MULTIPLE COMPARISON PROCEDURES
|
|
- Marlene Sims
- 5 years ago
- Views:
Transcription
1 MULTIPLE COMPARISON PROCEDURES RAJENDER PARSAD Indan Agrcultural Statstcs Research Insttute Lbrary Avenue, New Delh Introducton Analyss of varance s used to test for the real treatment dfferences. When the null hypothess that all the treatment means are equal s not reected, t may seem that no further questons need to be asked. However, n some expermental stuatons, t may be an oversmplfcaton of the problem. For example, consder an experment n rce weed control wth 15 treatments vz. 4 wth hand weedng, 10 wth herbcdes and 1 wth no weedng (control). The probable questons that may be rased and the specfc mean comparsons that can provde ther answers may be: ) Is any treatment effectve n controllng the weeds? Ths queston may be answered smply by comparng the mean of the control treatment wth the mean of each of the 14 weed-control treatments. ) Is there any dfference between the group of hand-weedng treatments and the group of herbcde treatments? The comparson of the combned mean of the four hand weedng treatment effects wth the combned mean of the 10-herbcde treatment effects may be able to answer the above queston. ) Are there dfferences between the 4 hand weedng treatments? To answer ths queston one should test the sgnfcant dfferences among the 4 hand weedng treatments. Smlar queston can be rased about the 10-herbcde treatments and can be answered n the above fashon. Ths llustrates the dversty n the types of treatment effects comparsons. Broadly speakng, these comparsons can be classfed ether as Par Comparson or Group Comparson. In par comparsons, we compare the treatment effects parwse whereas ngroup comparsons, the comparsons could be between group comparsons, wthn group comparsons, trend comparsons, and factoral comparsons. In the above example, queston () s the example of the par comparsons and queston () llustrates the between group comparson and queston () s wthn group comparson. Through trend comparsons, we can test the functonal relatonshp (lnear, quadratc, cubc, etc.) between treatment means and treatment levels usng orthogonal polynomals. Factoral comparsons are related to the testng of means of levels of a factor averaged over levels of all other factors or average of treatment combnatons of some factors averaged over all levels of other factors. For parwse treatment comparsons there are many test procedures, however, for the group comparsons, the most commonly used test procedure s to partton the treatment sum of squares nto meanngful comparsons. Ths can be done through contrast analyss ether usng sngle degrees of freedom contrasts or contrasts wth multple degrees of freedom. Further, the comparsons can be dvded nto two categores vz. planned comparsons and unplanned comparsons or data snoopng. These have the followng meanngs. Before the experment commences, the expermenter wll have wrtten out a checklst, hghlghtng the comparsons or contrasts that are of specal nterest, and desgned the experment n
2 Multple Comparson Procedures such a way as to ensure that these are estmable wth as small varances as possble. These are the planned comparsons. After the data have been collected, the expermenter usually looks carefully at the data to see whether anythng unexpected has occurred. One or more unplanned contrasts may turn out to be the most nterestng, and the conclusons of the experment may not be antcpated. Allowng the data to suggest addtonal nterestng contrasts s called data snoopng. The most useful analyss of expermental data nvolves the calculaton of a number of dfferent confdence ntervals, one for each of several contrasts or treatment means. The confdence level for a sngle confdence nterval s based on the probablty, that the random nterval wll be correct (meanng that the random nterval wll contan the true value of the contrast or functon). It s shown below that when several confdence ntervals are calculated, the probablty that they are all smultaneously correct can be alarmngly small. Smlarly, when several hypotheses are tested, the probablty that at least one hypothess s ncorrectly reected can be uncomfortably hgh. Much research has been done over the years to fnd ways around these problems. The resultng technques are known as methods of multple comparson, the ntervals are called smultaneous confdence ntervals, and the tests are called smultaneous hypothess test. Suppose an expermenter wshes to calculate m confdence ntervals, each havng a 100( 1 α *)% confdence level. Then each nterval wll be ndvdually correct wth probablty 1 α *. Let S be the event that the th confdence nterval wll be correct and = 1,...,m. Then, usng the standard rules for probabltes of unons and ntersectons of events, t follows that S the event that t wll be ncorrect ( ) P ( S S... S ) = 1 P( S S... ). 1 2 m 1 2 Sm Ths says that the probablty that all of the ntervals wll be correct s equal to one mnus the probablty that at least one wll be ncorrect. If 2 m=, P ( S1 S2) = P( S1) + P( S2) P( S1 S2) P( S ) + P( S ) 1 2 A smlar result, whch can be proved by mathematcal nducton, holds for any number m of events, that s, ( S S... S ) P( ) P, 1 2 m S wth equalty f the events P S are mutually exclusve. Consequently, 1, S2,..., Sm ( S S... S ) 1 P( S ) = 1 m * (1.1) 1 2 m α 2
