Basic Concepts
The mean of successive squared differences (MSSD) of the data sequence S = {x1, …, xn} is defined as
If a data sequence is random then MSSD is approximately equal to the variance. In fact, we can use the following statistic to test the null hypothesis that a data sequence is random
If the null hypothesis holds, then for large n this statistic has a standard normal distribution. The normal approximation is pretty good for α = .05 and n ≥ 10, or α = .025, .10, .25 and n ≥ 25, or α = .01, .005 and n ≥ 100.
Example
Example 1: Calculate the MSSD statistic for the data series in column A of Figure 1. Determine whether this data is random.
Figure 1 – MSSD calculation
Cell B4 contains the formula =A5-A4 and cell C4 contains =B4^2, to fill in the other cells, highlight range B4:C17 and press Ctrl-D. The formulas for the other cells are shown in column H. Using the normal distribution approximation, we see that p-value = .036 < .05 = α, demonstrating that the series is not random.
Note that the MSSD value can also be calculated by the Excel formula
=SUMXMY2(A5:A18,A4:A17)/(2*COUNT(A4:A17))
Worksheet Functions
Real Statistics Functions: The Real Statistics Resource Pack provides the following array function where R1 is a column array containing the data series.
MSSD(R1, lab): column array containing the values: MSSD, sample variance, z-statistic, and p-value; if lab = TRUE (default = FALSE), then a column of labels is appended to the output
For Example 1, the formula =MSSD(A4:A18) outputs the value 2.8214, the value shown in cell F5. The array formula =MSSD(A4:A18,TRUE) outputs the range E5:F10 with rows 7 and 8 omitted.
The MSSD function uses the normal distribution approximation to obtain the z-statistic and p-value. Especially for small samples, a better approximation can be obtained for the critical value of the z-statistic (using a table of critical values of the c statistic). From these critical values, we can estimate the p-values. These can be obtained for values of n from 8 to 150 by using the following functions:
MSSD_CRIT(n, α, interp) = critical value for the z-statistic based on a sample of size n and a significance level of α (default .05); when interpolation is required, then the recommended interpolation is used when interp = TRUE (default); otherwise linear interpolation is used.
MSSD_PROB(z, n, iter, interp, txt) = p-value for the z-statistic z based on a sample of size n; interp is as for MSSD_CRIT; iter = # of iterations required to make the approximation (default 40); when txt = FALSE (default), then a p-value less than .005 is rounded down to 0 and a p-value greater than .25 is rounded up to 1, while if txt = TRUE then such values are returned as “< .005” and “> .25” respectively.
For Example 1, MSSD_CRIT(F7) = 1.647945 and MSSD_PROB(F9,F7) = .034446.
Reference
Zar. J. H. (2010) Biostatistical analysis 5th Ed. Pearson
https://bayesmath.com/wp-content/uploads/2021/05/Jerrold-H.-Zar-Biostatistical-Analysis-5th-Edition-Prentice-Hall-2009.pdf