Dr.Ramadan. Jabr Page 1 Mutah University/Mech. Eng. Dept. Multifactor Analysis of Variance In many experimental situations, there are two or more factors that are of simultaneous interest. ANOVA 2 extends the methods of ANOVA 1 to investigate such multifactor situations. Example 2.1 In a study on automobile traffic and air pollution reported in the International Journal of Environmental Studies, air samples taken at four different times and at five different locations were analyzed to obtain the amount of particulate matter present in the air (mg/rn3). Let; Example 1, the number of levels of factor A is I = 4, the number of levels of factor Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 2 Mutah University/Mech. Eng. Dept. The Model Includingσ², there are now I + J + 1 model parameters, so if I ≥ 3 and J ≥ 3, then there will be fewer parameters than observations. The model specified above is called an additive model because Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 3 Mutah University/Mech. Eng. Dept. Figure 1. Mean responses for (a) an additive, and (b) a nonadditive model Example 2(Example 1 continued) If we plot the observed xij’s in a manner analogous to that of Figure 1, the result is shown in Figure 2. While there is some “crossing over” in the observed xij’s, the configuration is reasonably representative of what would be expected under additivity with just one observation per treatment. Figure 2 Plot of data from Example 2 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 4 Mutah University/Mech. Eng. Dept. ( For the The interpretation of the parameters of (3) is straightforward: μ is the true grand mean), αi is the effect of factor A at level i (measured as a deviation from μ ), and βj is the effect of factor B at level j. Unbiased (and maximum likelihood) estimators for these parameters are; ( Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 5 Mutah University/Mech. Eng. Dept. Example 3 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 6 Mutah University/Mech. Eng. Dept. ANOVA table (Table 2.1) summarizes further calculations. Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 7 Mutah University/Mech. Eng. Dept. Example 4 (Example 3 continued) Randomized Block Experiments It frequently happens, though, that subjects or experimental units exhibit heterogeneity with respect to other variables that may affect the observed responses. This was the reason for introducing a paired experiment in C.I section. The analogy to a paired experiment when I > 2 is Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 8 Mutah University/Mech. Eng. Dept. called a randomized block experiment. An extraneous factor, “blocks,” is constructed by dividing the IJ units into J groups with I units in each group. This grouping or blocking is done in such a way that within each block, the I units are homogeneous with respect to other factors thought to affect the responses. Then within each homogeneous block, the I treatments are randomly assigned to the I units or subjects in the block. Example 5 A consumer product-testing organization wished to compare the annual power consumption for five different brands of dehumidifier. Because power consumption depends on the prevailing humidity level, it was decided to monitor each brand at four different levels ranging from moderate to heavy humidity (thus blocking on humidity level). Within each level, brands were randomly assigned to the five selected locations. The resulting amount of power consumption (annual kwh) appears in Table 2. Table 2 Power consumption data for Example 5 Table .3 ANOVA table for Example .5 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 9 Mutah University/Mech. Eng. Dept. Figure 3 SAS output for power consumption data Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 10 Mutah University/Mech. Eng. Dept. Models for Random Effects In a two-factor situation, a random effects model is appropriate. The case in which the levels of one factor are the only ones of interest and the levels of the other factor are selected from a population of levels leads to a mixed effects model. The two-factor random effects model when Kij = 1 is Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Mutah University/Mech. Eng. Dept. Page 11 Exercises Continuation of question 4 Continuation of question 2 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 12 Mutah University/Mech. Eng. Dept. Solution for question 2 Solution for question 3 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 13 Mutah University/Mech. Eng. Dept. Solution for question 4 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 14 Mutah University/Mech. Eng. Dept. 2.2 To obtain valid test procedures, the μij’s were assumed to have an additive structure with Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 15 Mutah University/Mech. Eng. Dept. Parameters for the Fixed Effects Model with Interaction Let (2.7) (2.8) (2.9) Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 16 Mutah University/Mech. Eng. Dept. (2.10) Example 2.6 Three different varieties of tomato (Harvester, Pusa Early Dwarf, and Ife No. 1) and four different plant densities (10, 20, 30, and 40 thousand plants per hectare) are being considered for planting in a particular region. To see whether either variety or plant density affects yield, each combination of variety and plant density is used in three different plots, resulting in the data on yields in Table 2.6. Table 2.6 Yield data for Example 2.6 To test the hypotheses of interest, we again define sums of squares and present computing formulas: Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 17 Mutah University/Mech. Eng. Dept. Total variation is thus partitioned into four pieces: unexplained (SSE) and three pieces that may be explained by the truth or falsity of the three H0’s. Each of four mean squares is defined by MS = SS/df. The expected mean squares suggest that each set of hypotheses should be tested using the appropriate ratio of mean squares with MSE in the denominator: Each of the three mean square ratios can be shown to have an F distribution when the associated H0 is true, which yields the following level α test procedures: As before, the results of the analysis are summarized in an ANOVA table. Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 18 Mutah University/Mech. Eng. Dept. Example 2.7 (continuation of example 2.6) Table 2.7 ANOVA table for Example 2.7 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 19 Mutah University/Mech. Eng. Dept. Example 2.8 (continuation of example 2.7) Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 20 Mutah University/Mech. Eng. Dept. Models with Mixed and Random Effects In some problems, the levels of either factor may have been chosen from a large population of possible levels, so that the effects contributed by the factor are random rather than fixed. If both factors contribute random effects, the model is referred to as a random effects model, while if one factor is fixed and the other is random, a mixed effects model results. We will consider here the analysis for a mixed effects model in which factor A (rows) is the fixed factor and factor B (columns) is the random factor. The mixed effects model in this situation is Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 21 Mutah University/Mech. Eng. Dept. Example 2.9 A study was carried out to compare the writing lifetimes of four premium brands of pens. It was thought that the writing surface might affect lifetime, so three different surfaces were randomly selected. A writing machine was used to ensure that conditions were otherwise homogeneous (e.g., constant pressure and a fixed angle). Table 2.8 shows the two lifetimes (mm) obtained for each brand—surface combination. Table 2.8 Lifetime data for Example 2.9 Table 2.9 ANOVA table for Example 2.9 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 22 Mutah University/Mech. Eng. Dept. Exercises Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 23 Mutah University/Mech. Eng. Dept. Solution for question 15 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 24 Mutah University/Mech. Eng. Dept. Solution for question 16 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 25 Mutah University/Mech. Eng. Dept. e)- Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr 2.3 Page 26 Mutah University/Mech. Eng. Dept. Three-Factor ANOVA If we use dot subscripts on the μij’s to denote averaging (rather than summation), then (2.11) (2.12) (2 11) (2.13) (2 11) (2.13) (2.11) Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 27 Mutah University/Mech. Eng. Dept. factors, there would be corresponding higher-order interaction terms with analogous interpretations. The Analysis of a Three-Factor Experiment When L > 1, there is a sum of squares for each main effect, note that any of the model parameters in (2.13) can be estimated unbiasedly by averaging Xijkl over appropriate subscripts and taking differences. Thus, Even the computational formulas for these SSs are quite tedious to use, so we eschew them in favour of output from a statistical computer package. The current version of MINITAB, for example, will fit a three-factor model with fixed, mixed, or random effects. Each sum of squares (excepting SST) when divided by its df gives a mean square, with Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 28 Mutah University/Mech. Eng. Dept. Main effect and interaction hypotheses are tested by forming F ratios with MSE in each denominator: Example 2.10 The following observations (body temperature — 100°F) were reported in an experiment to study heat tolerance of cattle. Measurements were made at four different periods (factor A, with I = 4) on two different strains of cattle (factor B, with J = 2) having four different types of coat (factor C, with K = 4); L = 3 observations were made for each of the 4 x 2 x 4 = 32 combinations of levels of the three factors. Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 29 Mutah University/Mech. Eng. Dept. Figure 2.4 Figure 2.4 Plots of Xijk. for Example 2.10 Table 2.10 ANOVA table for Example 2.10 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 30 Mutah University/Mech. Eng. Dept. Latin Square Designs When several factors are to be studied simultaneously, an experiment in which there is at least one observation for every possible combination of levels is referred to as a complete layout. If the factors are A, B, and C with I, J, and K levels, respectively, a complete layout requires at least IJK observations. A three-factor experiment in which fewer than JfK observations are made is called an incomplete layout. There are some incomplete layouts in which the pattern of combinations of factors is such that the analysis is straightforward. One such three-factor design is called a Latin square. It is appropriate when I = J = K (e.g., four display configurations, four stores, and four time periods) and all two- and three-factor interaction effects are assumed absent. If the levels of factor A are identified with the rows of a two-way table and the levels of B with the columns of the table, then the defining characteristic of a Latin square design is that every level of factor C appears exactly once in each row and exactly once in each column. Pictured in Figure 2.5 are examples of 3 x 3, 4 x 4, and 5 x 5 Latin squares. There are 12 different 3 x 3 Latin squares, and the number of different N x N Latin squares increases rapidly with N (e.g., every permutation of rows of a given Latin square yields a Latin square, and similarly for column permutations). Figure 2.5 Examples of Latin squares Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 31 Mutah University/Mech. Eng. Dept. The Model and Analysis for Latin Squares Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 32 Mutah University/Mech. Eng. Dept. Example 2.11 In an experiment to investigate the effect of relative humidity on abrasion resistance of leather cut from a rectangular pattern, a 6 x 6 Latin square was used to control for possible variability due to row and column position in the pattern. The six levels of relative humidity studied were 1 = 25%, 2 = 37%, 3 = 50%, 4 = 62%, 5 = 75%, and 6 = 87%, with the following results: Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 33 Mutah University/Mech. Eng. Dept. Table 2.10 ANOVA table for Example 2.11 Exercises Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 34 Mutah University/Mech. Eng. Dept. Solution of question 27 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 35 Mutah University/Mech. Eng. Dept. Solution of question 28 Advanced Engineering Statistics The Analysis of Variance 2 Dr.Ramadan. Jabr Page 36 Mutah University/Mech. Eng. Dept. Advanced Engineering Statistics The Analysis of Variance 2