Least significant spacing for ‘one versus the rest’ normal populations

When comparing k normal populations, an investigator might want to know the probability that the population with the largest population mean will have the largest sample mean. Put another way, what is the probability of correctly identifying the most effective treatment? The paper describes and illustrates methods of approximating this probability when the variances are unknown and possibly unequal. The results described here can also be used to measure the extent to which the populations differ for one another.

A procedure is given for the construction of a fixed-width confidence interval of the largest mean of k(k≥1) normal populations with a common known variance, based on the largest sample mean. The sample size required to achieve the preassigned confidence coefficient can be computed by using the standard normal table only.

SummaryThis paper gives a two‐sample procedure for selecting the m populations with the largest means from k normal populations with unknown variances. The method is a generalization of a recent work by Ofosu [1973] and hence should find wider practical applications. The experimenter takes an initial sample of preset size N0 from each population and computes an unbiased estimate of its variance. From this estimate he determines the second sample size for the population according to a table presented for this purpose. The populations associated with the m largest overall sample means will be selected. The procedure is shown to satisfy a confidence requirement similar to that of Ofosu.

SYNOPTIC ABSTRACTWe propose a two-stage procedure for comparing k normal populations with a normal control, when the populations have a common, unknown variance. The population with the largest mean is selected, if it is larger than the control's mean; otherwise, the control is selected. The proposed procedure is shown to satisfy the requirements: (1) the probability of selecting the control treatment is at least P0*, whenever the control's mean is larger than the k population means, and (2) the probability of selecting the population with the largest mean is at least P1*, whenever its mean is larger than the control's mean and the means of the other populations under consideration. The procedure has the attractive feature that it allows for a different sample size to be assigned to the control than is assigned to the other populations.

To explore the relationship between sample size in the groups and statistical power of ANOVA and Kruskal-Wallis H test with an imbalanced design. The sample sizes of the two tests were estimated by SAS program with given parameter settings, and Monte Carlo simulation was used to examine the changes in power when the total sample size varied or remained fixed. In ANOVA, when the total sample size was fixed, increasing the sample size in the group with a larger mean square error improved the statistical power, but an excessively large difference in the sample sizes between groups led to reduced power. When the total sample size was not fixed, a larger mean square error in the group with increased sample size was associated with a greater increase of the statistical power. In Kruskal-wallis H test, when the total sample size was fixed, increasing the sample size in groups with large mean square errors increased the statistical power irrespective of the sample size difference between the groups; when total sample size was not fixed, a larger mean square error in the group with increased sample size resulted in an increased statistical power, and the increment was similar to that for a fixed total sample size. The relationship between statistical power and sample size in groups is affected by the mean square error, and increasing the sample size in a group with a large mean square error increases the statistical power. In Kruskal-Wallis H test, increasing the sample size in a group with a large mean square error is more cost- effective than increasing the total sample size to improve the statistical power.

In Chapter 6 the t test was used to decide whether the difference between two sample means was too great for them to come from the same population. We now describe the test for the case of more than two samples (Note: applying the t test to the difference between largest and smallest mean will give the wrong answer). This test is based on the fact that the mean of a sample of size n from a normal population with given σ, is itself normally distributed with variance σ2/n (see Section 5.7). If, then, we have m samples, and their means have a greater variance than would be expected from m individuals from a population whose variance is σ2/n, we conclude that, very probably, there are some real physical differences between the means. Unfortunately, σ2 is not usually known and so has to be estimated from the variance within each sample (it is assumed that all population variances are the same).

Several estimators for the variance components of the above model are derived. Biases and mean square errors of the estimators for small samples are examined. Results on the skewness and kurtosis coefficients and the large sample biases and mean square errors of these estimators are presented in detail.

AIM: To determine the minimum number of samples required to obtain a robust estimate of the Cu and Se status of dairy herds, as assessed by determining liver Cu and serum Se concentrations. METHODS: Results were collated from analyses of samples of liver from 18 dairy herds and serum from 19 herds, for concentrations of Cu and Se, respectively. All herds were in either the Manawatu or Rangitikei regions of the North Island of New Zealand. Data were used to determine the required sample size for each herd; firstly to estimate the population mean with 90% confidence with a precision of 27.5 nmol/L for Se in serum, and 100 μmol/kg fresh weight (FW) for Cu in liver; and secondly to ensure that the 90% CI of the sample mean did not include specified thresholds for concentrations of Se or Cu. RESULTS: For Se concentration in serum, the SD of each batch varied from 0.5–147 nmol/L, and for Cu concentration in liver, the SD varied from 173–829 μmol/kg FW. For Se, the minimum sample size required to estimate the population mean to within 27.5 nmol/L with 90% confidence was >10 for 13/19 batches. For Cu, the minimum sample size required to estimate the population mean to within 100 μmol/kg FW was >10 for 17/18 batches. When estimating required sample size based on 90% CI and a threshold value, the minimum sample size to confirm the population mean of Se was >140 nmol/L was four in 17/18 batches where the sample mean was>140 nmol/L. For concentrations of Cu in liver, ≤8 samples would have been sufficient for a threshold of 45 μmol/kg FW in 16/18 batches. For the 95 μmol/kg threshold, the minimum required was 12. For the threshold of 300 μmol/kg FW, 6/17 batches with a mean >300 μmol/kg FW required≤20 samples. CONCLUSIONS: From this dataset of 21 herds, the sample size recommendation for ensuring that the population mean of Se concentration was not below the marginal threshold was similar to previous recommendations. For Cu concentrations in liver, the estimated sample size recommendations for ensuring that the population mean was not below the marginal threshold was much larger than currently recommended. CLINICAL RELEVANCE: In dairy cattle, five to six blood samples per group should be taken to determine Se status, and to effectively monitor Cu status a minimum of 12 liver samples should be taken, preferably in the autumn.

