FAILURE OF STATISTICAL METHODS TO PROVE BIOEQUIVALENCE OF TWO MELOXICAM BIOEQUIVALENT FORMULATIONS. II. NON-PARAMETRIC METHODS

Size: px
Start display at page:

Download "FAILURE OF STATISTICAL METHODS TO PROVE BIOEQUIVALENCE OF TWO MELOXICAM BIOEQUIVALENT FORMULATIONS. II. NON-PARAMETRIC METHODS"

Transcription

1 FARMACIA, 0, Vol. 59, FAIURE OF STATISTICA METHODS TO PROVE BIOEQUIVAENCE OF TWO MEOXICAM BIOEQUIVAENT FORMUATIONS. II. NON-PARAMETRIC METHODS ROXANA SANDUOVICI, ANCA VATASESCU, FORIN ENACHE 3, CONSTANTIN MIRCIOIU * Bopharmacy & Pharmacol Res S.A., Bucharest Carol Davla Unversty of Medcne & Pharmacy, Bucharest 3 Insttute of Statstcs, Romanan Academy, Bucharest * correspondng author: constantn.mrcou@yahoo.com Abstract Applyng the offcal parametrc methods to analyze the results of a clncal boequvalence (BE) study concernng two suppostory formulatons contanng meloxcam as actve substance, the boequvalence couldn t be proved, tested drug appearng to have a greater boavalablty than the reference drug. Snce the reference drug presented an mportant ntervarablty and the tested drug proved a greater boavalablty than the reference drug, t was consdered that the products could be boequvalent, but the offcal statstcal test faled to prove ths. Followng manly the hgh varablty of reference drug and a dstrbuton of plasma levels of reference drug far from normalty, the falure was thought as a consequence of the applcaton of statstcal parametrc tests beyond the feld of ther valdty. Statstcal models for buldng non-parametrc confdence ntervals for the ratos of means of pharmacoknetc parameters were less restrctve that n the case of parametrc analyss. In a frst approxmaton there were neglected the sequence effects and further, both sequence and perod effects. The results lead to the same falure of provng BE, lke parametrc methods. The concluson was that non-parametrc methods lead to the same concluson concernng BE but are more effcent n rejectng the effects of outlers. Suspcon remans that even non-parametrc methods are not effcent n correctng the bas nduced by partton of data n some dfferent classes as n the case of pharmacoknetc parameters of the reference drug. Rezumat Aplcând metodele ofcale parametrce pentru a analza rezultatelor unu studu clnc de boechvalenţă (BE) a două formulăr de supoztoare care conţn ca substanţă actvă meloxcam, BE nu a putut f demonstrată, medcamentul testat dovednd o bodsponbltate (BD) ma mare decât medcamentul de refernţă. Deoarece medcamentul de refernţă a prezentat o ntervarabltate foarte mare, ar medcamentul testat a avut o BD ma mare decât cel de refernţă, s-a luat în consderare faptul că produsele ar putea f BE, dar testele statstce ofcale nu au demonstrat acest lucru. Urmărnd în prncpal varabltatea mare ş dstrbuţa nvelelor plasmatce a medcamentulu de refernţă, departe de normaltate, s-a consderat că eşuarea demonstrăr BE este consecnţa aplcăr testelor statstce parametrce în afara câmpulu lor de aplcare.

2 368 FARMACIA, 0, Vol. 59, 3 Modelele statstce pentru construrea ntervalelor de încredere non-parametrce pentru rapoartele medlor parametrlor farmacocnetc au fost ma puţn restrctve decât în cazul analze parametrce. Ca o prmă aproxmare, s-au negljat efectele de secvenţă, ar apo atât efectele de secvenţă cât ş efectele de peroadă. Rezultatele au dus la acelaş eşec în a demonstra BE ca ş metodele parametrce. Rezultatul a fost că metodele nonparametrce conduc la aceeaş concluze în ceea ce prveşte BE dar sunt ma efcente în respngerea efectelor outler-lor. Este posbl ca ş metodele non-parametrce să fe nefcente în deplasarea ndusă de partţa datelor în clase dferte cum a fost cazul parametrlor farmacocnetc a medcamentulu de refernţă. Keywords: boequvalence, meloxcam, non-parametrc methods. Introducton The present paper looked for alternatve non-parametrc methods n testng boequvalence (BE). Startng from both C (maxmum max concentraton) and total AUC0 (area under curve) data, non-parametrc methods appled [] to compare the dfferences n means wth offcal acceptance lmts ndcated that cannot conclude the boequvalence of tested and reference formulatons. Snce pharmacoknetc parameters found for the reference drug were far from a normal dstrbuton, beng shared n 3 dfferent classes, applcaton of parametrc methods was n fact not justfed and results were not relable. The problem of establshng BE s a problem of buldng a confdence nterval (CI) for the rato of means of man pharmacoknetc parameters. Statstcal model for the man pharmacoknetc parameters area under curve and maxmum concentraton ( AUC0 or C ) s: max Y = + S + P + F + C + e jk µ k j jk ( jk, ) ( j, k) where: µ = general mean, S k =the effect of the -th subject wthn the k-th sequence, whch, for the sake of testng hypotheses, we must assume to be a normally dstrbuted random varable wth mean 0 and varance σ, P j = the effect of the j-th perod, C ( j, k) F ( jk, ) =the drect effect of the drug, = the resdual effect of the drug, ε jk = the random fluctuaton whch s normally dstrbuted wth mean 0 and varance σ e, and s ndependent of the S. k The condtons for the applcaton of parametrc methods n testng boequvalence were not fulflled. Data concernng the reference drug (both untransformed and logarthmcally transformed) were not normally s

