SINGLE SYSTEM RESEARCH DESIGN

Interpreting Results of Single-Subject Research Designs

MARK WOLERY and SUSAN R. HARRIS

Although single-subject research design has been discussed at some length in the recent physical therapy and occupational therapy literature, there has been little attempt to describe the procedures used for interpreting the experimental effects of such research. The purpose of this paper is to present the use of two strategies for determining whether changes have occurred as a result of exper- imental manipulation. The first strategy, visual analysis of graphed data, relies on visual interpretation of changes in data patterns both within and between experimental conditions. The second strategy is the use of statistical procedures such as trend estimation, the Rn statistic, and time series analysis. Finally, the issues of both clinical and statistical significance in interpreting the results of single-subject research are discussed.

Key Words: Research design, Single-subject research, Data analysis.

Single-subject research seeks to explore the effects of independent variables on the behavior of individ- ual subjects. The designs used in single-subject re- search share three common characteristics: 1) mea- surement of the dependent variable (the subject's performance) continuously and repeatedly across time, 2) comparison of the subject's performance during treatment to his past performance during base- line measurement, enabling the subject to serve as his own control, and 3) establishment of evidence of treatment effectiveness or believability through rep- lication.1 Through the use of repeated measures taken over the course of experimental conditions, single- subject research allows the experimenter to examine changes in the data as they occur and thus may be considered "process research."2 In an effort to gen- eralize about the results of such research, the experi- ments may be replicated on other individual subjects, on the same subject across different settings, or on the same subject with various therapists or intervention- ists.

Recently, several authors have proposed the use of single-subject research for evaluation of the effects of physical therapy and occupational therapy.3"5 While these authors present accurate descriptions of the various single-subject designs and rationales for their use, they largely ignore the importance of graphically demonstrating and interpreting the results of such

research. Without adequate and accurate interpreta- tion of experimental results, the collection of data and selection of design are relatively meaningless. The purpose of this article is to address the crucial process of interpreting the results of single-subject research. We will present two strategies for determining whether changes have occurred. Particular attention is given to the first strategy described, the visual analysis of experimental data. Secondly, specific sta- tistical procedures for evaluating single-subject data are presented. Finally, the issues of both clinical and statistical significance in interpreting the results of single-subject research are discussed.

INTERPRETATION OF EXPERIMENTAL EFFECTS

Single-subject research involves the collection of data on a single subject under standard conditions for a period of time. Individual data points are frequently interdependent. Given several data points on a sub- ject's behavior, the investigator often can predict later data points with some degree of accuracy. This phe- nomenon is known as serial dependency. If present to a substantial degree, serial dependency violates the assumptions of independence made by most types of inferential statistics. Data may be tested for serial dependency, and, if it is absent, use of analysis of variance (ANOVA) or other similar statistics may be appropriate. Such testing, however, to be conserva- tive, requires a larger number of data points than are usually found in single-subject research.

Whereas ANOVA and other conventional statisti- cal procedures, such as t tests, evaluate differences in

Dr. Wolery is Assistant Professor, Department of Special Educa- tion, University of Kentucky, Lexington, KY 40506 (USA).

Dr. Harris is Instructor, Rehabilitation Medicine, Division of Physical Therapy, RJ-30, University of Washington, Seattle, WA 98195.

This article was submitted December 1, 1980, and accepted Septem- ber 14, 1981.

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Fig. 1. Hypothetical data of examples from single-subject research of a variable data pattern (Graph A), stable data patterns (Graphs B and C), change in level (Graph D), and change in trend (Graph E).

means of data between experimental conditions, it is the trends in data that are of primary concern in single-subject research. Trends are not evaluated by conventional statistical procedures. Thus, the recom- mendation is frequently made that single-subject re- searchers avoid the use of statistical procedures such as ANOVA.6 The existence of serial dependency and the importance of trends in single-subject research raise two important questions: 1) How should the results of single-subject designs be interpreted? and 2) What is the role of statistical procedures in single- subject research?

