APPENDIX 7.2 Daniel’s Test for Trend
Daniel’s test for trend is based on Spearman’s rank correlation coefficient, r s. The hypothesis test is as follows:
H0: The time series does not exhibit trend. H A : The time series does exhibit trend. To perform this test, the observations are ranked in ascending order. (If two or more observations have the same value, they are each given a rank equivalent to their average ranking.) After ranking the observations, r s, the rank correlation coefficient, is calculated as follows:
rs 1
6d2t n(n2 1)
where
dt t (rank of observation y t ) and
n the number of observations in the time series If n is less than or equal to 30, the following procedure is used to test H 0 at a significance level of :
Reject H0 if rs r /2 (values for r /2 are given in Appendix H).
In cases where the number of observations is greater than 30, the test statistic is calculated as follows:
z
rs
n 1
H0 is rejected if z z /2. (Values for z /2 are given in Appendix A.) For example, consider the following copy paper usage at Allied Industries.
Period (t ) 1 2 3 4 5 6 7 8 9 10 11 12 CD-366
Number of Reams 456 483 462 504 521 470 429 453 444 473 486 494
Rank 4 16 6 20.5 24 10 1 3 2 12 17.5 19
d t 3 14 3 16.5 19 4 6 5 7 2 6.5 7
d 2t 9 196 9 272.25 361 16 36 25 49 4 42.25 49
CD-367
Appendix 7.2
Period (t ) 13 14 15 16 17 18 19 20 21 22 23 24
Number of Reams 512 507 504 486 467 475 458 464 481 478 472 463
Rank 23 22 20.5 17.5 9 13 5 8 15 14 11 7 2 dt 2227
d t 10 8 5.5 1.5 8 5 14 12 6 8 12 17
d 2t 100 64 30.25 2.25 64 25 196 144 36 64 144 289
The lowest time series value is 429 which occurs at period 7. Therefore, this period is given a ranking of 1. The highest time series value occurs at period 5 (521); this period is given a ranking of 24. Note that, since there are identical data values for periods 11 and 16 as well as for periods 4 and 15, the rankings for these periods are equal to their average rankings. For example, the value for periods 11 and 16 (486) corresponds to the seventeenth and eighteenth highest values. Hence, each of these periods is given a ranking of 17.5 (the average of 17 and 18). The values for dt are then calculated by taking the difference between the period number, t, and the ranking y t. For example, period t 1 has a ranking of y 1 4. Therefore, d1 1 4 3 and d2t (3)2 9. The last column of the table gives the values of d2t. Summing the d2t values gives 2227, and the value of rs is:
rs 1 6(2227)/(24(242 1)) 1 .96826 .03174 Using .05 and n 24, from Appendix H, r .025 .4061. Hence, since rs .03174 .4061, H0 cannot be rejected, that is, the evidence is not strong enough to conclude that the time series exhibits trend. Now consider the time series data for year-end current assets of American Family Products given in Table 7.5 of the text. In this case, the following are obtained: Year (t ) 1 2 3 4 5 6 7 8 9 10
Current Assets 1990 2280 2328 2635 3249 3310 3256 3533 3826 4119
Rank 1 2 3 4 5 7 6 8 9 10 2 dt 2
dt 0 0 0 0 0 1 1 0 0 0
d 2t 0 0 0 0 0 1 1 0 0 0
From this data r s 1 6(2)/(10(100 1)) 1 .012 .988. Testing at .05 with n 10 gives (from Appendix H) a critical value of r .025 .6364. Since rs .988 .6364, H0 is rejected and one concludes that this time series does exhibit trend.
