How do I compute slopes of linear trend for single cases?
It is often useful to produce a summary measure of a variable measured over time (e.g. to assess learning). This can be later used, for example to correlate with IQ to assess if rate of learning is related to IQ. SPSS and other packages only produce these for groups, rather than individuals.
The table below gives regression estimates of slopes corresponding to individual linear trends involving three up to six points. These may be computed from the data assuming each time point is entered in a separate column.
The formula is obtained by multiplying the k observed responses over time, $$y_text{1}$$ to $$y_text{k}$$, (in chronological order with $$y_text{1}$$ being the first response and $$y_text{k}$$ the last) by orthogonal polynomial coefficients and dividing by the sum of their squared coefficients (These may be found on page 678 of Howell DC (1997)). It should, in fact, be found in the appendices of any of the five editions of this book. There are copies in the CBU library. Howell also illustrates how you can work out these trend coefficients for unequally spaced intervals.
No. Points |
Slope |
|
3 |
($$y_text{3} - y_text{1}$$)/2 |
|
4 |
($$-3y_text{1} -y_text{2} + y_text{3} + 3y_text{4}$$) / 20 |
|
5 |
($$-2y_text{1} -y_text{2} + y_text{4} + 2y_text{5}$$) / 10 |
|
6 |
($$-5y_text{1} -3y_text{2} - y_text{3} + y_text{4} + 3y_text{5} +5y_text{6}$$) / 70 |
|
An alternative approach compares trends in scores with a second measure. It uses first and second derivatives of the scores representing magnitude of change (differences between successive time points) and rate of change (differences of these differences between successive time points) on each individual over time respectively. Summary measures of skew and/or kurtosis of these derivatives are then correlated with a second measure to see how change and rate of change over time relate to this second measure. Plenty of time points are required (preferable stretching daily over weeks) and a second measure which is thought to relate to the score measured over time. Further details of this method including the theory, examples of applications and R code (in the appendices) for implementation is available in Deboeck, PR, Montpetit, MA, Bergeman, CS and Boker, SM (2009) available for free download (as of January 2010) to CBSUers via sciencedirect on the CBSU intranet library pages.
Since a Pearson correlation is analogous to fitting a best fitting linear trend (via simple regression) it follows that the strength of a linear trend may be gauged by the magnitude of the correlation. The relative strengths of linear trends may, therefore, by compared by comparing Pearson correlations (as suggested here).
These contrast coefficients can also be used to test for a linear trend in proportions (see Everitt and Wykes, 1999, p.31).
References
Deboeck, PR, Montpetit, MA, Bergeman, CS and Boker, SM (2009) Using derivative estimates to describe intraindividual variability at multiple time scales. Psychological Methods 14(4) 367-386.
Dallery J, Cassidy R and Raiff B (2013) Single-case experimental designs to evaluate novel technology-based health interventions. Journal of Medical Internet Research 15(2), e22. An excellent overview of single-case studies illustrating graphical analyses and assumptions such as lack of carry-over effects and a constant treatment effect. These trials usually feature fewer than 10 participants.
Everitt BS and Wykes T (1999) A dictionary of statistics for psychologists. Arnold:London.
Howell DC (1997) Statistical methods for psychologists. Fourth Edition. Duxbury Press:Belmont,CA
Lorch Jr RF and Myers JL (1990) Regression analyses of repeated measures data in cognitive research. Journal of Experimental Psychology: Learning, Memory and Cognition 16(1) 169-157.
Suzanne McDonald, Rute Vieira & Derek W. Johnston (2020) Analysing N-of-1 observational data in health psychology and behavioural medicine: a 10-step SPSS tutorial for beginners, Health Psychology and Behavioral Medicine, 8:1, 32-54, DOI: 10.1080/21642850.2019.171109. This brief article explains removing auto-correlations which can bias results e.g. scores may naturally increase over time due to practice effect rather than due to differences in conditions. Any auto-correlations can be removed using linear regression.