1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 13 Experimental Design and Analysis of Variance nIntroduction to Experimental Design and Analysis of Variance and Analysis of Variance nAnalysis of Variance and the Completely Randomized Design and the Completely Randomized Design nMultiple Comparison Procedures

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved nStatistical studies can be classified as being either experimental or observational. nIn an experimental study, one or more factors are controlled so that data can be obtained about how the factors influence the variables of interest. nIn an observational study, no attempt is made to control the factors. nCause-and-effect relationships are easier to establish in experimental studies than in observational studies. An Introduction to Experimental Design and Analysis of Variance nAnalysis of variance (ANOVA) can be used to analyze the data obtained from experimental or observational studies.

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved An Introduction to Experimental Design and Analysis of Variance nA factor is a variable that the experimenter has selected for investigation. nA treatment is a level of a factor. nExperimental units are the objects of interest in the experiment. nA completely randomized design is an experimental design in which the treatments are randomly assigned to the experimental units.

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Analysis of Variance: A Conceptual Overview Analysis of Variance (ANOVA) can be used to test Analysis of Variance (ANOVA) can be used to test for the equality of three or more population means. for the equality of three or more population means. Analysis of Variance (ANOVA) can be used to test Analysis of Variance (ANOVA) can be used to test for the equality of three or more population means. for the equality of three or more population means. Data obtained from observational or experimental Data obtained from observational or experimental studies can be used for the analysis. studies can be used for the analysis. Data obtained from observational or experimental Data obtained from observational or experimental studies can be used for the analysis. studies can be used for the analysis. We want to use the sample results to test the We want to use the sample results to test the following hypotheses: following hypotheses: We want to use the sample results to test the We want to use the sample results to test the following hypotheses: following hypotheses: H 0 :  1  =  2  =  3  = ... =  k H a : Not all population means are equal

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved H 0 :  1  =  2  =  3  = ... =  k H a : Not all population means are equal If H 0 is rejected, we cannot conclude that all If H 0 is rejected, we cannot conclude that all population means are different. population means are different. If H 0 is rejected, we cannot conclude that all If H 0 is rejected, we cannot conclude that all population means are different. population means are different. Rejecting H 0 means that at least two population Rejecting H 0 means that at least two population means have different values. means have different values. Rejecting H 0 means that at least two population Rejecting H 0 means that at least two population means have different values. means have different values. Analysis of Variance: A Conceptual Overview

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved For each population, the response (dependent) For each population, the response (dependent) variable is normally distributed. variable is normally distributed. For each population, the response (dependent) For each population, the response (dependent) variable is normally distributed. variable is normally distributed. The variance of the response variable, denoted  2, The variance of the response variable, denoted  2, is the same for all of the populations. is the same for all of the populations. The variance of the response variable, denoted  2, The variance of the response variable, denoted  2, is the same for all of the populations. is the same for all of the populations. The observations must be independent. The observations must be independent. nAssumptions for Analysis of Variance Analysis of Variance: A Conceptual Overview

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved nSampling Distribution of Given H 0 is True   Sample means are close together because there is only because there is only one sampling distribution one sampling distribution when H 0 is true. when H 0 is true. Analysis of Variance: A Conceptual Overview

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved nSampling Distribution of Given H 0 is False 33 33 11 11 22 22 Sample means come from different sampling distributions and are not as close together when H 0 is false. when H 0 is false. Analysis of Variance: A Conceptual Overview

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved Analysis of Variance nBetween-Treatments Estimate of Population Variance nWithin-Treatments Estimate of Population Variance nComparing the Variance Estimates: The F Test nANOVA Table

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved Between-Treatments Estimate of Population Variance  2 Denominator is the degrees of freedom associated with SSTR Numerator is called the sum of squares due to treatments (SSTR) The estimate of  2 based on the variation of the The estimate of  2 based on the variation of the sample means is called the mean square due to sample means is called the mean square due to treatments and is denoted by MSTR. treatments and is denoted by MSTR.

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved The estimate of  2 based on the variation of the sample observations within each sample is called the mean square error and is denoted by MSE. The estimate of  2 based on the variation of the sample observations within each sample is called the mean square error and is denoted by MSE. Within-Treatments Estimate of Population Variance  2 Denominator is the degrees of freedom associated with SSE Numerator is called the sum of squares due to error (SSE)

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved Comparing the Variance Estimates: The F Test If the null hypothesis is true and the ANOVA If the null hypothesis is true and the ANOVA assumptions are valid, the sampling distribution of assumptions are valid, the sampling distribution of MSTR/MSE is an F distribution with MSTR d.f. MSTR/MSE is an F distribution with MSTR d.f. equal to k - 1 and MSE d.f. equal to n T - k. equal to k - 1 and MSE d.f. equal to n T - k. If the means of the k populations are not equal, the If the means of the k populations are not equal, the value of MSTR/MSE will be inflated because MSTR value of MSTR/MSE will be inflated because MSTR overestimates  2. overestimates  2. Hence, we will reject H 0 if the resulting value of Hence, we will reject H 0 if the resulting value of MSTR/MSE appears to be too large to have been MSTR/MSE appears to be too large to have been selected at random from the appropriate F selected at random from the appropriate F distribution. distribution.

