Hence using (3.3-3) and (3.3-4) approximations to the moments of t R , when X
has a standard normal distribution, are given by:
E [ t R] = 0 - 0 (n - 3 / 2) V [ t R] - 1 + \ + 0 ( n - 2 )
( 3 . 3 - 5 ) y/3,(tR) - 0 - 0 (n-3 / 2)
^(32( t R) = 3 + 2 [3 + ^ / [ l + Y2] ] +
If the normal population is subject to no rounding we have:
E [ t ] = 0 . r r .. n - 1 .. 2 . V [ t ] = — =■ L J n-3 = 1 + — + 0 ( n 2 )n ( 3 .3 -6 ) t ) = 0
^ (t) - f ^ i r ' 3 + 1 +
0 ( n ' 2 )Study of the moments of tR (3.3-5) and t (3.3-6) shows that in terms of order n” 1, rounding affects only the kurtosis, and that the effect is clearly negligible for r < 2.0. These moment results suggest that the distribution of t and tR are similar. However the moment results are for large n and the discontinuities in tR have a serious effect on the significance level of the test for small n.
EXACT/SIMUL Results
n < 5
In Appendix B, Table (B .l) shows the range in values of o r for n = 5, where r = 2.0, 1.0 and 0.5.
Like the distributions of all functions of rounded observations, the distribution of tR is discontinuous. Generally for a given r, as n decreases in size the discontinuities in tR become more numerous in any given interval and the steps increase in size. A further complication as n becomes small is that the probability of S 2r = 0 increases. This causes t R to be either +oo or -a > , according to the sign of the
numerator of (3.3-2). The exception is where Xr equals the population mean, when tR = 0/0, which can be defined to be equal to zero. This seems a sensible
population mean and under H0 we would expect tR = 0. Table (3.3.1) shows that values t^ = +00 or 0 / 0 occur with annoying frequency in very small samples, especially for coarse rounding (r > 1). The value of c is important in determining the probability of such values of tr for r > 1.0. For such small values of n, the infinite values of tR caused an inbalance in the values of o r between the upper and lower tails. As expected increasing n or decreasing r caused the inbalance to become low. As shown by the tR values for n = 5 in Table (B .l), this inbalance is very noticeable for r = 2.0.
T a b l e 3 . 3 . 1 P r o b a b i l i t y t h a t t R = + < » o r 0 / 0 f o r s a m p l e s o f s i z e n d r a w n f r o m a r o u n d e d n o r m a l p o p u l a t i o n i n a s i n g l e s a m p l e t - t e s t . r 2 1 .5 1 . 0 0 .5 0.25
X
0 0.5 0 0.5 al 1 c a l l c al 1 c 2 0.511 0.457 0.391 0.384 0.271 0.140 0.070 3 0.326 0.271 0.183 0.163 0.085 0 . 0 2 2 0.006 4 0.218 0.104 0.094 0.071 0.028 0.004 0 . 0 0 0 5 0.148 0.049 0.051 0.031 0.009 0 . 0 0 1 6 0.125 0.024 0.028 0.013 0.003 0 . 0 0 0 7 0.069 0 . 0 1 1 0.015 0.006 0 . 0 0 1 8 0.047 0.005 0.008 0 . 0 0 2 0 . 0 0 0Note: The probabilities in Table (5.3.1) are also the probability that S 2r is equal to zero.
For small n, tR will have a large number of discontinuities, especially for coarse rounding. This will result in the distribution of tR being a poor approximation to the distribution of t. We would expect the a and o r values to be considerably different unless the value of r is low. The results for o r confirm this. For example, as shown by the results in Table (B .l), a and o r are only in close agreement for n = 5, where r < 0.5.
A clear pattern emerged in the values of o r, which were caused by the influence of the lattice. For c in the range [-0.5,0), the lower tail values of o r are the same as the upper tail values, for c in the range (0,+0.5]. For c = 0 and ± 0.5 the upper and lower tail values of o r are the same. An example of this pattern is seen in Table (B .l). For the one sample t-test, this pattern will always occur when the unrounded distribution is symmetrical.
n = 1 0
Table (B.2) shows the range in values of c r for n = 10, where r = 2.0, 1.5, 1.0
and 0.5.
For this size sample values of tR = +co or 0/0 are no real problem. Only for r > 1.5 is there a strong disagreement between the a and o r values.
n = 25
Table (B.3) shows the range in values of o r for n = 25, where r = 2.0 and 1.5. Also given in the corresponding range in the mean and variance of t R . Even for
be very similar. This confirmed the results from the moments (3.3-5), which indicated the mean and variance of tR will be very close to those for t for large n and r < 2.0. For n as large as 25, the distribution of tR will closely approximate that of t, even for coarse rounding. This is evident from the results in Table (B.3) where for practical purposes there is very little difference between the values of o r and a for rounding as coarse as r = 1.5.
Table (3.3.2) gives the values of the degree of precision r that may be regarded as acceptable for n = 5, 10 and 25 (definition of 'acceptable' in section 3.3). In this table, the column 0.1/1.0/5.0 gives the range of r which is acceptable for these three levels of significance. Column 0.1/5.0 gives the range of r which is acceptable at both these levels of significance. Column 5.0 gives the range of r which is acceptable for just this level of significance.
T a b l e 3 . 3 . 2 T h e v a l u e s o f t h e d e g r e e o f p r e c i s i o n r t h a t m a y b e r e g a r d e d a s a c c e p t a b l e f o r n = 5 , 1 0 a n d 2 5 i n a o n e s a m p l e t - t e s t . One t a i l e d t e s t Two t a i l e d t e s t a(°/o) 0 . 1 / 1 . 0 / 5 . 0 1 . 0 / 5 . 0 5 . 0 5 . 0 n LT UT LT UT LT UT 5 r < 0 . 5 r < 0 . 5 r < 0 . 5 r < 0 . 5 r < 0 . 5 r < 0 . 5 r < 0 . 5 10 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 25 r < 1 . 5 r < 1 . 5 r < 1 . 5 r < 1 . 5 r < 1 . 5 r < 1 . 5 r < 1 . 5
Note: LT = lower tail U T = upper tail 3.16
Immediately noticeable from Table (3.3.2) is that there is no difference in the ranges of r between lower and upper tails. As n increases in size, the degree of precision of the data can be decreased without any further deterioration in the significance level.
3 . 3 . 2 C h i - s q u a r e d t e s t f o r 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 ^ , . . . ^ ) be the rounded sample where X r j is the rounded value