# Extrapolation along the trend line

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Metrics details Abstract Single-case data often contain ready- made website for making money on the Internet. Accordingly, to account for baseline trend, several data-analytical techniques extrapolate it into the subsequent intervention phase.

Extrapolation Formula To extrapolate data by formula, we need to use two points of the linear chart that we plotted before. Picture 1- The linear extrapolation formula in Excel How to Extrapolate a Graph by Trendline Extrapolating a graph by trendline helps you represent visual data trends. Go to the Insert tab from the ribbon.

To avoid impossible predicted values, we propose extrapolating a damping trend, when necessary. Furthermore, we propose a criterion for determining whether extrapolation is warranted and, if so, how far out it is justified to extrapolate a baseline trend.

This criterion is based on the baseline phase length and the goodness of fit of the trend line to the data. These proposals were implemented in a modified version of an analytical technique called Mean phase difference. We used both real and generated data to illustrate how unjustified extrapolations may lead to inappropriate quantifications of effect, whereas our proposals help avoid these issues.

The new techniques are implemented in a user-friendly website via the Shiny application, offering both graphical and numerical information. Finally, we point to an alternative not requiring either trend line fitting or extrapolation. In the present article we focus on trends. Even when there is and my mom makes money on the internet intention to take the trend into account, several challenges arise.

Fourth, other techniques may extrapolate baseline trend regardless of the degree to which the trend line is a good representation of the baseline data, and despite the possibility of impossible values being predicted see Parker et al.

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The latter two challenges compromise the interpretation of results. Aim, focus, and organization of the article Aim The aim of the present article is to provide further discussion on four issues related to baseline trend extrapolation, based on the comments by Parker et al.

As part of this discussion, we propose tentative solutions to the issues identified. Focus Most single-case data-analytical techniques focus on linear trend, although there are certain exceptions. Another is Tau-U, developed by Parker et al. Moreover, this focus is well aligned with our willingness to improve the MPD, a procedure to make big money fitting a linear trend line to baseline data.

### Extrapolating baseline trend in single-case data: Problems and tentative solutions

Despite this focus, three of the four issues identified by Parker et al. Organization In the following sections, first we mention procedures that include extrapolating the trend line fitted in the baseline, and distinguish them from procedures that account for baseline trend but do not extrapolate it.

Second, we perform a review of published research in order to explore how frequently trend extrapolation leads to out-of-bounds predicted values for the outcome variable. Third, we deal separately with the four main issues of extrapolating a baseline trend, as identified by Parker et al. Fourth, on the basis of the proposals from the previous two points, we propose a modification of the MPD.

In the same section, we also provide examples, based on previously published data, of the extent to which our modification helps avoid misleading results.

## Trend Extrapolation

Fifth, we include a small proof-of-concept simulation study. Analytical techniques that entail extrapolating baseline trend Visual analysis When discussing how visual analysis should be carried out, Kratochwill et al. Apart from OLS regression, the generalized least squares proposal by Swaminathan et al.

The overall effect size described by the authors entails comparing the treatment data as estimated from the treatment-phase trend line to the treatment data as estimated from the baseline-phase trend line. Apart from the procedures based on the general linear model assuming normal errorsgeneralized linear models Fox, need to be mentioned as well in the present subsection.

Other useful models are based on the binomial distribution, specifying a logistic model Shadish et al. Despite dealing with certain issues arising from single-case data, these models are not flawless. Note that a Poisson model may present limitations when the data are more variable than expected i.

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Finally, what is most relevant to the topic of the present text is that none of these generalized linear models necessarily includes an extrapolation of baseline trend.

Nonregression procedures MPD involves estimating baseline trend and extrapolating it into the intervention phase in order to compare the predictions with the actual intervention-phase data. In one of the steps of the SLC, baseline trend is removed from the nA baseline measurements and the nB intervention-phase measurements by subtracting from each value yi the slope estimate b1multiplied by the measurement occasion i.

Therefore, we consider that it is more accurate to conceptualize this step as removing baseline trend from the intervention-phase trend for the purpose of comparison. Regarding Tau-U Parker et al. Therefore, no intercept or slope is estimated, and no trend line is fitted or extrapolated, either.

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The way in which trend is controlled for in Tau-U cannot be described as trend extrapolation in a strict sense. Two other nonoverlap indices also entail baseline trend control.

Therefore, as we discussed above for SLC, there is actually no trend extrapolation in the baseline-corrected Tau or percentage-of-nonoverlapping-corrected data. Procedures not extrapolating trend The analytical procedures included in the present subsection do not extrapolate baseline trend, but they do take baseline trend into account.

