** Line parallelism, graph **1

**LINE PARALLELISM, GRAPH MEMORY, AND DETECTING STATISTICAL INTERACTIONS**

*(Article in press; to be published in: Theoretical Issues in Ergonomics Science)*

Karel Hurts

April, 2009

This study attempts to explore in more detail than was done before:

1. ohw How behavioural effects of common line and bar graphs (especially when performing more complex, integrated tasks), including an unusual type of line graph, can be predicted using a framework for cognitive task analysis.

2. ow Thhhe How performing tasks involving the detection of statistical interaction in a visual style may be supported by various formats for data graphing.

3. ow ho

How variations in task structure and complexity may have effects on performance independent of the effects of graphical format and how these two types of effect may interact with each other.

Two types of line graph and one type of bargraphwere compared in two experiments,requiring participants to answer integrated questions from memory. Questions were visual in nature and were related to the concept of statistical interaction. Both types of line graphcontained the emergent feature of line parallelism, assumed to facilitate the task of detecting interactions. One type was unusual: it contained two separate, orthogonal value axes, instead of a single, integrated axis. In Experiment I, questions were relatively simple. Its results were fitted to a simple model of cognitive task execution. In Experiment II, the graphs were introduced as presenting the outcomes of an imaginary research study.In both experiments the line graphs resulted in superior performance, though memorizing the unusual line graphs remained difficult, despite extensive pre-experimental training. The discussion focuses on the importance of graph-related and task-relatedvariables in explaining performance on graph-based, memory-based tasks.

INTRODUCTION

In this article we will address the question how onetype of emergent feature (i.e., line parallelism) and one feature of graph syntax (i.e., integrated value axes) can be combined in simple line and bar graphs to affect people´s accuracy and speed in answering integrated questions about these graphs from memory. Specifically, the questions require participants to detect statistical interactions in these graphs. Participants only have to look at the (memory) image of a graph, whence no formal testing of interactions is required.

Emergent features are graph properties that visualize higher-order data patterns present in the plotted data which can be used for more rapid and more accurate data extraction(Garner, 1974; Pomerantz, 1981).Because of the importance of geometrical data patterns in these graphs, they are also called configuralgraphs. Line parallelismrefers to degree of parallelism that emerges from a particular way of plotting lines in a line graph. Often, this parallelism can be used in order to detect higher-order data patterns, e.g., assessing the presence of statistical interaction in line graphs between two independent variables (see the examples of Figure 4Figures 1 and 2).

Integratedvalue axes refers to the standard way of graphing data, using a single, commonvalueaxis if the data are measured on the same continuous scale. When talking about On the other hand, non-integrated value axes we refers to the (rather uncommon) use of more than one (usually two (or more) orthogonal) axes to depict values even if they are measured on identical scales. Mostly, this is done in order to visualize certain patterns in the data, for example to show differences or commonalities between data collected under different conditions. For example, Figure 4Figures 1and 3 contains an single integrated value (Y) axis, whereas and Figure 6Figure 2 contains two non-integrated axes for identical scales,each axis corresponding to a different condition of observation., though the two axes represent identical scales.

Insert Figures 1, 2, and 3 about here

Not much is known about the detailed cognitive steps people go through when they use common graphs for answering questions that also require mental integration of data values(Carswell, 1992; Ratwani, Trafton, & Boehm-Davis, 2008). Even less is known about these cognitive steps under memory-based circumstances (also referred to as retrospective viewing conditions) than under concurrent (direct) viewing conditions (for more details, see below under *Review of past research on configural displays and integrated value axes*).

Therefore, one objective of this study is to look more closely at various cognitive factors underlying the performance on relatively complex graph-based tasks under memory-based conditions. To this end, cognitive task analysiswill be is used in which the cognitive steps needed to perform a specific task using a specific graph will be are described in detail.

