Fuzzy Approach for Group Sequential Test

Buckley’s approach (Buckley (2004), (2005), (2006)) uses sets of confidence intervals by taking into consideration both of the uncertainty and impreciseness of concepts that produce triangular shaped fuzzy numbers for the estimator.This approach produces fuzzy test statistics and fuzzy critical values in hypothesis testing. In addition, the sample size is fixed for this test. When data comes sequentially, however, it is not suitable to study with a fixed sample size test. In such cases, sequential and group sequential tests are recommended. Unlike a sequential test, a group of sequential test provides substantial savings in sample and enables us to make decisions as early as possible. This intends paper to combine the benefits of group sequential test and Buckley’s approach using α-cuts. It attempts to show that using α-cuts can be used within the group sequential tests. To illustrate the test more explicitly a numerical example is also given.


Introduction
Estimation of unknown parameters of statistical models or testing of statistical hypothesis in fuzzy environments are interesting subjects for different approaches.So far Zadeh [1], Taheri and Behboodian [2,3], Taheri [4], Torabi and Behboodian [5], and Taheri and Arefi [6] have worked on these issues.Using a different point of view, Buckley [7][8][9] developed an approach which uses a set of confidence intervals.In addition, many studies have been done to combine several statistical methods and fuzzy sets, called fuzzy statistics, such as regression analysis, time series analysis, design of experiments, probability theory, conjoint analysis, and control charts [10].Since statistical tests based on fuzzy test statistics are more flexible than classical tests, they seemed to be competitive tools in certain situations; for example, when the observed value of the test statistic is close to the quantile of the test statistic [6].
Fixed sample size test is not useful where subjects enter the study sequentially.And thus the accumulated data can be analyzed sequentially.Wald [11] introduced a sequential probability ratio test (SPRT) which requires substantially fewer observations than a fixed sample size test.Several authors such as Torabi and Behboodian [12] have proposed fuzzy sequential probability ratio test.Talukdar and Baruah [13] have fuzzified the SPRT.Torabi and Mirhosseini [14] introduced the SPRT for fuzzy hypotheses testing.Jamkhaneh and Gildeh [15] presented a new approach for SPRT based on fuzzy hypothesis.
SPRT has no finite maximum number of observation; it is generally considered inappropriate for clinical trials [16].Group sequential tests (GST) are generally more practical and they give more possible savings than SPRT when it is impractical to perform an interim analysis after each new observation [17].GST are widely used in clinical trials.For ethical, scientific, and economic reasons, clinical trials are often repeatedly monitored for evidence of treatment benefit or harm.To achieve this, statisticians conduct interm analysis periodically on accumulating data [18].Various group sequential testing procedures have been proposed to achieve the desired levels of type-I error [18][19][20].Much of the development of GST are reviewed in detail by Jennison and Turnbull [17].
None of the studies mentioned above consider Buckley's approach for group sequential tests [7][8][9].In this study, our aim is to use a group sequential test using -cuts inspired by Buckley's approach when response variable has a normal distribution with known variance.
This paper is organized as follows: some preliminaries for fuzzy numbers are presented in Section 2. Buckley's approach 2 Advances in Fuzzy Systems using -cuts for hypothesis testing is briefly reviewed in Section 3. Then Pocock's group sequential test and using Buckley's approach within Pocock's group sequential test for a normal response with known variance are given in Section 4. A numerical example is given in Section 5. Results and discussions are given in Section 6.Finally, concluding remarks and some possible future perspectives are presented in Section 7.

Preliminaries
This section contains some definitions of fuzzy sets and fuzzy numbers defined by Dubois and Prade [21] and Buckley [7][8][9].
where  /2 is the  value that the probability of a (0, 1) random variable exceeding it is /2 ( is the type-I error).
Then inequality is solved to produce  that is given as follows: This leads directly to the (1 − ) 100% confidence interval for where and (0, 1) denotes the normal density with mean zero and unit variance.With putting these confidence intervals one on top of other, we obtain X that is the fuzzy estimator of  [7][8][9] whose -cuts are confidence intervals as for 0.01 ≤  ≤ 1. Hence we obtain the fuzzy estimator of .

