MOJ ISSN: 2379-6383MOJPH

Public Health
Review Article
Volume 4 Issue 4 - 2016
An Alternative Approach to Cochran Q Test for Dichotomous Data
Okeh UM1*, Oyeka ICA2 and Igwenagu CM3
1Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Nigeria
2Department of Statistics, Nnamdi Azikiwe University, Nigeria
3Department of Industrial Mathematics and Applied Statistics and Demography, Enugu State University of Science and Technology, Nigeria
Received:April 03, 2016 | Published: May 07, 2016
*Corresponding author: Okeh UM, Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Nigeria, Email:
Citation: Okeh UM, Oyeka ICA, Igwenagu CM (2016) An Alternative Approach to Cochran Q Test for Dichotomous Data. MOJ Public Health 4(4): 00086. DOI:10.15406/mojph.2016.04.00086

Abstract

This paper proposes, develops and presents a statistical method for the analysis of sample data where these data are dichotomous responses assuming only two possible mutually exclusive values such as 1, representing positive response say, and 0 representing negative response say. The Chi-square test statistic based on the Chi-square test for independence is developed as an alternative to the usual Cochran Q test for dichotomous data to test the null hypothesis of equal positive response rate by subjects to a set of test or treatment. The proposed method is illustrated with some sample data and shown to be at least as powerful as the usual Cochran Q test when applied to the same sample observations.

Keywords: Chi-square; Dichotomous data; Cochran Q test; Subjects; Responses; Success

Introduction

Sometimes a researcher may perform an experiment involving repeated observations or blocks, in which the variable of interest is dichotomous, meaning that it can assume only one of two possible mutually exclusive values. One of these two possible values is considered a ‘success’, positive response, present, well, etc. This is often coded as ‘1’,while the other value may be considered a failure, negative response, absent, bad, etc often coded a ‘0’ [1].

Research interest would then be to determine the proportions of subjects responding positive if the sampled blocks of subjects are the same across all treatments or tests. In this situation Cochran Q test [2-4] for the dichotomous data may be applied.

To adjust, and make allowance for some situations in which test or trial outcomes are not just dichotomous assuming only two mutually exclusive options such as 1 or 0,but when there may be some intermediate and third outcome such as unknown, indeterminate, non-definitive, etc, that may be code with say a minus sign(-). Oyeka, et al. [5] introduced a third category of response and developed a Chi-square ( χ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHhpWydaahaaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMca aaaa@3B5E@  test statistic for independence to test the null hypothesis of equal positive response rates under various treatments or test.

We will here however construct and alternative test statistic similar to Cochran Q test assuming that there are only two possible response outcomes or options that may be coded as either 1 or 0.

Methods

The proposed method would then be compared with the usual Cochran Q test as shown below.

The proposed method

Sometimes a researcher may be interested in comparing responses of blocks of subjects to a set of treatments in a diagnostic screened test or clinical trials. Specifically, suppose a researcher has collected a random sample of n block of subjects matched on some demographic characteristics such as age, sex, body weight, etc, where each block of subjects contain some c matched subjects and interest of the researcher is to administer each of these c subjects randomly one of c treatments. The subject responses to each of the treatments are all dichotomous assuming only one of two possible and mutually exclusive response options such as positive, or negative, present or absent, good or bad; success or failure, dead or alive, etc.

