Chapter 11 Estimation and Hypothesis Testing III: One-Way Analysis of Variance Learning Objectives: 1. Describe when to use a one-way analysis of variance ANOVA test. 2. Describe the concept of the ANOVA test. 3. Describe the assumptions of the ANOVA test. 4. Manually calculate the formulas for the test and populate the ANOVA table. 5. Describe the F-ratio and the F-distribution. 6. Test hypotheses using the one-way ANOVA test. 7. Briefly discuss the purpose of post-hoc tests. Chapter Summary In this chapter, students extend their understanding of statistical hypothesis testing to the concept of ANOVA. Specifically, students learn about the utility of the ANOVA for examining mean differences across three or more groups of interest, and the application of the ANOVA table for estimating the F-Ratio and making statistical inferences. Students are exposed to the concepts and formulas for estimating between group and within group variance; and between, within and total sums of squares. The chapter ends with a demonstration of how to operationalize all of the formulas involved in completing a one-way ANOVA test, and determining the F-ratio. Key Formulas The following represent the key formulas for this chapter. PowerPoint slides are provided for each chapter. In addition to these slides, a PDF file containing only the formulas are also provided. F-Ratio Between Group Sum of Squares Within Group Sum of Squares Total Sum of Squares Between Groups Degrees of Freedom Within Groups Degrees of Freedom Mean Sum of Squares for Between Groups Mean Sum of Squares for Within Groups Interactive Figures: The textbook contains interactive figures. You may wish to use these in a lecture. Students also have access to these. For this chapter, there are two interactive figures. 1. Figure 11.1 illustrates the plot of individuals by group. 2. Table 11.2 provides an interactive example of an ANOVA. These interactive figures can be found on in the eBook and the Library under Chapter 11 Resources. Typical Lecture Material We have provided two sample lectures below. You may wish to add in additional discipline specific information to make these more relevant to your students. Lecture 1: Objective: To understand the conceptual underpinnings of the ANOVA test, its assumptions, and when it is appropriate to use this measure. Review the following concept table with your students; and help them to fill in the definitions of each statistical concept. The definitions that the students should come up with are in italics. Statistical Concept Defintion The Grand Mean The total of all observations divided by the total sample size (x-double bar). Between Group Variance The total amount the group means vary from the grand mean (between group variability). Within Group Variance The total amount that all indivudal observations vary from their group mean (within group variability). F-Ratio The ratio of the between group variance to within group variance. Between Group Sum of Squares The sum of the squared deviations of each group mean from the grand mean. Within Group Sum of Squares Is the sum of the squared deviations of each observation from its group mean. Total Sum of Squares The sum of the squared deviations of each observation from the grand mean. Mean Sum of Squares for Between Groups Represents the between group variance Mean Sum of Squares for Within Groups Represents the within group variance. Example 1: As students evolve in their statistical knowledge, it is important to know what type of data they have, what statistical test(s) are appropriate for that data; and what assumptions are being made when applying particular statistical tests. Ask your students to identify whether the following necessitates an independent sample t-test, paired t-test or ANOVA. The correct response is in italics. 1. Does the mean level of math competency among 6th grade students significantly change following the implementation of an applied learning program? (paired-sample t-test) 2. Does the mean level of reading activity significantly vary between males and females? (independent sample t-test) 3. Does the mean level of overall academic competency vary according to low, medium and high socio-economic status? (one-way ANOVA). 4. Do mortality rates among individuals aged 80 years and above differ across those who are married, single, separated and divorced? (one-way ANOVA)? Example 2: Ask your students to answer the following: 1. What are the four assumptions of the ANOVA test? a. The dependent variable must be measured at either the interval or ratio level of measurement. b. The groups (and more specifically, the data) must be independent of one another. c. The data must be relatively normally distributed. d. The variance in the population must be equal for all groups (i.e. The assumption of the homogeneity of variance). 