3 Multple Comparson Procedures that s, the probablty that the m ntervals wll smultaneously be correct s at least 1 mα *. The probablty mα * s called the overall sgnfcance level or experment wse error rate or famly error rate. A typcal value for α * for a sngle confdence nterval s 0.05, so the probablty that sx confdence ntervals each calculated at a 95% ndvdual confdence level wll smultaneously be correct s at least 0.7. Although at least means bgger than or equal to, t s not known n practce how much bgger than 0.7 the probablty mght actually be. Ths s because the degree of overlap between the events S 1, S2,..., Sm s generally unknown. The probablty at least 0.7 translates nto an overall confdence level of at least 70% when the responses are observed. Smlarly, f an expermenter calculates ten confdence ntervals each havng ndvdual confdence level 95%, then the smultaneous confdence level for the ten ntervals s at least 50%, whch s not very nformatve. As m becomes larger the problem becomes worse, and when m 20, the overall confdence level s at least 0%, clearly a useless asserton! Smlar comments apply to the hypothess-testng stuaton. If m hypotheses are to be tested, each at sgnfcance level α *, then the probablty that at least one hypothess s ncorrectly reected s at most mα *. Varous methods have been developed to ensure that the overall confdence level s not too small and the overall sgnfcance level s not too hgh. Some methods are completely general, that s, they can be used for any set of estmable functons, whle others have been developed for very specalzed purposes such as comparng each treatment wth a control. Whch method s best depends on whch contrasts are of nterest and the number of contrasts to be nvestgated. Some of these methods can also be used for dentfyng the homogeneous subsets of treatment effects. Such procedures are called as multple range tests. Several methods are dscussed n the sequel some of them control the overall confdence level and overall sgnfcance level. Followng the lecture notes on Fundamentals of Desgn of Experments, let l t denote a treatment contrast, l 0 where t s the effect of treatment. The (BLUE) and the = confdence nterval for the above contrast can be obtaned as per procedure gven n the aforementoned lecture notes. However, besdes obtanng confdence ntervals one may be nterested n hypothess testng. The outcome of a hypothess test can be deduced from the correspondng confdence nterval n the followng way. The null hypothess H : l t = wll be reected at sgnfcance level α n favour of the two-sded 0 h alternatve hypothess fals to contan h. H : l t 1 h f the correspondng confdence nterval for In the followng secton, we dscuss the confdence ntervals and hypothess tests based on several methods of multple comparsons. A shorter confdence nterval corresponds to a more powerful hypothess test. l t 3
4 Multple Comparson Procedures 2. Multple Comparson Procedures The termnology a set of smultaneous 100( 1 α *)% confdence ntervals wll always refer to the fact that the overall confdence level for a set of contrasts or treatments means s (at least) 100( 1 α *)%. Each of the methods dscussed gves confdence ntervals of the form l t ltˆ ± w Vâr( ltˆ )...(2.1) where w,whch we call the crtcal coeffcent, depends on the method, the number of treatments v, on the number of confdence ntervals calculated, and on the number of error degrees of freedom. The term ( l tˆ ) msd = w Vâr (2.2) whch s added and subtracted from the least square estmate n (2.1) s called the mnmum sgnfcant dfference, because f the estmate s larger than msd, the confdence nterval excludes zero, and the contrast s sgnfcantly dfferent from zero. 2.1 The Least Sgnfcant Dfference (LSD) Method Suppose that, followng an analyss of varance F test where the null hypothess s reected, we wsh to test H : l t = aganst the alternatve hypothess H : l t. 0 0 For makng parwse comparsons, consder the contrasts of the type expermenters are often nterested, are obtanable from 1 0 t t n whch l t by puttng l = 1, l = 1 and zero for the other l 's. The 100 (1-α)% confdence nterval for ths contrast s ( ltˆ ± tedf, α / 2 Vâr( ltˆ ) ) l t (2.3) where edf denotes the error degrees of freedom. As we know that the outcome of a hypothess test can be deduced from the correspondng confdence nterval n the followng way. The null hypothess wll be reected at sgnfcance level α n favour of the two-sded alternatve hypothess f the correspondng confdence nterval for to contan 0. The nterval fals to contan 0 f the absolute value of l tˆ t Vâr l tˆ edf, α / 2 ( ) l t fals s bgger than. The crtcal dfference or the least sgnfcant dfference for testng the sgnfcance of the dfference of two treatment effects, say ( l tˆ ) t t s lsd = tedf, α / 2 Vâr, where t edf, α / 2 s the value of Student's t at the level of sgnfcance α and error degree of freedom. If the dfference of any two-treatment means s greater than the lsd value, the correspondng treatment effects are sgnfcantly dfferent. 4
