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Trend Lines

Trend lines, also called lines of progress or celeration lines, are drawn through charted data to summarize the relationship between two variables. The two variables usually represent changes in performance over time (see Figure 1, panels A and B), but the relationship between any two variables can be summarized with trend lines (panels C and D). Trend lines have been used to describe patterns within phases of a study and to predict values beyond the immediate data set (panels A and B). Because accepted “theoretical” or “mathematical” justifications do not exist for predicting values (extrapolating) beyond the data set used to estimate the trend, experience and empirically demonstrated success must be used in deciding whether the data are orderly enough to warrant predictions.

Figure 1 Examples of Trend Lines

Note: Panel A: Equal interval chart of an individual's disruptive behaviors under various schedules of time-out. DPH indicates disruptions per hour; MedSlp, median slope; QtrInt, use of the quarterintersect method; Smooth, use of the smoothing method. Panel B: The same individual's record presented on a ratio (semilogarithmic) chart. The various trend lines labeled in Panel B refer to the data in phase 3–VR3. Panel C: Free-hand curvilinear trend drawn through mean rates of disruptions as a function of the probability of being timed out. CRF indicates continuous reinforcement; VR, variable ratio. Panel D: The same means presented on a ratio chart, described with phase detail, using two straight-line trends.

All data were adapted from Clark, Rowbury, and Baer (1973).

General Approaches for Estimating Trends

Approaches used to estimate trends differ primarily in whether they attempt to describe a trend using straight lines or curves and whether they employ a freehand method or are based on mathematical models.

Describing Trends as Straight Lines

The following methods are generally based on the assumption that the overall trend in the data is best described by a simple linear (straight-line) function.

Freehand or Focus. The freehand method relies on simple visual inspection of the data. The process of finding an appropriate line can be “focused” by imposing a straight line over the data (perhaps printed on a plastic sheet), then moving the line up and down at different angles until what appears to be a satisfactory solution has been found. An attempt is usually made to find a line that evenly divides the data on or above the line (or on or below the line). The freehand method can also be used to draw curved lines through the data when a straight-line solution appears inappropriate. In such a case, a straight-line “focusing aid” can still be helpful by highlighting patterns in the data that deviate from a straightline model.

The freehand method allows the analyst to take into consideration factors (such as common patterns of learning) that might be overlooked by more explicit mathematical models. Several studies suggest that the freehand method can be faster to employ and more accurate in predictions than other methods, but the experience of the analyst can be an important factor.

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