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PSTAT 126 Multiple Linear Regression
创建日期：2024-05-14 17:14:00
所属分类：数学mathematics
标签： PSTAT 126
Multiple Linear Regression

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PSTAT 126

Multiple Linear Regression (MLR)
We can extend the Simple Linear Regression to a Multiple Linear Regression by incorporating more that one
predictor:
yi = β0 + β1xi1 + . . .+ βpxip + ϵi, i = 1, . . . , n
Using matrix notation, the model can be written as:
y = Xβ + ϵ, ϵ ∼ Nn(0, σ2In)
To get the LS solution of β we minimize the Sum of Squared Residuals:
SSR = (y−Xβ)T (y−Xβ)
We obtain the solution: βˆLSE = (XTX)−1XTy. Additionally, it can be proved that βˆ ∼
Np∗(β, (XTX)−1σ2).
Therefore, it is possible to obtain the Standard Error of βˆj :
SE(βˆj) =
√
σˆ2 [XTX]−1jj
Where σˆ2 = (y−Xβˆ)
T (y−Xβˆ)
n−p∗ , with p∗ = p+ 1.
Data Example
library(faraway)
data(diabetes)
model <- lm(weight~chol+ stab.glu+ hdl+height+waist+age, data=diabetes)
summary(model)
##
## Call:
## lm(formula = weight ~ chol + stab.glu + hdl + height + waist +
## age, data = diabetes)
##
## Residuals:
## Min 1Q Median 3Q Max
## -70.090 -11.415 0.022 11.952 58.537
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.540e+02 1.800e+01 -8.556 2.77e-16 ***
## chol 2.082e-02 2.232e-02 0.933 0.351
## stab.glu 8.289e-03 1.896e-02 0.437 0.662
1
## hdl -8.465e-02 5.790e-02 -1.462 0.145
## height 1.889e+00 2.392e-01 7.899 2.95e-14 ***
## waist 5.992e+00 1.741e-01 34.417 < 2e-16 ***
## age -4.466e-01 6.070e-02 -7.357 1.14e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 18.34 on 387 degrees of freedom
## (9 observations deleted due to missingness)
## Multiple R-squared: 0.7958, Adjusted R-squared: 0.7926
## F-statistic: 251.3 on 6 and 387 DF, p-value: < 2.2e-16
summary(model)$coefficients
Inference
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.539895e+02 17.99772561 -8.5560510 2.767558e-16
## chol 2.082080e-02 0.02231999 0.9328316 3.514885e-01
## stab.glu 8.289022e-03 0.01896535 0.4370614 6.623106e-01
## hdl -8.465368e-02 0.05789839 -1.4621077 1.445233e-01
## height 1.889310e+00 0.23919598 7.8985848 2.949709e-14
## waist 5.991746e+00 0.17409034 34.4174514 7.958028e-120
## age -4.465606e-01 0.06070061 -7.3567719 1.136818e-12
We get βˆ:
library(faraway)
betas = summary(model)$coefficients[,1]
betas
## (Intercept) chol stab.glu hdl height
## -1.539895e+02 2.082080e-02 8.289022e-03 -8.465368e-02 1.889310e+00
## waist age
## 5.991746e+00 -4.465606e-01
coef(model) # this returns the same results
## (Intercept) chol stab.glu hdl height
## -1.539895e+02 2.082080e-02 8.289022e-03 -8.465368e-02 1.889310e+00
## waist age
## 5.991746e+00 -4.465606e-01
We can get SE(βˆj) j = 0, . . . , p:
summary(model)$coefficients[,2]
## (Intercept) chol stab.glu hdl height waist
## 17.99772561 0.02231999 0.01896535 0.05789839 0.23919598 0.17409034
## age
## 0.06070061
Coefficient od determination R2:
summary(model)$r.squared
## [1] 0.7957582
Residuals:
2
Res=residuals(model)
Standard deviation of residuals (σˆ):
sqrt(sum(Resˆ2) /model$df.residual) # using formula
## [1] 18.33928
sigma(model) # using R built-in function
## [1] 18.33928
Confidence and Prediction Intervals:
# LSE of coefficients with CI
confint(model, level = .95)