In the present study, we will use common, static graphs (bar graphs and line graphs as used in the sciences and in social studies) and questions requiring the detection of statistical interactions (or a visual analog of this such a task) by remembering the graphsed values.Each

question that is to be answered (varying from trial to trial in a semi-random way) will be is presented in the test phase of a trial, after the corresponding graphhas been studied and memorized in the memorization phase of the same trial.

One type of graph which we will study has annon-traditional syntax and appearance, i.e., has non-integrated value axes (see the example of Figure 6Figure 2). Therefore, this study is also meant to explore the possibility to design new types of graph for detecting statistical interactions,informed by the outcomes of cognitive task analysis and experimentation.

We assume the graphs of our experiments are not presented and memorized long enough for transfer to long-term memory to take place, but are presented and memorized long enough (5-15 s) to reach visual short-term memory. As a result, the outcomes of this study do not generalize to longer retention intervals.

**Relevance of graph-based task performance under memory-based conditions**

Modern information society increasingly requires us to attend to multiple information sources, switching from one source to another, and returning to previously attended sources if information-related tasks are waiting to be completed. This interrupt-driven type of performance places an increasing load on human (short-term and long-term) memory (McFarlane& Latorella, 2002).

For example, during a conference presentation reporting an empirical research study, several related statistical graphs may be presented in rapid succession. This places ough placing a high demand on the short-term memoriesy of the people in the audience.,Of course, this successive style of presentation has the advantage of synchronizing focusing the visual attention of the audience with the oral part of the presentation.on important information at the right time.

Obviously, from a practical point of view, the use of displays and graphs in a concurrent fashion (i.e., all important, related information is present and visible simultaneously) is to be preferred, whenever possible.

**REVIEW OF PAST RESEARCH ON CONFIGURAL DISPLAYS AND **

**INTEGRATED VALUE AXES**

Under concurrent viewing conditions configural graphs have been shown to be beneficial for integrated tasks most of the time (Wickens& Carswell, 1995; Bennett & Flach, 1992; Bennett & Walters, 2001), provided the emergent features somehow correspond to a domain constraint or information need. Integratedtasks are tasks in which the viewer must combine most or all of the lower-order graphical information. The happy marriage between integrated tasks and configural displays has also been captured by the notion of proximity compatibility (Wickens & Carswell, 1995).

There is less certainty about the effectiveness of configural graphs under retrospective viewing conditions. However, as indicated by the Table 1 studies, the evidence points largely in the direction of better usefulness for configural displays than for non-configural (separable) displays.

Insert Figures 1Table 1 about here

Previous studies have found evidence for the special, emergent, character of the feature of line parallelism and associated features, such as slope patterns in multi-line graphs (Pomerantz & Pristach, 1989; Carpenter & Shah, 1998).Therefore, it makes sense to investigate the extent to which these features facilitate the answering of complex, integrated questions about graphs.

The advantage of having integrated (common) value axes has long been known in the literature on graph design (Kosslyn, 1989; Cleveland, 1985), because they facilitate pair-wise comparison of values plotted on the same axis. They can be said to yield the a kind of emergent features of height and lengthwith which task that can facilitate the performance of this task can be made easier. For example, the heights of the bars in Figure Figure 32 directly correspond to numeric values indicated by the single value axis Y, which facilitates pair-wise comparison of these values. In contrast, in Figures23and contains 6 examples are shown aof line graphs using two separable value axes (X-axis and Y-axis). , instead of a single value axis. In a previous study,it was shown these graphs were more difficult to use for answering integrated questions under concurrent viewing conditions than more traditional line graphs having a single value axis (Hurts, 1998).

On the other hand, uUsing integrated value axes also has disadvantages: for example, it is not easy to see that the second and fourth bar in Figure 5Figure 3both correspond to the condition “high function”. One needs to read the labels printed along the X-axis (below the bars)in order to understand this. In contrast, in the graph of in Figure 6Figure 2 (which uses separable value axes) this commonality can be seen directly, because the two values corresponding to “high function” are “bound” together by the same data point.

