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Figure 2.8 Scatter plot matrix of meta-analysis data from Stangor and McMillan (1992).

and have been since corrected. In this case, the graphics revealed structure and avoided error.

Going Deeper

Although the scatter plot matrix is valuable and informative, it is important that the reader recognize that a series of two-dimensional views is not as informative as a three-dimensional view. For example, when Stangor and McMillan computed a simultaneous regression model, the variables indicating the number of targets and traits used in each study reversed the direction of their slope, compared with their simple correlations. Although a classical "suppressor" interpretation was given, the exploratory analyst may wonder whether the simple constant and linear functions used to model these data were appropriate. One possibility is that the targets variable mediates other relationships. For example, it may be the case that some variables are highly related to effect size for certain levels of target, but have different relationships with effect size at other levels of targets.

To provide a quick and provisional evaluation of this possibility, we created a histogram of the target variables, selected those bins in the graphic that represent low levels of targets, and chose a unique color and symbol for the observations that had just been selected. From here, one can simply click on the pull-down menu on any scatter plot and choose "Add color regression lines." Because the observations have been colored by low and high levels of the target variable, the plots will be supplemented with regression lines between independent variables and the effect size-dependent variable separately for low and high levels of targets, as displayed in Figure 2.9.

Moving across the second row of Figure 2.9 (which corresponds to the response variable), first we see two regression lines with low identical slopes indicating little relationship between task and effect, which is constant across levels of target. The delay variable in the next column shows a similar pattern, whereas the next three variables show small indications of interaction. The interaction effect is very clear in the relationship between effect size and the congruent-incongruent ratio in the rightmost column. This relationship is positive for

Figure 2.9 View of computer screen using Data Desk software for selecting, brushing, and linking across multiple plots. Several plots have been enhanced with multiple regression lines that vary by subsets of selected data.

observations with high numbers of targets, but negative for low numbers of targets. Unfortunately, in failing to recognize this pattern, one may use a model with no interactions. In such a case the positive slope observations are averaged with the negative slope observations to create an estimate of 0 slope. This would typically lead the data analyst to conclude that no relationship exists at all, when in fact a clear story exists just below the surface (one variable down!).

Although the graphics employed so far have been helpful, we have essentially used numerous low-dimensional views of the data to try to develop a multidimensional conceptualization. This is analogous to the way many researchers develop regression models as a list of variables that are "related" or "not related" to the dependent variable, and then consider them altogether. Our brushing and coding of the scatter plot matrix has shown that this is a dangerous approach because "related" is usually operationalized as "linearly related"—an assumption that is often unwarranted. Moreover, in multidimensional space, variables may be related in one part of the space but not in the other.

Working in an exploratory mode, these experiences suggest we step back and ask a more general question about the meta-analytic data: In what way does the size and availability of effects vary across the variety of study characteristics? To begin to get such a view of the data, one may find three-dimensional plots to be useful. A graphic created using a nonlinear smoother for the effect size of each study as a function of the number of targets and presentation speed is presented in panel A of Figure 2.10. The general shape is similar to the "saddle" shape that characterizes a two-way interaction in continuous regression models (Aiken & West, 1991). The graphic also reveals that little empirical work has been undertaken with high presentation speed and a low number of targets, so it is difficult to assess the veracity of the smoothing function given the lack of data in that area. At a minimum, it suggests that future research should be conducted to assess those combinations of study characteristics. Panel B of Figure 2.10 shows an alternate representation of the data with a traditional linear surface function that is designed to provide a single additive prediction across all the data.

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