## 2.5 % 97.5 %
## (Intercept) -189.37501620 -118.60389976
## chol -0.02306283 0.06470442
## stab.glu -0.02899899 0.04557703
## hdl -0.19848845 0.02918109
## height 1.41902346 2.35959601
## waist 5.64946462 6.33402711
## age -0.56590480 -0.32721631
# 95% CI for the mean response w/ Age=34,chol=186,gluc=85,hdl=46,height=66,waist=46
new = data.frame(chol=186, stab.glu=85, hdl=46,height=66,waist=46,age=34)
ans1 = predict(model, new, se.fit = TRUE, interval = "confidence", level = 0.95, type = "response")
ans1$fit
## fit lwr upr
## 1 231.8254 227.9674 235.6834
# 95% PI for a new observation w/ Age=34,chol=186,gluc=85,hdl=46,height=66,waist=46
ans2 = predict(model, new, se.fit = TRUE, interval = "prediction", level = 0.95, type = "response")
ans2$fit
## fit lwr upr
## 1 231.8254 195.5625 268.0883
Normal Distribution Review
Let X1, X2, ..., Xn
iid∼ N(µ, σ2). Then the pdf is given for any µ ∈ R, x ∈ R, σ > 0 Then define the
following: E(Xi) = µ, V ar(Xi) = σ2 ∀i
The CDF is given by, FX(x) = P (X ≤ x) = 1 − P (X > x) For µ = 0, σ = 1, we have the standard
normal Z ∼ N(0, 1):
Some properties are as follows:
1) P (Z < z) = P (Z > −z)
2) P (|Z| > z) = 2P (Z > z) = 2P (Z < −z)
3) if P (Z > zα) = α, then
P (z1−α/2 < Z < zα/2)
= P (−zα/2 < Z < zα/2) = 1− α
3
# consider the x-values
x<- seq(-3.5, 3.5, length.out=100)
# pdf of N(0,1)
f<- dnorm(x)
f2<- dnorm(x, mean =0, sd = .5) # pdf of N(0,.5)
f3<- dnorm(x, mean = 1, sd = .75) # pdf of N(1,.75)
f4<- dnorm(x, mean= -2, sd = 1.5) # pdf of N(-2,1.5)
# plot of pdfs
plot(x, f, type = "l", lwd=1, col= 1, ylab="Density", ylim=c(0, .95))
lines(x, f2, type = "l", lwd=1, col= 2)
lines(x, f3, type = "l", lwd=1, col= 3)
lines(x, f4, type = "l", lwd=1, col= 4)
legend(2, .8, legend=c("N(0,1)", "N(0,0.5)","N(1,0.75)","N(-2,1.5)"),
col=c(1,2,3,4), lty=1, cex=0.8)
Computation:
−3 −2 −1 0 1 2 3
0.
0
0.
2
0.
4
0.
6
0.
8
x
D
en
si
ty
N(0,1)
N(0,0.5)
N(1,0.75)
N(−2,1.5)
#cdf
cf1<- pnorm(x) # pdf of N(0,1)
cf2<- pnorm(x, mean =0, sd = .5) # pdf of N(0,.5)
cf3<- pnorm(x, mean = 1, sd = .75) # pdf of N(1,.75)
cf4<- pnorm(x, mean= -2, sd = 1.5) # pdf of N(-2,1.5)
#quantiles
4
q<- seq(0,1, length.out=100)
qf1<- qnorm(q) # pdf of N(0,1)
qf2<- qnorm(q, mean =0, sd = .5) # pdf of N(0,.5)
qf3<- qnorm(q, mean = 1, sd = .75) # pdf of N(1,.75)
qf4<- qnorm(q, mean= -2, sd = 1.5) # pdf of N(-2,1.5)
par(mfrow= c(1,2))
# plot of cdfs
plot(x, cf1, type = "l", lwd=1, col= 1, ylab="CDF", ylim=c(0, 1))
lines(x, cf2, type = "l", lwd=1, col= 2)
lines(x, cf3, type = "l", lwd=1, col= 3)
lines(x, cf4, type = "l", lwd=1, col= 4)
legend(-3.6, 1.02, legend=c("N(0,1)", "N(0,0.5)","N(1,0.75)","N(-2,1.5)"),
col=c(1,2,3,4), lty=1, cex=0.6)
# plot of quantiles
plot(q, qf1, type = "l", lwd=1, col= 1, ylab="Quantile function", ylim=c(-3,3))
lines(q, qf2, type = "l", lwd=1, col= 2)
lines(q, qf3, type = "l", lwd=1, col= 3)
lines(q, qf4, type = "l", lwd=1, col= 4)
legend(0, 3, legend=c("N(0,1)", "N(0,0.5)","N(1,0.75)","N(-2,1.5)"),
col=c(1,2,3,4), lty=1, cex=0.6)
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MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 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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