Let x ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaaa@3A3B@ be the response by a randomly selected subject from the ith block of subjects administered treatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGQbaabeaaaaa@37EA@  for i=1,2,…,n;j=1,2,…,c where each x ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38FC@  is dichotomous, either positive or negative, present or absent etc. To develop a test statistic to help determine whether on the average subject responses are the same for all treatments or conditions we may
Let
u ij ={ 1,ifarandomlyselectedsubjectfromtheithblockofsubjectsrespond positive,indicatingconditionpresent,goodwhenadministeredtreatment T j 0,ifthesesamesubjectrespondsnegative,indicatingconditionabsent,badwhen adminiterdtreatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaaiabg2da9maaceaabaqb aeqabiqaaaabaeqabaGaaGymaiaacYcacaWGPbGaamOzaiaaysW7ca WGHbGaaGjbVlaadkhacaWGHbGaamOBaiaadsgacaWGVbGaamyBaiaa dYgacaWG5bGaaGjbVlaadohacaWGLbGaamiBaiaadwgacaWGJbGaam iDaiaadwgacaWGKbGaaGjbVlaadohacaWG1bGaamOyaiaadQgacaWG LbGaam4yaiaadshacaaMe8UaamOzaiaadkhacaWGVbGaamyBaiaayk W7caaMe8UaamiDaiaadIgacaWGLbGaaGjbVlaadMgacaWG0bGaamiA aiaaysW7caWGIbGaamiBaiaad+gacaWGJbGaam4AaiaaysW7caWGVb GaamOzaiaaysW7caWGZbGaamyDaiaadkgacaWGQbGaamyzaiaadoga caWG0bGaam4CaiaaysW7caWGYbGaamyzaiaadohacaWGWbGaam4Bai aad6gacaWGKbaabaGaaGjbVlaadchacaWGVbGaam4CaiaadMgacaWG 0bGaamyAaiaadAhacaWGLbGaaiilaiaadMgacaWGUbGaamizaiaadM gacaWGJbGaamyyaiaadshacaWGPbGaamOBaiaadEgacaaMe8Uaam4y aiaad+gacaWGUbGaamizaiaadMgacaWG0bGaamyAaiaad+gacaWGUb GaaGjbVlaadchacaWGYbGaamyzaiaadohacaWGLbGaamOBaiaadsha caGGSaGaam4zaiaad+gacaWGVbGaamizaiaaysW7caWG3bGaamiAai aadwgacaWGUbGaaGjbVlaadggacaWGKbGaciyBaiaacMgacaGGUbGa amyAaiaadohacaWG0bGaamyzaiaadkhacaWGLbGaamizaiaaysW7ca WG0bGaamOCaiaadwgacaWGHbGaamiDaiaad2gacaWGLbGaamOBaiaa dshacaaMc8UaamivamaaBaaajuaibaGaamOAaaqcfayabaaaaqaabe qaaiaaicdacaGGSaGaamyAaiaadAgacaaMc8UaamiDaiaadIgacaWG LbGaam4CaiaadwgacaaMe8Uaam4CaiaadggacaWGTbGaamyzaiaays W7caWGZbGaamyDaiaadkgacaWGQbGaamyzaiaadogacaWG0bGaaGjb VlaadkhacaWGLbGaam4CaiaadchacaWGVbGaamOBaiaadsgacaWGZb GaaGjbVlaad6gacaWGLbGaam4zaiaadggacaWG0bGaamyAaiaadAha caWGLbGaaiilaiaadMgacaWGUbGaamizaiaadMgacaWGJbGaamyyai aadshacaWGPbGaamOBaiaadEgacaaMe8Uaam4yaiaad+gacaWGUbGa amizaiaadMgacaWG0bGaamyAaiaad+gacaWGUbGaaGjbVlaadggaca WGIbGaam4CaiaadwgacaWGUbGaamiDaiaacYcacaWGIbGaamyyaiaa dsgacaaMc8Uaam4DaiaadIgacaWGLbGaamOBaaqaaiaaysW7caWGHb GaamizaiGac2gacaGGPbGaaiOBaiaadMgacaWG0bGaamyzaiaadkha caWGKbGaaGPaVlaadshacaWGYbGaamyzaiaadggacaWG0bGaamyBai aadwgacaWGUbGaamiDaiaaysW7caWGubWaaSbaaKqbGeaacaWGQbaa juaGbeaaaaaaaiaawUhaaaaa@3701@  (1)

For i=1,2,..,n;j=1,2,..c.

Let π j + =P( u ij =1)(2) and W j = f .j + = i=1 n u ij (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGmbGaamyzaiaadshacaaMe8oabaGaeqiWda3aa0baaKqbGeaacaWG QbaabaGaey4kaScaaKqbakabg2da9iaadcfacaGGOaGaamyDamaaBa aajuaibaGaamyAaiaadQgaaKqbagqaaiabg2da9iaaigdacaGGPaGa aGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7ca aMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaa ysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaG jbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaiikaiaaikdacaGG PaaabaGaamyyaiaad6gacaWGKbaakeaajuaGcaWGxbWaaSbaaKqbGe aacaWGQbaajuaGbeaacqGH9aqpcaWGMbWaa0baaKqbGeaacaGGUaGa amOAaaqaaiabgUcaRaaajuaGcqGH9aqpdaaeWbqaaiaadwhadaWgaa qcfasaaiaadMgacaWGQbaajuaGbeaaaKqbGeaacaWGPbGaeyypa0Ja aGymaaqaaiaad6gaaKqbakabggHiLdGaaGjbVlaaysW7caaMe8UaaG jbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaM e8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaays W7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjb VlaacIcacaaIZaGaaiykaaaaaa@BC96@