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved nSampling Distribution of MSTR/MSE Do Not Reject H 0 Reject H 0 MSTR/MSE Critical Value FF FF Sampling Distribution of MSTR/MSE  Comparing the Variance Estimates: The F Test

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved Source of Variation Sum of Squares Degrees of Freedom MeanSquare F Treatments Error Total k - 1 n T - 1 SSTR SSE SST n T - k SST is partitioned into SSTR and SSE. SST’s degrees of freedom (d.f.) are partitioned into SSTR’s d.f. and SSE’s d.f. ANOVA Table p - Value

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved ANOVA Table SST divided by its degrees of freedom n T – 1 is the SST divided by its degrees of freedom n T – 1 is the overall sample variance that would be obtained if we overall sample variance that would be obtained if we treated the entire set of observations as one data set. treated the entire set of observations as one data set. SST divided by its degrees of freedom n T – 1 is the SST divided by its degrees of freedom n T – 1 is the overall sample variance that would be obtained if we overall sample variance that would be obtained if we treated the entire set of observations as one data set. treated the entire set of observations as one data set. With the entire data set as one sample, the formula With the entire data set as one sample, the formula for computing the total sum of squares, SST, is: for computing the total sum of squares, SST, is: With the entire data set as one sample, the formula With the entire data set as one sample, the formula for computing the total sum of squares, SST, is: for computing the total sum of squares, SST, is:

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved ANOVA Table ANOVA can be viewed as the process of partitioning ANOVA can be viewed as the process of partitioning the total sum of squares and the degrees of freedom the total sum of squares and the degrees of freedom into their corresponding sources: treatments and error. into their corresponding sources: treatments and error. ANOVA can be viewed as the process of partitioning ANOVA can be viewed as the process of partitioning the total sum of squares and the degrees of freedom the total sum of squares and the degrees of freedom into their corresponding sources: treatments and error. into their corresponding sources: treatments and error. Dividing the sum of squares by the appropriate Dividing the sum of squares by the appropriate degrees of freedom provides the variance estimates degrees of freedom provides the variance estimates and the F value used to test the hypothesis of equal and the F value used to test the hypothesis of equal population means. population means. Dividing the sum of squares by the appropriate Dividing the sum of squares by the appropriate degrees of freedom provides the variance estimates degrees of freedom provides the variance estimates and the F value used to test the hypothesis of equal and the F value used to test the hypothesis of equal population means. population means.

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved Test for the Equality of k Population Means F = MSTR/MSE H 0 :  1  =  2  =  3  = ... =  k  H a : Not all population means are equal nHypotheses nTest Statistic

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved Test for the Equality of k Population Means nRejection Rule where the value of F  is based on an F distribution with k - 1 numerator d.f. and n T - k denominator d.f. Reject H 0 if p -value <  p -value Approach: Critical Value Approach: Reject H 0 if F > F 

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved Multiple Comparison Procedures nSuppose that analysis of variance has provided statistical evidence to reject the null hypothesis of equal population means. nFisher’s least significant difference (LSD) procedure can be used to determine where the differences occur.

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved Fisher’s LSD Procedure nTest Statistic nHypotheses

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved Fisher’s LSD Procedure where the value of t a /2 is based on a where the value of t a /2 is based on a t distribution with n T - k degrees of freedom. nRejection Rule Reject H 0 if p -value <  p -value Approach: Critical Value Approach: Reject H 0 if t t a /2

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved nTest Statistic Fisher’s LSD Procedure Based on the Test Statistic x i - x j __ where Reject H 0 if > LSD nHypotheses nRejection Rule

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved nThe experiment-wise Type I error rate gets larger for problems with more populations (larger k ). Type I Error Rates  EW = 1 – (1 –  ) ( k – 1)! The comparison-wise Type I error rate  indicates the level of significance associated with a single pairwise comparison. The comparison-wise Type I error rate  indicates the level of significance associated with a single pairwise comparison. The experiment-wise Type I error rate  EW is the probability of making a Type I error on at least one of the ( k – 1)! pairwise comparisons. The experiment-wise Type I error rate  EW is the probability of making a Type I error on at least one of the ( k – 1)! pairwise comparisons.