VISUAL ANALYSIS

Typically, single-subject researchers rely on visual analysis of graphed data for the interpretation of experimental effects. As with any interpretative pro- cedure, visual analysis must address two questions. First, Are there changes in the data patterns? Second, If changes do exist, do they correspond with the experimental manipulations? The experimenter must consider whether there are changes within experimen- tal conditions and between experimental conditions. In single-subject research, three types of changes may

be noted in data patterns. These changes are in vari- ability, trend, and level.

Changes Within Experimental Conditions

Dissimilarity of scores in a given experimental condition is known as variability. Variability of scores presents problems for both single-subject and tradi- tional group researchers. In group research, variabil- ity is controlled in two ways—by increasing sample size or by using statistical procedures such as analysis of covariance.7 Variability in single-subject research is controlled by searching for the sources causing the variability and then removing them.8 To control for single-subject variability, specific environmental var- iables may be identified and removed, and standard- ization of the measurement conditions may be in- creased. Hypothetical examples from single-subject research of variable and stable data patterns are displayed in Figure 1.

If the base-line data pattern is variable (Fig. 1: Graph A), it is unwise to implement treatment. The proper course of action is to search for the factors causing the variable performance. If the data pattern is unstable and treatment is implemented, interpre- tation of the experimental effects will be open to question and strong conclusions based on the data will not be possible.

If base-line data patterns are stable (Fig. 1: Graphs B and C), treatment may be implemented. Note that the data pattern in Graph C (Fig. 1) shows an increase in the behavior. If the treatment is designed to in- crease the rate of behavior, such as increasing the speed of independent ambulation by the patient, im- plementation of treatment should be withheld. The experimenter should wait until the behavior "levels o f f before implementing treatment. For example, if the physical therapist had been taking data daily on range of motion of knee flexion as a result of passive stretching and had achieved an increasing pattern, it would be unwise to implement an additional treat- ment technique, such as joint mobilization. The rea- son for this is that it may be impossible to determine whether a continued increase in range of motion was due to the passive stretching alone, or to the combi- nation of passive stretching and joint mobilization. Further, it should be noted that a base-line could be both stable and systematically decreasing. Such a pattern would be similar to that in Graph C (Fig. 1) but moving in the opposite direction.

If base-line data patterns show changes in level or trend (Fig. 1: Graphs D and E), treatment implemen- tation is not recommended. The proper course of action is to extend the base-line condition and attempt to identify change-producing factors in the data pat- tern. If such factors can be identified, and if the change produced was in the desired direction, those

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factors might be valuable as treatment procedures in and of themselves. If the changes are not in the desired direction, care should be taken to eliminate the possibility of the factors recurring.

Changes Between Experimental Conditions

Comparisons of data across experimental condi- tions are made in both large-group and single-subject research. In single-subject research, three types of change must be considered when comparing data across experimental conditions: changes in variability, level, and trend of data patterns. Figure 2 depicts data patterns that may be found across any two adjacent experimental conditions in single-subject re- search. Usually, single-subject researchers do not in- tend to produce changes that result in variability between conditions (Fig. 2: Graph A). In rare in- stances, however, introduction of change to stabilize a variable base-line is sometimes desirable (Fig. 2: Graph B). For example, if a subject is attempting to feed himself and performs inconsistently over a long base-line period, the experimenter may attempt a treatment such as providing adaptive equipment that would assist the subject in performing at a more stable rate. Changes in variability are usually difficult to interpret, however.

Level is defined as the relative value of the data pattern on the dependent variable. Changes in level represent changes in the value of the data series as measured on the dependent variable at the point of intervention (Fig. 2: Graphs C and D). Note that no change in trend occurs in Graphs C or D (Fig. 2).

