Chap 10-1 Analysis of Variance

Chap 10-2 Overview Analysis of Variance (ANOVA) F-test Tukey- Kramer test One-Way ANOVA Two-Way ANOVA Interaction Effects

Chap 10-3 General ANOVA Setting Investigator controls one or more independent variables Called factors (or treatment variables) Each factor contains two or more levels (or groups or categories/classifications) Observe effects on the dependent variable Response to levels of independent variable Experimental design: the plan used to collect the data

Chap 10-4 Completely Randomized Design Experimental units (subjects) are assigned randomly to treatments Subjects are assumed homogeneous Only one factor or independent variable With two or more treatment levels Analyzed by one-factor analysis of variance (one-way ANOVA)

Chap 10-5 One-Way Analysis of Variance Evaluate the difference among the means of three or more groups Examples: Accident rates for 1 st, 2 nd, and 3 rd shift Expected mileage for five brands of tires Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn

Chap 10-6 Hypotheses of One-Way ANOVA All population means are equal i.e., no treatment effect (no variation in means among groups) At least one population mean is different i.e., there is a treatment effect Does not mean that all population means are different (some pairs may be the same)

Chap 10-7 One-Factor ANOVA All Means are the same: The Null Hypothesis is True (No Treatment Effect)

Chap 10-8 One-Factor ANOVA At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or (continued)

Chap 10-9 Partitioning the Variation Total variation can be split into two parts: SST = Total Sum of Squares (Total variation) SSA = Sum of Squares Among Groups (Among-group variation) SSW = Sum of Squares Within Groups (Within-group variation) SST = SSA + SSW

Chap Partitioning the Variation Total Variation = the aggregate dispersion of the individual data values across the various factor levels (SST) Within-Group Variation = dispersion that exists among the data values within a particular factor level (SSW) Among-Group Variation = dispersion between the factor sample means (SSA) SST = SSA + SSW (continued)

Chap Partition of Total Variation Variation Due to Factor (SSA) Variation Due to Random Sampling (SSW) Total Variation (SST) Commonly referred to as:  Sum of Squares Within  Sum of Squares Error  Sum of Squares Unexplained  Within Groups Variation Commonly referred to as:  Sum of Squares Between  Sum of Squares Among  Sum of Squares Explained  Among Groups Variation = +

Chap Total Sum of Squares Where: SST = Total sum of squares c = number of groups (levels or treatments) n j = number of observations in group j X ij = i th observation from group j X = grand mean (mean of all data values) SST = SSA + SSW

Chap Total Variation (continued)

Chap Among-Group Variation Where: SSA = Sum of squares among groups c = number of groups or populations n j = sample size from group j X j = sample mean from group j X = grand mean (mean of all data values) SST = SSA + SSW

Chap Among-Group Variation Variation Due to Differences Among Groups Mean Square Among = SSA/degrees of freedom (continued)

Chap Among-Group Variation (continued)

Chap Within-Group Variation Where: SSW = Sum of squares within groups c = number of groups n j = sample size from group j X j = sample mean from group j X ij = i th observation in group j SST = SSA + SSW

Chap Within-Group Variation Summing the variation within each group and then adding over all groups Mean Square Within = SSW/degrees of freedom (continued)

Chap Within-Group Variation (continued)

Chap Obtaining the Mean Squares

Chap One-Way ANOVA Table Source of Variation dfSS MS (Variance) Among Groups SSAMSA = Within Groups n - cSSWMSW = Totaln - 1 SST = SSA+SSW c - 1 MSA MSW F ratio c = number of groups n = sum of the sample sizes from all groups df = degrees of freedom SSA c - 1 SSW n - c F =

Chap One-Factor ANOVA F Test Statistic Test statistic MSA is mean squares among variances MSW is mean squares within variances Degrees of freedom df 1 = c – 1 (c = number of groups) df 2 = n – c (n = sum of sample sizes from all populations) H 0 : μ 1 = μ 2 = … = μ c H 1 : At least two population means are different

Chap Interpreting One-Factor ANOVA F Statistic The F statistic is the ratio of the among estimate of variance and the within estimate of variance The ratio must always be positive df 1 = c -1 will typically be small df 2 = n - c will typically be large Decision Rule: Reject H 0 if F > F U, otherwise do not reject H 0 0  =.05 Reject H 0 Do not reject H 0 FUFU

Chap One-Factor ANOVA F Test Example You want to see if three different golf clubs yield different distances. You randomly select five measurements from trials on an automated driving machine for each club. At the.05 significance level, is there a difference in mean distance? Club 1 Club 2 Club

Chap One-Factor ANOVA Example: Scatter Diagram Distance Club 1 Club 2 Club Club 1 2 3

