Testing for significant changes in popularity

Meye , Denny H. A icle Tes ing o signi ican changes in popula i y Sou h A ican Jou nal o Business Managemen P o ided in Coope a ion wi h: Uni e si y o S ellenbosch Business School (USB), Bell ille, Sou h A ica Sugges ed Ci a ion: Meye , Denny H. (1988) : Tes ing o signi ican changes in popula i y, Sou h A ican Jou nal o Business Managemen , ISSN 2078-5976, A ican Online Scien i ic In o ma ion Sys ems (AOSIS), Cape Town, Vol. 19, Iss. 3, pp. 96-98, h ps://doi.o g/10.4102/sajbm. 19i3.978 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/217997 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ 96 S.-A .Tydsk .Bed y sl.1988,19(3) Tes ing o signi ican changes in popula i y Denny H. Meye Depa men o S a is ics, Uni e si y o he Wi wa e s and, Johannesbu g, 2050 Republic o Sou h A ica Recei ed 4 Janua y 1988; accep ed 15 Janua y 1988 A g aphic p ocedu e o de e mining whe he a s a is ically signi ican change in ele ision a ings has occu ed is p oposed. The me hod akes in o accoun he ac ha he a ings a e gene a ed om he same sample o iewe s a he han om wo independen samples. In addi ion i inco po a es he e ec o sample size. The me hod is applicable o he analysis o all ma ke esea ch da a collec ed by such panel me hods. 'n P osedu e om g a ies e bepaal o 'n be ekenis olle e ande ing in die pe sen asie ele isiekyke s oo gekom he , wo d oo ges el. Die ei da die pe sen asie kyke s bepaal wo d ui diesel de s eekp oe an kyke s, in s ede an ui wee ona hanklike s eekp oewe, wo d me die me ode in ag geneem. Daa benewens wo d die e ek an s eekp oe g oo e ook ingeslui . Die me ode is an oepassing op die analise an alle ma kna o singsda a wa deu sulke 'paneel'-me odes ingesamel wo d. In oduc ion Tele ision a ings a e due o appea on he Sou h A ican scene sho ly. These a ings will be based on in o ma ion collec ed om ele ision moni o s ins alled in 500 Sou h A ican homes, which ha e been ca e ully chosen so as o be ep esen a i e o Sou h A ican ele ision iewe ship. These a ings will be o conside able in e es o ad e ise s, and, in pa icula , ad e ise s will wan o es whe he changes in a ings o pa icula p og ammes e lec a eal change in popula i y o whe he he change is simply he esul o sampling e o . The s a is ical solu ion o his p oblem lends i sel o a simple g aphic analysis. The impo an pa ame e s a e ound o be sample size, he (lowe ) a ing, and he p opo ion o iewe ship common o bo h a ings. Me hod The solu ion o he abo e p oblem is based upon he McNema es . This is a nonpa ame ic es which can be iewed as a simple chi-squa ed goodness o i es . The da a a e gi en in Table 1. This able sugges s a o al sample o N = A + B + C + D indi iduals. In he case o p og amme 1 he a ing o iewing would ha e been lOO(B + C)/N% and in he case o p og amme 2 he a ing o iewing would ha e been lOO(A + C)/N%. We shall assume ha he o me a ing is he lowe . Tha is,. p og amme 1 was less popula . In es ing whe he p og amme 1 was signi ican ly less Table 1 Viewe ship equencies o wo ele ision p og ammes P og amme 2 No iewing Viewing P og amme 1 No iewing D A Viewing B C Table 2 Chi-squa ed goodness o i es Obse ed Expec ed equency Change cells equency o no a ing change Viewed 1 bu no 2 B (A + B)/2 Viewed 2 bu no 1 A (A + 8)/2 popula , a en ion ocuses on he B and A cells in Table 1, because hese a e he only cells which e lec di e ences be ween iewe ship o he wo p og ammes. I he e is no signi ican di e ence in popula i y be ween he wo p og ammes hen we would es ima e he equencies o hese wo cells o be (A + B)/2, gi ing he expec ed equencies o he goodness o i es in Table 2. The chi-squa ed es s a is ic has one deg ee o eedom, a c i ical alue o 3,841 when he p obabili y o w ongly de ec ing a signi ican di e ence is se a a maximum o 5%, and can be simpli ied as shown below: I (Obse ed-Expec ed)2 (A-B)2 ----------= Expec ed (A+B) I he di e ence be ween he wo a ings, 100(A + C)/N% -100(B + C)IN% is called d, his means ha a signi ican di e ence in a ings exis s a a 5% signi icance le el, whene e (Nd)2 Nd+2B > 3,841 . d 3,841 + /3,8412 + 30,7288 i.e. > ----------- 