Trend represents the direction in which the data pattern is progressing. A data series that is systemat- ically increasing or decreasing over time, even though it may be stable, is described as a trend. A change in trend is demonstrated by a change in the direction in which the data pattern is moving. Changes in trend are displayed in Figure 2: Graphs F, G, and H. In each case, there is no initial change in level when treatment conditions begin, but when the treatment is introduced the direction of the data pattern changes.

In some cases the base-line is increasing (Fig. 2: Graph I) when the treatment is implemented, but there is no change in either the level or trend of the data pattern. The pattern simply continues to progress in the same fashion as the base-line. This phenome- non is also shown in Graph E (Fig. 2).

Examples in which there are changes in level and trend following the introduction of treatment are found in Figure 2: Graphs J, K, and L. The base-line data pattern is level and stable in Graph J (Fig. 2), but the treatment results in a drop in level and a downward trend in the data pattern. In Graph K (Fig. 2), the base-line data pattern is increasing but

the introduction of treatment results in a drop in level and a leveling off of the trend. The base-line is level in Graph L (Fig. 2) and the treatment results in an increase in level and a change in trend.

When using visual analysis to interpret the effects of single-subject research, the experimenter must note whether there is variability in the data or changes in the level, trend, or both level and trend of the data pattern within experimental conditions. Variability or changes are not desired within experimental condi- tions and, if they exist, the internal validity of the research may be seriously threatened. If no such circumstances exist, the experimenter must determine whether there are changes between experimental con- ditions. An experimenter can conclude that a treat- ment is effective when three situations exist concur- rently. First, there must be either no changes or very minor changes within experimental conditions. Sec- ond, a clear change in level, trend, or both level and trend must occur when the treatment is introduced. Third, the changes between conditions must be rep- licated during additional phases of the experiment. Thus, based on data patterns such as those displayed in Graphs E and I of Figure 2, the experimenter must conclude that the treatment did not result in changes in the measured performance. It should also be noted that the use of only two experimental phases (base- line and treatment), such as shown in Figure 2, is not sufficient to conclude that a given treatment was

Fig. 2. Possible data patterns across two experimental conditions: Graphs A and B—changes in variability; Graphs C and D—changes in level; Graphs E and I—no change in data; Graphs F, G, and H—changes in trend; and Graphs J, K, and L—changes in level and trend.

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Fig. 3. Steps involved in calculating the split-middle line of trend estimation: Steps 3-5 (Graph A), steps 6-7 (Graph B).

effective or ineffective. These changes must be repli- cated in additional phases of the experiment.

ROLE OF STATISTICS IN SINGLE-SUBJECT RESEARCH

The use of statistical procedures is a controversial issue among single-subject researchers, ranging from considerable acceptance of such procedures to total rejection of their use in analyzing the effectiveness of single-subject research.9 We take the position that both visual analysis of graphed data and appropriate statistical procedures can serve as judgment aids that assist the experimenter in determining whether changes have occurred in the data, both within and between experimental conditions.

As noted above, when serial dependency is present, conventional statistical procedures are considered in- appropriate for analyzing the results of single-subject research. Certain statistical procedures have been de- veloped, however, that control for serial dependency and evaluate trends in data. This section will describe three such procedures: trend estimation, the Rn sta- tistic, and time-series analysis.

Trend Estimation Procedures

Several procedures have been described for calcu- lating estimates of trends in data. All trend estimation procedures are used to make judgments about changes in level and trend, but they do not produce statements of statistical significance. Examples of such procedures include the semiaverage method,10 the least squares method,10 the median slope proce- dure,11 and the split-middle procedure.12 The median slope and least squares procedures are complicated procedures requiring considerable space for descrip- tion and have been described in some detail in the literature.10,11 The predictive validity of the semiav- erage method is not known. None of these will be described here. The split-middle method of trend estimation is calculated easily (see below) without a computer and its predictive validity is known.13

Calculation of the split-middle method of trend estimation involves seven steps.12 These steps are described using the example presented in Figure 3. The split-middle is based on the medians of two halves of the data series for which the trend line is being determined.

