a.. = ( a11 + a12 + ... + ABC) / BC. Where a=1, B=3, and C=2, 1.. = ( 111 + 112 + 121 + 122 + 131 + 132 ) / 6 = ( 5 + 5 + 5 + 6 + 7 + 8 ) / 6 = 6. Likewise, 2.. = 6 and there would be no main effect of A, because these values are similar. Multivariate Statistics: Concepts, Models, and Applications
David W. Stockburger
This chapter will focus on four designs that serve the same function, to test the effects of three factors simultaneously. The designs that will be studied include:
S X A X B X C
S ( A ) X B X C
S ( A X B ) X C
A X B X C
Since the naming of the factors is arbitrary, these designs include all possible three factor designs. In a departure from the last few chapters, the similarities of these designs will first be studied, followed by the differences. The advantages and disadvantages of each will be then be presented.
The function of the four designs given above is to test for the reality of three kinds of effects, main, two-way interaction, and three-way interaction. Although the first two have been described in detail in earlier chapters, the different forms of the effects will be discussed. The three-way interaction will be discussed in detail.
A study of effects begins with a table of means. This table might be constructed by averaging over subjects in any number of ways, depending upon the design. An example follows.
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a1 |
6 |
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7 |
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6 |
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5 |
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5 |
5 |
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a2 |
7 |
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6 |
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Main effects are found in a manner analogous to finding main effects in a two factor design, except that the data must be collapsed over two other effects rather than one. In the case of a three factor experiment, there will be three main effects, one for each factor, A, B, and C.
For example, in order to find the main effect of factor A, one must find the mean of each level of A, collapsing over levels of B and C. In the above example a.. = ( a11 + a12 + ... + ABC) / BC. Where a=1, B=3, and C=2, 1.. = ( 111 + 112 + 121 + 122 + 131 + 132 ) / 6 = ( 5 + 5 + 5 + 6 + 7 + 8 ) / 6 = 6. Likewise, 2.. = 6 and there would be no main effect of A, because these values are similar.
In a like manner, .1. = .2. = .3. = 6 and there would be no main effect of B. From the table above it can be seen that ..1 = ..2 = 6 and there would be no main effect of C. Thus, this table is an example of a three factor experiment where no main effects would be found.
Each combination of two factors produces a two-way interaction by collapsing over the third factor. The three two-way interactions are interpreted just like the single two-way interaction would be in an A X B design.
By collapsing over the C factor, the AB interaction yields the following table and graph. Note that an AB interaction is present because the simple main effect of B does changes over levels of A, in one instance increasing with B and the other decreasing. This table also clearly illustrates the lack of an A or B main effect.
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5.5 |
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6.5 |
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6.5 |
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5.5 |
6 |
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6 |
By collapsing over the B factor, the AC interaction produces the following table and graph. The cells in the table reproduce the numbers which appeared as row means in the full table. In this case there is an AC interaction present.
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By collapsing over the A factor, the BC table and graph are produced. The numbers in the graph appear as row means on the separate tables in the original data. In this case the interaction is absent.
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The three-way interaction, ABC, is a change in the simple two-way interaction over levels of the third factor. A simple two-way interaction is a two-way interaction at a single level of a third factor. For example, going back to the original table of means in this example, the simple interaction effect of AB at c1 would be given in the means in the left-hand boxes. The same simple interaction at c2 would be given in the right-hand boxes.
A change in the simple two-way interaction refers a change in the relationship of the lines. If in both simple two-way interactions the lines were parallel, no matter what the orientation, there would be no three-way interaction. Similarity, if the lines in the simple two-way interactions intersected at the same angle, again no matter what the orientation, there would be no three-way interaction.
In the case of the example data, graphed below, the orientation of the lines comprising the simple interactions changes from parallel to non-parallel from one graph to the other. In this case a three-way interaction would exist. It may or may not be significant depending upon the size of the error term.
The following table of means was constructed such that all effects would be significant, given that the error terms were small relative to the size of the effects.
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The A X B interaction in table and graph form follow:
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4.5 |
5 |
5.5 |
5 |
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6.5 |
6.5 |
6.5 |
7 |
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5.5 |
5.75 |
6 |
6 |
The same is now done for the A X C interaction:
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c2 |
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a1 |
5 |
5 |
5 |
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a2 |
8 |
5 |
6.5 |
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5.5 |
5 |
5.75 |
In a similar fashion the table and graph for the B X C interaction:
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5.5 |
6.5 |
7.5 |
6.5 |
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5.5 |
5 |
4.5 |
5 |
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5.5 |
5.75 |
6 |
5.75 |
Finally the graph of the three-way interaction is given.
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The presence of a B main effect and the lack of an A main effect and AB interaction is seen in the following table and graph.
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5.5 |
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6.5 |
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5.5 |
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6.5 |
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5.5 |
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6.5 |
6 |
The AC interaction is seen in the following.
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5.5 |
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6 |
The ABC three-way interaction is not significant because the simple interaction of AB does not change over levels of C. In this case the lines are parallel in both cases.
Because a three-way interaction does not always appear as intuitive to students, two additional examples three-way interactions are now given. In the first case, the three-way interaction is not significant because the relationships between the lines in the simple interactions do not change. In the second example, only the three-way interaction is significant.
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The reader should verify that in the above example there might be a significant main effect of B, an AB interaction, and an AC interaction, but no other effects could be significant.
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In the above example, only the three-way interaction could be significant. There could be no other significant effects.
In its full form, the experiment designed to test the Lombard effect in choral singers neatly fulfill this design. Factor A was choral singing experience - None, Some, or Lots. Each subject sang the national anthem twice, Factor B, the first time with no instructions and the second with instructions to try to control vocal intensity. Singing loudness (in decibels) was measured at three different points (Factor C) in the song.
The data has been presented in its full form in a previous chapter (Design SXAXB), so it is not necessary to present it again. The MANOVA commands necessary to do the analysis is given below. Note that the only difference between this set of commands and the set of commands do the SXAXB analysis the BY EXPER (1,3) on the first line following the MANOVA command.
The means and standard deviations are presented first.
The results of the hypothesis tests from the preceding set of commands is presented below in its full form.
From the preceding analysis the results are reasonably clear. Main effects were found for the INSTRUCT and PLACE factors. When instructed, the singers sang with less intensity. In different places in the song the singers also sang with different intensity levels over both levels of instructions.
The significant interactions of EXPER BY PLACE and INSTRUCT BY PLACE are best seen in the three-way interaction, even though the three-way interaction was not significant. The graph of the EXPER BY INSTRUCT BY PLACE interaction is presented below.
From the above graphs it can be seen that the more experienced the singer, the more loudly he or she sang. In addition, experienced singers sang relatively more loudly than inexperienced singers at song positions 2 and 3. Also, the instructions to sing at an even level seemed to work best at position 2, combined over all groups.
For an explanation of what all this means, it is perhaps best to refer to the original author (Tonkinson,1990, p. 25)
The original problem was: Is the Lombard effect, to a significant degree, and unconscious response in choral singing at different levels of experience and training, and can it be consciously avoided. Most of the choral singers in this study, regardless of experience, tended to succumb to a Lombard effect when faced with increasing loss of auditory feedback. They were, however, able to control the level of vocal intensity with some brief instructions. It appears that simple training in awareness would be enough for a member of an amateur choir to begin regulating the intensity of their voice in a healthy manner.