Data Reduction Introduction Once you have designed an experiment, collected the data, and begun thinking about how to communicate the results to other people, your next step will almost always be to construct some kind of graphical representation of your data. In some cases, deciding what to plot on graphs will be fairly straightforward. More often than not, however, it will not be so obvious what the most effective representation might be. In perception experiments, it is common to investigate the effects of more than one variable at a time and to obtain several measurements per condition. Although we won’t be running such extensive experiments in this class, some researchers manipulate several variables at once and collect hundreds of measurements per condition. But even if we restrict ourselves to fairly simple designs and only ask our observers to make five or ten estimates for each type of trial, you can see that there are some fundamental decisions we must make when constructing our graphs. For one thing, should we plot all the data for each subject (that is, all the measurements per condition), or a condensed version of it? If we condense it, how should we convey how spread out the data are? Should we show individual subject data, or average across subjects? If there is more than one independent variable, which one should we plot on the x-axis? To illustrate and discuss some of these issues, we will refer to hypothetical data from a magnitude estimation experiment investigating the Müller-Lyer Illusion. An Extended Example: The Müller-Lyer Illusion Imagine that we ran an experiment to investigate the Müller-Lyer illusion and that we manipulated two independent variables: arrow angle and line length. Below is a set of hypothetical data from that experiment: In column A, you can see that there are three values of line length (80, 100, and 120). Likewise, there are three values of arrow angle (45, 90 and 135 degrees) in column B. For each combination of line length and arrow angle, the subject made four estimates for each condition. These estimates show up in columns C through F. Finally, the means and standard errors of the mean (SEM) across the four estimates are shown in columns G and H, respectively. Because there are two independent variables, line length and arrow angle, we can make graphs with either of these variables plotted on the x-axis. (In principle, we could plot all the raw data in these graphs—that is, each of the four estimates made for every combination of line length and arrow angle. However, for the purposes of this handout, we will only plot the mean and SEM for each condition. SeePsychology 0044 Data Reduction Page 2 the SEM handout for a more in-depth discussion of this issue.) If we plot the mean data in column G above, this is what the two graphs could look like: 1301201101009080706080100120140Arrows: <>Arrows: | |Arrows: >< Figure 1a:Plot by Line LengthPhysical Line Length (pixels)Mean Magnitude Estimate1401201008060406080100120140Length: 80Length: 100Length: 120 Figure 1b: Plot by AngleArrow Angle (degrees) Which of these two figures would be most informative in communicating the results of the experiment? The answer is, “it depends”. Each figure empasizes a different aspect of the data according to which independent variable is plotted on the x-axis. So, before choosing which figure would be most informative, we must first think about which results we feel are the most interesting. Figure 1a plots magnitude estimates as a function of line length. The lines have a positive slope, which indicates that as we increase line length, magnitude estimates of line length also increase. This result is not too surprising, but if you really were interested in how estimates of line length changed with changes in physical line length, this would be the graph to present in a paper. Figure 1b plots magnitude estimates as a function of arrow angle. The lines connecting the plot symbols show the effect of changing arrow angle while holding line length constant. The positive slope of these lines shows that even when we kept the physical line length constant, the subject made larger or smaller magnitude estimates depending on what the arrow angle was. This is the real heart of the Müller-Lyer illusion—we can make lines look longer or shorter just by putting arrows at different angles on the ends of the lines. If this is the aspect of the data you want to emphasize, Figure 1b would be the graph to present in a paper. Thus, you can see that when there is more than one independent variable, we can emphasize different aspects of the data simply by putting one variable or the other on the x-axis.Psychology 0044 Data Reduction Page 3 Averaging Across an Independent Variable If we decide, then, that we want to emphasize the effect of angle, would it be “legal” to summarize our data one step further by averaging across line length, and produce a figure that has only a single line? You probably can tell that we must be very careful when summarizing data—the goal is NOT to sweep results we find uninteresting under the carpet, but rather to produce a summary that accurately reflects the data. When we’re deciding whether or not to average across line lengths in this example, we must first assess how strongly line length influences the effect of arrow angle. In other words, do we need to know what line length was presented in order to understand the effect of arrow angle? If arrow angle only affects magnitude estimates for certain values of center line lengths, we should not collapse across line length. (You might recognize here that there is a statistical test we can perform to find out whether or not there is a significant interaction between arrow angle and line length—the analysis of variance, or ANOVA. If the variables interact, we should not collapse across one of the variables. For the purposes of this class, however, we will perform a visual test for interactions by looking at graphs of the data.) Looking at Figure 1b above, we find that arrow angle had about the same effect on magnitude estimates regardless of the line length. That is, as we increase the angle from 45 to 135 degrees, the mean magnitude estimates increase by roughly 20 units, and this is true for all three line lengths (80, 100, and 120). The result is that the lines connecting the three plot symbol types are