**DESIGNING THE GRAPHS**

In this section we will use the simpler graphs of The two graphical properties studied in this article can be illustrated by referring to the graphs of Figures 1Figures 4-63 as illustrations, because these will be used in the cognitive task analysis and in Experiment I, both to be discussed later. We started out with the task of having pParticipants were always asked to judge(from their memory)the extent to which a difference between two plotted values is smaller than, larger than, or equal to the difference between two other values. For example, participants might be asked to answer the following question about the graph shown in Figure 1: Is the difference between A and B in Figure 1 larger than that between C and D?(In Experiment II, similar graphs and questions were used, but both were presented in terms relevant to the task of visually detecting statistical interactions.)

Notice that the graphs shown in Figure 1 Figures 4 and Figure 3 5 share several emergent features related to line parallelism(line convergence,line divergence, and line intersection).Therefore, we call Figure 1 Figure 4 an example of a 2-line configural graph and Figure 3 Figure 5 an example of a 1-line configural graph.Noticee that the graph shown in Figure 2 Figure 6 does not have such features, at least not to the same extent (see below for a check of this assumption). Therefore, we will call this graph an example of a separable graph. Line parallelism defines the first graphical property studied in this article (present in Figures 1 Figures 4 and 53, absent in Figure 62).

Insert Figures 1Figures 4, 52, and 63 about here

The 1-line configural graph shown in Figure 3 Figure 5 differs from both Figure 1 Figure 4 and Figure 2 Figure 6 in the sense that it uses two(instead of one) axes for plotting values of the same dependent variable, the two axes corresponding to different conditions of observation.Therefore,we will say this graph contains*non-integrated (or separable) value axes*, and the graphs in the other two figures are said to contain *integratedvalue axes.Axis integration*defines the second graphical property of this article(present in Figures 1 Figures 4 and 62 and absent in Figure 3Figure 5).

In summary, the design of the study enabled us to make the following three comparisons among between the three types of graph:

- The comparison between 1-line configural and 2-line configural graphs allowed an estimation of the size of the effect of axis integration.
- The comparison between 2-line configural and separable graphs allowed an estimation of the size of the effect of line parallelism.
- The comparison between 1-line configural and separable graphs allowed an estimation of the size of the
*combined effect of both axis integrationandline parallelism*.

Note that thisdesign did not allow us to obtain information about the way these two display factors interact in a statistical sense. This was caused by the fact that we were not able to (easily) design separable graphs not having integrated axes. Therefore, the two display factors, considered as independent variables, were not completely crossed in the experimental design.

**COGNITIVE TASK ANALYSIS **AND PREDICTIONS**OF GRAPH COMPREHENSION**

Though the features of line parallelism and value axis integration have been investigated before separately, not much research attention has been paid so far to their combined presence in graphs and the way in which they both determine the same task performance. This justifies an overall approach to studying these features and their effects,which we will provide below in the form of a cognitive task analysis (CTA).As will be seen, this analysis will only be conducted at a global, functional level, to the point of allowing us to derive predictions of the relative (empirical) effectiveness of the three graph types for answering the integrated question mentioned in the previous section. These predictions will be tested in Experiment I.

**Analysis framework**

From an analysis of the literature on graph comprehension (Lohse, 1993; Gillan & Lewis, 1994; Ratwani et al., 2008), and fromtwo previous empirical studiesusing similar graphs and questions (Hurts, 1998; Hurts & Van Leeuwen, 1998), we arrived at the list of ten elementary operations listed mentioned in Table 2. For example, evidence for the psychological relevance of operations such as DI (determine intersection), ELP (establish line parallelism), and ESP (establish slope pattern) can be derived from previous research into the importance of these features in controlling human pattern recognition and graph perception (Pomerantz & Pristach, 1989; Carpenter & Shah, 1998; Kimchi, & Bloch, 1998).

We assume most questions extracting and integrating information from simple line and bar graphs (such as the ones used in this study) can be answered by applying these operations sequentially. Each operation may occur more than once in each sequence, each time instantiated to a different part (point, line, text label) of the graph.