3 FARMACIA, 0, Vol. 59, dstrbuted and the varablty of the reference drug looked to be much hgher than that of the tested drug. Snce the decson concernng boequvalence or non-boequvalence s based on statstcal calculus, a soluton s to look for n ths feld. Current regulatons allow the use of alternatve non-parametrc methods. An mportant change appeared n the regulatons concernng boequvalence rules [] of European Medcne Agency (EMEA). The new regulatons specfy a non-parametrc analyss s not acceptable, no other alternatve beng ndcated n cases of clear alteraton of the normalty and assumptons. Amercan gudelnes and statstcans recommend crtera scaled wth the varance of the reference drug for falure of provng BE clearly followng reference drug defcences. Even EMEA ntated n 008 a dscusson about scaled crtera for Hghly Varable Drugs, but the 00 gudelnes mssed completely the subject. A soluton to ths problem would be to run another study wth a greater number of subjects, but ths s dffcult to be accepted by the Ethcs Commttees. Consequently, a further statstcal analyss had to be undertaken. Materals and methods In the study, there were enrolled 4 healthy volunteers, 8 of them completng the study. The data obtaned for areas under curve (AUC) and ther ratos are presented n table I. Table I Area under curve (AUC) for Reference ( R ) and Tested ( T ) drug Subject secv RT RT RT RT RT RT RT RT RT P P T/R Subject secv TR TR TR TR TR TR TR TR TR P P T/R secv. and secv. the frst sequence of drug admnstraton (the order of drug admnstraton: RT (reference drug and afterwards the tested drug); TR (the tested drug frst and afterwards the reference drug)

4 370 FARMACIA, 0, Vol. 59, 3 One subject had to be elmnated snce he showed zero absorpton after admnstraton of the reference drug. Also some concentratons n the fnal curve of reference drug were under the lmt of quantfcaton. Snce neglectng such data nduces a bas estmaton of area under curve [3] we shortened the tme nterval taken nto calculus. Results and dscusson Non - parametrc analyss Wlcoxon- Man Whtney two one-sded test for boequvalence [4] For the standard x crossover desgn consstng of a par of dual sequences (.e. RT and TR), the dstrbuton free rank sum test can be appled drectly to the two one-sded tests concernng boequvalence [5,6,7] Usng the above standard notatons t results are: θ = µ T µ R. The usual set of unlateral hypotheses concernng boequvalence H : θ 0 vs H : θ 0 where θ = θ θ and 0 A H : θ 0 vs H : θ 0 where θ = θ θ 0 U A U U U θ θ The estmated values of and U can be obtaned from a lnear combnaton (contrast) of perod dfferences d k, =,.., n; k =,. b hk dk θh; h=, U forsubjects n sequence =, d ; forsubjects n sequence k where: =,n, k =, dk = Y k Y k /, and h= or U ndex for lower respectvely upper lmt of confdence nterval When there are no carry-over effects, the expected value and varance of b, are gven by the equatons: hk k, ( ) ( ) E b hk ( P P) + ( θ θh ) for k = = ( P P) + θ for k =

