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# INTROTOLINEARSTATMOD STAT551

Penn

GPA 3.53

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This 11 page Class Notes was uploaded by Orval Funk on Monday September 28, 2015. The Class Notes belongs to STAT551 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/215441/stat551-university-of-pennsylvania in Statistics at University of Pennsylvania.

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Date Created: 09/28/15

Further Analysis of 3rd Grade Data Stepwise Fit Response pctpassmath 4 rows not used due to missing values NOTE 4 outliers have been deleted here and in the subsequent analyses Here s a snapshot of the Current Estimates table from the Stepwise Fit platform Current Estimates SSE DFE MSE 479016 3405 140680 Lock Entered Parameter X X Intercept X avgteachersalary avgclasssize avgteacherexperience pctltdengish pctecondisadv totalenrollment grade3enroment pctspecialed pctgifted pctblack pcthispanic perpupilexpend gtlt gtltgtltgtltgtltgtltgtltgtltgtltgtlt RSquare 02406 Estimate 81112 000033 00978 02969 005076 02150 00060 002040 017128 007057 01119 0 0000358 CID 11000506 SS 0 1993543 3785024 1460818 1233165 6352481 1815659 1166069 1268901 5748455 7183462 0071197 3094438 AIC 1691414 F Ratio 0000 14171 2691 10384 8766 451554 12906 8289 9020 4086 51062 0001 2200 ProbgtF 10000 00002 01010 00013 00031 00000 00003 00040 00027 00433 00000 09821 01381 Here st Step Hi Step AOOmNmUlhOONA he Step History table story Parameter pctecondisadv pctblack avgteacherexperience avgteachersalary totalenrollment grade3enrollment pctltdenglish pctspecialed pctgifted avgclasssize perpupilexpend Action Entered Entered Entered Entered Entered Entered Entered Entered Entered Entered Entered quotSig Probquot 00000 00000 00000 00017 00337 00092 00255 00103 00615 01797 01381 Seq SS 1 24631 1 1294239 8528387 1 392151 6387887 9590316 7043103 9277665 4924745 2533836 3094438 RSquare 01976 02181 02316 02338 02348 02364 02375 02389 02397 02401 02406 CID 18392 9395 35345 27452 24913 20098 17093 125 11 11199 AC 1692805 1692326 1692026 1691566 1691415 1691435 11001 1691414 One might consider using a model with 11 predictors as above The ANOVA table is as follows AIC values are not automatically provided in this JMP table but are available elsewhere in the JMP analysis The minimum CD and AIC values are shown in red They don t quite agree OmNCDU39IAOON39O Response pctpassmath Summary of Fit RSquare 02406 Root Mean Square Error 11860 Mean of Response 82528 Observations or Sum Wgts 3417 Analysis of Variance Source DF Sum of Squares Mean Square Model 11 15177927 137981 Error 3405 47901692 1407 C Total 3416 63079618 Parameter Estimates Term Estimate Std Error t Ratio Intercept 81112087 284358 2852 avgteachersaary 00003379 000009 376 avgclasssize 0097883 0059675 1 64 avgteacherexperience 02969464 009215 322 pctltdengish 00507609 0017145 296 pctecondisadv 0215031 0010119 2125 totalenrollment 0006066 0001689 359 grade3enroment 00204037 0007087 288 pctspecialed 0171289 0057034 300 pctgifted 00705721 0034912 202 pctblack 0111982 0015671 715 perpupilexpend 0000358 0000241 148 The Residual by Predicted plot is very interesting F Ratio 980813 Prob gt F lt0001 Probgtt lt0001 00002 01010 00013 00031 lt0001 00003 00040 00027 00433 lt0001 01381 Parenthetically before going further I note that I might have decided to use a 9 factor model rather than an 11 factor model The ANOVA table would then be as follows Summary of Fit RSquare 02397 Root Mean Square Error 11864 Mean of Response 82528 Observations or Sum Wgts 3417 Analysis of Variance Source DF Sum of Squares Mean Square F Ratio Model 9 15121644 168018 1193625 Error 3407 47957974 1408 Prob gt F C Total 3416 63079618 lt0001 Parameter Estimates Term Estimate Std Error t Ratio Probgtt Intercept 79420 2707507 2933 lt0001 avgteachersalary 000029 0000085 341 00007 avgteacherexperience 03114 0091443 341 00007 pctltdenglish 005031 0017138 294 00033 pctecondisadv 02151 0010122 2125 lt0001 totalenrollment 000568 0001635 347 00005 grade3enrolment 001926 0006975 276 00058 pctspecialed 01555 0055458 280 00051 pctgifted 006510 0034804 187 00615 pctblack 011237 0015629 719 lt0001 Using this model ratherthan the 11factor one would not have made much difference in the considerations that follow Residual by Predicted Plot S 40 quot 3 20 CE 39 9 0 395 ElZO 340 3 83960 Iquot39Iquot39Iquot39Iquot39Iquot39Iquot39Iquot39Iquot39IIII Q 10 20 30 4O 5O 6O 7O 8O 90 pctpassmath Predicted The data is not homoscedastic Ifyou think about what it is this is not surprising Here is a plot of the squared residuals