Step 1. Plot the data on semilogarithmic graph paper.

Step 2. Count the datum points in the phase for which a trend line is being drawn, divide by 2, and add 0.5 to the quotient. For the data in Figure 3, there are 6 datum points. Dividing by 2 produces a quotient of 3 and the addition of 0.5 produces a final answer of 3.5.

Step 3. The final answer from Step 2 is used in Step 3. Count over from the left, beginning with the datum point nearest the ordinate, the number of datum points in the final answer of Step 2 and draw a hash line through the data series. In Figure 3, the experimenter would count over three datum points and draw a hash line halfway between the third and fourth datum points (ie, 3.5). If the answer from Step 2 were a whole number, such as 4, the experimenter would count over, from the left, that number of datum points and draw a hash line through the datum point equal to the whole number, in this case the fourth datum point. This datum point is ignored in calcula- tions of Steps 4 through 6. The result of Step 3 is a division of the data series into two equal sets (halves) of data.

Step 4. For each equal set or half of data from Step 3, calculate the mid-date and draw a dashed vertical line through it. The mid-date is calculated by repeat- ing the process described in Steps 2 and 3 for each half of the data (for each of the two sets in the data series). In Figure 3, for the first half of the data there are 3 datum points; 3 divided by 2 equals 1.5, 1.5 plus 0.5 equals 2. The experimenter would start with the first datum point on the left and count over two

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datum points and draw a dashed vertical line through the second datum point in the first half of the data. This process would be repeated for the second half of the data. In Figure 3, the mid-date for the first half of the data is the second day of data, and for the second half of the data it is the fifth day of data.

Step 5. Calculate the mid-rate for each half of the data series and draw a horizontal dashed line through the mid-rate that intersects with the dashed vertical line through the mid-date. The mid-rate is calculated by counting the number of datum points in each half, dividing by 2, and adding 0.5 to the quotient. (For Fig. 3: 3 2 = 1.5 + 0.5 = 2.) The answer, of course, is the same as in Step 4 for calculating the mid-date. To identify which rate is the mid-rate, the experimen- ter does not count datum points from the left of the graph, but rather counts them from the bottom or abscissa of the graph. Thus, for the first half of the data in Figure 3, the datum point on the first day of data is the mid-rate. When data occur that are similar to those in Graph A of Figure 4, the mid-rate is determined by the following rule. The datum point on which 50 percent of the datum points are equal to or lower than it and 50 percent are equal to or higher than it is the mid-rate. For example, in Graph A of Figure 4 there are seven datum points. Calculations as described above would result in an answer of 4, because 7 ÷ 2 = 3.5 + 0.5 = 4. The fourth rate is the mid-rate, which is also equal to the second, third, and fifth rates. This process is repeated for the second half of the data series. NOTE: Frequently, when calculat- ing the mid-date and mid-rate, the answer to calcu- lations will not be a whole number such as in Graph B of Figure 4. In this case, the mid-date and mid-rate (4 + 2 = 2 + 0.5 = 2.5) are halfway between the middlemost two dates and rates, respectively.

Step 6. Draw a straight line through the two sets of intersecting dashed lines at their points of intersec- tion. This straight line is known as the quarter-inter- sect line.

Step 7. Draw a line parallel to the quarter-intersect line that has 50 percent of the datum points on or above it, and 50 percent of the datum points on or below it. The resulting line is the split-middle line of trend estimation.

Rn Statistic

The Rn statistic is used only with the multiple base- line design.14 It will provide a statement of statistical significance at the .05 level if at least four base-lines are used. It assumes that the base-line selected for treatment will be done randomly.

