Under H 0, for values of r within the recommended ranges given in Table (3.3.2), the distribution of tR (3.3-2) closely approximated that of a t distribution. Under H , we would expect a similar situation, namely that the distribution of tR (4.3-2) will be in close agreement with a non-central t distribution for these recommended values of r. As the population is subject to rounding the non-centrality parameter will be:
^ 3
R <rR
As the population is normal, good approximation to /ir and o r are given by Sheppard's corrections. Applying Sheppard's corrections to approximate j i r and <j r
we have
5r -
Jn
< 8 ( 4 . 3 - 4 )
From (4.3-4) we would expect the distribution of tR to be in close agreement with a non-central t distribution, t(5R), where 6 r is less than 5. This reduction in the
non-centrality parameter caused by rounding will result in the test becoming less powerful for rounded data. If values of r satisfy the recommendations given in
This can be illustrated as follows.
For various values of r within the recommended ranges given in Table (3.3.2) for a = 0.05 (one tailed)
(i) the mean and variance of the distribution of tR were obtained by simulation and compared with those for a t(5j^) distribution.
(ii) the powers of the test for values of H , corresponding to powers of 0.3, 0.5, 0.7 and 0.95 under normal theory conditions were obtained by simulation and compared with those given by the distribution of t(5p).
T a b l e 4 . 3 . 1
R a n g e o f v a l u e s o f P r a t a = 0 . 0 5 , m e a n a n d v a r i a n c e o f t R w h e n n = 1 0 a n d r = 1 . 0 f o r a o n e s a m p l e t - t e s t
Power Pr (s i m u l a t io n ) P R ( t ( 5 R ) ) eI 'r)* V [tR]*
P lower t a i l upper t a i l lower & upper
t a i 1 sim u l- at iont a t ion^s im u l- 0.30 0.279-0.299 0.276-0.2 98 0.285 1.27 1.28 1.40 1.39 0.50 0.474-0.489 0.464-0.4 89 0.474 1.87 1.87 1.52 1.55 0.70 0.663-0.684 0.663-0.683 0.669 2.47 2.47 1.78 1.74 0.95 0.933-0.940 0.935-0.941 0.931 3.75 3.76 2.38 2.32
The mean and variance are given only for the upper tail. The values for lower tail were very similar with a change in sign for E[tR].
t The simulated values of the mean and variance differ only in their third decimal place for values of c.
Table 4.3.2
R a n g e o f v a l u e s o f P r a t a = 0. 0 5 , m e a n a n d v a r i a n c e o f t R w h e n n = 2 5 a n d r = 1 . 5 f o r a o n e s a m p l e t - t e s t
Power Pr (sim ul ati on) E l t R l * V [ t R] *
P lower t a i l upper t a i l lower & upper simu l- slmul-
t a i l at ion t ( « R > 1at ion K «r)
0.30 0.259-0.281 0 . 2 5 7- 0 .2 84 0.265 1 . 0 9 - 1 . 1 0 1.09 1 . 1 2 - 1 . 1 3 1.12 0.50. 0.435-0 .466 0 . 4 4 3- 0 .4 64 0.446 1. 6 0 - 1 . 6 1 1.61 1. 1 4 - 1 . 1 5 1.13 0.70 0.624-0 .654 0.62 3-0 .65 1 0.635 2. 1 0 - 2 . 1 1 2.12 1 . 1 9 - 1 . 2 0 1.18 0.95 0.911-0 .924 0. 91 0-0 .92 1 0.915 3 . 2 0 - 3 . 2 1 3.21 1 . 3 2 - 1 . 3 3 1.33
The mean and variance are given only for the upper tail. The values for lower tail were very similar with a change of sign for E[tR].
Tables (4.3.1) and (4.3.2) show a selection of results for a = 0.05. They illustrate the close agreement between the simulated results and those obtained from the distribution of t(5R). The value of the means and variances indicate that the distribution of t(5R) will closely approximate that of tR. This is also evident by the close agreement of the Pr values from simulation and the distribution of U Sr ) .
To obtain the Pr values from the distribution of t(d > R ), tables given by Owen (1965) were used.
The range in the simulated values of the mean, variance and Pr values is caused by the lattice position c.
significance, if the value of r was in the recommended range. It is reasonable to conclude that if the value of r is in the recommended range given in Table (3.3.2)
the distribution of t( 5r) will be a good approximation to tR. For values of r
outside the range of values given in Table (3.3.2) the simulation results showed that the distribution of tR became progressively unlike a non-central t distribution.
Simulation Results
In section (3.3.1) the recommended ranges of r for which the significance level of the test was found acceptable under rounding for a = 0.05 (one tailed) were: r < 0.5 when n = 5, r < 1.0 when n = 10 and r < 1.5 when n = 25. The power of the test will be generally more affected by the rounding process at the maximum value of r allowed. As a result P r values for the maximum value of r are of most interest. This would indicate the worst possible effect that rounding can have on the power of a test within the recommended range of r. For a = 0.05, values of P r for n = 5 at r = 0.5, n = 10 at r = 1.0 and n = 25 at r = 1.5 are shown in Tables (4.3.1 to 4.3.3).
Table 4.3.3
Range of values of Pr for a single sample t-test for a = 0.05, n = 5, r = 0.5
p PR lower t a i l upper t a i l 0.30 0.50 0.70 0.95 0 .294-0.306 0.494-0.505 0.693-0.701 0 .946-0.949 0 .292-0.304 0 .491-0.502 0 .692-0.699 0 .947-0.949 4.9
As expected, rounding has caused a reduction in the power of the test. As indicated by (4.3-4), as the value of r increased, the reduction in the power becomes greater. In general the results indicate that the four power levels are not adversely affected by rounding if the values of r are in the recommended ranges for a = 0.05 given in Table (3.3.2).
Although in this section the power of tR test has been found by simulation, an estimate of this power can be obtained from tables or from an approximation for values of r. Assuming that values of r are within the ranges given in Table (3.3.1) then tables or a suitable approximation can be used to estimate the power Pr, which is given by P[t(5R) > tn_ 1>a]. For example, tables given by Owen
(1965) or an approximation to the cumulative distribution of the non-central t in Johnson and Kotz (1970) may be used.
4 . 3 . 2 C h i - s q u a r e d t e s t f o r a v a r i a n c e
Let X = (X1 ,...,X n) be a random sample of size n from a normal population X.
Let X r = (X rp .^ X rh ) be the rounded sample where Xrj is the value of Xj