Compensation for Rounding

The chi-squared test statistic has been found to be the least robust to rounding. Both the significance level and power of the test were found to be very sensitive to rounding. This sensitivity was a result of the mean and variance of the test statistic being increased for rounded data. The increases in the mean and variance are approximately of the same order for all n; thus it may be possible to use a standard correction to the test statistic to compensate for the rounding effect. If the denominator in (3.3-8) is changed to c2 + w 2/12 then the adjusted test

statistic, denoted by x 2C

(n -l)S R

2 , w 2

O'o + 1 2

( 4 .3 - 1 8 )

From (3.3-10) the approximate mean and variance of the distribution of x 2C

are

respectively (n -1) and 2(n -l).

This adjustment to the test statistic will compensate for the effect of rounding. For large enough n (4.3-18) will be in closer agreement with a chi-squared distribution than is (3.3-8). For small n, the distribution of x 2C

may

combinations of (n,r).

Simulation Results

For n < 5 the distribution of x 2C was f°und to be in poor agreement with a chi-squared sistribution. The probability of x 2C ~ 0 was st^ high for this size of n and resulted in the a values being severely distorted by rounding. For n > 5 the x 2C test statistic gave a closer agreement between the a and o r values than did

x 2R

x2c

Table 4.3.18

Minimum and maximum values of G r(% ) found for 11 values of c in the x 2R and

X2c test statistics

n = 1 0

a(°/o) lower t a i l a(°/o) upper t a i l

r s t a t i s t i ctest 0 . 1 1 . 0 2.5 5.0 5.0 2.5 1 . 0 0 . 1 1.5 _X2C min 0.05 0.58 0.58 2.72 3.06 2.19 0.69 0.03 max 0.24 2 . 1 1 2 . 1 1 5.43 3.76 2.54 0.95 0.07 X 2R min 0.05 0.58 0.58 0.58 11.58 4.76 2.52 0.40 max 0.24 2 . 1 1 2 . 1 1 2 . 1 1 12.84 6.04 2.77 0.43 1 . 0 _{x 2c} min 0.06 0.99 3.32 5.04 5.03 2 . 2 1 1.04 0.09 max 0 . 1 0 1.07 3.33 5.26 5.18 2.35 1.06 0.09 - x2r min 0.06 0.44 2.05 3.32 6.72 3.74 1.75 0.23 max 0.09 0.60 2.47 3.33 6.75 3.77 1.77 0.23 n = 25

a(°/o) lower t a i l a(°/o) upper t a i l r s tat is t ictest 0 . 1 1 . 0 2.5 5 .0 5.0 2 .5 1 . 0 0 . 1 1.5 _{X 2C} min 0 . 0 2 0.33 2.13 3.68 4.80 2.27 0.85 0.09 max 0 . 2 0 0.95 2.90 4.81 5.21 2.47 0.94 0 . 1 0 x2r min 0 . 0 0 0.06 0.33 1.31 15.24 9.71 4.81 0.85 max 0.06 0.37 0.95 2.17 15.46 10.72 5.21 0.94 1 . 0 _{x 2c} min 0.09 0.09 2 . 2 2 4.83 4.94 2.36 0.90 0.09 max 0.13 0.98 2.55 5.21 5.12 2.39 1.04 0 . 1 1 x2r min 0.05 0.57 1.43 2 . 8 8 8.74 5.04 2.23 0.30 i max 0.07 0.63 1.72 3.17 9.39 5.34 2.35 0.32 4.31

Table (4.3.18) shows that the difference between a and o r values is less for the X 2c test statistic. This is especially true in the upper tail. As n increased in value, x 2C became a better approximation to a chi-squared distribution with n-1 degrees of freedom. This is apparent by comparing the values of a and aR for n = 10 and 25 in Table (4.3.18).

In using the adjusted test statistic x2C> ^ was possible to extend the range of r for which the significance level of the test will be acceptable. Table (4.3.19) shows the values of r for which or was found acceptable for n = 5, 10 and 25.