The Rn statistic is based on the underlying logic of the multiple base-line design. Specifically, if an in- dependent variable is successful in changing the de- pendent variable, its effectiveness will be recognized

in two ways. A change in level or trend, or both, will be present in the treated base-lines, but untreated base-lines will be characterized by stability. If no change is present in the treated base-line, or if there are concomitant changes in the untreated base-lines, experimental control is in serious question. In using the Rn statistic, the experimenter compares the level of the data in the treated base-line to all the untreated base-lines whenever a treatment is applied to any base-line. The comparison can be with the first treat- ment day, or, if the treatment is expected to demon- strate its effects only after a few treatment days, the comparison can be made with the mean of the first few treatment days.

Once the level of each base-line is noted, the base- lines are ranked. The rank of 1 is assigned to the base-line with the highest level in the expected direc-

Fig. 4. Graph A: Mid-date and mid-rate of data having more than one datum point on the same rate. Graph B: Mid-date and mid-rate of data series where the mid-date and mid-rate calculations result in fractional numbers.

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tion of the treatment. A rank of 2 is assigned to the base-line with the next highest level in the expected direction and each subsequent base-line is similarly ranked. These rankings are made each time treatment begins with a new base-line. However, once a treated base-line has been ranked, it is not considered in subsequent rankings. When all rankings are com- pleted for the entire experiment, the ranks of treated base-lines (at the point at which they were treated) are summed. This sum is then compared to the tables of significance to determine the statistical signifi- cance.

A major weakness of the Rn statistic as described above is its attention only to relative difference of levels between base-lines. However, an expanded ap- plication of the statistic to changes in both level and trend has been described.15 The expanded application uses a trend line (split-middle line of trend estimation) to determine changes in level and trend of the treated and untreated base-lines. The numerical values of these changes are then ranked as in the original Rn statistic. The expanded application of the Rn provides statements of statistical significance for level and trend.

Time-Series Analysis

Time-series analysis (TSA) is an inferential statistic designed to analyze data collected repeatedly over time on the same subject. Unlike ANOVA, TSA controls for the serial dependency in the data series. Because of this control for serial dependency, TSA has been recommended for use in applied research using single-subject designs.16 Briefly, TSA involves four steps.17 First, autocorrelations and partial auto- correlations are performed to identify the structure of the serial dependency in the data series. An appro- priate model that describes the serial dependency can then be identified. The most common model is the autoregressive integrated moving average (ARIMA) model. Identification of the appropriate model allows for transformation to standard regression models, which in turn allows for testing of differences between conditions.

The second step is to estimate the parameters of the proposed model. The third step serves to confirm or disprove whether the proposed model is appropriate for the given data series. If it is appropriate, transfer functions are applied and conventional tests of sig- nificance, such as t tests, can be employed.

This description is too brief to provide a complete explanation of TSA or of how to compute the statistic. However, several points should be made. Time-series analysis requires assistance from a computer and considerable statistical sophistication, but sources of computer programs, informational articles, and more expanded descriptions are available.16-18 Further,

while the recommendation to use TSA has been made, one particular problem remains. At issue is the number of data points needed to identify the ARIMA model.19 Some estimates are as high as 100 per con- dition while others are as low as 10 per condition. Nonetheless, when the model is identified, TSA is a very useful judgment aid to the single-subject re- searcher.

CLINICAL AND STATISTICAL SIGNIFICANCE

Many factors must be considered when interpreting experimental data. First, the data must be collected in a reliable, consistent manner. In single-subject research, estimates of reliability are usually derived from comparisons of two independent observers' measures of the same behavioral events and are pre- sented as the percent of agreement between observ- ers.20 Given that reliability estimates are sufficiently high, the experimenter can proceed to the second step in the process of interpretation. This step involves determining whether there were changes in the data series. If changes exist, the experimenter can proceed to the third step, which involves determining whether the changes in data coincide with the introduction of the treatment and whether the changes continued to occur in a consistent manner across replications of the experimental manipulations. We have attempted to address the second and third steps in the interpre- tation process. If an experiment is internally valid (ie, changes occurred in the data consistent with the introduction of the treatment and can be attributed to the treatment), then the experimenter must address the fourth step in the process. This involves determin- ing whether the changes in performance resulting from the treatment are important and useful. The importance and usefulness of the change has been called clinical significance or, more recently, social validity.21 At issue is whether the change in the sub- ject's performance results in changes that are clini- cally important or are perceived by others, including the subject, as being worthwhile.