vs the pred values actually vs 100 pred value Bivariate Fit of residquot2 By 100pred 3000 39 N2000 39 lt E U I a L I 1000 0 quotquotlquotquotlquotquotlquotquotl39 0 10 20 30 40 100pred Polynomial Fit Degree2 Smoothing Spline Fit lambda1 The plot also shows a best quadratic t and a smoothing spline t Note that the two are similar which suggests that the quadratic fit is a good representation ofthe general pattern in the scatterplot This is the equation for the polynomial t residquot2 101604 12538 gtlt100pred 05116gtlt100pred174719quot2 In the interest of parsimony and some common sense I decided to approximate this by 1 O5gtltpredquot2 Here is a plot comparing the ts Overlay Plot 1500 1000 gt I 500 O I I I I I I I O 10 2O 30 4O 100pred Y 15100predquot2 quadr Predicted residquot2 splinePredicted residquot2 2 Hence the data appears heteroscedastic with error variance approximately given by the formula 02 X 100 quotpredquotx02 Because ofthis an analysis via Generalized Least Squares seems appropriate The observations are still assumed to be independent so the matrix V on p 122 of the text is diagonal with diagonal elements 02 The corresponding GLS analysis is equivalent to an analysis with the observations weighted by V391 ie with individual weights 02 X71 JMP easily provides such an analysis though I m not sure of the meaning of all the entries in the resulting tables Here is the analysis Response pctpassmath Weight weight Summary of Fit RSquare 0296475 RSquare Adj 0294824 Root Mean Square Error 323943 Mean of Response 858147 Observations or Sum Wgts 341109 Analysis of Variance Source DF Sum of Squares Mean Square F Ratio Model 8 15071117 188389 1795223 Error 3408 35763228 1049 Prob gt F C Total 3416 50834345 lt0001 Parameter Estimates Term Estimate Std Error t Ratio Probgtt Intercept 92047885 1355284 6792 00000 pctltdengish 00272368 0016342 167 00957 pctecondisadv 0207679 0008757 2371 lt0001 totalenrollment 0001598 0001378 116 02463 grade3enrolment 00094497 0005848 162 01062 pctspecialed 01082677 0049151 220 00277 pctgifted 00790371 0023108 342 00006 pctblack 0128056 0017013 753 lt0001 perpupilexpend 00003334 0000195 171 00873 The remaining independent variables are not signi cant with Pvalues above 08 Note that this table differs in some qualitative respects from that for the unweighted analysis Lecture Notes on Random Mixed Effects Models General Form and Results Examples General Structure Observe Y D n gtltl with the structure 1 YX ZKleuke where X D ngtlt pkrllown each Zk D ngtlt pk known 11k D N003927k1pkxpk e D N0O39ZIW all indep Here uk and e are unobserved hidden or latent variables of7k are the variance components parameters to be estimated or tested and of course 039 denotes the error variance parameter Examples A OneWay random effects model X 1 Kl Z is usual oneway design matrix BR 2way additive Random effects model X 1 K2 21 is usual oneway design matrix for roweffects Z 2 is usual oneway design matrix for columneffects 039 are the variance components for row and column effects respectively Note1 The first columns of21 and Z are each 1 szand 21 and Z 2 are not linearly independent That s OK in this type of structure Notez can also be versions of this model that include another variance component for the interaction terms If so the usual way to do this would be to add a third variance component with the Z 3 matrix describing the row and column and the corresponding 113 being an gnu vector BM 2way additive Mixed effects model X is the usual oneway design matrix for roweffects 21 is usual oneway design matrix for columneffects There is only one variance component Note3 There can also be versions of this model that include another variance component for the interaction terms As in Notez above this third variance component could involve a Z 3 matrix describing the row and column with the corresponding 113 being an gnu vector Or the model could involve the random effects nested within the levels of the fixed effect This nested model would normally be written to have only one variance component parameter and this would describe the common variance for all the nested effects Sampling Distribution It follows from 1 that 2 Y D NX V where V 021Za ykaZg Note that Vinvolves the unknown random effects variance parameters as well as the error variance parameter Maximum likelihood estimation The normal density in 2 describes the likelihood for the unknown parameters From there one can differentiate and get the likelihood equations A considerable amount of algebraic manipulation is needed but th

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