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STA302H1: Methods of Data Analysis
创建日期：2024-05-21 15:49:26
所属分类：统计学statistics
标签： STA302H1
Methods of Data Analysis

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STA302H1: Methods of Data Analysis I

(Lecture 5)
Distribution of the regression parameters
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 2 / 33
Distribution of βˆ
• Since, βˆ is linear combination of Y, thus βˆ also follows normal and
E (βˆ|X) = E ((X′X)−1X′Y | X)
= (X′X)−1X′Xβ
= β
• Thus, βˆ|X is an unbiased estimator of β
• The variance,
Var(βˆ|X) = Var((X′X)−1X′Y | X)
= (X′X)−1X′σ2IX(X′X)−1
= σ2(X′X)−1X′X(X′X)−1
= (X′X)−1σ2
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 3 / 33
Distribution of βˆ
• Thus, assume C = (X′X)−1. Then variance of βj will be Cjjσ2 and
Cov(βk , βj) = σ2Ckj
• The least squares estimates are the Best Linear Unbiased Estimator (BLUE)
according to the Gauss-Markov theorem
• The proof of Gauss-Markov will be skipped for now
• However, we need to know that the Gauss-Markov assumes:
• E (ϵ) = 0, that is the error mean is zero
• Var(ϵ) = σ2, homoscedasticity
• It does not assume normality
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 4 / 33
Distribution of βˆ
• Like in the simple linear regression case the the βˆjs also follow normal distribution
• That is βˆj ∼ N(βj , σ2Cjj)
• We can test hypothesis,
H0 : βj = β(0)j
H1 : βj ̸= β(0)j
with a z−test, when σ2 is known, using the test statistic,
Z =
βˆj − β(0)j
σ
√
Cjj
• Or we can replace σ with S, where, S =
√
1
n − p − 1
∑n
i=1 e2i , and perform a
t−test with,
T =
βˆj − β(0)j
S
√
Cjj
where T ∼ tn−p−1
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 5 / 33
Interpretation
• These subset examples show us that the relationship between predictors and
response depends on the value that the other predictors take in our multiple
regression model.
• So when we interpret a coefficient from a multiple linear model, we must reflect
this concept
• For a one inch increase in height, we see on average a decrease of 0.55% body fat
when abdomen size is fixed at a constant value.
• We always need to make such conditional interpretation
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 6 / 33
ANOVA
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 7 / 33
The RSS for Multiple Linear Regression
• The RSS for multiple regression can be given as,
RSS =
n∑
i=1
(yi − yˆi)2 = e′e
• Again recall that e = (I−H)y, where H = X(X′X)−1X′
• Thus, RSS = y′ [I− X(X′X)−1X′] y
• What is E (RSS)?
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 8 / 33
The RSS for Multiple Linear Regression
Theorem
If y is a n × 1 random vector with mean vector µ variance-covariance matrix V (non
singular) and A is a n × n matrix of constants then,
E (y′Ay) = tr(AV) + µ′Aµ
• In the case of RSS we have A = I− X(X′X)−1X′ and V = σ2I.
