ANOVA with Summary Statistics: A STATA Macro

Transcription

1 ANOVA wth Summary Stattc: A STATA Macro Nadeem Shafque Butt Departmet of Socal ad Prevetve Pedatrc Kg Edward Medcal College, Lahore, Pata Shahd Kamal Ittute of Stattc, Uverty of the Puab Lahore, Pata Muhammad Qaer Shahbaz GC Uverty, Lahore, Pata Abtract Almot all avalable tattcal pacage are capable of performg Aaly of Varace (ANOVA) from raw data. Some of tattcal pacage have capablty to perform depedet ample t-tet, ad ome other tet of gfcace o ummary data, but eldom would you come acro a oftware that ha the capablty to perform ANOVA drectly o ummary data. However ome pacage ca perform oe-way ANOVA after geeratg urrogate data from ummary tattc. I th hort ote we have gve STATA program to perform oe-way ANOVA o ummary data; addto th program alo perform Bartllet tet of equalty of varace. The dea ca be exteded to two-way ad hgher way ANOVA. Example have bee gve for llutrato. Key Word: ANOVA, Summary data.. Itroducto I ome cae, where oly ummary data avalable, a reearcher may wat to perform Aaly of Varace (ANOVA) from the avalable formato. Mot tattcal program are deged to perform ANOVA o raw data oly. Davd A. Laro (99) decrbe a method to geerate urrogate data from the ummary tattc that ca be ued to perform oe way ANOVA. Accordg to Laro oe ha to geerate two ew colum amely X ' ad X ' a X ' X + ad ( ) X ' X X ' Ad after ome data mapulato data ready to perform ANOVA uual way. We ue a alteratve method that ca be ued to perform oe way, two way ad hgher way ANOVA by ug ummary meaure, x ad for,,...,, where, x ad are, repectvely, the ze, mea ad tadard devato of - th treatmet. I th ote we have gve a STATA program that ca be ued otly to perform oe way ANOVA ad equalty of varace f oly ummary meaure are avalable. However, th program ca be ealy exteded for twoway ad hgher way ANOVA. Pa.. tat. oper. re. Vol.II No. 006 pp57-6

2 Nadeem Shafque Butt, Shahd Kamal, Muhammad Qaer Shahbaz. Methodology The oe-way, two-way ad hgher-way ANOVA ue certa tattcal model for operato. Specfcally, the oe-way fxed effect ANOVA model gve a:,,..., y µ + τ + ε (.),,..., Th model ca be ued to compare the gfcace of treatmet or to tet the equalty of everal mea. The ull hypothe of teret th model ca be tated ay oe of the followg way: H : τ 0 for,,..., (.) or 0 0 H : µ µ... µ Th hypothe ca be teted by ug the F rato gve a: where ( ) ( ) SST / F SSE / SST x x wth x x.... ; (.3) (.4) SSE (.5) The gfcace of the group mea ca be teted by ug the p value of computed F tattc. STATA program to perform thee aalye gve Appedx - A. We have alo gve the STATA commad to perform Bartlett Tet of equalty of varace o ummary data. The tet tattc to be ued th tet gve a: χ MC (.6) M l ˆ σ ad C + 3 ( ) ˆ σ ˆ σ wth ˆ σ alo The two-way fxed effect ANOVA model gve a: y µ + β + τ + ε,,..., r,,..., c (.7) The ull hypothee of teret two-way ANOVA are: H : β 0 for,,..., r or H : µ µ... µ / / r. H : τ 0 for,,..., c or H : µ µ... µ // // c (.8) 58 Pa.. tat. oper. re. Vol.II No. 006 pp57-6

3 ANOVA wth Summary Stattc: A STATA Macro but geerally we are more cocered wth the hypothe of colum (treatmet). The eceary um of quare to tet the above hypothee two-way ANOVA are gve a: c c r SSTr r x. x.. wth x.. x. x. (.9) c r... (.0) SSB c x x c or SSE r SSB SSE c SST r (.) where x. mea of -th row (bloc), x. mea of -th colum (treatmet), varace of -th row ad varace of -th colum (treatmet). The F rato to tet the gfcace of bloc ad treatmet are gve a: SSB ( r ) SSTr ( c ) F ad F (.) SSE r c SSE r c The dea ca be ealy exteded to three-way ANOVA wth model: y µ + α + β + τ + ε,,,,... p (.3) The ull hypothee three -way ANOVA are gve a: H : α 0 for,,..., p or H : µ µ... µ / / p.. H : β 0 for,,..., p or H : µ µ... µ // // p. H : τ 0 for,,..., p or H : µ µ... µ /// /// ( ) ( p) (.4) but aga we are more cocer wth the lat hypothe of treatmet. The um of quare to tet the gfcace of above hypothee are gve a: p p p p SSB p x.. x... wth x... x.. x.. x.. ; p (.5) p p p p..... (.6) SSC p x x p..... (.7) SSTr p x x p p p (.8) SSE p SSR SSC p SSC SSTr p SSTr SSC the above um of quare x.., x.. ad x.. are, repectvely, the row, colum ad treatmet mea. Alo, ad are the row, colum ad treatmet varace, repectvely. Pa.. tat. oper. re. Vol.II No. 006 pp

4 Nadeem Shafque Butt, Shahd Kamal, Muhammad Qaer Shahbaz The F rato to tet the gfcace of hypothee three-way ANOVA are gve a: SSR p SSC p SSTr p F, F, F3 SSE p p SSE p p SSE p p (.9) 3. Numercal Example I th ecto we have gve two umercal example to demotrate the uefule of the program to perform ANOVA o ummary tattc. Example : Data ha bee tae from Moore ad McCabe (998). The varable of teret lead cocetrato recorded a mllgram per quare meter ad reearch queto to ee gfcat mea dfferece over the year. Here are ome ummary data for fve year: Year The reult of the example by ug the STATA program aova are gve below:. aova mea d SOV df SS MS F P Betwee Group Wth Group Total Bartllet' tet for equalty of varace Ch-Sq df 4 P Example : Data ha bee tae from Haf et al (004). Study to compare three dfferet drug, groupg the ubect to bloc o the ba of age (becaue t ow that age affect ytolc blood preure ytematcally). Summary data gve the followg table. x 60 Pa.. tat. oper. re. Vol.II No. 006 pp57-6

5 ANOVA wth Summary Stattc: A STATA Macro Drug Mea SD Drug A Drug B Drug C Age Group > The reult of the example by ug the STATA program aova are gve below:. aova c cmea rmea cd SOV df SS MS F P Treatmet Bloc Error Total Appedx A: STATA program to perform to perform oe-way ANOVA I th ecto we have gve a STATA program amed aova.ado that ca be ued to perform the ANOVA o ummary data. The program aova.ado requre three colum a put, colum of ample ze, colum of mea ad colum of tadard devato repectvely. Th program ca be ealy modfed for two-way ad hgher-way ANOVA, further th methodology ca be ealy corporated ay pread-heet baed oftware (e.g. MS excel, SPSS, Stattca etc) aova.ado program defe aova /* Requre three col a put ample ze mea ad d */ arg `' `' `3' ge v`'*`' ge v`'*`'^ ge v3 (`'-)*`3'^ ege umum(`') ege umvum(v) ege umvum(v) ege umv3um(v3) ge gmeaumv/um ge bumv-um*gmea^ ge eumv3 ge dfum- Pa.. tat. oper. re. Vol.II No. 006 pp57-6 6

6 Nadeem Shafque Butt, Shahd Kamal, Muhammad Qaer Shahbaz ge df[_n]- ge dfdf-df ge f(b/df)/(e/df) ge p- F(df,df,f) /*calculato for Bartllet' tet*/ ge v/(`'-) ge m`'- ege ummum(m) ege umvum(v) ge c+((umv-(/umm)))/(3*(_n+)) ge qumv3/umm ge t(`'-)*l(q) ge t (`'-)*l(`3'^) ege umtum(t) ege umtum(t) ge b(umt-umt)/c ge pch chtal(_n-,b) /*commad to cotrol output for aova*/ dplay gree "" dplay gree " " dplay gree " SOV df SS MS F P" dplay gree " " dplay yellow" Betwee Group" %6.0f df[] %0.f b[] %0.f b[]/df[] %9.f f[] %.4f p[] dplay yellow" Wth Group" %7.0f df[] %0.f e[] %0.f e[]/df[] dplay gree " " dplay gree " Total" %5.0f df[] %0.f b[]+e[] dplay gree " " dplay yellow" " dplay yellow"bartllet' tet for equalty of varace" dplay gree "Ch-Sq " %4.3f b[] " df " %.0f df[] " P " %5.4f pch[] /*drop all geerated col*/ drop v - p ed aova.ado Referece. Bartlett, M. S., (937), Properte of uffcecy ad tattcal tet, Proceedg of Royal Socety, Sere A 60: Haf, M., Ahmad M. ad Ahmad A. M., (004), Botattc for Health Studet, ISOSS Publcato. 3. Laro, Davd A., (99), Aaly of Varace wth ut Summary Stattc a Iput, Amerca Stattca, 46, Moore, D. S. ad McCabe, G. P., (998), Itroducto to the practce of Stattc, Freema Publher. 6 Pa.. tat. oper. re. Vol.II No. 006 pp57-6