Be the total number of positive responses, that is total number of 1’s by subjects administered treatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGQbaabeaakiaaysW7aaa@3981@ and

f .j 0 =n f .j + (4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaaiOlaiaadQgaaeaacaaIWaaaaKqbakabg2da9iaa d6gacqGHsislcaWGMbWaa0baaKqbGeaacaGGUaGaamOAaaqaaiabgU caRaaajuaGcaaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaM e8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaays W7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjb VlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caGGOa GaaGinaiaacMcaaaa@7277@

Be the total number of negative response that is number of 0’s obtained when subjects are administered treatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGQbaabeaakiaaysW7aaa@3981@ .

Now the total number of positive responses, that is the total number of 1’s and the total number of negative responses, that is the total number of 0’s for all the c treatments are respectively

f .. + = j=1 c f .j + ; f .. 0 = j=1 c ( n f .j + )=nc f .. + (5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaabaGaaiOlaiaac6caaKqbGeaacqGHRaWkaaqcfaOaeyypa0Za aabCaeaacaWGMbWaa0baaKqbGeaacaGGUaGaamOAaaqaaiabgUcaRa aajuaGcaGG7aGaamOzamaaDaaabaGaaiOlaiaac6caaKqbGeaacaaI WaaaaKqbakabg2da9maaqahabaWaaeWaaeaacaWGUbGaeyOeI0Iaam OzamaaDaaajuaibaGaaiOlaiaadQgaaeaacqGHRaWkaaaajuaGcaGL OaGaayzkaaGaeyypa0JaamOBaiaadogacqGHsislcaWGMbWaa0baae aacaGGUaGaaiOlaaqcfasaaiabgUcaRaaajuaGcaaMe8UaaGjbVlaa ysW7caaMe8UaaGjbVdqcfasaaiaadQgacqGH9aqpcaaIXaaabaGaam 4yaaqcfaOaeyyeIuoaaKqbGeaacaWGQbGaeyypa0JaaGymaaqaaiaa dogaaKqbakabggHiLdGaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7ca aMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaa ysW7caaMe8UaaiikaiaaiwdacaGGPaaaaa@833B@

Note that the expected value and variance of u ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38F9@ are respectively

E( u ij )= π .j + ,Var( u ij )= π .j + ( 1 π .j + )(6) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGaamyDamaaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaGa ayjkaiaawMcaaiabg2da9iabec8aWnaaDaaajuaibaGaaiOlaiaadQ gaaeaacqGHRaWkaaqcfaOaaiilaiaadAfacaWGHbGaamOCamaabmaa baGaamyDamaaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaGaayjkai aawMcaaiabg2da9iabec8aWnaaDaaajuaibaGaaiOlaiaadQgaaeaa cqGHRaWkaaqcfa4aaeWaaeaacaaIXaGaeyOeI0IaeqiWda3aa0baaK qbGeaacaGGUaGaamOAaaqaaiabgUcaRaaaaKqbakaawIcacaGLPaaa caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVl aaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8Ua aGjbVlaaysW7caGGOaGaaGOnaiaacMcaaaa@77CA@

The sample estimate of π j + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aa0 baaSqaaiaadQgaaeaacqGHRaWkaaaaaa@39B1@  is

π ^ .j + = p .j = f .j + n (7) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaadaqhaaqcfasaaiaac6cacaWGQbaabaGaey4kaScaaKqbakab g2da9iaadchadaWgaaqcfasaaiaac6cacaWGQbaajuaGbeaacqGH9a qpdaWcaaqaaiaadAgadaqhaaqcfasaaiaac6cacaWGQbaabaGaey4k aScaaaqcfayaaiaad6gaaaGaaGjbVlaaysW7caaMe8UaaGjbVlaays W7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjb VlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8 UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaiikaiaaiEda caGGPaaaaa@7279@