T a b l e 4 . 3 . 1 9

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 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 u s i n g t h e

x2C

One t a i l e d test Two

t a i l e d test 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 < 0.25 r < 0.5 r < 0.25 r < 0.5 r < 0.5 r < 0.5 1 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 25 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0 r < 1 . 0

In comparing Table (3.3.4) and (4.3.19), using x 2C a^ows the range of r to be extended to 1.0 for n = 10 and 25. For x2C» values of P r are shown in Table (4.3.20) for a = 0.05 (one tailed), for r = 1.0 where n = 10 and 25. These values of Pr indicate what loss in power to expect if the maximum recommended

reduction will be less.

T a b l e 4. 3 . 2 0

R a n g e o f v a l u e s o f P r a t a = 0 . 0 5 f o r t h e t e s t s t a t i s t i c s \ 2C ^ o r r = 1 . 0 w h e r e n = 1 0 a n d 2 5

n = _{1 0} r = _{1 . 0} n = 25 r = _{1 . 0}

p 1<Dwer ta i 1 upper ta i 1 lower t a i 1 upper t.ai 1 0 30 0 .237-- 0 .274 0 276-- 0.278 0 .259-- 0 .274 0 270-- 0.275 0 50 0 .389-- 0 .446 0 464-- 0.465 0 .424-- 0 .444 0 457-- 0.459 0 70 0 .554-- 0 .614 0 662-- 0.663 0 .604-- 0 .619 0 654-- 0.657 0 95 0 .836-- 0 .882 0 936-- 0.937 0 .880-- 0 .883 0 932-- 0.933

4 . 4 D i s c u s s i o n a n d C o n c l u s i o n s

In Chapter 3 recommended ranges or r were given for n = 5, 10 and 25, in which the significance level of the test may be considered acceptable. It is natural to investigate what the power of the test will be using these recommendations. Although the power of each test was considered mainly for a = 0.05, it provided a clear indication of the level of power we should expect under rounding.

For hypothesis tests concerned with means, rounding resulted in a loss of power. The main result of this section was that by adjusting the non-centrality parameter in the non-central distribution, an estimate of the power under rounding could be obtained. For values of r for which the significance level was found acceptable, this estimate of power for rounded data was found to be reasonably accurate. This estimate of power can be useful in practice, in providing an idea of the expected loss in power under rounding.

For the one sample t-test, for values of r for which the significance level was found acceptable, the power is still of an appreciable magnitude. However for the two sample t-test, one and two-way analysis of variance, a lower value of r than that recommended for the significance level may be necessary to limit the loss in power caused by rounding. Although for the analysis of variance only two layouts were considered, the results suggest the likely level of power to expect with such statistical procedures for rounded data.

Unlike hypothesis tests for means, those concerned with variances will not have a constant degree of precision r under H 1. This was found to make the power more sensitive to rounding. With the chi-squared test its lack of robustness to rounding meant that acceptable levels of significance were obtainable only for low values of r. For these low values of r, the power was not found to be adversely affected for a = 0.05. In general, the results indicate that the chi-squared test will be only slightly less powerful for values of r for which the significance level was found acceptable. However for the F-test, the reduction in the power could be severe for values of r for which the significance level was found acceptable. In order to maintain a more suitable level of power the recommended ranges for r had to be reduced.

The chi-squared test could be made more robust to rounding by making a simple adjustment to the test statistic. Test of hypothesis regarding the value of a 2 should be based on the adjusted test statistic, x2C- using this adjusted test statistic it was possible to extend the range of r for which the level of significance of the test is acceptable. However by extending the range of r there will be a corresponding loss in power.

In this chapter we have confined our attention to values of n < 25. However the power levels for n = 25 provide a good indication of what level of power to expect for larger sample sizes.

The results of Chapter 4 have provided a 'good feel' for the robustness of the tests considered in Chapter 3, with respect to power. More importantly we now know what level of power a test may have for the values of r for which the significance level was found acceptable.

CHAPTER 5

TH E EFFECT OF ROUNDING ON TH E SIGNIFICANCE LEVEL