We decided to mention these techniques for three reasons. First, we wanted to provide a broader overview of analytical techniques applicable to single-case data.

Second, we wanted to make it explicit that not all analytical procedures entail baseline trend extrapolation, and therefore, such extrapolation is not an indispensable step in single-case data analysis. Stated in other words, it is possible to deal with baseline trend without extrapolating it.

## Extrapolation - Wikipedia

Third, the procedures mentioned here were those more recently developed or suggested for single-case data analysis, and so they may be less widely known. Moreover, they can be deemed more sophisticated and more strongly grounded on statistical theory than is MPD, which is the focus of the present article.

It is not clear that a regression model using time and its interaction with a dummy variable representing phase entails baseline trend extrapolation.

Moreover, a different approach was suggested by Pustejovsky, Hedges, and Shadish for obtaining a d statistic—namely, in relation to multilevel analysis. Another statistical option is to use generalized additive models GAMs; Sullivan et al. GAMs that have been specifically suggested include the use of cubic polynomial curves fitted to different portions of the data and joined at the specific places called knots that divide the data into portions.

Just like when using multilevel models, trend lines are fitted separately to each phase, without the need to extrapolate baseline trend. A review of research published in Aim of the review It has already been stated Parker extrapolation along the trend line al.

Accordingly, the research question we chose was the percentage of studies in which extrapolating the baseline trend of the data set across several different techniques for fitting the trend line leads to values that are below the lower bound or above the upper bound of the outcome variable.

Procedure Search strategy We focused on the four journals that have published most SCED research, according to the review by Shadish and Sullivan Given that the bibliographic search was performed in Septemberwe focused on the year and looked for any articles using phase designs AB designs, variations, or extensions or alternation designs with a baseline phase and providing a graphical representation of the data, with at least three measurements in the initial baseline condition.

Techniques for finding a best fitting straight line For the present review, we selected five techniques for finding a best-fitting straight line: OLS, split-middle, tri-split, Theil—Sen, and differencing. The motivation for this choice was that these five techniques are included in single-case data-analytical procedures Manolov,and therefore, applied researchers can potentially use them.

We counted the number and percentage of studies in which values out of logical bounds were obtained after extrapolating the baseline trend, estimated either from an initial baseline phase or from a subsequent withdrawal phase e. Additional upper bounds included the maximal scores obtainable for an exam e.

We chose a conservative approach, and did not to speculate Footnote 1 about upper bounds for behaviors that were expressed as either a frequency e.

Footnote 2 Results of the review The numbers of articles included per journal are as follows.

The results of this review are as follows. Extrapolation led to impossibly extrapolation along the trend line values for all five trend estimators in 27 studies Complementarily, extrapolation led to impossibly large values for all five trend estimators in eight studies In terms of when the extrapolation led to an impossible value, a summary is provided in Table 1.

Note that this table refers to the data set in each article, including the earliest out-of-bounds forecast. Thus, it can be seen that for all trend-line-fitting techniques, it was most common to have out-of-bounds forecasts already extrapolation along the trend line the third intervention phase measurement occasion. This is relevant, considering that an immediate effect can be understood to refer to the first three intervention data points Kratochwill et al. Table 1 Absolute frequencies of articles out of a total of 68 in which an out-of-bounds forecast was obtained at earliest before the ith intervention phase measurement occasion Full size table These results suggest that researchers using techniques to extrapolate baseline trend should be cautious about downward trends that would apparently lead to negative values, if continued.

We do not claim that the four journals and the year are representative of all published SCED research, but the evidence obtained suggests that trend extrapolation may affect the meaningfulness of the quantitative operations performed with the predicted data frequently enough for it to be considered an issue worth investigation. Main issues when extrapolating baseline trend, and tentative solutions The main issues when extrapolating baseline trend that were identified by Parker et al.

In this section, we comment on each of these four issues identified by Parker et al. However, we extrapolation along the extrapolation along the trend line line by discussing in brief how these issues could be avoided rather than simply addressed.

Avoiding the issues Three decisions can be made in relation to trend extrapolation. First, the extrapolation along the trend line may wonder whether there is any clear trend at all.

We consider that either of these descriptive approaches is likely to be more reasonable than testing the statistical significance of the baseline trend before deciding whether or not to take it into account, because such a statistical test might lack power for short baselines Tarlow, Second, if the data show considerable variability and no clear trend, it is possible to use a quantification that does not rely on a linear trend, b any specific nonlinear trend, or c any average level whatsoever, by using a nonoverlap index.