Similarly, the overall sample proportion of positive responses for all the c treatments is

p .. = f .. + nc (8) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiCam aaBaaabaGaaiOlaiaac6caaeqaaiabg2da9maalaaabaGaamOzamaa DaaabaGaaiOlaiaac6caaKqbGeaacqGHRaWkaaaajuaGbaGaamOBai aadogaaaGaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjb VlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8 UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7 caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVl aacIcacaaI4aGaaiykaaaa@722C@

To obtain a test statistic for the null hypothesis that subjects on the average do not differ in their positive responses to all the c treatments, it would be easier to use the Chi-square test for independence.

Under this approach, we note that the observed frequencies of positive and negative responses by subjects at treatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGQbaabeaaaaa@37EA@  that is the total number of 1’s and 0’s at treatment T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGQbaabeaaaaa@37EA@  are respectively

o 1j = f .j + ; o 2j = f .j 0 =n f .j + (9) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Bam aaBaaajuaibaGaaGymaiaadQgaaKqbagqaaiabg2da9iaadAgadaqh aaqcfasaaiaac6cacaWGQbaabaGaey4kaScaaKqbakaacUdacaWGVb WaaSbaaKqbGeaacaaIYaGaamOAaaqcfayabaGaeyypa0JaamOzamaa DaaajuaibaGaaiOlaiaadQgaaeaacaaIWaaaaKqbakabg2da9iaad6 gacqGHsislcaWGMbWaa0baaKqbGeaacaGGUaGaamOAaaqaaiabgUca RaaajuaGcaaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8 UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7 caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaacIcacaaI5aGaaiykaa aa@7108@

Now, under the null hypothesis of equal positive responses for all the c treatments, the expected frequencies of positive and negative responses for T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGQbaabeaaaaa@37EA@ are respectively

E 1j = n. f .. + nc = f .. + c ; O 1j = n. f .. 0 nc = f .. 0 c ; O 2j = nc f .. 0 c ; E 2j = nc f .. + c (10) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aaBaaajuaibaGaaGymaiaadQgaaKqbagqaaiabg2da9maalaaabaGa amOBaiaac6cacaWGMbWaa0baaeaacaGGUaGaaiOlaaqcfasaaiabgU caRaaaaKqbagaacaWGUbGaam4yaaaacqGH9aqpdaWcaaqaaiaadAga daqhaaqaaiaac6cacaGGUaaajuaibaGaey4kaScaaaqcfayaaiaado gaaaGaai4oaiaad+eadaWgaaqcfasaaiaaigdacaWGQbaajuaGbeaa cqGH9aqpdaWcaaqaaiaad6gacaGGUaGaamOzamaaDaaabaGaaiOlai aac6caaKqbGeaacaaIWaaaaaqcfayaaiaad6gacaWGJbaaaiabg2da 9maalaaabaGaamOzamaaDaaabaGaaiOlaiaac6caaKqbGeaacaaIWa aaaaqcfayaaiaadogaaaGaai4oaiaad+eadaWgaaqcfasaaiaaikda caWGQbaajuaGbeaacqGH9aqpdaWcaaqaaiaad6gacaWGJbGaeyOeI0 IaamOzamaaDaaabaGaaiOlaiaac6caaKqbGeaacaaIWaaaaaqcfaya aiaadogaaaGaai4oaiaadweadaWgaaqcfasaaiaaikdacaWGQbaaju aGbeaacqGH9aqpdaWcaaqaaiaad6gacaWGJbGaeyOeI0IaamOzamaa DaaabaGaaiOlaiaac6caaKqbGeaacqGHRaWkaaaajuaGbaGaam4yaa aacaaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjb VlaacIcacaaIXaGaaGimaiaacMcaaaa@83E7@

Hence,the required χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@ test statistic is

