In either case, the first test we should do is the test on the interaction effects. If there is interaction and it is significant, i. If the interaction term is significant that tells us that the effect of A is different at each level of B.
Or you can say it the other way, the effect of B differs at each level of A. Therefore, when we have significant interaction, it is not very sensible to even be talking about the main effect of A and B , because these change depending on the level of the other factor. If the interaction is significant then we want to estimate and focus our attention on the cell means.
If the interaction is not significant, then we can test the main effects and focus on the main effect means. Just the form of these variances tells us something about the efficiency of the two factor design.
The factorial structure, when you do not have interactions, gives us the efficiency benefit of having additional replication, the number of observations per cell times the number of levels of the other factor. This benefit arises from factorial experiments rather than single factor experiments with n observations per cell. An alternative design choice could have been to do two one-way experiments, one with a treatments and the other with b treatments, with n observations per cell.
However, these two experiments would not have provided the same level of precision, nor the ability to test for interactions. Do we get remove the interaction term in the model? You might consider dropping that term from the model. If n is very small and your df for error are small, then this may be a critical issue. If the p-value for the interaction test is greater than 0. This is not an exact cut off but a general rule.
Remember, if you drop the interaction term, then a variation accounted for by SSab would become part of the error and increasing the SSE, however your error df would also become larger in some cases enough to increase the power of the tests for the main effects.
Statistical theory shows that in general dropping the interaction term increases your false rejection rate for subsequent tests. Hence we usually do not drop nonsignificant terms when there are adequate sample sizes. However, if we are doing an independent experiment with the same factors we might not include interaction in the model for that experiment.
What should we do in order to test our hypothesis? We obviously cannot perform the test for interaction because we have no error term. If you are willing to assume, and if it is true that there is no interaction, then you can use the interaction as your F -test denominator for testing the main effects. It is a fairly safe and conservative thing to do. If it is not true then the MSab will tend to be larger than it should be, so the F -test is conservative. We extend the model in the same way.
Our analysis of variance has three main effects, three two-way interactions, a three-way interaction and error. The non-centrality parameter for calculating sample size for the A factor is:.
Actually at the beginning of our design process we should decide how many observations we should take, if we want to find a difference of D , between the maximum and the minimum of the true means for the factor A. There is a similar equation for factor B. In the two factor case this is just an extension of what we did in the one factor case.
But now we have the marginal means benefiting from a number of observations per cell and the number of levels of the other factor. In this case we have n observations per cell, and we have b cells. So, we have nb observations. Eberly College of Science.
Introduction to Factorial Designs. Printer-friendly version For now we will just consider two treatment factors of interest. It looks almost the same as the randomized block design model only now we are including an interaction term: The Effects Model vs. We will define our marginal means as the simple average over our cell means as shown below: But first we want to look at the effects model and define more carefully what the interactions are We can write the cell means in terms of the full effects model: Now consider the non-additive case.
We illustrate this with Example 2 which follows. Example 2 This example was constructed so that the marginal means and the overall means are the same as in Example 1. Using the definition of interaction: How we do this, in what order, and how do we interpret these tests? Once the mean squares are known the test statistics can be calculated. For example, the test statistic to test the significance of factor or the hypothesis can then be obtained as:.
Similarly the test statistic to test significance of factor and the interaction can be respectively obtained as:. It is recommended to conduct the test for interactions before conducting the test for the main effects. This is because, if an interaction is present, then the main effect of the factor depends on the level of the other factors and looking at the main effect is of little value.
However, if the interaction is absent then the main effects become important. Consider an experiment to investigate the effect of speed and type of fuel additive used on the mileage of a sport utility vehicle. Three speeds and two types of fuel additives are investigated.
Each of the treatment combinations are replicated three times. The mileage values observed are displayed in the table below. In the figure, the factor Speed is represented as factor and the factor Fuel Additive is represented as factor. The experimenter would like to investigate if speed, fuel additive or the interaction between speed and fuel additive affects the mileage of the sport utility vehicle. In other words, the following hypotheses need to be tested:. In order to calculate the test statistics, it is convenient to express the ANOVA model of the equation given above in the form.
This can be done as explained next. Since the effects , and represent deviations from the overall mean, the following constraints exist. Therefore, only two of the effects are independent. Assuming that and are independent,.
The null hypothesis to test the significance of factor can be rewritten using only the independent effects as. The DOE folio displays only the independent effects because only these effects are important to the analysis.
The independent effects, and , are displayed as A and A respectively because these are the effects associated with factor speed. Therefore, only one of the effects are independent. Assuming that is independent,.
The null hypothesis to test the significance of factor can be rewritten using only the independent effect as. The independent effect is displayed as B: B in the DOE folio. The last five equations given above represent four constraints, as only four of these five equations are independent.
Therefore, only two out of the six effects are independent. Assuming that and are independent, the other four effects can be expressed in terms of these effects. The null hypothesis to test the significance of interaction can be rewritten using only the independent effects as.
Since factor has three levels, two indicator variables, and , are required which need to be coded as shown next:. Factor has two levels and can be represented using one indicator variable, , as follows:. The interaction will be represented by all possible terms resulting from the product of the indicator variables representing factors and. There are two such terms here - and. The vector can be substituted with the response values from the above table to get:.
Knowing , and , the sum of squares for the ANOVA model and the extra sum of squares for each of the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since five effect terms , , , and are used in the model, the number of degrees of freedom associated with is five. The total sum of squares, , can be calculated as:. Since there are 18 observed response values, the number of degrees of freedom associated with the total sum of squares is The error sum of squares can now be obtained:.
Since there are three replicates of the full factorial experiment, all of the error sum of squares is pure error. This can also be seen from the preceding figure, where each treatment combination of the full factorial design is repeated three times.
The number of degrees of freedom associated with the error sum of squares is:. The sequential sum of squares for factor can be calculated as:. Since there are two independent effects , for factor , the degrees of freedom associated with are two.
Similarly, the sum of squares for factor can be calculated as:. Since there is one independent effect, , for factor , the number of degrees of freedom associated with is one. The sum of squares for the interaction is:. Since there are two independent interaction effects, and , the number of degrees of freedom associated with is two.
Knowing the sum of squares, the test statistic for each of the factors can be calculated. Analyzing the interaction first, the test statistic for interaction is:. The value corresponding to this statistic, based on the distribution with 2 degrees of freedom in the numerator and 12 degrees of freedom in the denominator, is:.
Assuming that the desired significance level is 0. In the absence of the interaction, the analysis of main effects becomes important. The test statistic for factor is:. The value corresponding to this statistic based on the distribution with 2 degrees of freedom in the numerator and 12 degrees of freedom in the denominator is:.
Therefore, it can be concluded that speed and fuel additive type affect the mileage of the vehicle significantly. Results for the effect coefficients of the model of the regression version of the ANOVA model are displayed in the Regression Information table in the following figure.
Calculations of the results in this table are discussed next.
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