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Generalized Linear Models MATH 523

Creation date：2024-04-27 16:38:01

Belonging category：数学mathematics

Tag MATH 523

Generalized Linear Models

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Generalized Linear Models MATH 523

Consider a binomial GLM with an arbitrary link function g and n responses that have
been entered in a grouped format. Using the same notation as in the lecture notes,
show that:
(1) The maximum likelihood estimates of β do not depend on whether the data have
been entered in a grouped or ungrouped format.
(2) The Fisher information matrix does not depend on whether the data have been
entered in a grouped or ungrouped format. Conclude that the asymptotic covari-
ance matrix of βˆ (and consequently the standard errors of βˆj, j = 1, . . . , p) does
not depend on the data entry format. Hint: It is easiest if you verify the entry
at position (j, k) of the Fisher information for arbitrary j, k, rather than doing
the matrix multiplication.
Q2 Suppose that miYi is binomial (mi, pii), where g(pii) = Xiβ and i = 1, . . . , n. Consider
the null model, for which pi1 = . . . = pin. Show that
pˆi =
∑n
i=1miyi∑n
i=1mi
.
When mi = 1 for all i ∈ {1, . . . , n}, show that in this case, the Pearson X2 statistic,
which is defined as the sum of the squared Pearson residuals, equals n. Decide whether
or not X2 is useful for testing whether a Binomial GLM model fits the data well when
the response is binary.
Q3 R exercise
Consider the following data on home-well contamination in 3020 households in Ara-
hazar upazila, Bangladesh. The response variable is switch (binary variable whether
or not the household switched to another well from an unsafe well). Other variables
collected for each household were arsenic (the level of arsenic contamination in the
household’s original well, in hundreds of micrograms per liter), dist100 (distance in
100-meter units to the closest known safe well), educ (years of education of the head of
the household) and assoc (whether or not any members of the household participated
in any community organizations: no or yes). The data is available in MyCourses under
Datasets. Load the data and compute dist100 as follows.
wells <- read.table("../Datasets/wells.dat")
attach(wells)
dist100 <- dist/100
Johanna G. Nešlehová
Generalized Linear Models MATH 523
McGill University, Winter Term 2022
Assignment 2 due on March 25 at noon.
(1) Report whether the data have been entered in a grouped or ungrouped form, and
which explanatory variables are continuous and which are factors.
(2) Fit a logistic regression model with the intercept and arsenic. Assess the fit
of this model graphically as follows: divide arsenic into 30 approximately filled
categories, group the data accordingly, and display the empirical logits of switch-
ing to a safe well for each category and display the fitted regression line. Do you
think the model is adequate? Perform an approximate goodness-of-fit test of the
model using the above binning and Pearson’s X2 statistic; conclude at the 5%
level.

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LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 FINC5090 MATH1052 SciComp Elec4622 FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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