χ 2 = j=1 c i=1 2 ( O ij E ij ) 2 E ij (11) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaaeWbqaamaaqaha baWaaSaaaeaadaqadaqaaiaad+eadaWgaaqcfasaaiaadMgacaWGQb aajuaGbeaacqGHsislcaWGfbWaaSbaaKqbGeaacaWGPbGaamOAaaqc fayabaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOmaaaaaKqbag aacaWGfbWaaSbaaKqbGeaacaWGPbGaamOAaaqcfayabaaaaaqcfasa aiaadMgacqGH9aqpcaaIXaaabaGaaGOmaaqcfaOaeyyeIuoaaKqbGe aacaWGQbGaeyypa0JaaGymaaqaaiaadogaaKqbakabggHiLdGaaGjb VlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8 UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7 caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caGGOaGaaGymai aaigdacaGGPaaaaa@7CDA@

With c-1 degrees of freedom under the null hypothesis H0

Substituting equations (9) and (10) in equation (11), we obtain

χ 2 = j=1 c ( f .j + f .. + c )/ f .. + c + j=1 c ( (n f .j + ) (nc f .. + ) c ) 2 nc f .. + c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaaeWbqaamaalyaa baWaaeWaaeaacaWGMbWaa0baaKqbGeaacaGGUaGaamOAaaqaaiabgU caRaaajuaGcqGHsisldaWcaaqaaiaadAgadaqhaaqaaiaac6cacaGG UaaajuaibaGaey4kaScaaaqcfayaaiaadogaaaaacaGLOaGaayzkaa aabaWaaSaaaeaacaWGMbWaa0baaeaacaGGUaGaaiOlaaqaaiabgUca RaaaaeaacaWGJbaaaaaaaKqbGeaacaWGQbGaeyypa0JaaGymaaqaai aadogaaKqbakabggHiLdGaey4kaSYaaSaaaeaadaaeWbqaamaabmaa baGaaiikaiaad6gacqGHsislcaWGMbWaa0baaKqbGeaacaGGUaGaam OAaaqaaiabgUcaRaaajuaGcaGGPaGaeyOeI0YaaSaaaeaacaGGOaGa amOBaiaadogacqGHsislcaWGMbWaa0baaeaacaGGUaGaaiOlaaqcfa saaiabgUcaRaaajuaGcaGGPaaabaGaam4yaaaaaiaawIcacaGLPaaa daahaaqabKqbGeaacaaIYaaaaaqaaiaadQgacqGH9aqpcaaIXaaaba Gaam4yaaqcfaOaeyyeIuoaaeaadaWcaaqaaiaad6gacaWGJbGaeyOe I0IaamOzamaaDaaabaGaaiOlaiaac6caaKqbGeaacqGHRaWkaaaaju aGbaGaam4yaaaaaaaaaa@74C3@

Which when further simplified becomes

χ 2 = n. c 2 f .. + ( nc f .. + ) j=1 c ( f .j + f .. + c ) (12) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaWcaaqaaiaad6ga caGGUaGaam4yamaaCaaabeqcfasaaiaaikdaaaaajuaGbaGaamOzam aaDaaabaGaaiOlaiaac6caaKqbGeaacqGHRaWkaaqcfa4aaeWaaeaa caWGUbGaam4yaiabgkHiTiaadAgadaqhaaqaaiaac6cacaGGUaaaju aibaGaey4kaScaaaqcfaOaayjkaiaawMcaaaaadaaeWbqaamaabmaa baGaamOzamaaDaaajuaibaGaaiOlaiaadQgaaeaacqGHRaWkaaqcfa OaeyOeI0YaaSaaaeaacaWGMbWaa0baaeaacaGGUaGaaiOlaaqcfasa aiabgUcaRaaaaKqbagaacaWGJbaaaaGaayjkaiaawMcaaaqcfasaai aadQgacqGH9aqpcaaIXaaabaGaam4yaaqcfaOaeyyeIuoacaaMe8Ua aGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7ca aMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaa ysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaacIcacaaIXaGaaG OmaiaacMcaaaa@8443@

Which under the null hypothesis H0 of equal positive response rate has approximately the χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@  distribution with c-1 degrees of freedom for sufficiently large n and c. Equation (12) when expressed in terms of sampled proportion becomes