Statistical significance is important in the second and third steps but not necessarily in the fourth step. Statistical significance and clinical significance do not have to be concurrently present in any given experi- ment. Experiments can show changes in data that are statistically significant but not clinically significant. For example, if a developmentally disabled child can consistently maintain a "head-up" position for 2 per- cent of the observation intervals in base-line and a given treatment increases his head-up behavior to 10 percent of the intervals, the difference may be statis- tically significant. However, he may not be able to receive enough information from the environment to learn new concepts or behaviors even though he is holding his head up for a statistically significant

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longer period of time. Thus, the treatment does not demonstrate clinical significance. Further, some ex- periments may produce differences in data that are clinically significant but not statistically significant. It should be noted, however, that some experiments will produce differences that are both clinically and sta- tistically significant, although it is also possible for experiments to produce differences that are neither clinically nor statistically significant.

Clinical significance is concerned chiefly with changes the treatment makes in the subject's "real world" setting, particularly if they concern useful or desirable changes in abilities and performance. Sta- tistical significance involves the amount of difference in the data series in the analogue experimental setting regardless of whether such change is useful or desir- able.

If a given treatment produces clinically significant differences in the performance of a subject, the inves- tigator should address the fifth step in the interpre- tation process. This step involves determining the external validity of the experiment. External validity refers to the generalizability of the experimental ef- fects across other subjects, settings, and behaviors. One cannot be confident that an experiment is exter- nally valid until attempts at replication are made.22

Logically, however, given treatments are more likely to produce similar results if the subjects, settings, and behaviors are similar. Thus, investigators should take great care in accurately describing subjects, settings, behaviors, and treatments.

SUMMARY

Interpretation of results of single-subject research is accomplished by using either of two different strat- egies: visual analysis of graphed data or statistical analysis. Through visual analysis, the more com- monly used strategy, changes in the data series within and between experimental conditions are noted. Four

types of changes can be observed: changes in varia- bility, level, trend, or both level and trend. Changes within experimental conditions may demonstrate threats to the internal validity of the research. Changes between conditions are desirable but must be consistently replicated across several experimental manipulations to demonstrate that the treatment is effective.

Conventional statistical procedures, such as AN- OVA and t tests, are seldom used in analyzing the effects of single-subject research. Because of their failure to examine trends in the data and because of the presence of serial dependency of data points, conventional statistical procedures are usually inap- propriate for use in single-subject research. Three statistical procedures that are appropriate in that they can accommodate for serial dependency and examine trends are the split-middle method of trend estima- tion, the Rn statistic, and TSA. The function of the split-middle line of trend estimation is to explicate the movement in the data series rather than provide a statement of statistical significance. The Rn statistic provides a statement of statistical significance but must be used with multiple base-line designs. Time- series analysis is an inferential statistic requiring con- siderable statistical sophistication and a computer for calculations.

In interpreting single-subject research, the investi- gator must determine whether the data were reliably collected, whether there were changes within experi- mental conditions, and whether there were changes between experimental conditions. In addition, the investigator must determine whether the changes were clinically significant—whether they were impor- tant and worthwhile in the subject's real world envi- ronment. With the increasing emphasis in the physical therapy profession on accountability for effecting ob- jective and meaningful changes in patients' condi- tions, the issue of clinical significance is particularly relevant.

REFERENCES

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bell DT (eds): Quasi-Experimentation: Design and Analysis Issues for Field Settings. Chicago, IL, Rand McNally & Co, 1979, pp 207-233

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19. Bernstein GS: Time series analysis and research in behavior modification: Some unanswered questions. Behavior Ther- apy 8:503-504, 1977

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