5 Multple Comparson Procedures The above formula s qute general and partcular cases can be obtaned for dfferent expermental desgns. For example, the least sgnfcant dfference between two treatment effects for a randomzed complete block (RCB) desgn, wth v treatments and r replcatons s t ( v 1)(r 1), α / 2 2MSE / r, where t (v 1)(r 1), α / 2 s the value of Student's t at the level of sgnfcance α and degree of freedom (v - 1)(r - 1). For a completely randomzed desgn wth v treatments such that th treatment s replcated r tmes and v r = n, the total number of expermental unts, the least sgnfcant dfference between = two treatment effects s t ( n v), / 2 MSE. r r α + It may be worthwhle mentonng here that the least sgnfcant dfference method s sutable only for planned par comparsons. Ths test s based on ndvdual error rate. However, for those who wsh to use t for all possble parwse comparsons, should apply only after the F test n the analyss of varance s sgnfcant at desred level of sgnfcance. Ths procedure s often referred as Fsher's protected lsd. Snce F calls for us to accept or reect a hypothess smultaneously nvolvng means. If we arrange the treatment means n ascendng or descendng order of ther magntude and keep the, means n one group for whch the dfference between the smallest and largest mean s less than the lsd, we can dentfy the homogeneous subsets of treatments. For example, consder an experment that was conducted n completely randomzed desgn to compare the fve treatments and each treatment was replcated 5 tmes. F-test reects the null hypothess regardng the equalty of treatment means. The mean square error (MSE) s The means of fve treatments tred n experment are 9.8, 15.4, 17.6, 21.6 and 10.8 respectvely. The lsd for the above comparsons s 3.75, then the homogeneous subsets of treatments are Group1: Treatment 1 and 5, group 2: treatment 2 and 3 and group 3: treatment 4. Treatments wthn the same homogeneous subset are dentfed wth the same alphabet n the output from SAS. 2.2 Duncan's Multple Range Test A wdely used procedure for comparng all pars of means s the multple range test developed by Duncan (1955). The applcaton of Duncan's multple range test (DMRT) s smlar to that of lsd test. DMRT nvolves the computaton of numercal boundares that allow for the classfcaton of the dfference between any two treatment means as sgnfcant or non-sgnfcant. DMRT requres computaton of a seres of values each correspondng to a specfc set of par comparsons unlke a sngle value for all parwse comparsons n case of lsd. It prmarly depends on the standard error of the mean dfference as n case of lsd. Ths can easly be worked out usng the estmate of varance of an estmated elementary treatment contrast through the desgn. For applcaton of the DMRT rank all the treatment means n decreasng or ncreasng order based on the preference of the character under study. For example for the yeld data, the rank 1 s gven to the treatment wth hghest yeld and for the pest ncdence the treatment wth the least nfestaton should get the rank as 1. Consder the same example as n case of lsd. The ranks of the treatments are gven below: 5
6 Multple Comparson Procedures Treatments T1 T5 T2 T3 T4 Treatment Means Rank Compute the standard error of the dfference of means (SE d ) that s same as that of square root of the estmate of the varance of the estmated elementary contrast through the desgn. In the present example ths s gven by 2 (8.06) / 5 = Now obtan the value rα (p, edf ) *SEd of the least sgnfcant range R p =, where α s the desred sgnfcance 2 level, edf s the error degrees of freedom and p = 2,, v s one more than the dstance n rank between the pars of the treatment means to be compared. If the two treatment means have consecutve rankngs, then p = 2 and for the hghest and lowest means t s v. The r α p,edf can be obtaned from Duncan's table of sgnfcant ranges. values of ( ) For the above example the values of r α ( p,edf) at 20 degrees of freedom and 5% level of sgnfcance are r ( 2,20) = 2. 95, r ( 3,20) = 3. 10, r. 05 ( 4,20) ( 5,20) R are r =. Now the least sgnfcant ranges p R 2 R 3 R 4 R = and Then, the observed dfferences between means are tested, begnnng wth largest versus smallest, whch would be compared wth the least sgnfcant range R v. Next, the dfference of the largest and the second smallest s computed and compared wth the least sgnfcant range R v 1. These comparsons are contnued untl all means have been compared wth the largest mean. Fnally, the dfference of the second largest mean and the smallest s computed and compared aganst the least sgnfcant range R v 1. Ths process s contnued untl the dfferences of all possble v(v 1) 2 pars of means have been consdered. If an observed dfference s greater than the correspondng least sgnfcant range, then we conclude that the par of means n queston s sgnfcantly dfferent. To prevent contradctons, no dfferences between a par of means are consdered sgnfcant f the two means nvolved fall between two other means that do not dffer sgnfcantly. For our case the comparsons wll yeld 4 vs 1: = 11.8 > 4.13( R 5 ); 4 vs 5: = 10.8 > 4.04( R 4 ); 4 vs 2: = 6.2 > 3.94( R 3 ); 4 vs 3: = 4.0 > 3.75( R 2 ); 3 vs 1: = 7.8 > 4.04( R 4 ); 3 vs 5: = 6.8 > 3.94( R 3 ); 3 vs 2: = 2.2 < 3.75( R 2 ); 2 vs 1: = 5.6 > 3.94( R 3 ); 6