χ 2 = n p .. ( 1 p .. ) j=1 c ( p .j + p .. ) 2 = n.( j=1 c ( p .j +2 p .. 2 ) ) p .. ( 1 p .. ) (13) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaWcaaqaaiaad6ga aeaacaWGWbWaaSbaaeaacaGGUaGaaiOlaaqabaWaaeWaaeaacaaIXa GaeyOeI0IaamiCamaaBaaabaGaaiOlaiaac6caaeqaaaGaayjkaiaa wMcaaaaadaaeWbqaamaabmaabaGaamiCamaaDaaajuaibaGaaiOlai aadQgaaKqbagaacqGHRaWkaaGaeyOeI0IaamiCamaaBaaabaGaaiOl aiaac6caaeqaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaa aabaGaamOAaiabg2da9iaaigdaaeaacaWGJbaajuaGcqGHris5aiab g2da9maalaaabaGaamOBaiaac6cadaqadaqaamaaqahabaWaaeWaae aacaWGWbWaa0baaKqbGeaacaGGUaGaamOAaaqaaiabgUcaRiaaikda aaqcfaOaeyOeI0IaamiCamaaDaaabaGaaiOlaiaac6caaKqbGeaaca aIYaaaaaqcfaOaayjkaiaawMcaaaqcfasaaiaadQgacqGH9aqpcaaI XaaabaGaam4yaaqcfaOaeyyeIuoaaiaawIcacaGLPaaaaeaacaWGWb WaaSbaaeaacaGGUaGaaiOlaaqabaWaaeWaaeaacaaIXaGaeyOeI0Ia amiCamaaBaaabaGaaiOlaiaac6caaeqaaaGaayjkaiaawMcaaaaaca aMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaa ysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaacIcacaaIXaGaaG 4maiaacMcaaaa@8B8E@

Illustrative example: The effects of four drug presentations on patients are to be studied. Interest is to determine whether or not the four drugs equally improved patients’ condition. Sixty patients are selected and grouped into 15 blocks so that the four patients in each block are approximately identical in age, initial condition, sex, etc. Patients in each block are randomly selected for treatment with only one of the four experimental drugs. After the specified medication period, the patients are classified as either improved (success) or not improved (failure) under a given drug and coded with a 1 or 0 respectively. The results are shown in Table 1. Can it be concluded on the basis of these data that patients improve equally on all the four drugs?

An illustration of proposed alternative to Cochran Q test: To illustrate the proposed method we apply equation 1 to obtain values of u ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38F9@ for the data of Table 1,for i=1,2,…,15;j=1,2,…,4.The summary values of u ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38F9@ that is of f .j + and f .j 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaDa aaleaacaGGUaGaamOAaaqaaiabgUcaRaaakiaaysW7caWGHbGaamOB aiaadsgacaaMe8UaamOzamaaDaaaleaacaGGUaGaamOAaaqaaiaaic daaaaaaa@42EA@ with the corresponding sample proportions p .j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaGGUaGaamOAaaqabaaaaa@38B8@  of 1’s are shown at the bottom of Table 1, for j=1,2,3,4. Using the values of p .j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaGGUaGaamOAaaqabaaaaa@38B8@ in equation 13 we obtain the Chi-square test statistic for the null hypothesis of no difference in possible response rates, that is the proportion of subjects, or patients improving under the four drugs as

