# All pairwise comparisons among levels of fertilizer

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All Pairwise Comparisons among Levels of Fertilizer Fertilizer = 1 subtracted from: Difference SE of Adjusted Fertilizer of Means Difference T-Value P-Value 2 0.4000 0.3648 1.097 0.7003 3 1.4750 0.3648 4.044 0.0127 4 2.7250 0.3648 7.471 0.0002 Fertilizer = 2 subtracted from: Difference SE of Adjusted Fertilizer of Means Difference T-Value P-Value 3 1.075 0.3648 2.947 0.0650 4 2.325 0.3648 6.374 0.0006 Fertilizer = 3 subtracted from: Difference SE of Adjusted Fertilizer of Means Difference T-Value P-Value 4 1.250 0.3648 3.427 0.0316 13-66
Topic 11a, part 4 Two-Way Analysis of Variance
Objectives Objectives Objectives Objectives 1. Analyze a two-way ANOVA design 2. Draw interaction plots 3. Perform the Tukey test 13-68
Objective 1 Objective 1 Objective 1 Objective 1 • Analyze a Two-Way Analysis of Variance Design 13-69
Recall, there are two ways to deal with factors: (1) control the factors by fixing them at a single level or by fixing them at different levels, and (2) randomize so that their effect on the response variable is minimized. In both the completely randomized design and the randomized complete block design, we manipulated one factor to see how varying it affected the response variable. 13-70
In a Two-Way Analysis of Variance design, two factors are used to explain the variability in the response variable. We deal with the two factors by fixing them at different levels. We refer to the two factors as factor A and factor B. If factor A has n levels and factor B has m levels, we refer to the design as an factorial design. n × m 13-71
Parallel Example 3: A 2 x 4 Factorial Design Suppose the rice farmer is interested in comparing the fruiting period for not only the four fertilizer types, but for two different seed types as well. The farmer divides his plot into 16 identical subplots. He randomly assigns each seed/fertilizer combination to two of the subplots and obtains the fruiting periods shown on the following slide. Identify the main effects. What does it mean to say there is an interaction effect between the two factors? 13-72
13-73 Fertilizer 1 Fertilizer 2 Fertilizer 3 Fertilizer 4 Seed Type A 13.5 13.9 13.5 14.1 15.2 14.7 17.1 16.4 Seed Type B 14.4 15.0 14.7 15.4 15.3 15.9 16.9 17.3
Solution The two factors are A: fertilizer type and B: seed type. Since all levels of factor A are combined with all levels of factor B, we say that the factors are crossed . The main effect of factor A is the change in fruiting period that results from changing the fertilizer type. The main effect of factor B is the change in fruiting period that results from changing the seed type. We say that there is an interaction effect if the effect of fertilizer on fruiting period varies with seed type. 13-74
1. The populations from which the samples are drawn must be normal. 2. The samples are independent. 3. The populations all have the same variance. Requirements to Perform the Two-Way Analysis of Variance
In a two-way ANOVA, we test three separate hypotheses.