Footnote 3 A different approach could be to quantify the difference in level e. Thus, there would be no trend line fitting and no trend extrapolation. Third, if the trend looks clear visually or according to a formal rule and the researcher decides to take it into account, it is also possible not to extrapolate trend lines.

Although these potential solutions seem reasonable, here we deal with another option: namely, the case in which baseline extrapolation is desired because it is part of the analytical procedure chosen prior to data collectionbut the researcher is willing to improve the way in which such extrapolation is performed. First extrapolation along the trend line Unreliable trend lines fitted If an unreliable linear trend is fitted e.

If the fit of the baseline trend line to the data is poor, its extrapolation would also be problematic.

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It is expected that, if the amount of variability were the same, shorter baselines would result in more uncertain estimates. In that sense, this issue is related to the next one. Focusing specifically on reliability, we advocate quantifying the amount of fit of the trend line and using this information when deciding on baseline trend extrapolation.

Thus, values of MASE greater than one could be indicative that a general trend e. Second issue: Assuming that trend continues unabated This issue refers to treating baseline trend as if it were always the same for the whole period of extrapolation.

Similarly, when discussing the gradual-effects model, Swan and Pustejovksy also cautioned against long extrapolations, although their focus was on the intervention phase and not on the baseline phase.

- Extrapolating baseline trend in single-case data: Problems and tentative solutions | SpringerLink
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An initial approach could be to select how far out to extrapolate baseline trend prior to gathering and plotting the data, by selecting a number that would be the same across studies. It is possible to extend this recommendation to the present situation and state that the baseline trend should be extrapolated until the fifth intervention-phase measurement occasion. The choice of five measurements is arbitrary, but it is well-aligned with the minimal phase length required in the What Works Clearinghouse Standards Kratochwill et al.

Nonetheless, our review Table 1 suggests that impossible extrapolations are common even before the fifth intervention-phase measurement occasion, extrapolation along the trend line thus a comparison at that point might not avoid comparison with an impossible projection extrapolation along the trend line the baseline. Similarly, when presenting the gradual-effects model, Swan and Pustejovsky defined the calculation of the effect size for an a priori set number of intervention-phase measurement occasions.

In their study, this number depends on the actually observed intervention-phase lengths. Moreover, Swan and Pustejovsky suggested a sensitivity analysis, comparing the results of several possible a-priori-set numbers.

It could be argued that a fixed choice would avoid making data-driven decisions that could favor finding results in line with the expectations of the researchers Wicherts et al. A second approach would be to choose how far away to extrapolate on the basis of both a design feature baseline phase length; see the next section and a data feature the amount of fit of the trend line to the data, expressed as the MASE.

### What is Interpolation and Extrapolation?

In the following discussion, we present a tentative solution including both these aspects. Third issue: No consideration of baseline-phase length Parker et al.

The problem is that a short baseline is potentially related to unreliable trend, and it could also entail predicting many values i. To take baseline length nA into account, one approach would be to limit the extrapolation of baseline trend to the first nA treatment-phase measurement occasions. This approach introduces an objective criterion based on a characteristic of the design. Thus, the extrapolation is determined by both the number of baseline measurements nA and the goodness of fit of the trend line to the data.

What is it about? Be very careful about over-fitting. There can be a huge difference between fitting well a model and obtaining good forecasts. Again see more about it here.

When Parker et al. This procedure could be useful for statistically controlling baseline trend, although the evidence provided by Tarlow suggests that the trend control incorporated in Tau-U is insufficient i. An additional limitation of this trend correction procedure is that it cannot be used when extrapolating baseline trend. Therefore, we consider other options in the following text.

Nonlinear models One option, suggested by Rindskopf and Ferronis to use nonlinear models for representing situations in which a stable and low initial level during the baseline phase experiences a extrapolation along the trend line due to the intervention e. Rindskopf and Ferron suggested using logistic regression with an additional term for identifying the moment at which the response has gone halfway between the floor and the ceiling.

Similarly, Shadish et al. Shadish et al. Focusing on the need to improve MPD, the proposals by Rindskopf and Ferron and Verboon and Peters are not applicable, since the logistic model they present deals with considering the data of a baseline phase and an intervention phase jointly, whereas in MPD baseline trend is estimated and extrapolated in order to allow for a comparison between projected and observed patterns of the outcome variable as suggested by Kratochwill et al.

In contrast, Shadish et al.