5 FARMACIA, 0, Vol. 59, 3 37 σ e d hk k = σ = d The dfference between the means of and D( b ) = D( ) sequences equals the formulaton effect as follows: E b E = θ θ Consderng n n sequence : R( ) ( ) ( ) ( ) = θ h b h h h b n the two hk R as the sum of the ranks of the responses for subjects ( n ) n + R = b and W = R. = For testng the second hypothess we consder smlarly n n ( n + ) R U = R( b U ) and WU = RU = w α We reject H f 0 W w α, where α thquantle of the dstrbuton of W whch can be found n tables for Mann Whtney test, and H f 0 WU w α ( w = n n w ). α α Hence, boequvalence s concluded f both H and 0 H are rejected; 0 that s: W w α W w α s the ( ) and ( ) U Analyss of AUC tot data for the two MEOXICAM formulatons We estmate the θ and θ as 0% of the mean AUC for R U (estmaton of µ ) R AUCR AUC = R n+ n = ;!! =! U = 0." AUC R = 8980 It results: b = d θ = = 3088, etc. The complete set of obtaned values are gven n table II. We ordered the values of b k ( b Uk ) n n( n+ ) R = R( b ) = 54 ; W = R = 9 = n n( n+ ) RU = R( bu ) = 8 ; WU = RU = 36 =

6 37 FARMACIA, 0, Vol. 59, 3 P P d k Table II Calculus of ranks of perod dfferences R b ( ) R b k U b Uk b ( ) secv RT RT RT RT RT RT RT RT RT P P d b ( ) R b k b R ( ) secv k U TR TR TR TR TR TR TR TR TR In the tables t was found w = W = α 9;9;0.05 and w α = nn wα = 59 Snce 36 wα W and U 59 9 w α W, t means that the products are not boequvalent. Furthermore, we checked the hypothess concernng boequvalence, based on enlarged acceptance nterval 0.67 ;.33 ( ). In ths case!! =! U = 0.33" AUC R = 487 and b = d θ = = 895, etc Orderng descendng the absolute values of results the followng: n n( n+ ) R = R( b ) = 54, W = R = 9 = n n( n+ ) RU = R( bu ) = ; WU = RU = 66 = b Uk b k, respectve b Uk t

7 FARMACIA, 0, Vol. 59, Boequvalence s concluded f WU wα w α W 66 and whch s not the case. W U w α w - α W C Fgure Order of quantles and results of test requred to accept BE Once agan we couldn t establsh boequvalence. In order to understand better the statstcal phenomena appearng n comparson of the two drugs we determned 90% confdence ntervals for the rato of the mean values of areas under curve for the tested and the reference drugs. Dstrbuton-free confdence ntervals An estmator and confdence nterval assocated wth Wlcoxon sgned rank statstc. et us consder the Wlcoxon sgned rank test [8]. We can compare n pars of values( X, Y ), =,,N. Put Z = ( X Y). We computed the statstcs T + and the absolute values Z,..., Z and ordered them n n ascendng order. et r denote the rank of d = f Z > 0 and d = 0 f Z < 0 et be T + the sum of postve ranks. ThenT + = d. The mean of T + + N N s E ( T ) = E ( d ) = E ( d ) But E d ( ) ( ) =! + 0! = N N( N ) = = and consequently Z and d the random varable ET 4 We can consder two sded test for the effect of treatment θ : H : 0 : 0 0 θ = H a θ at the α level of sgnfcance, by comparson of T + wth w and w, quantles of Wlcoxon repartton functon. α α + +

8 374 FARMACIA, 0, Vol. 59, 3 An estmator of the treatment effect θ and confdence nterval for dfferences of means was gven by Hollander and Wolfe [9,0] based on the the Hodges-ehmann [] estmator!, defned by the followng equaton:! " = medan Z + Z j % $,! j =,...,n # ' & nn+ averages ( Z Z ) / The ( / ) +, j =,..., n, are called Walsh [] averages. If we defne W + the number of postve Walsh averages, then (when there are no tes among the [Z] s and none of the Z s s zero) W + s dentcal wth T +. Ths result s n accordance wth Tukey [3]. Whenθ = 0, the dstrbuton of the statstc T + s symmetrc about ts mean, nn+ ( ) be subtracted from each j /4. A natural estmator of θ s the amount! that should Z so that the value of T +, when appled to the shfted sample Z!!!,...,Z n!!!, s as close to nn+ ( /4 ) as possble. Roughly speakng, we estmate θ by the amount (!! ) that the Z sample should be shfted n order that Z!!!,...,Z n!!! appears (when vewed by the sgned rank statstc T + ) as a sample from a populaton wth medan 0. The estmator!! s relatvely nsenstve to outlers. Ths s not the case wth the classcal estmator n Z = Z / n. Thus the use of! provdes protecton aganst mportant errors. Ths procedure was appled by Stenjens and Dlett [4] n order to dfferentate the pharmacoknetc data. In fact ther statstcal model s very smple, neglectng both sequence and perod effects. We appled the method for AUC. We can calculate for each subject = AUC T d = ln ( AUC T) ln ( AUC R) = ln ln r = and AUC R d + dj = ln r + ln rj = ln rr j The method calculates the geometrc means of ratos for all possble N N + pars of subjects, d est for pars, ncludng the par (R, R), for the ( )