• Thus, tr(AV) = tr((I− X(X′X)−1X′)σ2I) = σ2tr(I− X(X′X)−1X′)
• Recall that tr(A− B) = tr(A)− tr(B)
• Thus tr(I− X(X′X)−1X′) = tr(I)− tr(X(X′X)−1X′)
• Obviously tr(I) = n
• Recall that tr(ABC) = tr(CAB). Thus,
tr(X(X′X)−1X′) = tr(X′X(X′X)−1) = tr(Ip+1) = p + 1
• Thus, σ2tr(I− X(X′X)−1X′) = σ2(n − p − 1)
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 9 / 33
The RSS for Multiple Linear Regression
• Now we know, µ = Xβ
• Thus, µ′Aµ = (Xβ)′(I− X(X′X)−1X′)Xβ = β′X′Xβ − β′X′X(X′X)−1X′Xβ = 0
• Which implies, that E (RSS) = E (e′e) = E (∑ni=1 e2i ) = (n − p − 1)σ2
• Thus, E
( ∑n
i=1 e2i
n − p − 1
)
= E (MSR) = σ2
• Where, p is the number of covariates. One can easily obtain the result for simple
linear regression when p = 1
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 10 / 33
The RSS and SSreg for Multiple Linear Regression
• The SSreg =∑ni=1(yˆi − (¯y))2 can be constructed similarly to the SLR and thus we
have,
SSreg/p
RSS/(n − p − 1) ∼ F0(p, n − p − 1)
• Since RSS and SSreg both can be represented in a vectorized form if we can show
that their dot products are zero then we know that they are independent
• We can perform a F-test. The null hypothesis is,
H0 : β1 = β2 = · · · = βp = 0
The alternative is H1 : at least one beta is ̸= 0
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 11 / 33
ANOVA table
• The ANOVA table for multiple regression looks like as following,
Sources of variation Sum Squares DF Mean Squares F value
Regression SSreg p MSreg =
SSreg
p F0 =
MSreg
MRSS
Residuals RSS n-p-1 MRSS = RSSn − p − 1
Total SST n-1
• We can create the ANOVA table using the anova command
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 12 / 33
Adjusted R2
• Recall that the coefficient of determination is defined as R2 = SSregSST
• However, as the number of variables increase in the model then R2 also increases
even if the model is not right (Why?)
• This is very closely related to the concept of over fitting (Later)
• We can correct for this increase by using an adjusted R2
• The adjusted R2 is given by,
R2adjusted = 1−
RSS/(n − p − 1)
SST/n − 1
• This accounts for the addition of multiple predictors
• If we are comparing models with different number of predictors, we should use
R2adjusted and R2 (WHY?)
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 13 / 33
Partial F-test
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 14 / 33
Testing a subset of predictors
• Sometimes our interest could be to investigate the significance of a group (subset)
of predictors
• When we say we are trying to test a subset of predictors for their relationship with
the response, we are really testing which of two possible models are better.
• Consider the full model, which includes all the p predictors we think represent the
true relationship with the response
Y = β0 + β1X1 + β2X2 + ...+ βpXp + ϵ
• We fit this model and notice that the first k predictors, k < p, don’t have
significant t-tests.
• Then we can just remove the first k predictors and refit the model
Y = β0 + βk+1Xk+1 + βk+2Xk+2 + ...+ βpXp + ϵ
• The second model is called the reduced model
• We can test whether the reduced model is a better fit
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 15 / 33
Example: nyc dataset
• To perform a partial F test we are going to use the nyc.csv dataset. The file is
uploaded in Quercus
• Data from surveys of customers of 168 Italian restaurants in the target area are
available
• The data has the following variables:
1. Y : Price = the price (in $US) of dinner (including 1 drink & a tip)
2. x1 : Food = customer rating of the food (out of 30)
3. x2 : Décor = customer rating of the décor (out of 30)
4. x3 : Service = customer rating of the service (out of 30)
5. x4 : East = dummy variable = 1 (0) if the restaurant is east (west) of Fifth Avenue
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 16 / 33
Diagnostic Checking
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 17 / 33
Diagnostic Checking
• Recall the assumptions of linear regression:
• Linearity
• Homoscedasticity in the error terms
• Normaility of the errors
• One of the important tasks before we move on with our analyses is to check these
assumptions
• These are often referred to as diagnostic checking
• In this lecture we are going to discuss how to check!
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 18 / 33
Diagnostic Checking
• Recall a simple linear regression model
Y = β0 + β1X + ϵ
E (Y |X ) = β0 + β1X
ϵ = Y − E (Y |X )
• Obviously we don’t know the true relationship between Y and X
• We can only estimate the relationship
• The fiited regression yˆ = βˆ0 + βˆ1X produces the estimate for E (Y |X )
• e is an unbiased estimate of ϵ
• Thus e can be used to check the validity of the model
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 19 / 33
Anscombe’s Four Data Sets
• Anscombe (1973) constructed 4 small toy datasets, to illustrate how blindly
tting a simple linear regression model can lead to very misleading conclusions
about the data. (anscombe.txt from the textbook website)
• Each dataset contains 11 observations, with a single predictor variable and
response.