7 Multple Comparson Procedures 2 vs 5: = 4.6 > 3.75( R 2 ); 4 vs 1: = 1.0 < 3.75( R 2 ); We see that there are sgnfcant dfferences between all pars of treatments except T3 and T2 and T5 and T1. A graph underlnng those means that are not sgnfcantly dfferent s shown below: T1 T5 T2 T3 T It can easly be seen that the confdence ntervals of the desred parwse comparsons followng (2.1) s ( p, edf) rα l t l ± tˆ Vâr ltˆ (2.4) 2 and least sgnfcant range n general s ( p,edf) rα lsr = Vâr l 2 tˆ. The methods of multple comparson gven n Sectons 2.1 and 2.2 uses ndvdual error rates (probablty that a gven confdence nterval wll not contan the true dfference n level means). Ths may be msleadng as s clear from nequalty (1.1),,e., f m smultaneous confdence ntervals are calculated for preplanned contrasts, and f each confdence nterval has confdence level 100( 1 α *)% then the overall confdence level s greater than or equal to 100( 1 mα *)%. Therefore, the methods of multple comparsons that utlze experment wse error rate or famly error rate (Maxmum probablty of obtanng one or more confdence ntervals that do not contan the true dfference between level means) may be qute useful. In the sequel, we descrbe some methods of multple comparsons that are based on famly error rates. 2.3 Bonferron Method for Preplanned Comparsons In ths method the overall confdence level of 100( 1 α *)% for m smultaneous confdence ntervals can be ensured by settng α * = α / m. Replacng α by α / m n the formula (2.3) for an ndvdual confdence nterval, we obtan a formula for a set of smultaneous 100( 1 α *)% confdence ntervals for m preplanned contrasts l t s l t l ± tˆ tedf, α / 2m Vâr ltˆ (2.5) Therefore, f the contrast estmate s greater than correspondng contrast s sgnfcantly dfferent from zero. t edf, α / 2m Vâr ltˆ the It can easly be seen that ths method s same as that of least sgnfcant dfference wth α n least sgnfcant dfference to be replaced by / m α / 2m s lkely to be a α. Snce ( ) 7
8 Multple Comparson Procedures typcal value, the percentles tedf, α /( 2m) may need to be obtaned by use of a computer package. When m s very large, α /( 2m) s very small, possbly resultng n extremely wde smultaneous confdence ntervals. In ths case the Scheffe or Tukey methods descrbed n the sequel would be preferred. Note that ths method can be used only for preplanned contrasts or any m preplanned estmable contrasts or functons of the parameters. It gves shorter confdence ntervals than the other methods lsted here f m s small. It can be used for any desgn. However, t cannot be used for data snoopng. An expermenter who looks at the data and then proceeds to calculate smultaneous confdence ntervals for the few contrasts that look nterestng has effectvely calculated a very large number of ntervals. Ths s because the nterestng contrasts are usually those that seem to be sgnfcantly dfferent from zero, and a rough mental calculaton of the estmates of a large number of contrasts has to be done to dentfy these nterestng contrasts. Scheffe s method should be used for contrasts that were selected after the data were examned. 2.4 Scheffe Method of Multple Comparsons In the Bonferron method of multple comparsons, the maor problem s that the m contrasts to be examned must be preplanned and the confdence ntervals can become very wde f m s large. Scheffe's method, on the other hand, provdes a set of smultaneous 100( 1 α *)% confdence ntervals whose wdths are determned only by the number of treatments and the number of observatons n the experment. It s not dependent on the number of contrasts are of nterest. It utlzes the fact that every possble contrast l t can be wrtten as a lnear combnaton of the set of ( v 1) treatment - versus - control contrasts, t2 t1,t3 t1,...,tv t1. Once the expermental data have been collected, t s possble to fnd a 100( 1 α *)% confdence regon for these v 1 treatment - versus - control contrasts. The confdence regon not only determnes confdence bounds for each treatment - versus - control contrasts, t determnes bounds for every possble contrast l t and, n fact, for any number of contrasts, whle the overall confdence level remans fxed. For mathematcal detals, one my refer to Scheffe (1959) and Dean and l can be Voss (1999). Smultaneous confdence ntervals for all the contrasts obtaned from the general formula (2.1) by replacng the crtcal coeffcent w by where w wth a as the dmenson of the space of lnear estmable functons s = af a,edf, α beng consdered, or equvalently, a s the number of degrees of freedom assocated wth the lnear estmable functons beng consdered. The Scheffe's method apples to any m estmable contrasts or functons of the parameters. It gves shorter ntervals than Bonferron method when m s large and allows data snoopng. It can be used for any desgn. 2.5 Tukey Method for All Parwse Comparsons Tukey (1953) proposed a method for makng all possble parwse treatment comparsons. The test compares the dfference between each par of treatment effects wth approprate adustment for multple testng. Ths test s also known as Tukey s honestly sgnfcant dfference test or Tukey s HSD. The confdence ntervals obtaned usng ths method are shorter than those obtaned from Bonferron and Scheffe methods. Followng the formula t w s 8