Chap One-Factor ANOVA Example Computations Club 1 Club 2 Club X 1 = X 2 = X 3 = X = n 1 = 5 n 2 = 5 n 3 = 5 n = 15 c = 3 SSA = 5 (249.2 – 227) (226 – 227) (205.8 – 227) 2 = SSW = (254 – 249.2) 2 + (263 – 249.2) 2 +…+ (204 – 205.8) 2 = MSA = / (3-1) = MSW = / (15-3) = 93.3

Chap F = One-Factor ANOVA Example Solution H 0 : μ 1 = μ 2 = μ 3 H 1 : μ i not all equal  =.05 df 1 = 2 df 2 = 12 Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence that at least one μ i differs from the rest 0  =.05 F U = 3.89 Reject H 0 Do not reject H 0 Critical Value: F U = 3.89

Chap SUMMARY GroupsCountSumAverageVariance Club Club Club ANOVA Source of Variation SSdfMSFP-valueF crit Between Groups E Within Groups Total ANOVA -- Single Factor: Excel Output EXCEL: tools | data analysis | ANOVA: single factor

Chap The Tukey-Kramer Procedure Tells which population means are significantly different e.g.: μ 1 = μ 2  μ 3 Done after rejection of equal means in ANOVA Allows pair-wise comparisons Compare absolute mean differences with critical range x μ 1 = μ 2 μ 3

Chap Tukey-Kramer Critical Range where: Q U = Value from Studentized Range Distribution with c and n - c degrees of freedom for the desired level of  (see appendix E.9 table) MSW = Mean Square Within n i and n j = Sample sizes from groups j and j ’

Chap The Tukey-Kramer Procedure: Example 1. Compute absolute mean differences: Club 1 Club 2 Club Find the Q U value from the table in appendix E.9 with c = 3 and (n – c) = (15 – 3) = 12 degrees of freedom for the desired level of  (  =.05 used here):

Chap The Tukey-Kramer Procedure: Example 5. All of the absolute mean differences are greater than critical range. Therefore there is a significant difference between each pair of means at 5% level of significance. 3. Compute Critical Range: 4. Compare: (continued)

Chap Two-Way ANOVA Examines the effect of Two factors of interest on the dependent variable e.g., Percent carbonation and line speed on soft drink bottling process Interaction between the different levels of these two factors e.g., Does the effect of one particular carbonation level depend on which level the line speed is set?

Chap Two-Way ANOVA Assumptions Populations are normally distributed Populations have equal variances Independent random samples are drawn (continued)

Chap Two-Way ANOVA Sources of Variation Two Factors of interest: A and B r = number of levels of factor A c = number of levels of factor B n ’ = number of replications for each cell n = total number of observations in all cells (n = rcn ’ ) X ijk = value of the k th observation of level i of factor A and level j of factor B

Chap Two-Way ANOVA Sources of Variation SST Total Variation SSA Factor A Variation SSB Factor B Variation SSAB Variation due to interaction between A and B SSE Random variation (Error) Degrees of Freedom: r – 1 c – 1 (r – 1)(c – 1) rc(n’ – 1) n - 1 SST = SSA + SSB + SSAB + SSE (continued)

Chap Two Factor ANOVA Equations Total Variation: Factor A Variation: Factor B Variation:

Chap Two Factor ANOVA Equations Interaction Variation: Sum of Squares Error: (continued)

Chap Two Factor ANOVA Equations where: r = number of levels of factor A c = number of levels of factor B n ’ = number of replications in each cell (continued)

Chap Mean Square Calculations

Chap Two-Way ANOVA: The F Test Statistic F Test for Factor B Effect F Test for Interaction Effect H 0 : μ 1.. = μ 2.. = μ 3.. = H 1 : Not all μ i.. are equal H 0 : the interaction of A and B is equal to zero H 1 : interaction of A and B is not zero F Test for Factor A Effect H 0 : μ.1. = μ.2. = μ.3. = H 1 : Not all μ.j. are equal Reject H 0 if F > F U

Chap Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Statistic Factor ASSAr – 1 MSA = SSA /(r – 1) MSA MSE Factor BSSBc – 1 MSB = SSB /(c – 1) MSB MSE AB (Interaction) SSAB(r – 1)(c – 1) MSAB = SSAB / (r – 1)(c – 1) MSAB MSE ErrorSSE rc(n ’ – 1) MSE = SSE/rc(n’ – 1) TotalSSTn – 1

Chap Features of Two-Way ANOVA F Test Degrees of freedom always add up n-1 = rc(n ’ -1) + (r-1) + (c-1) + (r-1)(c-1) Total = error + factor A + factor B + interaction The denominator of the F Test is always the same but the numerator is different The sums of squares always add up SST = SSE + SSA + SSB + SSAB Total = error + factor A + factor B + interaction

Chap Examples: Interaction vs. No Interaction No interaction: Factor B Level 1 Factor B Level 3 Factor B Level 2 Factor A Levels Factor B Level 1 Factor B Level 3 Factor B Level 2 Factor A Levels Mean Response Interaction is present: