Chapter 8. Ifereces about More Tha Two Populato Cetral Values
Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers ) To vestgate whether treatmet at a youg age would yeld better results tha treatmet at a older age. Data classfed by age:
Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers )
Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers 3) From the boxplots, we ca observe that the four groups do ot appear to have that great a dfferece mprovemet. We ca ow develop the aalyss of varace procedure to cofrm whether a statstcally sgfcat dfferece exsts betwee the four age groups.
Wth-ample Varato Because the varablty amog the sample meas s large comparso to the wth-sample varato, we mght coclude tutvely that the correspodg populato meas are dfferet.
Betwee-ample Varato I ths table, the sample meas are the same as gve the prevous table, but the varablty wth a sample s much larger, ad the betwee-sample varato s small relatve to the wth-sample varablty. We would be less lkely to coclude that the correspodg populato meas dffer based o these data.
A Aalyss of Varace ) A statstcal test about more tha two populato meas T test s used to test the equalty of two populato meas. t = s y y p ) ) s p = ) s ) ) s ) = ) s ) s p s a pooled estmate of the commo populato varace. Now suppose that we wsh to exted ths method to test the equalty of more tha two populato meas. A more geeral method of data aalyss s the aalyss of varace. σ
A Aalyss of Varace ) ummary of the samples results for fve populatos If we are terested testg the equalty of the populato meas.e., µ = µ = µ 3 = µ 4 = µ 5 ) we mght be tempted to ru all possble parwse comparsos of two populato meas. If we cofrm that the fve dstrbutos are approxmately ormal wth the same varace σ, we could ru 0 t tests comparg all pars of meas. Although we may have probablty of a Type error fxed at α=0.05 for each dvdual test, the probablty of falsely rejectg at least oe of those tests s larger tha 0.05.
A Aalyss of Varace 3) The aalyss of varace procedures are developed uder the followg codtos: Each of the fve populatos has a ormal dstrbuto. The varaces of the fve populatos are equal: that s, σ = σ = σ = σ = σ =. 3 4 5 σ The fve sets of measuremets are depedet radom samples form ther respectve populatos.
A Aalyss of Varace 4) Wth-sample varace Note that ths quatty s merely a exteso of represets a combed estmate of the commo varace σ, ad t measures the varablty of the observatos wth the fve populatos. 5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 5 4 3 5 5 4 4 3 3 5 4 3 5 5 4 4 3 3 = = W ) ) ) ) ) ) = = s s s s s p W
A Aalyss of Varace 5) If the ull hypothess µ = µ = µ 3 = µ 4 = µ 5 s true, the the populatos are detcal, wth mea µ ad varace σ. Drawg sgle samples from the fve populatos s the equvalet to drawg fve dfferet samples from the same populato. To evaluate the varato the fve sample meas, we eed to kow the samplg dstrbuto of the sample mea computed from a radom sample of 5 observatos from a ormal populato. We ca estmate the varace of the dstrbuto of sample meas σ /5, usg the formula:
A Aalyss of Varace 6) Betwee-sample varace The quatty estmates σ /5,ad hece 5 sample varace of the meas) estmates σ. The quatty as B. The subscrpt B deotes a measure of the varablty amog the sample meas for the fve populatos. Uder the ull hypothess that all fve populato meas are detcal, we have two estmates of σ ---amely, B ad W. uppose the rato B W s used as the test statstcs to test the hypothess that µ = µ = µ = µ =. 3 4 µ 5 B follows a F dstrbuto wth degrees of freedom df =4 for ad B W df =0 for. W
A Aalyss of Varace 7) The test statstc used to test equalty of the populato meas s F = B W Whe the ull hypothess s true, both B ad estmate σ W, ad expect F to assume a value ear F =. Whe the hypothess of equalty s false, B wll ted to be large tha W due to the dffereces amog the populato meas. If the calculated value of F falls the rejecto rego, we coclude that ot all fve populato meas are detcal.
Aalyss of Varace --- Completely Radomzed Desg ) ummary of sample data for a completely radomzed desg
Aalyss of Varace --- Completely Radomzed Desg ) Total sum of squares T) Let be the sample varace of the T measuremets. T T = t = j= y y.. ) = ) It s possble to partto the total sum of squares as follows: Wth-sample sum of squares W; ) Betwee-sample sum of squares B; ) j yj y ) = yj y. ) y. j W = j y B = y T.. y.. ) j j T W y.) = ) )... ). y.. ) B
Aalyss of Varace --- Completely Radomzed Desg 3) Although the formulas for T, W ad B are easly terpreted, they are ot easy to use for calculatos. Istead, we recommed usg a computer software program. A aalyss of varace for a completely radomzed desg wth t populatos has the followg ull ad alteratve hypotheses: H µ = µ = µ =... = µ H 0 : 3 a : At least oe of the t populato meas dffers from the rest. The quattes ad B ca be computed usg the shortcut formula W t B = B t W = W t T
Aalyss of Varace Table
Example 8. A clcal psychologst wshed to compare three methods for reducg hostlty levels uversty studets, ad used a certa test HLT) to measure the degree of hostlty.
