Two way ANOVA with replication - Portal双向方差分析与复制-门.ppt

1、Two Way ANOVA With Replication,Also called a Factorial Experiment. Factorial Experiment is used to evaluate 2 or more factors simultaneously. Replication means an independent repeat of each factor combination. The purpose of factorial experiment is to examine: 1. The effect of factor A on the depend

2、ent variable, y. 2. The effect of factor B on the dependent variable, y along with 3. The effects of the interactions between different levels of the factors on the dependent variable, y. Interaction exists when the effect of a level for one factor depends on which level of the other factor is prese

3、nt. Advantages of Factorial Experiment over one factor at a time (one-way ANOVA) more efficient & allow interactions to be detected.,The effect model for a factorial experiment can be written as:,There are three sets of hypothesis: 1. Factor A effect: 2. Factor B effect: 3. Interaction effect:,The r

4、esults obtained in this analysis are summarized in the following ANOVA table:,Two way Factorial Treatment Structure,where,Example: The two-way table gives data for a 2x2 factorial experiment with two observations per factor level combination. Construct the ANOVA table for this experiment and do a co

5、mplete analysis at a level of significance 0.05.,Solution:,154.2,81.6,95.3,140.5,235.8,Solution: 1. Set up hypothesis Factor A effect: Factor B effect: Interaction effect:,2. Calculation (given the ANOVA table is as follows): 3. With = 0.05 we reject if :,4. From ANOVA/table F, the critical and F ef

6、fects are given as follow: 5. Factor A : since , thus we reject We conclude that the difference level of A effect the response Factor B : since , thus we reject We conclude that the difference level of B effect the response Interaction: since , thus we failed to reject We conclude that no interactio

7、n between factor A and factor B.,Exercise: In a study to determine which are the important source of variation in an industrial process, 3 measurements are taken on yield for 3 operators chosen randomly and 4 batches a raw materials chosen randomly. It was decided that a significance test should be made at the 0.05 level of significance to determine if the variance components due to batches, operators, and interaction are significant. In addition, estimates of variance components are to be computed. The data are as follows, with the response being p