continued until all means have been compared with the largest mean. 7. Repeat steps 5-6 but compare all means (except the largest) with the second largest mean. Use to compare the difference between the second largest and smallest meansand then use for the next pair. Continue until all pairs have been considered. 8. If the difference between means is greater than the corresponding least significant range , it can be concluded that the pair of means is significantly different.

33. Residual Analysis and Model Checking The one-way ANOVA assumes that the observations are described by the model and that the errors are normally and independently distributed with mean zero and constant but unknown variance To check if the errors are normally and independently distributed, the residuals are examined where

34. A check of the normality assumption Compute residuals and arrange them in increasing order.

35. Use normal probability paper to plot the residuals against their cumulative frequency (j-0.5)/n . Figure 1 Normalprobability plot and dot diagram

36. A check of the independence assumption Plot the residuals in time order of data collection to detect correlation between the residuals. Figure 2 Plot of residuals versus time

37. The Random-Effects Model ANOVA and Variance Components The linear statistical model is The variance of the response is Where each term on the right hand side is called a variance component.

38. The Random-Effects Model ANOVA and Variance Components For a random-effects model, the appropriate hypotheses to test are: The ANOVA decomposition of total variability is still valid:

39. The Random-Effects Model ANOVA and Variance Components The expected values of the mean squares are

40. The Random-Effects Model ANOVA and Variance Components The estimators of the variance components are

41. The Random-Effects Model Example

42. The Random-Effects Model Example

43. Choice of sample size Operating Characteristic Curves (OC Curves) can be used to guide the experimenter in selecting the number of replicates. OC curves plot the probability of type II error against a parameter . In using the OC Curves, the parameter must be specified. Method 1 The parameter can be computed from

44. To obtain , the actual values of treatment means must be chosen for which we would like to reject the null hypothesis with high probability (1- ) . Then, can be found from where It is also required to estimate . Method 2 This approach is to select a sample size such that if the difference between any two treatment means exceeds a specific value, the hypothesis should be rejected.

45. If the difference between any two treatment means is as large as , then the minimum can be found from

47. For the random effect model, OC curves can be used to determine the sample size. The OC curves plot the probability of type II error against the parameter , where To obtain the sample size n , we need to specify the probability of rejecting the null hypothesis 1- and the ratio .