• The responses in each dataset are different, but the predictors in 3 of the 4
datasets are identical.
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 20 / 33
Anscombe’s Four Data Set
• The dataset is as following
case x1 x2 x3 x4 y1 y2 y3 y4
1 10 10 10 8 8.04 9.14 7.46 6.58
2 8 8 8 8 6.95 8.14 6.77 5.76
3 13 13 13 8 7.58 8.74 12.74 7.71
4 9 9 9 8 8.81 8.77 7.11 8.84
5 11 11 11 8 8.33 9.26 7.81 8.47
6 14 14 14 8 9.96 8.10 8.84 7.04
7 6 6 6 8 7.24 6.13 6.08 5.25
8 4 4 4 19 4.26 3.10 5.39 12.50
9 12 12 12 8 10.84 9.13 8.15 5.56
10 7 7 7 8 4.82 7.26 6.42 7.91
11 5 5 5 8 5.68 4.74 5.73 6.89
• It’s not obvious by looking at the raw data, but a linear model would only be
appropriate for one of these datasets.
• How should we check that?
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 21 / 33
Anscombe’s Four Data Set
4 6 8 10 12 14
4
5
6
7
8
9
10
11
x1
y 1
4 6 8 10 12 14
3
4
5
6
7
8
9
x2
y 2
4 6 8 10 12 14
6
8
10
12
x3
y 3
8 10 12 14 16 18
6
8
10
12
x4
y 4
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 22 / 33
Anscombe’s Four Data Set
• Now let’s look into the outputs of the coefficients (rounding up to two decimal
points)
βˆ0 βˆ1
x1 3.00 0.50
x2 3.00 0.50
x3 3.00 0.50
x4 3.00 0.50
• Does this mean that linear model is appropriate for all the data?
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 23 / 33
Anscombe’s Four Data Sets
4 6 8 10 12 14
4
5
6
7
8
9
10
11
x1
y 1
4 6 8 10 12 14
3
4
5
6
7
8
9
x2
y 2
4 6 8 10 12 14
6
8
10
12
x3
y 3
8 10 12 14 16 18
6
8
10
12
x4
y 4
• Dataset 1: The linear model is
appropriate
• Dataset 2: Relationship
quadratic. So not appropriate
• Dataset 3: The line is influenced
by one outlier. Not appropriate
• Dataset 4, the slope of the
regression is determined by a
single point. Not appropriate
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 24 / 33
Anscombe’s Four Data Sets
• Anscombe’s dataset was meant to illustrate how one should always supplement
one’s modelling with an investigation of the modelling assumptions
• We first need to check whether the relationship between X and Y are linear
• Scatterplot does not always help. For example when we have more then one
predictor (multiple regression)
• An individual observation can have massive impact on model fit
• What would be an easy way to check model assumptions
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 25 / 33
Residual Plots
• One way to check the assumptions is through residual plot
• Plotting residuals allows us to visually inspect the model assumptions (WHY?)
• Residuals measure the remaining variability in the data after fitting a model.
Thus, it is more sensitive to any irregularities
• There are two main types of residual scatter plots that we use:
• Residual vs predictor plot, i.e., plot e against the observed values of X . Not
appropriate for multiple regression
• Residual vs fitted plot i.e., plot e against yˆ
• Both plots are usefull to check whether the assumptions hold
Mohammad Kaviul Anam Khan Data Analysis 1 Lecture 5 26 / 33
Residual Plots
• Assumptions hold if there is no evident pattern seen in the residual plots
• specifically, we hope that the residuals are uniformly scattered around 0 for the
full range of predictor values
• Important patterns to be aware of for each of the assumptions are:
• Linearity of the relationship: any systematic pattern in the residuals, such as a
curve
• Independence of the errors: clusters of residuals that have obvious separation
from the rest
• Homoscedasticity: any pattern, but especially a fanning pattern, where residuals
gradually become more spread out

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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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