Runs test for comparing to curves

Comparison of PET TACs is necessary for instance when verifying the goodness of compartmental model or function fits, or in image clustering and segmentation.

Plotting the curve of residuals, that is, fitted TAC subtracted from the measure noisy TAC, or simply the difference between two measured TACs, helps visual comparison of two TACs. If TACs are similar, then the residuals should oscillate randomly below and above zero. Runs test is a non-parametric method to test the (non)randomness in result sequences (Wald & Wolfowitz, 1940; Bradley, 1960), but it can be used to test the (non)randomness in residuals of time series, too (Cobelli et al., 2002; Wackerly et al., 2008). It has been used in modelling PET data for instance by Syvänen et al (2006).

Let's say that we have the two TACs, consisting of N samples, measured at the same time points. From each sample value in TACA the corresponding sample value in TACB is subtracted, giving the residuals of each sample. A residual is 0 in case the sample value of both curves is the same. A run is defined as a subsequence of residuals having the same sign. The number of runs (R) can be calculated by counting the number of times the residuals change sign, plus one. If the curves are similar except for the noise, we would expect that the number of runs is large; small number of runs means that one TAC is above the other for a relatively long time. Then we calculate the number of negative residuals (n1) and positive residuals (n2). For reliable results, both n1 and n2 should be >10. Then we calculate:

, and Z statistic as:

According to a lower tail test, the residuals are considered to be nonrandom (the two TACs are different) if at Z the value of density function Φ(Z) is lower than given significance level α (usually 0.05).

Simple and very fast method to use for comparing TACs of image voxels is to calculate maximum run length, MRL, i.e. the largest number of subsequent residuals with the same sign (Herholz et al., 1989). If a critical MRL value (dependent on N) is exceeded (Grant, 1946 and 1947; Child, 1946), then we can assume that the TACs represent different physiological kinetics.

See also:


Bradley JV. Distribution-Free Statistical Tests. Wright Air Development Division, 1960.

Child IL. A note on Grant's "New Statistical Criteria for Learning and Problem Solution." Psychol Bull. 1946; 43(6): 558-561. doi: 10.1037/h0060570.

Cobelli C, Forster D, Toffolo G: (2002) Tracer Kinetics in Biomedical Research: From Data to Model. Kluwer Academic Publishers.

Grant DA. New statistical criteria for learning and problem solution in experiments involving repeated trials. Psychol Bull. 1946; 43: 272-282. doi: 10.1037/h0058516.

Grant DA. Additional tables of the probability of runs of correct responses in learning and problem-solving. Psychol Bull. 1947; 44(3): 276-279. doi: 10.1037/h0054957.

Herholz K, Heiss WD, Pietrzyk U, Wienhard K. Pixel-by-pixel fits of blood volume, transport, and metabolic processes: principle, normal values, and brain tumor studies with dynamic FDG-PET. In: Beckers C, Goffinet A, Bol A (eds.): Positron Emission Tomography in Clinical Research and Clinical Diagnosis: Tracer Modelling and Radioreceptors, pp 148-161. Kluwer, 1989. ISBN: 0-7923-0254-0.

Motulsky HJ, Ransnas LA. Fitting curves to data using nonlinear regression: a practical and nonmathematical review. FASEB J. 1987; 1: 365-374. doi: 10.1096/fasebj.1.5.3315805.

Muzic RF Jr, Christian BT. Evaluation of objective functions for estimation of kinetic parameters. Med Phys. 2006; 32(2): 342-353. doi: 10.1118/1.2135907.

NIST/SEMATECH e-Handbook of Statistical Methods,

Wackerly D, Mendenhall W, Scheaffer RL. Mathematical Statistics with Applications, 7th ed., Cengage Learning, 2008. ISBN-13: 978-0495110811.

Young P: Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask. Springer, 2005. ISBN 978-3-319-19051-8.

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Updated at: 2019-01-19
Created at: 2007-02-06
Written by: Vesa Oikonen, Kaisa Liukko