χ 2 = (15)( (0.60) 2 + (0.67) 2 + (0.33) 2 + (0.33) 2 40 (0.48) 2 ) (0.48)(0.52) χ 2 = (15)( 1.0270.922 ) 0.250 = (15)(0.105) 0.25 = 1.575 0.25 =6.300(pvalue=0.0984), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGcq aHhpWydaahaaqabKqbGeaacaaIYaaaaKqbakabg2da9maalaaabaGa aiikaiaaigdacaaI1aGaaiykamaabmaabaGaaiikaiaaicdacaGGUa GaaGOnaiaaicdacaGGPaWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcaGGOaGaaGimaiaac6cacaaI2aGaaG4naiaacMcadaahaaqabK qbGeaacaaIYaaaaKqbakabgUcaRiaacIcacaaIWaGaaiOlaiaaioda caaIZaGaaiykamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaai ikaiaaicdacaGGUaGaaG4maiaaiodacaGGPaWaaWbaaeqajuaibaGa aGOmaaaajuaGcqGHsislcaaI0aGaaGimaiaacIcacaaIWaGaaiOlai aaisdacaaI4aGaaiykamaaCaaabeqcfasaaiaaikdaaaaajuaGcaGL OaGaayzkaaaabaGaaiikaiaaicdacaGGUaGaaGinaiaaiIdacaGGPa GaaiikaiaaicdacaGGUaGaaGynaiaaikdacaGGPaaaaaGcbaqcfaOa eq4Xdm2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaWcaaqaai aacIcacaaIXaGaaGynaiaacMcadaqadaqaaiaaigdacaGGUaGaaGim aiaaikdacaaI3aGaeyOeI0IaaGimaiaac6cacaaI5aGaaGOmaiaaik daaiaawIcacaGLPaaaaeaacaaIWaGaaiOlaiaaikdacaaI1aGaaGim aaaacqGH9aqpdaWcaaqaaiaacIcacaaIXaGaaGynaiaacMcacaGGOa GaaGimaiaac6cacaaIXaGaaGimaiaaiwdacaGGPaaabaGaaGimaiaa c6cacaaIYaGaaGynaaaacqGH9aqpdaWcaaqaaiaaigdacaGGUaGaaG ynaiaaiEdacaaI1aaabaGaaGimaiaac6cacaaIYaGaaGynaaaacqGH 9aqpcaaI2aGaaiOlaiaaiodacaaIWaGaaGimaiaacIcacaWGWbGaey OeI0IaamODaiaadggacaWGSbGaamyDaiaadwgacqGH9aqpcaaIWaGa aiOlaiaaicdacaaI5aGaaGioaiaaisdacaGGPaGaaiilaaaaaa@A587@

Which with 4-1=3 degrees of freedom is not statistically significant ( χ 0.99;3 2 =11.35 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHhpWydaqhaaqcfasaaiaaicdacaGGUaGaaGyoaiaaiMdacaGG 7aGaaG4maaqaaiaaikdaaaqcfaOaeyypa0JaaGymaiaaigdacaGGUa GaaG4maiaaiwdaaiaawIcacaGLPaaaaaa@4476@ , leading to the conclusion that patients may not have deferred in their improvement rates over the four drugs.

Now that if one had instead applied the usual Cochran Q test to analyze the same data, we would obtain the required test statistic as

Q= (c1)[ j=1 4 T .j 2 ( j=1 4 T .j ) 2 /c ] i=1 15 B i ( i=1 15 B i 2 /c ) = ( 41 )( (9) 2 + (10) 2 + (5) 2 + (5) 2 )/4 ( 2+3...+2 ) ( (2) 2 + (3) 2 +...+(2) ) 2 /4 = (3)(231210.25) 2918.75 = 62.25 10.25 =6.073(pvalue=0.1057) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGrbGaeyypa0ZaaSaaaeaacaGGOaGaam4yaiabgkHiTiaaigdacaGG PaWaamWaaeaadaaeWbqaaiaadsfadaqhaaqcfasaaiaac6cacaWGQb aabaGaaGOmaaaajuaGcqGHsisldaWcgaqaaiaacIcadaaeWbqaaiaa dsfadaWgaaqcfasaaiaac6cacaWGQbaajuaGbeaaaKqbGeaacaWGQb Gaeyypa0JaaGymaaqaaiaaisdaaKqbakabggHiLdGaaiykamaaCaaa beqcfasaaiaaikdaaaaajuaGbaGaam4yaaaaaKqbGeaacaWGQbGaey ypa0JaaGymaaqaaiaaisdaaKqbakabggHiLdaacaGLBbGaayzxaaaa baWaaabCaeaacaWGcbWaaSbaaKqbGeaacaWGPbaajuaGbeaacqGHsi sldaqadaqaamaalyaabaWaaabCaeaacaWGcbWaa0baaKqbGeaacaWG PbaabaGaaGOmaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaaigdaca aI1aaajuaGcqGHris5aaqaaiaadogaaaaacaGLOaGaayzkaaaajuai baGaamyAaiabg2da9iaaigdaaeaacaaIXaGaaGynaaqcfaOaeyyeIu oaaaaabaGaeyypa0ZaaSaaaeaadaWcgaqaamaabmaabaGaaGinaiab gkHiTiaaigdaaiaawIcacaGLPaaadaqadaqaaiaacIcacaaI5aGaai ykamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaiikaiaaigda caaIWaGaaiykamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaai ikaiaaiwdacaGGPaWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWk caGGOaGaaGynaiaacMcadaahaaqabKqbGeaacaaIYaaaaaqcfaOaay jkaiaawMcaaaqaaiaaisdaaaaabaWaaeWaaeaacaaIYaGaey4kaSIa aG4maiaac6cacaGGUaGaaiOlaiabgUcaRiaaikdaaiaawIcacaGLPa aacqGHsisldaWcgaqaamaabmaabaGaaiikaiaaikdacaGGPaWaaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaGGOaGaaG4maiaacMcada ahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaac6cacaGGUaGaaiOl aiabgUcaRiaacIcacaaIYaGaaiykaaGaayjkaiaawMcaamaaCaaabe qcfasaaiaaikdaaaaajuaGbaGaaGinaaaaaaaakeaajuaGcqGH9aqp daWcaaqaaiaacIcacaaIZaGaaiykaiaacIcacaaIYaGaaG4maiaaig dacqGHsislcaaIYaGaaGymaiaaicdacaGGUaGaaGOmaiaaiwdacaGG PaaabaGaaGOmaiaaiMdacqGHsislcaaIXaGaaGioaiaac6cacaaI3a GaaGynaaaacqGH9aqpdaWcaaqaaiaaiAdacaaIYaGaaiOlaiaaikda caaI1aaabaGaaGymaiaaicdacaGGUaGaaGOmaiaaiwdaaaGaeyypa0 JaaGOnaiaac6cacaaIWaGaaG4naiaaiodacaGGOaGaamiCaiabgkHi TiaadAhacaWGHbGaamiBaiaadwhacaWGLbGaeyypa0JaaGimaiaac6 cacaaIXaGaaGimaiaaiwdacaaI3aGaaiykaaaaaa@CF51@