9 FARMACIA, 0, Vol. 59, same subject. The values of geometrc means are ordered. The nferor and superor lmts for 90 % confdence ntervals are found n Wlcoxon tables. Rato T for the subject number s R AUC T ( ) AUC R ( ) = =. Further more, we need the geometrc means of the par of ratos. For the frst subject combned wth tself, t results.!. ". For subject combned wth subject, t results.397!.067 ". For the entre group of 8 subjects there are N N +) = 8!9 = 7 combnatons presented n table III. Table III Geometrc means of pars of T/R ratos Table IV Non-parametrc confdence ntervals based on Wlcoxon s test upper and lower ranks Number of subjects Rank for lower lmt Rank for upper lmt (N) 95% 90% 95% 90%

10 376 FARMACIA, 0, Vol. 59, 3 As t can be seen n table IV, the lower and upper cutoff ponts for a 90% confdence nterval (CI) are the values ranked 48 and 4. For our data these values correspond to the ratos.8 and.54. Fgure ower and upper ranks for CIs We used the Excell functon small(array;k) for calculatng the k rankng value n a gven set (array). =small(f3:q0;48)=.5 =small(f3:q0;4)=.54 CI 90% =.8 ;.54 ( ) Consequently we had to reject the hypothess of boequvalence Hauschke Stenjans Dlett method. The prevous method was crtczed by the same authors later [5] snce t s based on the assumpton of equal perod effects. Both methods lead to the reject of boequvalence and the result s correct. Consderng the many possble outlers n the set of data (subject outlers, subject-by-formulaton outlers, sngle data pont outlers [6]) concernng the reference drug, the results are not relable and not ethcally correct. Followng hgh varablty of the reference drug, a more general approach should be consdered, regardng the dfferences n manufacturng processes [7], but we had no nformaton about the manufacturng of the reference drug. Another problem of the above method s that the confdence nterval s based on a theorem of ehmann [8] n obtanng the confdence nterval, whch ncludes the comparson of all possble pars between subjects. But boequvalence s essentally connected wth ntravarablty of the pharmacoknetc parameters. Interchangeablty means that we are nterested n comparng the plasma levels reached after admnstraton of dfferent drugs to the same subject. We further appled another non-parametrc test, based on the comparson of the plasma levels to the same subject.

11 FARMACIA, 0, Vol. 59, The a method whch the dfferences of o varables from sequences k and. Eo ( ) = Ed ( ) = ( P P) + ( FT FR) = ( FT FR) Eo ( ) = E( d) = ( P P) ( FR FT) = ( FT FR) Eo ( + o ) = ( F F) j T R Snce o = d the sums equal the dfferences d d,( =,..., n, j =,..., n ). j In ths case we don t need to suppose equal perod effects. Procedure comes from methods of Moses and Wolfe, once agan by ntermedate of Hollander and Wolfe. The examnaton of ndvdual profles reveals a clear splttng n three classes of reference drug curves (a subject wth aberrant small values, fve subjects wth hgh values and the homogeneous dstrbuted group), whch cannot be consdered as a normal stuaton [9]. Fgure 3 Indvdual concentraton profles for the reference drug The fact that plasma levels of the tested drug are greater than plasma levels of the reference drug represents a sgn that the tested drug s better than the reference drug or somethng s out of order wth the reference

12 378 FARMACIA, 0, Vol. 59, 3 drug but, snce the varablty of the reference s very hgh, the second hypothess s much more relable. In ths condtons, the rejectng of the tested drug s both ncorrect and unfar snce the defcences belong to the reference drug. Comparatve analyss of results obtaned wth parametrc and nonparametrc methods Both parametrc and non-parametrc methods ndcate a lack of boequvalence. The result of the nonparametrc order tests s an expected one, snce for almost all subjects. R T AUC0 AUC0 The result s essentally ncorrect snce varablty and dstrbuton of the reference drug data nfluence substantally the estmaton of ntravarablty and, consequently, the result of the tests concernng boequvalence. For such separaton, t s necessary to repeat the admnstraton of the drugs, whch s possble, but less acceptable by the Ethc Commttees. Table V 90 % Confdence ntervals (CI) calculated for all subjects (HSD - Hauschke, Stenjans, Dlett method, SD - Stenjans, Dlett method, - lower lmt, U - upper lmt) Method CI 90 U CI 90 ength Parametrc HSD SD Table VI 90 % Confdence ntervals calculated after excluson of an outler subject Method CI 90 U CI 90 ength Parametrc HSD SD It can be observed that all three methods are nfluenced by outlers but the most senstve seems to be the parametrc one. It can also be observed that the addton of an outler subject ncreases the length of ntervals. Table VII Decrease of the length of CIs followng elmnaton of an outler subject Method Parametrc 0.09 HSD 0.08 SD 0.03 Δ