Aswer to Example 8.
The Model for Observato a Completely Radomzed Desg Oe-Way Classfcato) Assumptos Idepedet radom samples Each sample s selected from a ormal populato. The mea ad varace for populato are, respectvely, ad. Four dstrbutos
Model for Aalyss of Varace ) y j : the jth sample measuremet selected from populato, s the sum of three terms. : deotes a overall mea that s a ukow costat. : deotes a effect due to populato. s a ukow costat. y = µ α j ε j : represets the radom devato of y j about the th populato mea,. The j s are ofte referred to as error terms. The term error smply refers to the fact that the observatos from the t populatos dffer by more tha just ther meas. j
Model for Aalyss of Varace ) µ = E y ) = E µ α ε ) = µ α E ε ) = µ j j j α The j s are ormally dstrbuted wth mea 0 ad varace. Also, the varace for each of the t populatos ca be show to be. σ ε σ ε ummary of some of the assumptos for a completely radomzed desg
Model for Aalyss of Varace 3) Usg the model, the ull hypothess s: H H 0 a : α = α =... = α : At least oe of t = 0 the α s dffers from 0. We eed to verfy that these codtos are satsfed pror to makg fereces from the aalyss of varace table. The ormalty codto s ot as crtcal as the equal varace assumpto whe we have large sample szes uless the populatos are severely skewed or have very heavy tals. The assumpto of homogeety equalty) of populato varaces s less crtcal whe the sample szes are early equal, where the varaces ca be markedly dfferet ad the p-values for a aalyss of varace wll stll be oly mldly dstorted.
Model for Aalyss of Varace 4) --- Resdual Aalyss The evaluato of the ormal codto wll be evaluated usg resdual aalyss. ε j = yj µ The f the codto of equal varaces s vald, the j s are a radom sample from a ormal populato, However, s a ukow costat, but f we estmate wth, ad let y. e j = y j y. The we ca use the e j s to evaluate the ormalty assumpto. Eve whe the dvdual s are small, we would have T resduals, whch would provde a suffcet umber of values to evaluate the ormalty codto. We ca plot the e j s a boxplot or a ormalty plot to evaluate whether the data appear to have bee geerate from ormal populatos.
Example 8.3 A teratoal orgazato wated to determe whether the clercs from dfferet relgos have dfferet levels of awareess wth respect to the causes of metal lless. Three radom samples were draw, oe cotag the Methodst msters, a secod cotag te catholc prests, ad a thrd cotag te Petecostal msters. Each of the 30 clercs was the examed, usg a stadard wrtte test, to measure hs or her kowledge about causes of metal lless. The test scores are lsted the followg table.
Aswer to Example 8.3 ) Resduals e j for clercs kowledge of metal lless e j = y j y.
Aswer to Example 8.3 ) Normal probablty plot for resduals A lack of cocetrato of the resduals about the straght le.
Aswer to Example 8.3 3)
Aswer to Example 8.3 4) Equal varace test Leve s test statstcs from L = MB/MW =78.3/86.9 = 0.95 < the crtcal value 3.35). Thus we fal to reject the ull hypothess that the stadard devatos are equal. The Kruskal-Walls test ca be used whe the populatos are oormal but have detcal dstrbutos uder the ull hypothess.
Trasformatos of the Data --- A Alteratve Aalyss ) A trasformato of the sample data s defed to be a process whch the measuremets o the orgal scale are systematcally coverted to a ew scale of measuremet. Trasformato to acheve uform varace
Trasformatos of the Data --- A Alteratve Aalyss ) Whe t appear that σ = kµ wth k. The trasformato s approprate. y T = y 0. 375 The logarthmc trasformato y T = logy)) s approprate ay tme the coeffcet of varato / s costat across the populatos of terest. The trasformato y T = arcs y ) s partcular approprate for data recorded as percetages or proportos.
A Noparametrc Alteratve : The Kruskal-Walls Test ) Exteso of the rak sum test for more tha two populatos H 0 :The k dstrbutos are detcal. H a : Not all the dstrbutos are the same. H T = 3 T ) ) T T deotes thesum of : the umber of observatos from sample =,,..., k) T : the combedtotal)samplesze; that s, T = adt the raks for the measuremets sample after thecombedsamplemeasuremets have bee raked.
A Noparametrc Alteratve : The Kruskal-Walls Test ) Note: Whe there are a large umber of tes the raks of the sample measuremets, use H ' = [ j t 3 j H t ) / j 3 T T )] where t j s the umber of observatos the jth group of ted raks.
A Noparametrc Alteratve : The Kruskal-Walls Test 3)