Hypothesis Testing Using 𝛼-Cuts
Buckley's approach [7][8][9] uses set of confidence intervals producing a triangular shaped fuzzy number for the estimator.Therefore this approach produces a fuzzy test statistic and fuzzy critical values in fuzzy hypothesis testing.
In this section the classical hypothesis test, based on fixed sample size of  from (,  2 ) mean and variance known, is given.The following hypothesis  0 :  =  0 against  1 :  ̸ =  0 is going to be tested at significance level .
From the random sample its mean is computed as  and then the test statistic is determined as Let , 0 <  < 1, denote the significance level of the test.Now under the null hypothesis  0 ,  0 is (0, 1) and decision rule is (1) reject  0 if  0 ≥  /2 ; (2) reject  0 if  0 ≤ − /2 , and (3) do not reject  0 when − /2 <  0 <  /2 .In the above decision rule  /2 is the  value so that the probability of a random variable having the (0, 1) probability density exceeding  is /2 ( is the type-I error rate) [7].
If the uncertainty is taken into account for parameter , triangular shaped fuzzy number X and its -cuts ( ∈ [0.01, 1]) can be given with Definition 4 in ( 7) Calculations are performed by -cuts and interval arithmetic.Substituting the bounds of X[] into (7), all -cuts of Z can be given with ( 8) Advances in Fuzzy Systems Each -cut is put one over the other in order to get a triangular fuzzy test statistic Z[] which is given in Figure 1.
Since the test statistics are fuzzy the critical values will also be fuzzy.There will be two fuzzy critical value sets: (1) let The end points of an -cut of CV 2 are computed from the end points of the corresponding -cut of Z with the following equations: Hence under  0 ,  0 is (0, 1) so By using the left end point of Z[] in (9), we have Hence -cuts of CV 2 and CV 1 are given by respectively [7,8].Both CV 1 and CV 2 are triangular shaped fuzzy numbers.Since the crisp test statistics has a normal distribution, CV 1 = − CV 2 because this density is symmetric with respect to zero [7][8][9].The final decision depends on the relationship between Z and CV 1 and CV 2 : (1) Z > CV 2 reject  0 (Figure 2 ).These situations are explained in detail as follows.For example, if Z > CV 2 , draw Z to the right of CV 2 , then find the height of the intersection as which measures how much CV 2 () is less than or equal to Z().Thus,  2(d) and 2(e)) then the final decision is "no decision" on  0 .This is because of the fuzzy numbers that incorporate all uncertainty in confidence intervals [7][8][9].Consequently, hypothesis testing based on fuzzy test statistic and fuzzy critical values that is described above is more realistic and provides more benefits when value of the test statistic is very near to the quantile of the test statistic.