13 FARMACIA, 0, Vol. 59, The parametrc method s actually less senstve than the nonparametrc method, but the dfference s not mportant. Conclusons Startng from AUC0 data, non-parametrc methods ndcated that cannot conclude boequvalence of tested and reference formulatons. T R Confdence nterval for dfferences of means µ AUC µ AUC was calculated usng Walsh means (method gven by Hollander and Wolfe [9, 0] based on the Hodges-ehmann [] estmator),.e. comparson of perod dfferences both between sequences and nsde sequences whch s a vald procedure n the hypothess of equal perod effects. The lack of perod effects was evdenced usng ANOVA calculus. Snce ANOVA s also based on normal dstrbuton, the results become relatve. It s largely accepted that non-parametrc tests are less senstve to outlers, whch was confrmed, but the dfferences were not mportant. The rejecton of non-parametrc methods by the EMEA recent gudelnes s not justfed. When data are normally or log-normally dstrbuted there are not sgnfcant dfferences between the results. When normalty hypothess s altered parametrc methods are no more relable and no other alternatve s ndcated. Applcaton of non-parametrc methods s fully justfed f sequence effects are absent. The lack of sequence effects can be assured by an adequate washng perod and verfed by the assay of plasma concentratons at the begnnng of the second perod. References. Vatasescu A., Enache F., Mrcou C., Mron D.S., Sandulovc R., Falure of statstcal methods to prove boequvalence of two meloxcam boequvalent formulatons. I. Parametrc methods, Farmaca, 0 (to be publshed). Gudelne of the nvestgaton of boequvalence, (row 504) 3. Tomuţă I., Iovanov R., Bodok E., eucuţa S.E., Quantfcaton of meloxcam and excpents on ntact tablets by near nfrared spectrometry and chemometry, Farmaca, 00, 58 (5), Chow S.C., u J.P., Desgn and Analyss of Boavalablty and Boequvalence Studes, nd edton, Marcel Dekker, 999, Hauschke D., Stenjans V.W., Dlet E., A dstrbuton free procedure for the statstcal analyses of boequvalence studes. Int J Cln Pharmacol Ther Toxcol, 990, 8, Cornell R.G., The evaluaton of boequvalence usng nonparametrc procedures, Commun Stat Theory Methods, 990, 9, u J.P., Boequvalence and Intrasubject varablty, J Bopharm Stat, 99,, Mrcou C., Sandulovc R., Statstcă aplcată în farmace ş stud clnce, ed.ii, Ed. Unv Carol Davla, Bucureşt 009,

14 380 FARMACIA, 0, Vol. 59, 3 9. Hollander M., Wolfe D.A., Non-parametrc Statstcal Methods, Wley, Hollander M., Wolfe D.A., Non-parametrc Statstcal Methods, nd edton,wley, New York, 999, Hodges J.., ehman E.., Hodges ehmann estmators - In Encclopeda of Statstcal Scences, volume 3, J. Wley, 983, Walsh J.E., Some sgnfcance tests for the medan whch are vald under very general condtons, Ann Mat Stat, 949, 0, Tukey J.W., The smplest sgned-rank tests. Memo 7, Statstcal Reseach Group, Prnceton Unversty, Stenjens V.W., Dlett E., Statstcal Analyss of Boavalablty Studes: Parametrc and Non-parametrc Confdence Intervals, Eur. J. Cln. Pharmacol, 983, 4, Hauschke D., Stenjans V.W., Dlett E., A dstrbuton-free procedure for the statstcal analyss of boequvalence studes, Cln Pharmacol Ther Toxcol, 990, 8(), Schall R., Endreny., Rng A., Resduals and outlers n replcate desgn crossover studes, J Bopharm Stat., 00, 0(4), Chow S.C., u J.P., Statstcal assessment of bosmlar products, J Bopharm Stat., 00, 0(), ehman E.., Non-parametrcs: Statstcal Methods Based on Ranks, Holden-Day, San Francsco, 975, Medvedovc A., Albu F., Georgta C., Mrcou C., Davd V., A non-extractng procedure for the determnaton of meloxcam n plasma samples by HPC-dode array detecton, Arznemttel Forschung/Drug Research, 005, 55(6), Manuscrpt receved: December 0 th 00