9 Multple Comparson Procedures (2.1), one can obtan the smultaneous confdence ntervals for all the contrasts of the type l t by replacng the crtcal coeffcent w by w t = qv, edf, α / 2 where v s the number of treatments and edf s the error degree of freedom and values can be seen as the percentle correspondng to a probablty level α n the rght hand tal of the studentzed range dstrbuton tables. For the completely randomzed desgn or the one-way analyss of varance model, Vâr(tˆ tˆ ) = 1 1 MSE +, where r denotes the replcaton number of treatment r r ( = 1,2,, v ). Then Tukey's smultaneous confdence ntervals for all parwse comparsons t t, wth overall confdence level at least 100( 1 α *)% s obtaned by takng w t = qv, n-v, α / 2 and Vâr(tˆ tˆ ) = 1 1 MSE +. The values of q v,n v, α r r can be seen the studentzed range dstrbuton tables. When the sample szes are equal ( r r;= 1,...,v) =, the overall confdence level s exactly 100( 1 α *)%. When the sample szes are unequal, the confdence level s at least 100( 1 α *)%. It may be mentoned here that Tukey's method s the best for all parwse treatment comparsons. It can be used for completely randomzed desgns, randomzed complete block desgns and balanced ncomplete block desgns. It s beleved to be applcable (conservatve, true α level lower than stated) for other ncomplete block desgns as well, but ths has not yet been proven. It can be extended to nclude all contrasts but Scheffe's method s generally better for these types of contrasts. 2.6 Dunnett Method for Treatment-Versus-Control Comparsons Dunnett (1955) developed a method of multple comparsons for obtanng a set of smultaneous confdence ntervals for preplanned treatment-versus-control contrasts t t1( = 2,...,v) where level 1 corresponds to the control treatment. The ntervals are shorter than those gven by the Scheffe, Tukey and Bonferron methods, but the method should not be used for any other type of contrasts. For detals on ths method, a reference may be made to Dunnett (1955, 1964) and Hochberg and Tamhane (1987). In general ths procedure s, therefore, best for all treatment-versus-control comparsons. It can be used for completely randomzed desgns, randomzed complete block desgns. It can also be used for balanced ncomplete block desgns but not n other ncomplete block desgns wthout modfcatons to the correspondng multvarate t-dstrbuton tables gven n Hochberg and Tamhane (1987). However, not much lterature s avalable for multple comparson procedures for makng smultaneous confdence statement about several test treatments wth several control treatments comparsons. A partal soluton to the above problem has been gven by Hoover (1991). 9
10 Multple Comparson Procedures 2.7 Hsu Method for Multple Comparsons wth the Best Treatment Multple comparsons wth the best treatment s smlar to multple comparsons wth a control, except that snce t s unknown pror to the experment whch treatment s the best, a control treatment has not been desgnated. Hsu (1984) developed a method n whch each treatment sample mean s compared wth the best of the others, allowng some treatments to be elmnated as worse than best, and allowng one treatment to be dentfed as best f all others are elmnated. Hsu calls ths method RSMCB, whch stands for Rankng, Selecton and Multple Comparsons wth the Best treatment. Suppose, frst, that the best treatment s the treatment that gves the largest response on th t max t denote the effect of the treatment mnus the effect of the average. Let ( ) best of the other v 1 treatments. When the be the effect of the second-best treatment. So, ( ) s the best, zero f the s worse than best. th treatment s the best, ( t )( ) max wll t max t wll be postve f treatment th treatment s ted for beng the best, or negatve f the treatment If the best treatment factor level s the level that gves the smallest response rather than the t mn t n place of largest, then Hsu s procedure has to be modfed by takng ( ) t max( t ). To summarze, Hsu's method for multple comparsons selects the best treatment and dentfes those treatments that are sgnfcantly worse than the best. It can be used for completely randomzed desgns, randomzed block desgns and balanced ncomplete block desgns. For usng t n other ncomplete block desgns, modfcatons of the tables s requred. 3. Multple Comparson Procedures usng SAS/ SPSS The MEANS statement n PROC GLM or PROC ANOVA can be used to generate the observed means of each level of a treatment factor. The TUKEY, BON, SCHEFFE, LSD, DUNCAN, etc. optons under MEANS statement causes the SAS to use Tukey, Bonferron, Scheffe's, least sgnfcant dfference, Duncan's Multple Range Test methods to compare the effects of each par of levels. The opton CLDIFF asks the results of above methods be presented n the form of confdence ntervals. The opton DUNNETT ('1') requests Dunnett's 2-sded method of comparng all treatments wth a control, specfyng level '1' as the control treatment. Smlarly the optons DUNNETTL ('1') and DUNNETTU ('1') can be used for rght hand and left hand method of comparng all treatments wth a control. To Specfy Post Hoc Tests for GLM Procedures n SPSS: From the menus choose: Analyze General Lnear Model From the menu, choose Unvarate, Multvarate, or Repeated Measures In the dalog box, clck Post Hoc Select the factors to analyze and move them to the Post Hoc Tests For lst Select the desred tests. Please note that Post hoc tests are not avalable when covarates have been specfed n the model. GLM 10
