
MEAN SUCCESSIVE DIFFERENCES TESTName:
The numerator term is a measure of variance adjusted for trend while the denominator is the standard variance. For N > 20, Dataplot computes critical values based on the following formula from Dixon:
T is compared to a standard normal distribution. For N ≤ 20, critical values are taken from tables given by Neubauer. If the data are random and from an underlying normal distribution, the average value of M is 2. Large values of M indicate excessive fluctuations in the data. Small values of M indicate long term trend. There are several variations of this test in the literature. For example, the numerator term is sometimes given as an absolute value rather than a square. Early versions of the test used N rather than N  1 in the denominator. There are also been a number of different approximations proposed for the critical values for this test. The approximation used here should be adequate for practical purposes. The Durbin Watson test is a variant of this test that is commonly used to test for serial correlation in regression problems. The mean successive differences test is applied to the residuals. Since the mean of the residuals is zero, the in the formula above drops out. This test is also sometimes referred to as the adjacency test.
<SUBSET/EXCEPT/FOR qualification> where <y> is a response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
<SUBSET/EXCEPT/FOR qualification> where <y1> ... <yk> is a list of 1 to 30 response variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax will perform a mean successive differences test for each of the response variables. For example,
is equivalent to
MEAN SUCCESSIVE DIFFERENCES TEST Y2 MEAN SUCCESSIVE DIFFERENCES TEST Y3 MEAN SUCCESSIVE DIFFERENCES TEST Y4
<SUBSET/EXCEPT/FOR qualification> where <y> is a response variable; <x1> ... <xk> is a list of 1 to 6 groupid variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax will compute the test for each unique combination of the groupid variables.
MEAN SUCCESSIVE DIFFERENCES TEST Y1 TO Y5 REPLICATED MEAN SUCCESSIVE DIFFERENCES TEST Y X1 X2
For N ≤ 20, critical values are obtained from tabulated values and some of these parameters are not defined. In this case, these parameters will be set to the minimum machine value. You can retrieve this value with the commands
LET CPUMIN = PROBVEVAL
LET A = MEAN SUCCESSIVE DIFFERENCES TEST NORMALIZED Y LET A = MEAN SUCCESSIVE DIFFERENCES TEST CDF Y LET A = MEAN SUCCESSIVE DIFFERENCES TEST PVALUE Y The NORMALIZED form returns the DixonMassey transformation of the statistic described above. Note that the CDF and PVALUE are not computed for N < 20 (they will be set to the minimum machine value in this case). Enter HELP STATISTICS for a list of commands that can be used with Dataplot supported statistics. See Program 2 for an example.
The run sequence plot, the lag plot, and the autocorrelation plots can be used to graphically assess whether or not there is trend or autocorrelationin the data. The 4plot can be used to assess the more general assumption of "independent, identically distributed" data. The Cox Stuart test is a nonparametric test for trend. The Ljung Box test is a test for randomness based on the autocorrelation for a number of lags (i.e., more than first order autocorrelation). The runs test is a test for randomness based on the number of runs. The frequency test, the frequency within a block test, and the cusum test can be used to test the randomness of sequence of zeros and ones.
DURBIN WATSON TEST MEAN SUCCESSIVE DIFFERENCES ADJACENCY RANDOMNESS TEST ADJACENCY RANDOMNESS ADJACENCY TEST ADJACENCY
John V. Neumann (1941), "Distribution of the Ratio of the Mean Successive Difference to the Variance", Annals of Mathematical Statistics, 12, 367395. Dean Neubauer, "Testing for Randomness: The Mean Successive Differences Test", ASTM Standardization News, September/October 2012, pp. 1213. Dixon and Massey (1957), "Introduction to Statistical Analysis", McGraw Hill, p. xxx.