Chapter 13: Multiple Regression

Chapter 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 information

Statistics II Final Exam 26/6/18

Statistics 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 information

Chapter 11: Simple Linear Regression and Correlation

Chapter 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 information

Comparison of Regression Lines

Comparison 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 information

Psychology 282 Lecture #24 Outline Regression Diagnostics: Outliers

Psychology 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 information

4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA

4 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 information

Department of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution

Department 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 information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. 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 information

Durban Watson for Testing the Lack-of-Fit of Polynomial Regression Models without Replications

Durban 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 information

Statistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation

Statistics 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

ECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics

ECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott

More information

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands

1. 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 information

/ n ) are compared. The logic is: if the two

/ 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 information

2016 Wiley. Study Session 2: Ethical and Professional Standards Application

2016 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 information

Department of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6

Department 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 information

Statistics for Economics & Business

Statistics 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 information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. 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

Topic 23 - Randomized Complete Block Designs (RCBD)

Topic 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 information

Chapter 14 Simple Linear Regression

Chapter 14 Simple Linear Regression Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng

More information

STAT 3008 Applied Regression Analysis

STAT 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 information

Chapter 11: I = 2 samples independent samples paired samples Chapter 12: I 3 samples of equal size J one-way layout two-way layout

Chapter 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 information

UCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Chapter 11 Analysis of Variance - ANOVA. Instructor: Ivo Dinov,

UCLA 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 information

Modeling and Simulation NETW 707

Modeling and Simulation NETW 707 Modelng and Smulaton NETW 707 Lecture 5 Tests for Random Numbers Course Instructor: Dr.-Ing. Magge Mashaly magge.ezzat@guc.edu.eg C3.220 1 Propertes of Random Numbers Random Number Generators (RNGs) must

More information

STATISTICS QUESTIONS. Step by Step Solutions.

STATISTICS 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 information

Chapter 12 Analysis of Covariance

Chapter 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 information

Economics 130. Lecture 4 Simple Linear Regression Continued

Economics 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 information

Statistical tables are provided Two Hours UNIVERSITY OF MANCHESTER. Date: Wednesday 4 th June 2008 Time: 1400 to 1600

Statistical 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 information

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9

Correlation 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 information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. 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 information

Statistics for Business and Economics

Statistics for Business and Economics Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR 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 information

Chapter 3 Describing Data Using Numerical Measures

Chapter 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 information

BOOTSTRAP 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. 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 information

Lecture 6: Introduction to Linear Regression

Lecture 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 information

x i1 =1 for all i (the constant ).

x 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 information

Comparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method

Comparison 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 information

UNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours

UNIVERSITY 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 information

x = , so that calculated

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 information

Lecture 4 Hypothesis Testing

Lecture 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 information

Negative Binomial Regression

Negative 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 information

F statistic = s2 1 s 2 ( F for Fisher )

F 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 information

Systematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal

Systematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal 9/3/009 Sstematc Error Illustraton of Bas Sources of Sstematc Errors Instrument Errors Method Errors Personal Prejudce Preconceved noton of true value umber bas Prefer 0/5 Small over large Even over odd

More information

A Robust Method for Calculating the Correlation Coefficient

A Robust Method for Calculating the Correlation Coefficient A Robust Method for Calculatng the Correlaton Coeffcent E.B. Nven and C. V. Deutsch Relatonshps between prmary and secondary data are frequently quantfed usng the correlaton coeffcent; however, the tradtonal

More information

Statistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models

Statistics 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 information

Answers Problem Set 2 Chem 314A Williamsen Spring 2000

Answers 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 information

ANOVA. The Observations y ij

ANOVA. 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 information

STAT 511 FINAL EXAM NAME Spring 2001

STAT 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 information

Statistical Evaluation of WATFLOOD

Statistical Evaluation of WATFLOOD tatstcal Evaluaton of WATFLD By: Angela MacLean, Dept. of Cvl & Envronmental Engneerng, Unversty of Waterloo, n. ctober, 005 The statstcs program assocated wth WATFLD uses spl.csv fle that s produced wth

More information

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)

Econ107 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 information

Statistics Chapter 4

Statistics 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 information

Assignment 5. Simulation for Logistics. Monti, N.E. Yunita, T.