2. What can and can’t we conclude from a one-way ANOVA test? (i.e. Students should identify that we can determine if there is a significant difference in sample means in at least one of the groups, but that ANOVA itself doesn’t tell us which group is different.). Example 3: Read the following scenario to your students: You are interested in studying levels of mathematical competency among grade 6 students in Manitoba. From the population of Manitoba Grade 6 students, you have randomly selected 15 students in order to answer your research questions. Specifically, you are interested in whether there are any significant differences in mathematical competency across the students who have been taught using three different curriculum and methods (Curriculum A, Curriculum B, and, Curriculum C). Math scores are assessed on a scale from 0 to 10. The following is the data, the group means and variances, and the grand mean. Curriculum A Curriculum B Curriculum C 1 10 1 4 1 4 2 6 2 7 2 6 3 7 3 5 3 7 4 9 4 6 4 7 5 9 5 7 5 2 𝑥̅1= 8.20 𝑥̅2= 5.80 𝑥̅3= 5.20 𝑠12 = 2.70 𝑠22 = 1.70 𝑠32 = 4.70 𝑛1 = 5 𝑛2 = 5 𝑛3 = 5 𝑥̅̅= 6.40 If you find it useful to plot the data, the follow diagram represents the individual values in each group along with their means and the grand mean. Ask your students to determine the following: 1. What is the Between Group Degrees of Freedom? Answer: dfB = k – 1 = 3 - 1 = 2 2. What is the Within Group Degrees of Freedom? Answer: dfW = n – k = 15 - 3 = 12 3. What would be the critical value of F if we were to use a 95% confidence level? Answer: The critical value of F(2,12) is 3.89 Lecture 2: Objective: To test a hypothesis utilizing a one-way ANOVA; and to learn how to formulate an ANOVA table. Example 1: Read the following to your students: In our previous lecture, we were given data pertaining to mathematical achievement among 6th graders; and indicated that we were interested in whether there were any differences in student’s mean mathematical competency across three different curriculum types. In that lecture, we calculated the between and within group degrees of freedom. Given that we have that data, the group means and variances, and the grand mean, we are ready to complete the one-way ANOVA test to determine if there are any significant differences in the competency scores across the three groups. Use a confidence level of 95% when testing the hypothesis. (Note: You can use the interactive Table 11.2 to do this in class) The hypothesis you are testing is: 𝐻0: 𝜇1 = 𝜇2 = 𝜇3 𝐻𝐴: 𝜇1 ≠ 𝜇2 ≠ 𝜇3 1. Draw a blank ANOVA table on the board (shown below) and ask your students to complete the table. The formulas and answers are below. The calculation for each cell is provided on the next page. 2. Once the students have completed the calculations, draw the F-distribution and write in the critical value and the value of F (black dot) as shown below: 3. Now ask the students to state what the result is of their hypothesis test. Answer: We reject the null hypothesis of no difference across the groups and conclude that there is evidence to suggest that at least one group is different than the others. Solutions to End-of-Chapter Problems Problem 11-1 a) F(4, 25); F0.05(4, 25) = 2.76 (LO5) b) F(7, 7); F0.05(7, 7) = 3.79 (LO5) c) F(3, 20); F0.05(3, 20) = 3.10 (LO5) Problem 11-2 a) Null hypothesis: the means for each groups are the same. Alternative hypothesis: at least one of the means is not the same as the others. (LO3) b) D.f.N = k-1=2; d.f.D = n-k = 15 (LO4) c) between group sum of squares = 459.18; within group sum of squares = 682.5 (LO4) d) mean between group sum of squares = 229.59; mean within group sum of squares = 45.5 (LO4) e) F(2, 15) = 5.05; F0.05(2, 15) = 3.68. Based on this information the null hypothesis would be rejected. There is enough evidence to believe the means are different. (LO6) f) Given below from a through e Source Sum of Squares df Mean Sum of Squares F-Ratio Between Groups 459.18 2 229.59 5.05 Within Groups 682.5 15 45.5 Total 1141.68 17 Problem 11-3 a) See table below (LO4/LO5/LO6) Source Sum of Squares df Mean Sum of Squares F-Ratio Between Groups 101.10 2 50.55 7.74 Within Groups 71.83 11 6.53 Total 172.93 13 b) Computed statistic = 7.74; critical value of F0.05(2, 11) = 3.98; conclusion: the null hypotheses would be rejected. There is not sufficient evidence. (LO4/LO5/LO6)) Source Sum of Squares df Mean Sum of Squares F-Ratio Between Groups 101.10 2 50.55 7.74 Within Groups 71.83 11 6.53 Total 172.93 13 Solutions to Interactive Exercises Exercise 11-1 A study of sodium content in 3 brands of cereals (which claims 9% of sodium) provided the following incomplete ANOVA table. If 4 packets of each brand are selected, what is your conclusion of the study? (Hint: Complete the below given ANOVA table) Source DF SS MS F Groups 8.01 Error 111. 