2013/1 . Purpose: Mean Successive Difference Test for Randomness . . Step 1: Read the Data . . The ZAR110.DAT file contains the data from the Neubauer . article. . skip 25 read zarr110.dat y1 read lew.dat y2 skip 0 . . Sample data from example 2 on page 171 of Conover. . let y3 = data 45.25 45.83 41.77 36.26 45.37 52.25 35.37 57.16 35.37 ... 58.32 41.05 33.72 45.73 37.90 41.72 36.07 49.83 36.24 ... 39.90 . let y x = stack y1 y2 y3 . set write decimals 4 mean successive difference test y1 y2 y3 replicated mean successive difference test y xThe following output is generated. Mean Successive Differences Test for Randomness Response Variable: Y1 H0: The Data Are Random Ha: The Data Are Not Random Summary Statistics: Number of Observations: 25 Sample Mean: 80.0855 Sample Standard Deviation: 1.0472 Sample Minimum: 77.0900 Sample Maximum: 81.5900 Test Statistic: 1.2034 Normalized Test Statistic: 2.0745 CDF Value: 0.9809 Test Based on Normal Approximation Conclusions (TwoTailed Test)  Null Significance Test Critical Hypothesis Level Statistic Value (+/) Conclusion  50.0% 2.0745 0.6744 REJECT 80.0% 2.0745 1.2815 REJECT 90.0% 2.0745 1.6448 REJECT 95.0% 2.0745 1.9599 REJECT 99.0% 2.0745 2.5758 ACCEPT 99.9% 2.0745 3.2905 ACCEPT Mean Successive Differences Test for Randomness Response Variable: Y2 H0: The Data Are Random Ha: The Data Are Not Random Summary Statistics: Number of Observations: 200 Sample Mean: 177.4350 Sample Standard Deviation: 277.3321 Sample Minimum: 579.0000 Sample Maximum: 300.0000 Test Statistic: 2.6096 Normalized Test Statistic: 4.3324 CDF Value: 0.0000 Test Based on Normal Approximation Conclusions (TwoTailed Test)  Null Significance Test Critical Hypothesis Level Statistic Value (+/) Conclusion  50.0% 4.3324 0.6744 REJECT 80.0% 4.3324 1.2815 REJECT 90.0% 4.3324 1.6448 REJECT 95.0% 4.3324 1.9599 REJECT 99.0% 4.3324 2.5758 REJECT 99.9% 4.3324 3.2905 REJECT Mean Successive Differences Test for Randomness Response Variable: Y3 H0: The Data Are Random Ha: The Data Are Not Random Summary Statistics: Number of Observations: 19 Sample Mean: 42.9005 Sample Standard Deviation: 7.3697 Sample Minimum: 33.7199 Sample Maximum: 58.3200 Test Statistic: 2.9972 Normalized Test Statistic: 2.2944 Test Based on Tabulated Values Conclusions (TwoSided Test)  Lower Upper Null Significance Test Critical Critical Hypothesis Level Statistic Value (<) Value (>) Conclusion  90% 2.9972 1.4339 2.5659 REJECT 95% 2.9972 1.2829 2.7170 REJECT 99% 2.9972 1.0200 2.9800 REJECT Mean Successive Differences Test for Randomness Response Variable: Y Factor Variable 1: X 1.0000 H0: The Data Are Random Ha: The Data Are Not Random Summary Statistics: Number of Observations: 25 Sample Mean: 80.0855 Sample Standard Deviation: 1.0472 Sample Minimum: 77.0900 Sample Maximum: 81.5900 Test Statistic: 1.2034 Normalized Test Statistic: 2.0745 CDF Value: 0.9809 Test Based on Normal Approximation Conclusions (TwoTailed Test)  Null Significance Test Critical Hypothesis Level Statistic Value (+/) Conclusion  50.0% 2.0745 0.6744 REJECT 80.0% 2.0745 1.2815 REJECT 90.0% 2.0745 1.6448 REJECT 95.0% 2.0745 1.9599 REJECT 99.0% 2.0745 2.5758 ACCEPT 99.9% 2.0745 3.2905 ACCEPT Mean Successive Differences Test for Randomness Response Variable: Y Factor Variable 1: X 2.0000 H0: The Data Are Random Ha: The Data Are Not Random Summary Statistics: Number of Observations: 200 Sample Mean: 177.4350 Sample Standard Deviation: 277.3321 Sample Minimum: 579.0000 Sample Maximum: 300.0000 Test Statistic: 2.6096 Normalized Test Statistic: 4.3324 CDF Value: 0.0000 Test Based on Normal Approximation Conclusions (TwoTailed Test)  Null Significance Test Critical Hypothesis Level Statistic Value (+/) Conclusion  50.0% 4.3324 0.6744 REJECT 80.0% 4.3324 1.2815 REJECT 90.0% 4.3324 1.6448 REJECT 95.0% 4.3324 1.9599 REJECT 99.0% 4.3324 2.5758 REJECT 99.9% 4.3324 3.2905 REJECT Mean Successive Differences Test for Randomness Response Variable: Y Factor Variable 1: X 3.0000 H0: The Data Are Random Ha: The Data Are Not Random Summary Statistics: Number of Observations: 19 Sample Mean: 42.9005 Sample Standard Deviation: 7.3697 Sample Minimum: 33.7199 Sample Maximum: 58.3200 Test Statistic: 2.9972 Normalized Test Statistic: 2.2944 Test Based on Tabulated Values Conclusions (TwoSided Test)  Lower Upper Null Significance Test Critical Critical Hypothesis Level Statistic Value (<) Value (>) Conclusion  90% 2.9972 1.4339 2.5659 REJECT 95% 2.9972 1.2829 2.7170 REJECT 99% 2.9972 1.0200 2.9800 REJECTProgram 2: skip 25 read splett2.dat y x skip 0 . title case asis title offset 2 label case asis x1label displacement 12 multiplot scale factor 2 multiplot corner coordinates 5 5 95 95 multiplot 2 2 . let ntemp = size y let xseq = sequence 1 1 ntemp char 1 2 3 4 line blank blank blank blank y1label Absorbed Energy x1label Sequence title Raw Data plot y xseq x char blank all line solid all . xlimits 1 4 major xtic mark number 4 minor xtic mark number 0 tic mark offset units data x1tic mark offset 0.5 0.5 tic mark label case asis x1tic mark label format alpha x1tic mark label content Tinius1 Tinius2 Satec Tokyo x1label Manufacterer . char X line blank y1label title MSD Test Statistic mean successive differences test normalized plot y x title MSD Test Statistic CDF mean successive differences test cdf plot y x title MSD Test Statistic PValue mean successive differences test pvalue plot y x . end of multiplot . case asis justification center move 50 97 text Mean Successive Differences Test for SPLETT2.DAT  
Date created: 02/15/2013 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 