Group Sequential Test Using 𝛼-Cuts
In Pocock's group sequential test, subject entry is divided into  equally sized groups containing  subject on each treatment and the data are analyzed after each new group [20].Consider the response variables to be normal with unknown means  1 and  2 with a common variance  2 .Subjects are randomized sequentially into two treatment groups.
The paper is planned as a test of the null hypothesis  0 :  1 =  2 against the two sided alternative  1 :  1 ̸ =  2 .Let  1 and  2 be the observed mean responses in the th group of 2 subjects, then the statistics, are normally distributed with mean Δ and unit variance, where Δ = (  ) = √( 1 − 2 )/ √ 2 2 .Therefore test statistic is defined with   = ∑  =1   and  = 1, 2, . . ., , where   , under  0 ,   ∼ (0, ), is a partial sum process of independent identically distributed (i.i.d.) standard normal random variables   ∼ (0, 1),  = 1, 2, . . ., .Under  1 ,   is again partial sum of i.i.d.normal random variables,   ∼ (Δ, 1),  = 1, 2, . . ., ,   ∼ (Δ, ).Critical values of Pococks's GST (  (, )) are given in Table 1 for  = 1, . . ., 5; 1 −  = 0.95, and  = 0.10.Then GST process is as follows: ( The value of noncentrality parameter, Δ, can be determined to achieve a given value of 1 −  [17].Corresponding to a specific Δ under  1 to be detected, the required sample size per treatment per stage, , at given  and 1 −  with a maximum of  stages, is obtained as The maximum sample size, , to find Δ for maximum of  stages is simply Group sequential tests can also be used in an experiment with only one treatment in which the response results are compared with a known standard.For a normal response with a known variance and  0 hypothesis mean  0 , the critical values   and Δ become √( 1 − 0 )/ [17,22].When  = 1, that is, for fixed sample size test, (17) becomes the classical hypothesis test for normal response.
The test statistic defined with ( 14) can also be given as follows: Suppose that   = ( 1 −  2 )/ √ 2  = 1, 2, . . .,  and   ∼ (,  2 ), where  = ( 1 −  2 )/ √ 2 [23].Therefore the mean of  group of  subject   can be defined as   = (∑  =1   )/.Now proceed to the fuzzy situation of X that is explained for each group of  subject in (5) as triangular shaped fuzzy number is defined as Substituting the -cuts of X into (19) and using interval arithmetic -cuts of the fuzzy test statistics are obtained as follows: In the sense of this fuzzification fuzzy group sequential test statistic, S can be defined as The defined test statistic sample size fuzzy test statistic for  = 1 which is given with (7).Afterwards, -cuts of Pococks's fuzzy critical value ( PCV  ,  = 1, 2) can be calculated.The following equations include calculations for PCV 2 [] and PCV 1 [], respectively: Hence pcv 22 () = (  (, ) +  /2 ) .
By using the left end point of Z [] given with (21) it is possible to have As a result we obtain -cuts of PCV 2 [] with (27) In the above equation for PCV 2 ,  is fixed and  ranges in the interval Both PCV 1 and PCV 2 are triangular shaped fuzzy numbers.
The final decision depends on the relationship between S and PCV 1 and PCV 2 .Therefore fuzzy group sequential test process is as follows: (1) after group  = 1, 2, . . .,  − 1 if S < PCV 1 or S > PCV 2 ; stop reject  0 , otherwise; continue to  + 1, (2) after group  if S < PCV 1 or S > PCV 2 ; stop, reject  0 , if PCV 1 < S < PCV 2 ; accept  0 , if S ≈ PCV  for  = 1, 2; no decision. (30) These situations are explained in detail in Figure 3.The final decision depends on the relationship between S and PCV 1 and PCV 2 : (1) S > PCV 2 reject  0 (Figure 3 3(d) and 3(e)).We take the height of the intersection  = 0.8, which measures how much PCV  ,  = 1, 2 is less than, bigger than, or equal to S .The advantage of -cuts approach to GST is that, instead of generating and processing a single confidence interval, all the confidence intervals at the same time are calculated in the process of corresponding fuzzy test statistic.Therefore in this study we showed that this advantage is also valid for the process of group sequential tests using -cuts.

Numerical Example
As an illustration we consider a real data in McCleve and Sincich [24] on page 381 given with Table 2.The content of data is as follows: a new developed diet, that is, low in fats, carbonhydrates, and cholesterol, is intended to be used by people with heart disease.Furthermore the dietitian wishes to examine the effect that this diet has on the weights of obese people.Hence this data set concerns two groups which are called low fat diet (A) and regular diet (B), respectively.We want to test  0 :  A −  B = 0 with  =  =0.10,  2 = 50.063,and  = 3 at  1 :  A −  B = ±5; sample size which is calculated from (17) is  A =  B = 17 per treatment per stage with maximum sample size ( max ) = 102 [23].The fuzzy group sequential test statistics S1 , S2 , and S3 and fuzzy Pocock's critical values PCV  ,  = 1, 2, 3 are given in Table 3.
The test statistic and critical value that is calculated in each stage is recalculated using Buckley's approach.Hence each -cut value is obtained for test statistic and critical value (for  = 0.01, 0.2, 0.4, 0.6, 0.8, and 1.00).-cuts, base and peak values of S1 , S2 , and S3 , and critical values that are calculated in each stage are detailed in Table 4. Thus, as suggested in Buckley's approach, not only one value but also more than one confidence interval is used, in order to test hypothesis, so that more information is included in group sequential test process.The results are obtained using the Maple 9 [25].

Table 2 :
Diet study data.