Which with 4-1=3 degrees of freedom is also not statistically significant ( χ 0.99;3 2 =11.35 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHhpWydaqhaaqcfasaaiaaicdacaGGUaGaaGyoaiaaiMdacaGG 7aGaaG4maaqaaiaaikdaaaqcfaOaeyypa0JaaGymaiaaigdacaGGUa GaaG4maiaaiwdaaiaawIcacaGLPaaaaaa@4476@ .However the Chi-square value of 6.073 obtained using the usual Cochran Q test is slightly less than the corresponding Chi-square value of 6.300 obtained using the proposed method. Hence the usual Cochran Q test statistic is likely to lead to an acceptance of false null hypothesis (Type II error) more frequently and is therefore likely to be less powerful than the present method at least for the present data.

Summary and Conclusion

We have discussed and presented above an alternative modified and probably easier method for the analysis of sample data that may be appropriate for use with the usual Cochran Q test.

A test statistic based on the Chi-square test for independence is developed for testing the null hypothesis that subjects or blocks of matched subjects on the average do not differ in proportions responding positive when administered a number of tests or treatments in a diagnostic screening test or clinical trial.

The proposed test method is illustrated with some sample data and shown to be at least as powerful as the usual Cochran Q test when applied to data of equal sizes. This is because the Chi-square value of 6.073 obtained using the usual Cochran Q test is slightly less than the corresponding Chi-square value of 6.300 obtained using the proposed method. Hence the usual Cochran Q test statistic is likely to lead to an acceptance of false null hypothesis (Type II error) more frequently and is therefore likely to be less powerful than the present method at least for the present data.

References

  1. Oyeka ICA, Nnanatu CC (2014) Pairwise Comparison in Repeated Measures. Journalof Modern Applied Statistical Methods 13(2): Article 8.
  2. Oyeka CA (2010) An Introduction to Applied Statistical Methods. (8th edn), Nobern Avocation Publishing Company, Enugu, Nigeria.
  3. Cochran WG (1950) The comparison of percentages in matched samples. Biometrika 37(3-4): 256-266.
  4. Freund JE (1992) Mathematical Statistics. (5th edn), Prentice-Hall Internal Editions, New York, USA.
  5. Oyeka CA, Utazi CE, Nwosu CR, Uwawunkonye GE, Ikpegbu PA, et al. (2010) A Statistical Comparison of Test Scores: A Non-Parametric Approach. Journal of Mathematical Sciences 21(1): 77-87.
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