42 Total 127. 44 Level of significance = 0.05 Answer: H0 : µ1 = µ2 = µ3 HA: At least one mean is different from the rest. Source DF SS MS F Groups 2 16.02 8.01 0.647 Error 9 111.42 12.38 Total 11 127.44 F-Ratio = 0.646 F(2, 9) critical = 4.26 F critical is greater than F ratio. Hence fail to reject Ho. There isn’t enough evidence to believe that at least one mean is different from the rest. Exercise 11-2 A researcher conducts an experiment comparing 4 treatment conditions with a sample of 6 from each treatment. An analysis of variance is used to evaluate the data and the results of ANOVA are presented in the table below, complete all missing values in the table below. Source DF SS MS F Groups 18 3 Error Total What is your conclusion? Answer: H0: µ1 = µ2 = µ3 = µ4 HA: At least one mean is different from the rest. Source DF SS MS F Groups 3 18 6 3 Error 20 40 2 Total 23 58 F-Ratio = 3 F(3, 20) critical = 2.38 F ratio is greater than F critical. Hence, reject Ho. There is enough evidence to believe that at least one mean is different from the rest. Exercise 11-3 The data below reflects the outcome of a one-way table. SS groups = 140 SS error=1300 n=100 k=5 a) What is MS group? b) What is MS error? c) Determine F-ratio. d) Determine F critical value @ 0.05 level of significance. e) What is your decision? Answer: a) H0: µ1 = µ2 = µ3 = µ4 b) HA: At least one mean is different from the rest. Source DF SS MS F Groups 4 140 35 2.56 Error 95 1300 13.68 Total 99 1440 c) F-Ratio = 2.56 d) F(4, 100) critical ≈ F(4, 95) critical = 2.46 e) F ratio is greater than F critical. Hence, reject Ho. There is enough evidence to believe that at least one mean is different from the rest. Solutions to SPSS Exercises Exercise 11-1 An experimenter is interested in evaluating the effectiveness of three different diet methods for weight loss. A group of 16 subjects is available for the experiment. Two subgroups of eight subjects each are formed at random; the subgroups are then given one of the three diet plans. Upon completion of the diet, each of the subgroups measured for the number of pounds lost in a two week period. The resulting test scores are given in the following table. Diet 1 Diet 2 3 4 5 4 2 3 4 8 8 7 4 4 3 2 9 5 Use SPSS to perform the ANOVA. Is there a significant difference between the different diet plans? Start by looking at analyze – compare means – One-Way Anova and the following result ensues. ANOVA VAR00001 Sum of Squares df Mean Square F Sig. Between Groups 41.500 5 8.300 8.300 .111 Within Groups 2.000 2 1.000 Total 43.500 7 LO: 6 Page: 318-322 Equations from Chapter 11 Scott R. Colwell and Edward M. Carter c 2012 Equation 11.1: F-Ratio F-Ratio = Between Group Variance Within Group Variance Equation 11.2: Between Group Sum of Squares SSB = Xni(x¯i −x¯¯)2 Where: SSB = between group sum of squares ni = sample size for each group x¯i = mean for each group x¯¯ = grand mean Equation 11.4: Within Group Sum of Squares SSW = X(xi −x¯i)2 Where: SSW = within group sum of squares xi = individual observation for each group x¯i = mean for each group Equation 11.6: Total Sum of Squares SST = X(xi −x¯¯i)2 Where: SST = total sum of squares xi = individual observation for each group x¯¯ = grand mean Equation 11.9: Between Groups Degrees of Freedom dfB = k −1 Where: dfB = between groups degrees of freedom k = number of groups Equation 11.11: Within Groups Degrees of Freedom dfW = n −k Where: dfW = within groups degrees of freedom n = total sample size k = number of groups Equation 11.14: Mean Sum of Squares for Between Groups MSB = SSB dfB Where: MSB = mean sum of squares for between groups SSB = between groups sum of squares dfB = between groups degrees of freedom Equation 11.17: Mean Sum of Squares for Within Groups MSW = SSW dfW Where: MSW = mean sum of squares for within groups SSW = within groups sum of squares dfW = within groups degrees of freedom Equation 11.20: F-Ratio F −Ratio = Between Group Variance Within Group Variance = MSB MSW Where: MSB = mean sum of squares for between groups MSW = mean sum of squares for within groups Solution Manual for Introduction to Statistics for Social Sciences Scott R. Colwell, Edward M. Carter 9780071319126
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