The problem of selecting the component having the largest population mean in a k-variate normal population is discussed when all the means, variances and correlations are unknown. The indifference-zone formulation has been adopted, and we propose a two-stage procedure (PS) that guarantees the nominal value (P∗) of probability of correct selection under the assumption that all the correlations are non-negative. The procedure PS is the only method thus far proposed that will accomplish the desired objective. It is also of some interest that PS is asymptotically (as δ∗ ↛ 0) better than the procedure PR of Dudewicz and Dalal (1975) for k = 2, P∗≤.95, while for k≥3 the procedure PR is better than PS when ρij = 0, i≠j. This comparison is naturally valid when both PR and PS are applicable, and of course there are situations where only PS is applicable.

This paper is a follow-up to an earlier article by the authors in which they proposed a two-stage procedure with screening to select the normal population with the largest population mean when the populations have a common known variance. The two-stage procedure has the highly desirable property that the expected total number of observations required by the procedure is always less than the total number of observations required by the corresponding single-stage procedure of Bechhofer (1954), regardless of the configuration of the population means. The present paper contains new results which make possible the more efficient implementation of the two-stage procedure. Tables for this purpose are given, and the improvements achieved (which are substantial) are assessed.

In a Bayesian factor analysis model proposed by Press and Shigemasu (1989), the sample size was assumed to be large enough to estimate the overall population mean by the sample mean. In this paper, the procedure of estimating the population mean by the sample mean is compared to estimating it along with the other parameters both by Gibbs sampling and Iterated Conditional Modes. Results show that even in small samples, the Gibbs sampling and Iterated Conditional Modes estimates of the mean are for practical purposes identical to the sample mean. Thus, the population mean is adequately estimated by its sample value.

This paper is divided into two parts. The significance tests investigated in Part I concern the population mean and are based on the quantity [(sample midrange)-(hypothetical mean)]/(sample range). The case in which the observations are a sample from a normal population is considered in detail. The tests investigated are summarized in Table 1. These tests are found to be very efficient for small samples (see Table 4, power efficiency is defined in section 3). An investigation of several extremely non-normal populations using the values of $D\alpha$ obtained for normality indicates that the significance level of the range-midrange test is not very sensitive to the requirement of normality for small samples (see Table 6). Also the tests of Table 1 can be applied without computation through the use of an easily constructed graph (see section 4). These properties suggest that the range-midrange test is preferable to the Student $t$-test and the analogue of the Student $t$-test using the sample range (see [1] and [2]) whenever the sample size is sufficiently small. Use of the range-midrange test for the case of normality was proposed by E. S. Pearson in [3], where properties of the test were experimentally investigated for the normal and certain non-normal populations. In Part II several significance tests for the mean are developed which have a specified significance level for the case of a sample from a normal population but whose significance level is bounded near the specified value under very general conditions, one of which is that the observations are from continuous symmetrical populations. Some of these tests are range-midrange tests. Table 2 contains a summary of the tests and their properties ($x_i = i$th largest observation, $i = 1, \cdots, n$; conditions $(D)$ are given in section 7).

In this paper, we consider the problem of making inferences on the common mean of several normal populations when sample sizes and population variances are possibly unequal. We are mainly concerned with testing hypothesis and constructing confidence interval for the common normal mean. Several researchers have considered this problem and many methods have been proposed based on the asymptotic or approximation results, generalized inferences, and exact pivotal methods. In addition, Chang and Pal (Comput Stat Data Anal 53:321–333, 2008) proposed a parametric bootstrap (PB) approach for this problem based on the maximum likelihood estimators. We also propose a PB approach for making inferences on the common normal mean under heteroscedasticity. The advantages of our method are: (i) it is much simpler than the PB test proposed by Chang and Pal (Comput Stat Data Anal 53:321–333, 2008) since our test statistic is not based on the maximum likelihood estimators which do not have explicit forms, (ii) inverting the acceptance region of test yields a genuine confidence interval in contrast to some exact methods such as the Fisher’s method, (iii) it works well in terms of controlling the Type I error rate for small sample sizes and the large number of populations in contrast to Chang and Pal (Comput Stat Data Anal 53:321–333, 2008) method, (iv) finally, it has higher power than recommended methods such as the Fisher’s exact method.

Given k normal populations with unknown means and a common known variance a two-stage procedure p1 with screening in the first stage to find the population with the largest mean using the indifference-zone approach is under concern. It was proposed and studied previously by Cohen (1959), Alam (1970) and Tamhane and Bechhofer (1977, 1979). But up to now a conjecture concerning the least favorable parameter configuration of p1remained unproved for k ≥ 3. In this paper we give a non-standard proof of the conjecture in case of k = 3 for p1. which (under minor changes) works also for a simplified version of p1,. Besides, a counterexample is provided to show that another (more intuitive) method of proof fails to work.

SUMMARY For some configurations of the means of k normal populations the probability of selecting the normal population with the largest mean does not increase when the number of observations from a given population is increased. We prove the statement and indicate some incorrect proofs in literature.