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Unformatted text preview: Lecture Notes for 10 th lecture First, a few notes concerning one-way ANOVA with blocking as compared to one-way without blocking using the same arrangement of data. Increasing Blocking Effect (but with no blocking) F-value DOWN, P-Value UP. The reason that this would be the case is that the increased block effect would be assigned to error variance which would increase the denominator of the F ratio. Decreasing Blocking Effect (with blocking) F-Value associated with blocking would be down, and therefore the P-value would be up associated with blocking. If there were absolutely no blocking effect, that is, if one blocked on a variable that had no impact on the dependent variable, and one used a blocking one-way ANOVA approach, one could actually lose power associated with finding significant treatment effects. The reason for this loss of power is that the degrees of freedom associated with the error variance would be less. If Standard Error σe = 0 Then the F-Values & P-Values are undefined. In reality this would be completely unrealistic. I t would not happen. However, in a simulation experiment, when we set σe = 0, the formulas for all the treatment sums of squares and block sum of squares are very easy to calculate. The student can almost visualize the mathematical relationships. This is the benefit. Problem-1 (2-Way Anova with Interaction) The following is the model. = + + + + xijk μ αi βj αβij eijk In a two way ANOVA with Interaction, we have two factors of Interest, and each factor has multiple treatments. Also there may be interaction between the treatments associated with one factor and the treatments associated with another factor. A fairly clear example that we may comprehend rather easily is associated with agriculture. Imagine that we have three different types (treatments) of chemical fertilizers (factor ), and two different types β (treatments) of land (factor ). α In this model there may be interaction between the type of fertilizer and the type of land. That is fertilizer j (where j goes from 1 to b) may interact with fertilizer i (where i goes from 1 to a). Let us get our subscripts straightened out: Alpha ( α i ) is typically associated with the factor associated with rows in the data matrix. The subscript i goes from 1 to a, in this case, from 1 to 2. Beta ( β j ) is associated with the column factor. The subscript J goes from 1 to b, in this case from 1 to 3....
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- Summer '10
- Statistics, Dr. Lamb