11 Multple Comparson Procedures Multvarate and GLM Repeated Measures are avalable only f you have the Advanced Models opton nstalled. 4. Conclusons Each of the methods of multple comparsons at subsectons 2.3 to 2.7 allows the expermenter to control the overall confdence level, and the same methods can be used to control the experment wse error rate when multple hypotheses are to be tested. There exst other multple comparson procedures that are more powerful (.e. that more easly detect a nonzero contrast) but do not control the overall confdence level nor the experment wse error rate. Whle some of these are used qute commonly, however, we don't advocate ther use. The selecton of the approprate multple comparson method depends on the desred nference. As dscussed n Secton 3 that for makng all possble parwse treatment comparsons, the Tukey s method s not conservatve and gves smaller confdence ntervals as compared to Bonferron, Sdak and Scheffe s methods. Therefore, one may choose Tukey s method for makng all possble parwse comparsons. For more detals on methods of multple comparsons, one may refer to Steel and Torre (1981), Gomez and Gomez (1984) and Montgomery (1991), Hsu (1996), Dean and Voss (1999). References and Suggested Readng Dean, A. and Voss, D.(1999). Desgn and Analyss of Experments. Sprnger Texts n Statstcs, Sprnger, New York Duncan, D.B. (1955). Multple range and multple F-Tests. Bometrcs, 11, Dunnett, C.W.(1955). A multple comparsons procedure for comparng several treatments wth a control. J. Am. Statst. Assoc., 50, Dunnett, C.W.(1964). New tables for multple comparsons wth a control. Bometrcs, 20, Gomez, K.A. and Gomez, A.A. (1984). Statstcal Procedures for Agrcultural Research, 2 nd Edton. John Wley and Sons, New York. Hayter, A.J. (1984). A proof of the conecture that the Tukey-Cramer multple comparson procedure s conservatve. Ann. Statst., 12, Hochberg, Y. and Tamhane, A.C.(1987). Multple Comparson Procedures. John Wley and Sons, New York. Hoover, D.R.(1991). Smultaneous comparsons of multple treatments to two (or more) controls. Bom. J., 33, Hsu, J.C. (1984). Rankng and Selecton and Multple Comparsons wth the Best. Desgn of Experments: Rankng and Selecton (Essays n Honour of Robert E.Bechhofer). Edtors: T.J.Santner and A.C.Tamhane , Marcel Dekker, New York. Hsu, J.C. (1996). Multple Comparsons: Theory and Methods. Chapman & Hall. London. Montgomery, D.C.(1991). Desgn and Analyss of Experments, 3 rd edton. John Wley & Sons. New York Peser, A.M. (1943). Asymptotc formulas for sgnfcance levels of certan dstrbutons. Ann. Math. Statst., 14, (Correcton 1949, Ann.Math. Statst., 20, ). Scheffe, H.(1959). The Analyss of Varance. John Wley & Sons. New York. Steel, R.G.D. and Torre,J.H.(1981). Prncples and Procedures of Statstcs: A Bometrcal Approach. McGraw-Hll Book Company, Sngapore, Tukey, J.W.(1953). The Problem of Multple Comparsons. Dttoed Manuscrpt of 396 Pages, Department of Statstcs, Prnceton Unversty. 11
x = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationLecture 6 More on Complete Randomized Block Design (RBD)
Lecture 6 More on Complete Randomzed Block Desgn (RBD) Multple test Multple test The multple comparsons or multple testng problem occurs when one consders a set of statstcal nferences smultaneously. For
More informationDepartment of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution
Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VIII LECTURE - 34 ANALYSIS OF VARIANCE IN RANDOM-EFFECTS MODEL AND MIXED-EFFECTS EFFECTS MODEL Dr Shalabh Department of Mathematcs and Statstcs Indan
More informationSimulated Power of the Discrete Cramér-von Mises Goodness-of-Fit Tests
Smulated of the Cramér-von Mses Goodness-of-Ft Tests Steele, M., Chaselng, J. and 3 Hurst, C. School of Mathematcal and Physcal Scences, James Cook Unversty, Australan School of Envronmental Studes, Grffth
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE II LECTURE - GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 3.
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More information# c i. INFERENCE FOR CONTRASTS (Chapter 4) It's unbiased: Recall: A contrast is a linear combination of effects with coefficients summing to zero:
1 INFERENCE FOR CONTRASTS (Chapter 4 Recall: A contrast s a lnear combnaton of effects wth coeffcents summng to zero: " where " = 0. Specfc types of contrasts of nterest nclude: Dfferences n effects Dfferences
More informationBOOTSTRAP METHOD FOR TESTING OF EQUALITY OF SEVERAL MEANS. M. Krishna Reddy, B. Naveen Kumar and Y. Ramu
BOOTSTRAP METHOD FOR TESTING OF EQUALITY OF SEVERAL MEANS M. Krshna Reddy, B. Naveen Kumar and Y. Ramu Department of Statstcs, Osmana Unversty, Hyderabad -500 007, Inda. nanbyrozu@gmal.com, ramu0@gmal.com
More informationJoint Statistical Meetings - Biopharmaceutical Section
Iteratve Ch-Square Test for Equvalence of Multple Treatment Groups Te-Hua Ng*, U.S. Food and Drug Admnstraton 1401 Rockvlle Pke, #200S, HFM-217, Rockvlle, MD 20852-1448 Key Words: Equvalence Testng; Actve
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationTopic- 11 The Analysis of Variance
Topc- 11 The Analyss of Varance Expermental Desgn The samplng plan or expermental desgn determnes the way that a sample s selected. In an observatonal study, the expermenter observes data that already
More informationSTATISTICS QUESTIONS. Step by Step Solutions.
STATISTICS QUESTIONS Step by Step Solutons www.mathcracker.com 9//016 Problem 1: A researcher s nterested n the effects of famly sze on delnquency for a group of offenders and examnes famles wth one to
More informationexperimenteel en correlationeel onderzoek
expermenteel en correlatoneel onderzoek lecture 6: one-way analyss of varance Leary. Introducton to Behavoral Research Methods. pages 246 271 (chapters 10 and 11): conceptual statstcs Moore, McCabe, and
More informationChapter 6. Supplemental Text Material
Chapter 6. Supplemental Text Materal S6-. actor Effect Estmates are Least Squares Estmates We have gven heurstc or ntutve explanatons of how the estmates of the factor effects are obtaned n the textboo.
