investigates the effect of rounding on a test statistic where the population is assumed to be exponential.
5.2 Description of the Investigation
As in Chapter 2, the family of Johnson distributions is taken to represent the family of non-normal distributions, and we use the same set of 29 Johnson distributions as were used by Pearson and Please (1975). The four added to this set in Chapter 2 to represent U shaped distributions will not be considered. This study deals with only moderate departures from normality; where the population is very non-normal one may argue that such tests as those in Chapter 3 should not be used.
In this present study the significance and power level were evaluated for each statistical test in Chapter 3. The following results were obtained by simulation, with sample sizes of n = 10 and 25:
(i) for each Johnson distribution, the significance level of the test under rounding was evaluated for values corresponding to the lower and upper 5% points under normal theory conditions, with no rounding;
(ii) for a selection of Johnson distributions, the power of the test statistic under rounding was evaluated for values of the alternative hypothesis H1
The usual lattice positions c = -0.5, -0.4, ..., 0.4, 0.5 were considered for each value of r. These eleven lattice values will indicate the effect of the position of the rounding lattice on the significance level and power of a test.
The value of a = 0.05 was chosen as this is normally the first level of significance at which the null hypothesis is rejected. To keep the computing within reasonable bounds, the study of the power was restricted. The power was evaluated at a 'low' and 'high' level for a = 0.05 for a selection of Johnson distributions. Sample sizes larger than 25 were not considered, as there is then a reasonable chance that non-normality can be detected and some corrective action taken.
The emphasis of this chapter is to provide guidance on what happens to the significance level and power of the tests in Chapter 3, if the values of r which were recommended for normal populations are applied when the population is non-normal. Essentially how far can the degree of precision r recommended for normal populations be applied to the non-normal situation? Hence the values of r considered will be in the vicinity of those which were recommended for sample sizes 10 and 25, when the population is normal. In this study the following approaches were used:
(a) The effect of rounding on the significance level of a test
(i) Approximations to the sampling moments of the test statistics were examined to provide a rough outline of what characteristics are to be expected when sampling from rounded non-normal data.
(ii) Estimation of the sampling distribution of the test statistic for rounded data was obtained by Monte Carlo methods.
(b) The effect of rounding on the power of a test
As in Chapter 4 for normal populations, only Monte Carlo methods were used to estimate the power of a test for rounded non-normal data.
In this investigation the significance level and power for each test for samples drawn from unrounded Johnson distributions were also obtained. Pearson and Please (1975) in their work on the robustness of normal test statistics for unrounded data, considered the same set of Johnson distributions used in this study. They used simulation to consider the effect of non-normality on the significance level of a test statistic. The results were presented in the form of charts.
In addition to test statistics investigated by Pearson and Please, the one and two-way analysis of variance are considered. Also the robustness of a test statistic is looked at from both a level of significance and power aspect for unrounded data.
To perform the necessary analysis two previous FORTRAN programs were adapted. The programs SIMUL and PSIMUL were modified to allow samples to be drawn from Johnson distributions as well as from normal distributions. As the significance and power level for each test for samples drawn from unrounded Johnson distributions were also required, a further program USIMUL was written.
The results of the SIMUL, USIMUL and PSIMUL programs were based on 10,000 iterations, which gave adequate precision for the 0.05 level of significance, and the
subject to sampling errors. However, these errors will be small for 10,000 iterations.
Quality of Results
The SIMUL and PSIMUL programs were tested to check validity of their results. As both programs were adapted from earlier programs only the generation of deviates from the Johnson distributions had to be checked. The USIMUL program was checked by comparing the results with those of Pearson and Please (1975). Although their results were in diagrammatic form, there was no apparent disagreement between these results and those given by the USIMUL program. A final check was established by comparing the results from SIMUL and PSIMUL programs, where the Johnson distributions are not subject to rounding, with the results from the USIMUL program.
5.3 Test Statistics
In this section it is assumed without any loss of generality that in the null case (H 0) the Johnson distributions have a mean of zero and variance one. The non-null case (H 1) was handled by adjusting the parameters in the standardised distributions to give the required power under normal theory conditions. Throughout this section a will denote the level of significance of the test for samples drawn from normal populations subject to no rounding, while a j and ojr
will be the resulting levels of significance of the test where the samples have been drawn fron non-rounded and rounded Johnson populations respectively. Thus a j and a jR are simply the probabilities that the test statistic fell above or below the a significance level limits for non-rounded and rounded data respectively. A