Assignment 5. Simulation for Logistics. Monti, N.E. Yunita, T. Assgnment 5 Smulaton for Logstcs Mont, N.E. Yunta, T. November 26, 2007 1. Smulaton Desgn The frst objectve of ths assgnment s to derve a 90% two-sded Confdence Interval (CI) for the average watng tme

More information

Chapter 15 - Multiple Regression

Chapter 15 - Multiple Regression Chapter - Multple Regresson Chapter - Multple Regresson Multple Regresson Model The equaton that descrbes how the dependent varable y s related to the ndependent varables x, x,... x p and an error term

More information

Chapter 8 Indicator Variables

Chapter 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 information

A Note on Test of Homogeneity Against Umbrella Scale Alternative Based on U-Statistics

A Note on Test of Homogeneity Against Umbrella Scale Alternative Based on U-Statistics J Stat Appl Pro No 3 93- () 93 NSP Journal of Statstcs Applcatons & Probablty --- An Internatonal Journal @ NSP Natural Scences Publshng Cor A Note on Test of Homogenety Aganst Umbrella Scale Alternatve

More information

LINEAR REGRESSION ANALYSIS. MODULE VIII Lecture Indicator Variables

LINEAR REGRESSION ANALYSIS. MODULE VIII Lecture Indicator Variables LINEAR REGRESSION ANALYSIS MODULE VIII Lecture - 7 Indcator Varables Dr. Shalabh Department of Maematcs and Statstcs Indan Insttute of Technology Kanpur Indcator varables versus quanttatve explanatory

More information

First Year Examination Department of Statistics, University of Florida

First 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 information

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor Taylor Enterprses, Inc. Control Lmts for P Charts Copyrght 2017 by Taylor Enterprses, Inc., All Rghts Reserved. Control Lmts for P Charts Dr. Wayne A. Taylor Abstract: P charts are used for count data

More information

Comparison of Outlier Detection Methods in Crossover Design Bioequivalence Studies

Comparison of Outlier Detection Methods in Crossover Design Bioequivalence Studies Journal of Pharmacy and Nutrton Scences, 01,, 16-170 16 Comarson of Outler Detecton Methods n Crossover Desgn Boequvalence Studes A. Rasheed 1,*, T. Ahmad,# and J.S. Sddq,# 1 Deartment of Research, Dow

More information

Using the estimated penetrances to determine the range of the underlying genetic model in casecontrol

Using the estimated penetrances to determine the range of the underlying genetic model in casecontrol Georgetown Unversty From the SelectedWorks of Mark J Meyer 8 Usng the estmated penetrances to determne the range of the underlyng genetc model n casecontrol desgn Mark J Meyer Neal Jeffres Gang Zheng Avalable

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

DERIVATION OF THE PROBABILITY PLOT CORRELATION COEFFICIENT TEST STATISTICS FOR THE GENERALIZED LOGISTIC DISTRIBUTION

DERIVATION OF THE PROBABILITY PLOT CORRELATION COEFFICIENT TEST STATISTICS FOR THE GENERALIZED LOGISTIC DISTRIBUTION Internatonal Worshop ADVANCES IN STATISTICAL HYDROLOGY May 3-5, Taormna, Italy DERIVATION OF THE PROBABILITY PLOT CORRELATION COEFFICIENT TEST STATISTICS FOR THE GENERALIZED LOGISTIC DISTRIBUTION by Sooyoung

More information

where I = (n x n) diagonal identity matrix with diagonal elements = 1 and off-diagonal elements = 0; and σ 2 e = variance of (Y X).

where I = (n x n) diagonal identity matrix with diagonal elements = 1 and off-diagonal elements = 0; and σ 2 e = variance of (Y X). 11.4.1 Estmaton of Multple Regresson Coeffcents In multple lnear regresson, we essentally solve n equatons for the p unnown parameters. hus n must e equal to or greater than p and n practce n should e

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. 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 information

Simulated Power of the Discrete Cramér-von Mises Goodness-of-Fit Tests

Simulated 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 information

Basic Business Statistics, 10/e

Basic 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 information

Joint Statistical Meetings - Biopharmaceutical Section

Joint 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 information

7.1. Single classification analysis of variance (ANOVA) Why not use multiple 2-sample 2. When to use ANOVA

7.1. 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 information

18. SIMPLE LINEAR REGRESSION III

18. SIMPLE LINEAR REGRESSION III 8. SIMPLE LINEAR REGRESSION III US Domestc Beers: Calores vs. % Alcohol Ftted Values and Resduals To each observed x, there corresponds a y-value on the ftted lne, y ˆ ˆ = α + x. The are called ftted values.