More informationChapter 12 Analysis of Covariance
Chapter Analyss of Covarance Any scentfc experment s performed to know somethng that s unknown about a group of treatments and to test certan hypothess about the correspondng treatment effect When varablty
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationMultiple Contrasts (Simulation)
Chapter 590 Multple Contrasts (Smulaton) Introducton Ths procedure uses smulaton to analyze the power and sgnfcance level of two multple-comparson procedures that perform two-sded hypothess tests of contrasts
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationStatistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationUCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Chapter 11 Analysis of Variance - ANOVA. Instructor: Ivo Dinov,
UCLA STAT 3 ntroducton to Statstcal Methods for the Lfe and Health Scences nstructor: vo Dnov, Asst. Prof. of Statstcs and Neurology Chapter Analyss of Varance - ANOVA Teachng Assstants: Fred Phoa, Anwer
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationLecture 4 Hypothesis Testing
Lecture 4 Hypothess Testng We may wsh to test pror hypotheses about the coeffcents we estmate. We can use the estmates to test whether the data rejects our hypothess. An example mght be that we wsh to
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experments- MODULE LECTURE - 6 EXPERMENTAL DESGN MODELS Dr. Shalabh Department of Mathematcs and Statstcs ndan nsttute of Technology Kanpur Two-way classfcaton wth nteractons
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationFirst Year Examination Department of Statistics, University of Florida
Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve
More informationCHAPTER 6 GOODNESS OF FIT AND CONTINGENCY TABLE PREPARED BY: DR SITI ZANARIAH SATARI & FARAHANIM MISNI
CHAPTER 6 GOODNESS OF FIT AND CONTINGENCY TABLE Expected Outcomes Able to test the goodness of ft for categorcal data. Able to test whether the categorcal data ft to the certan dstrbuton such as Bnomal,
More informationLecture 16 Statistical Analysis in Biomaterials Research (Part II)
3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan
More informationStatistics II Final Exam 26/6/18
Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationDurban Watson for Testing the Lack-of-Fit of Polynomial Regression Models without Replications
Durban Watson for Testng the Lack-of-Ft of Polynomal Regresson Models wthout Replcatons Ruba A. Alyaf, Maha A. Omar, Abdullah A. Al-Shha ralyaf@ksu.edu.sa, maomar@ksu.edu.sa, aalshha@ksu.edu.sa Department
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationChapter 11: I = 2 samples independent samples paired samples Chapter 12: I 3 samples of equal size J one-way layout two-way layout
Serk Sagtov, Chalmers and GU, February 0, 018 Chapter 1. Analyss of varance Chapter 11: I = samples ndependent samples pared samples Chapter 1: I 3 samples of equal sze one-way layout two-way layout 1
More informationChapter 3 Describing Data Using Numerical Measures
Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The
More information7.1. Single classification analysis of variance (ANOVA) Why not use multiple 2-sample 2. When to use ANOVA
Sngle classfcaton analyss of varance (ANOVA) When to use ANOVA ANOVA models and parttonng sums of squares ANOVA: hypothess testng ANOVA: assumptons A non-parametrc alternatve: Kruskal-Walls ANOVA Power
More informationTopic 23 - Randomized Complete Block Designs (RCBD)
Topc 3 ANOVA (III) 3-1 Topc 3 - Randomzed Complete Block Desgns (RCBD) Defn: A Randomzed Complete Block Desgn s a varant of the completely randomzed desgn (CRD) that we recently learned. In ths desgn,
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationCHAPTER 8. Exercise Solutions
CHAPTER 8 Exercse Solutons 77 Chapter 8, Exercse Solutons, Prncples of Econometrcs, 3e 78 EXERCISE 8. When = N N N ( x x) ( x x) ( x x) = = = N = = = N N N ( x ) ( ) ( ) ( x x ) x x x x x = = = = Chapter
More informationStatistics Chapter 4
Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment
More informationChapter 5 Multilevel Models
Chapter 5 Multlevel Models 5.1 Cross-sectonal multlevel models 5.1.1 Two-level models 5.1.2 Multple level models 5.1.3 Multple level modelng n other felds 5.2 Longtudnal multlevel models 5.2.1 Two-level
More informationCopyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
yes to (3) two-sample problem? no to (4) underlyng dstrbuton normal or can centrallmt theorem be assumed to hold? and yes to (5) underlyng dstrbuton bnomal? We now refer to the flowchart at the end of
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationChapter 15 Student Lecture Notes 15-1
Chapter 15 Student Lecture Notes 15-1 Basc Busness Statstcs (9 th Edton) Chapter 15 Multple Regresson Model Buldng 004 Prentce-Hall, Inc. Chap 15-1 Chapter Topcs The Quadratc Regresson Model Usng Transformatons
More informationComparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More informationANOVA. The Observations y ij
ANOVA Stands for ANalyss Of VArance But t s a test of dfferences n means The dea: The Observatons y j Treatment group = 1 = 2 = k y 11 y 21 y k,1 y 12 y 22 y k,2 y 1, n1 y 2, n2 y k, nk means: m 1 m 2
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationLOGIT ANALYSIS. A.K. VASISHT Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi
LOGIT ANALYSIS A.K. VASISHT Indan Agrcultural Statstcs Research Insttute, Lbrary Avenue, New Delh-0 02 amtvassht@asr.res.n. Introducton In dummy regresson varable models, t s assumed mplctly that the dependent
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationLECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More information/ n ) are compared. The logic is: if the two
STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence
More informationResource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis
Resource Allocaton and Decson Analss (ECON 800) Sprng 04 Foundatons of Regresson Analss Readng: Regresson Analss (ECON 800 Coursepak, Page 3) Defntons and Concepts: Regresson Analss statstcal technques
More informationBayesian predictive Configural Frequency Analysis
Psychologcal Test and Assessment Modelng, Volume 54, 2012 (3), 285-292 Bayesan predctve Confgural Frequency Analyss Eduardo Gutérrez-Peña 1 Abstract Confgural Frequency Analyss s a method for cell-wse
More informationBasic Business Statistics, 10/e
Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson
More informationSPANC -- SPlitpole ANalysis Code User Manual
Functonal Descrpton of Code SPANC -- SPltpole ANalyss Code User Manual Author: Dale Vsser Date: 14 January 00 Spanc s a code created by Dale Vsser for easer calbratons of poston spectra from magnetc spectrometer
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationJanuary Examinations 2015
24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More informationDO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes
25/6 Canddates Only January Examnatons 26 Student Number: Desk Number:...... DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department Module Code Module Ttle Exam Duraton
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More informationAnswers Problem Set 2 Chem 314A Williamsen Spring 2000
Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationTest for Intraclass Correlation Coefficient under Unequal Family Sizes
Journal of Modern Appled Statstcal Methods Volume Issue Artcle 9 --03 Test for Intraclass Correlaton Coeffcent under Unequal Famly Szes Madhusudan Bhandary Columbus State Unversty, Columbus, GA, bhandary_madhusudan@colstate.edu
More informationF statistic = s2 1 s 2 ( F for Fisher )
Stat 4 ANOVA Analyss of Varance /6/04 Comparng Two varances: F dstrbuton Typcal Data Sets One way analyss of varance : example Notaton for one way ANOVA Comparng Two varances: F dstrbuton We saw that the
More informationHere is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)
Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationInterval Estimation in the Classical Normal Linear Regression Model. 1. Introduction
ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationSTAT 511 FINAL EXAM NAME Spring 2001
STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte
More informationFinding Dense Subgraphs in G(n, 1/2)
Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng
More informationSTAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression
STAT 45 BIOSTATISTICS (Fall 26) Handout 5 Introducton to Logstc Regresson Ths handout covers materal found n Secton 3.7 of your text. You may also want to revew regresson technques n Chapter. In ths handout,
More informationStatistical tables are provided Two Hours UNIVERSITY OF MANCHESTER. Date: Wednesday 4 th June 2008 Time: 1400 to 1600
Statstcal tables are provded Two Hours UNIVERSITY OF MNCHESTER Medcal Statstcs Date: Wednesday 4 th June 008 Tme: 1400 to 1600 MT3807 Electronc calculators may be used provded that they conform to Unversty
More informationLecture 6: Introduction to Linear Regression
Lecture 6: Introducton to Lnear Regresson An Manchakul amancha@jhsph.edu 24 Aprl 27 Lnear regresson: man dea Lnear regresson can be used to study an outcome as a lnear functon of a predctor Example: 6
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 14 Multple Regresson Models 1999 Prentce-Hall, Inc. Chap. 14-1 Chapter Topcs The Multple Regresson Model Contrbuton of Indvdual Independent Varables
More informationISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 1, July 2013
ISSN: 2277-375 Constructon of Trend Free Run Orders for Orthogonal rrays Usng Codes bstract: Sometmes when the expermental runs are carred out n a tme order sequence, the response can depend on the run
More informationANOMALIES OF THE MAGNITUDE OF THE BIAS OF THE MAXIMUM LIKELIHOOD ESTIMATOR OF THE REGRESSION SLOPE
P a g e ANOMALIES OF THE MAGNITUDE OF THE BIAS OF THE MAXIMUM LIKELIHOOD ESTIMATOR OF THE REGRESSION SLOPE Darmud O Drscoll ¹, Donald E. Ramrez ² ¹ Head of Department of Mathematcs and Computer Studes
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationTopic 10: ANOVA models for random and mixed effects Fixed and Random Models in One-way Classification Experiments
Topc 10: ANOVA models for random and mxed effects eferences: ST&D Topc 7.5 (15-153), Topc 9.9 (5-7), Topc 15.5 (379-384); rules for expected on ST&D page 381 replaced by Chapter 8 from Montgomery, 1991.
More informationCase A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k.
THE CELLULAR METHOD In ths lecture, we ntroduce the cellular method as an approach to ncdence geometry theorems lke the Szemeréd-Trotter theorem. The method was ntroduced n the paper Combnatoral complexty
More informationOnline Appendix to: Axiomatization and measurement of Quasi-hyperbolic Discounting
Onlne Appendx to: Axomatzaton and measurement of Quas-hyperbolc Dscountng José Lus Montel Olea Tomasz Strzaleck 1 Sample Selecton As dscussed before our ntal sample conssts of two groups of subjects. Group
More informationSampling Theory MODULE VII LECTURE - 23 VARYING PROBABILITY SAMPLING
Samplng heory MODULE VII LECURE - 3 VARYIG PROBABILIY SAMPLIG DR. SHALABH DEPARME OF MAHEMAICS AD SAISICS IDIA ISIUE OF ECHOLOGY KAPUR he smple random samplng scheme provdes a random sample where every
More informationCHAPTER IV RESEARCH FINDING AND DISCUSSIONS
CHAPTER IV RESEARCH FINDING AND DISCUSSIONS A. Descrpton of Research Fndng. The Implementaton of Learnng Havng ganed the whole needed data, the researcher then dd analyss whch refers to the statstcal data
More information