2N S.A .J.Bus.Mgm .1988,19(3) In e p e a ion The equency B, which is he numbe o people who iewed p og amme 1 bu no p og amme 2, is c ucial in he abo e o mula. I e e yone who iewed p og amme 1 also iewed p og amme 2, hen B would be ze o, and a signi ican di e ence could be claimed whene e he a ing di e ence exceeded 3,841/N. Al e na i ely, i none o he iewe s o p og amme 1 iewed p og amme 2 hen C would be ze o. I k is he a ing o p og amme 1, his means ha a signi ican di e ence in iewe ship can be claimed in such a si ua ion only when he a ing di e ence exceeds he le el: 3,841 + '3,8412 + 30,728Nk 2N The abo e cases, B equal o ze o and C equal o ze o ep esen he ex emes. These wo cases a e shown in Figu es 1 and 2 below o sample sizes o 400 and 600 espec i ely. Any di e ence in a ings less han 3,841/N can ne e be conside ed o be signi ican , e en when e e yone who wa ches p og amme 1 also wa ches p og amme 2. On he o he hand, any a ing di e ence o mo e han 3,841 + /3,8412 + 30,728Nk 2N will always be signi ican , e en when nobody who wa ches p og amme 1 also wa ches p og amme 2. Clea ly, he la ge he size o he andom sample, he smalle he 'NEVER' a ea becomes. This means ha inc easing he sample size means ha smalle a ing di e ences will become signi ican . Simila ly, inc easing he sample size inc eases he a ea o he 'ALWAYS' egion, meaning ha a la ge sample size educes he a ing di e ence ha can be ega ded as signi ican unde all ci cums ances. In p ac ice, nei he B no Ca e ze o. This means ha signi ican a ing di e ences will all in he egion 3 S41/N· 3,841 + /3,8412 + 30,728Nk ' ' 2N In Figu es 1 and 2 his egion has been di ided by a line po aying he case B = Nk/2. This case po ays he leas signi ican di e ence which can be de ec ed when hal he people who iewed p og amme 1 also iewed p og amme 2. Ame ican s udies (Nielsen) indica e ha i is unusual o mo e han hal he iewe s o iew he second p og amme i hey iewed he i s . Consequen ly, he a ea abo e his line deno es hose 97 2D • a • 17 • I 115 14 D~ 12 - Al.NAYS ! I , IO " C 9 j "' .. • j ~ i 7 • s 4 3 1 1'- 0 0 '° 20 ~ 40 so ~ o ~ 90 m LOWDI RA TNl FOR SAWL£ SIZE - Figu e 1 5% Signi ican a ing di e ences (N = 400) 2D ." -~ 17 " I 18" ALWAYS ~ "' 115 J I ~ 14" "' a - i 12 " ! . - ------------ IO < C ~-------- "' I" .. • ~~ ------------ i ~ i 7 I • I s ;;;~ OCCASIOOALLY 4 3 1 I ~ I ,J IIEYPI 0 IO 2D ~ 40 50 ~ o ~ 90 m LOWDI RA TNl FOR SAWL£ SIZE~ Figu e 2 5% Signi ican a ing di e ences (N = 600) di e ences which a e o en signi ican , whils he a ea below his line indica es di e ences which a e only occasionally signi ican . Thus when he andom sample sizes a e 400 and 600, Figu es 1 and 2 can be used o de e mine whe he any gi en di e ence in ele ision a ing is likely o be s a is ically signi ican . I a a ing di e ence alls in he 'OCCASIONALLY' egion, an ad e ise would be ad ised no o make any d as ic changes o his ad e ising p og amme, bu o moni o u u e pe o mance ca e ully. I a a ing di e ence alls in he 'OFTEN' egion i can gene ally be sa ely assumed ha a signi ican change is in p og ess. Howe e , only when he a ing di e ence alls in he 'ALWAYS' egion can an ad e ise be con iden ha a change has occu ed. Conclusion The abo e g aphic me hod has a lo o appeal o ad e ising manage s who equi e an indica ion as o how wisely hei ele ision ad e ising budge is being spen , bu do no wan o ge in ol ed in s a is ical analysis. The me hod has many o he possible applica ions. Mos ma ke esea ch is p ac ised on a 98 panel o indi iduals ha emains ela i ely unchanged o e ime. I is inco ec o ea such epea samples o da a as being independen . A McNema ype o es , such as ha desc ibed abo e, he e o e has gene al applicabili y in he analysis o ma ke esea ch da a. Acknowledgemen My hanks o Leslie Pels om IBIS and o Michael S.-A .Tydsk .Bed y sl.1988,19(3) B own om SAARF o no only p o iding he p oblem bu also o gi ing some e y use ul ideas. Re e ences Re e ence Supplemen . 1986--87. Nielsen S a ion Index Me hodology Techniques and Da a In e p e a ion, p.53. Siegel, S. 1956. Nonpa ame ic S a is ics o he Beha io al Sciences. McG aw-Hill. p.63-67.