It can be seen that six of the ten operations pertain to visual arithmetic (visually estimating and comparing line slopes; comparing horizontal or vertical orientations of data points; determining whether lines intersect or not).This was done in order to better capture the visual nature of higher cognitive processes that are invoked when more complex., integrated questions are answered of graphs (Ratwani et al., 2008).

Additionally, each operation acts on certain graphical input information and/or computational information obtained in previous steps. In turn, it generates itself computational information with which to control future processing steps.

Finally, based on the literature about visuospatial images (Reisberg & Heuer, 2005),we assumed the memories of the graphical input resemble the graphical input when this is perceived directly.operations would operate on a kind of “memory image” of the graph in a way not much different from the way in which graph reading behavior is realized when the graph is continuously available for direct perception.

We distinguished the following types of operation:visual search, mental arithmetic (including logical reasoning), and visual arithmetic. Table 2 lists the proper classification of each operation.

It can be seen that six of the ten operations pertain to visual arithmetic (visually estimating and comparing line slopes; comparing horizontal or vertical orientations of data points; determining whether lines intersect or not).This reflects the importance of the visual nature of higher cognitive processes that are invoked when answering more complex questions about graphs (Ratwani et al., 2008).

The mental and visual arithmetic types of operation correspond to the component processes called *arithmetic operations and spatial comparisons* in the Mixed-Arithmetic Perceptual (MA-P) model (Gillan & Lewis, 1994). However, as can be seen, some of the operations listed in Table 2 capture more detailed aspects of the graph reading behavior. F, for example, by differentiating between the operation of detecting an intersection point (DI) and the operation of visually comparing the heights of two data points (DVO) both are examples within the group of visual arithmetic operations. In addition, unlike the five component processes identified in the MA-P model, our the operations of our CTA-framework are, to a certain extent, also graph-dependent. For example, the operation of detecting an intersection point (DI) and that of comparing line slopes (ESP) only apply to line graphs.

Insert Tables 2 and 3 about here

**Task and stimulus constraints**

The following task and stimulus constraints applied to the CTA we conducted. (Theyse also held for the design of Experiment I. Later, in the discussion of this experiment, we will come back to the plausibility of the assumption underlying the last constraint.)

- The labels A-D appeared at arbitrary locations in the general question
*“Is the difference between A and B larger than the difference between C and D?”*. - For the 2-line configural graphs the data belonging to each pair in the question either belonged to the same data column (column-wise way of organizing pairs) or belonged to the same line-segment (line-segment-wise way of organizing pairs). For example, in Figure 1 Figure 4 the pairs A,B and C,D were organized by line segment.
- For the 1-line configural graphs the paired data were organized either by axis or by data point. For example, in Figure 3 Figure 5 the pairsA,B and C,D were organized by axis.
- For the separable graphs (for example, see Figure 62) the data values (corresponding to bar heights) always appeared in the same sequence, indicatedby the labels A, B, C, and Dprinted along the X-axis.
- Participants were asked to compare the absolute (rather than the algebraic) differences mentioned in the question.
- If the lines in the 2-line configural graphs were intersecting, the participant wasere asked to compare differences across data columns (e.g., points A - and C vs. constitute a data column inC-D in Figure 41). In that case, the corresponding 1-line configural graphs would have the differences only organized by data line-end point (e.g. values A - and C vs. C – D in Figure 53). are bound to the same data point). The same is true if the lines in the 2-line configural graphs had different slope signs (one line descending and the other line ascending, going from left to right).
- Questions were not presented (and time measurements were not started) until the participant had indicated the graph had been seen and memorized. Therefore, we decided to leave out the “graph-orientation” phase from the cognitive analysis of the participant’s question-answering behavior.

7.All graphs had been memorized by the time the corresponding question was presented (retrospective viewing). Therefore, for question answering it was not necessary to first perceive (or remember) the graph in an overall sense, as this had already been done in the memorization phase immediately preceding the question answering phase.