More information

This column is a continuation of our previous column

This column is a continuation of our previous column Comparson of Goodness of Ft Statstcs for Lnear Regresson, Part II The authors contnue ther dscusson of the correlaton coeffcent n developng a calbraton for quanttatve analyss. Jerome Workman Jr. and Howard

More information

28. SIMPLE LINEAR REGRESSION III

28. SIMPLE LINEAR REGRESSION III 8. SIMPLE LINEAR REGRESSION III Ftted Values and Resduals US Domestc Beers: Calores vs. % Alcohol To each observed x, there corresponds a y-value on the ftted lne, y ˆ = βˆ + βˆ x. The are called ftted

More information

Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore

Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased

More information

Learning Objectives for Chapter 11

Learning Objectives for Chapter 11 Chapter : Lnear Regresson and Correlaton Methods Hldebrand, Ott and Gray Basc Statstcal Ideas for Managers Second Edton Learnng Objectves for Chapter Usng the scatterplot n regresson analyss Usng the method

More information

DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes

DO 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 information

A Comparative Study for Estimation Parameters in Panel Data Model

A Comparative Study for Estimation Parameters in Panel Data Model A Comparatve Study for Estmaton Parameters n Panel Data Model Ahmed H. Youssef and Mohamed R. Abonazel hs paper examnes the panel data models when the regresson coeffcents are fxed random and mxed and

More information

Composite Hypotheses testing

Composite 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 information

THE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE

THE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE THE ROYAL STATISTICAL SOCIETY 6 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutons to assst canddates preparng for the eamnatons n future years and for

More information

Lecture 6 More on Complete Randomized Block Design (RBD)

Lecture 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 information

Regression. The Simple Linear Regression Model

Regression. The Simple Linear Regression Model Regresson Smple Lnear Regresson Model Least Squares Method Coeffcent of Determnaton Model Assumptons Testng for Sgnfcance Usng the Estmated Regresson Equaton for Estmaton and Predcton Resdual Analss: Valdatng

More information

a. (All your answers should be in the letter!

a. (All your answers should be in the letter! Econ 301 Blkent Unversty Taskn Econometrcs Department of Economcs Md Term Exam I November 8, 015 Name For each hypothess testng n the exam complete the followng steps: Indcate the test statstc, ts crtcal

More information

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton

More information

Chapter 5 Multilevel Models

Chapter 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 information

is the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors

is the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors Multple Lnear and Polynomal Regresson wth Statstcal Analyss Gven a set of data of measured (or observed) values of a dependent varable: y versus n ndependent varables x 1, x, x n, multple lnear regresson

More information

One-sided finite-difference approximations suitable for use with Richardson extrapolation

One-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 information

COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS

COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

Chapter 15 Student Lecture Notes 15-1

Chapter 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 information

A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) ,

A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) , A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS Dr. Derald E. Wentzen, Wesley College, (302) 736-2574, wentzde@wesley.edu ABSTRACT A lnear programmng model s developed and used to compare

More information

Parametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010

Parametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010 Parametrc fractonal mputaton for mssng data analyss Jae Kwang Km Survey Workng Group Semnar March 29, 2010 1 Outlne Introducton Proposed method Fractonal mputaton Approxmaton Varance estmaton Multple mputaton

More information

Interval Estimation in the Classical Normal Linear Regression Model. 1. Introduction

Interval 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 information

Lecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding

Lecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding Recall: man dea of lnear regresson Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 8 Lnear regresson can be used to study an

More information

Lecture 16 Statistical Analysis in Biomaterials Research (Part II)

Lecture 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 information

Lecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding

Lecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 008 Recall: man dea of lnear regresson Lnear regresson can be used to study

More information

January Examinations 2015

January 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 information

Multivariate Ratio Estimator of the Population Total under Stratified Random Sampling

Multivariate Ratio Estimator of the Population Total under Stratified Random Sampling Open Journal of Statstcs, 0,, 300-304 ttp://dx.do.org/0.436/ojs.0.3036 Publsed Onlne July 0 (ttp://www.scrp.org/journal/ojs) Multvarate Rato Estmator of te Populaton Total under Stratfed Random Samplng

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. 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 information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The 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 information

A nonparametric two-sample wald test of equality of variances

A nonparametric two-sample wald test of equality of variances Unversty of Wollongong Research Onlne Centre for Statstcal & Survey Methodology Workng Paper Seres Faculty of Engneerng and Informaton Scences 0 A nonparametrc two-sample wald test of equalty of varances

More information

ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE)

ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) June 7, 016 15:30 Frst famly name: Name: DNI/ID: Moble: Second famly Name: GECO/GADE: Instructor: E-mal: Queston 1 A B C Blank Queston A B C Blank Queston

More information

Introduction to Regression

Introduction to Regression Introducton to Regresson Dr Tom Ilvento Department of Food and Resource Economcs Overvew The last part of the course wll focus on Regresson Analyss Ths s one of the more powerful statstcal technques Provdes

More information