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# Applied Multivariate Statistical Analysis ST 731

NCS

GPA 3.79

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This 26 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 731 at North Carolina State University taught by Peter Bloomfield in Fall. Since its upload, it has received 12 views. For similar materials see /class/223931/st-731-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15

Principal Factor Solution o Recall the Orthogonal Factor Model X 2 LF e which implies 2 2 LL II o The m factor Principal Component solution is to approxi mate 2 or if we standardize the variables R by a rank m matrix using the spectral decomposition 2 Alele1 Amemegl Am1em1en1 Apepeg o The first m terms give the best rank m approximation to 2 1 c We can sometimes achieve higher communalities 2 diag LL by either specifying an initial estimate of the communalities iterating the solution or both 0 Suppose we are working with R Given initial communalities hf form the reduced correlation matrix 2 h T1722 T1729 Rr T271 h T2729 2 T1971 7392 quot hp 0 Now use the spectral decomposition of Rr to find its best rank m approximation Rr Li Liquot 0 New communalities are Ms W 2 z 1m 1 J 0 Find III by equating the diagonal terms a 1 32 or iiquot I diag LrLr o This is the Principal Factor solution 0 The Principal Component solution is the special case where the initial communalities are all 1 o In proc factor use method prin as for the Principal Com ponent solution but also specify the initial communalities the priors option on the proc factor statement specifies a method such as squared multiple correlations priors SMC the priors statement provides explicit numerical values 0 SAS program and output proc factor data all method prin priors smc title Method Principal Factors var cvx xom o In this case the communalities are smaller than for the Prin cipal Component solution Other choices for the priors option include MAX maximum absolute correlation with any other vari able ASMC gt Adjusted SMC adjusted to make their sum equal to the sum of the maximum absolute correlations 0NEgt 1 RANDOM gt uniform on 01 Iterated Principal Factors 0 One issue with both Principal Components and Principal Fac tors if S or R is exactly in the form LL III or more likely approximately in that form neither method produces L and III unless you specify the true communalities 0 Solution iterate Use the new communalities as initial communalities to get another set of Principal Factors Repeat until nothing much changes o In proc factor use method prinit may also specify the initial communalities default 2 ONE o SAS program and output proc factor data all method prinit title Method Iterated Principal Factors var cvx xom o The communalities are still smaller than for the Principal Component solution but larger than for Principal Factors Likelihood Methods 0 If we assume that X N Npu2 with E 2 LL III we can fit by maximum likelihood L is not identified without a constraint uniqueness con dition such as L xIx lL diagonal still no closed form equation for E numerical optimization required c We can also test hypotheses about m with the likelihood ratio test Bartlett s correction improves the X2 approximation H02mm0 HAImgtmO 2 x log likelihood ratio N X2 with 19 mo2 p m0 degrees of freedom Degrees of freedom gt O ltgt m0 lt 2p 1 1819 1 o Eg for p 5 m0 lt 2298 gt mg g 2 p m0 deg rees of freedom 5 O 10 5 1 5 5 2 1 10 o In proc factor use method m1 may also specify the initial communalities default SMC SAS program and output proc factor data all method ml var cvx xom title Method Maximum Likelihood proc factor data all method ml heywood plot var cvx xom title Method Maximum Likelihood with Heywood fixup proc factor data all method ml ultraheywood plot var cvx xom title Method Maximum Likelihood with UltraHeywood fixup 11 0 Note that the iteration can produce communalities gt 1 a Two fixes use the Heywood option on the prOC factor statement caps the communalities at 1 use the UltraHeywood option on the proc factor statement allows the iteration to continue with communalities gt 1 12 Scaling and the Likelihood o If the maximum likelihood estimates for a data matrix X are L and III and Y X D nxp XPXP is a scaled data matrix with the columns of X scaled by the entries of the diagonal matrix D then the maximum likelihood estimates for Y are DE and D2 c That is the mle s are invariant to scaling SY DSXD 13 Proof Lyl2LXD1JD1ED1 o No distinction between covariance and correlation matrices 14 Weighting and the Likelihood o Recall the uniqueness condition L xIx lL Adiagona 0 Write 1 1 2 2 11 52117 1 1 III 5LL imp 5 1 1 xv 5L III 5L 11 2 LL 1p o 2 is the weighted covariance matrix 15 0 Here 1 L III 3L and L L L III1L A a Note 2L LL L L LA L LA Im so the columns of L are the unhormalized eigenvectors of 2 the weighted covariance matrix 16 0 Also 2 IpL LA so the columns of L are also the eigenvectors of 1 1 2 Ip III XE IIIII the weighted reduced covariance matrix 0 Since the likelihood analysis is transparent to scaling the weighted reduced covariance matrix is the same as the weighted reduced covariance matrix 17 Confidence Statements About the Mean 0 Suppose as before that X1X2Xn is a random sample from Npu2 o A confidence ellipsoid for u is the set of u satisfying n X LOIS 192 u g Ian jppwpm 0 That is the probability that this random set contains the true value no is 1 a It is a 1001 00 confidence set for no o For example for the microwave oven data using R oven cbindreadtablequotJandWTO401datquot readtablequotJandWTO405datquot ellipseoven 025 o Load the ellipse function using sourcequothttpwwwstatncsuedupeoplebloomfieldcoursesst731ovenrquot o The fourth root transformation oven 025 was chosen using Box Cox methods Linear Combinations o An elliptical or in higher dimensions ellipsoidal confidence region is hard to interpret c We can change it into an equivalent set of confidence inter vals for linear combinations of p o Recall T2 nX LgtIS1X Lgt mcaxtg 2 max wt N2 C C Sc oSO n X gtS 1 X 1 g Ian jFpmpoz implies that n 6 X Mgtl2 pm 1 C Sc n p Fun Ma for every c 0 Conversely if the latter inequality holds for every c then the former inequality also holds 0 Therefore the probability that 1 c X j J C SC gtlt Kn Fp7npa n n p contains the true value c uo for every c is 1 a 0 These are 1001 a simultaneous confidence intervals for all linear combinations of MO One at a Time Intervals o For a given c we would use the t interval CX I tn1 C SC TL o In particular for W we would use Oz Xi it15 7 0 These intervals are narrower than the T2 intervals but the probability that they are all correct is less than 1 a 6 Bonferroni Simultaneous Intervals o If we want simultaneous confidence intervals for some but not all combinations c uo we can sometimes do better than use the T2 intervals o Eg for the components of MO use a 3 intn1 2p 71 o Now the probability that each interval is correct is 1 ap so the probability that alp are correct is at least 1 04 Bon ferroni s Inequality These Bonferroni intervals are narrower than the T2 intervals o For example for the microwave oven data again ellipseoven 025 intervalType Cquotququot quotBonquot a Note that the Bonferroni intervals for M1 and 112 are narrower than the T2 intervals 0 But if we wanted also to provide a confidence interval for M1 112 we could use the T2 interval S S 2S 2 1 X1 X2 j 11 22 12 X n n n 2 and the three T2 confidence intervals would be simultane ously valid Try ellipseoven 025 c C1 1 intervalType quotquot add TRUE Fan 2a a To add such an interval to the Bonferroni intervals we would have to Replace a2p 044 by 046 in each interval Be sure that prior to seeing the data we had decided to construct these three intervals 0 That is to use the Bonferroni method the list of intervals must be decided before seeing the data whereas the T2 intervals are wide enough for all to be valid

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