Problem Set 3: Learning from Data
Question 1
library("Sleuth3") # Importing the data set
birthData <- ex0724
View(birthData) # Look at data frame
## Warning: running command ''/usr/bin/otool' -L '/Library/Frameworks/
## R.framework/Resources/modules/R_de.so'' had status 69
summary(birthData) # Look at statistical summary
## Year Denmark Netherlands Canada
## Min. :1950 Min. :0.5108 Min. :0.5087 Min. :0.5120
## 1st Qu.:1961 1st Qu.:0.5127 1st Qu.:0.5120 1st Qu.:0.5128
## Median :1972 Median :0.5141 Median :0.5129 Median :0.5136
## Mean :1972 Mean :0.5142 Mean :0.5130 Mean :0.5137
## 3rd Qu.:1983 3rd Qu.:0.5153 3rd Qu.:0.5139 3rd Qu.:0.5145
## Max. :1994 Max. :0.5175 Max. :0.5160 Max. :0.5153
## NA's :24
## USA
## Min. :0.5120
## 1st Qu.:0.5122
## Median :0.5126
## Mean :0.5126
## 3rd Qu.:0.5128
## Max. :0.5134
## NA's :24
(a) Use the lm function in R to fit four (one per country) simple linear
regression models of the yearly proportion of males births as a function of
the year and obtain the least squares fits. Write down the estimated linear
model for each country.
attach(birthData) # attach the data set
rateDenmark <- lm(Denmark ~ Year) # fit a linear regression model for Denmark
rateDenmark # learning about least squares regression
##
## Call:
## lm(formula = Denmark ~ Year)
##
## Coefficients:
## (Intercept) Year
## 5.987e-01 -4.289e-05
plot(Denmark ~ Year)
abline(rateDenmark) # draw the line of best fit
Male birth ratefor Denmark can be defined this way:
birthrateDenmark = (-4.289e-05)year + 5.987e-01
rateNetherlands <- lm(Netherlands ~ Year) # fit a linear regression
rateNetherlands # learning about least squares regression
##
## Call:
## lm(formula = Netherlands ~ Year)
##
## Coefficients:
## (Intercept) Year
## 6.724e-01 -8.084e-05
plot(Netherlands ~ Year)
abline(rateNetherlands) # draw the line of best fit
Male birth ratefor Netherlands can be defined this way:
birthrateNetherlands = (-8.084e-05)year + 6.724e-01
rateCanada <- lm(Canada ~ Year) # fit a linear regression
rateCanada # learning about least squares regression
##
## Call:
## lm(formula = Canada ~ Year)
##
## Coefficients:
## (Intercept) Year
## 0.7337857 -0.0001112
plot(Canada ~ Year)
abline(rateCanada) # draw the line of best fit
Male birth ratefor Canada can be defined this way:
birthrateCanada=(-0.0001112)year + 0.7337857
rateUSA <- lm(USA ~ Year) # fit a linear regression
rateUSA # learning about least squares regression
##
## Call:
## lm(formula = USA ~ Year)
##
## Coefficients:
## (Intercept) Year
## 6.201e-01 -5.429e-05
plot(USA ~ Year)
abline(rateUSA) # draw the line of best fit
Male birth rate for USA can be defined this way:
birthrateUSA=(-5.429e-05)year + 6.201e-01
(b) Obtain the t-statistic for the test that the slopes of the regression lines
are zero, for each of the four countries. Is there evidence that the proportion
of births that are male is truly declining over this period?
summary(rateDenmark) # get t-value -2.073, p-value 0.04424
##
## Call:
## lm(formula = Denmark ~ Year)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.003225 -0.001339 0.000089 0.001119 0.003790
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.987e-01 4.080e-02 14.673 <2e-16 ***
## Year -4.289e-05 2.069e-05 -2.073 0.0442 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.001803 on 43 degrees of freedom
## Multiple R-squared: 0.09083, Adjusted R-squared: 0.06968
## F-statistic: 4.296 on 1 and 43 DF, p-value: 0.04424
summary(rateNetherlands) # get t-value -5.71, p-value 9.637e-07
##
## Call:
## lm(formula = Netherlands ~ Year)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0031437 -0.0008246 0.0002819 0.0009287 0.0021478
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.724e-01 2.792e-02 24.08 < 2e-16 ***
## Year -8.084e-05 1.416e-05 -5.71 9.64e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.001233 on 43 degrees of freedom
## Multiple R-squared: 0.4313, Adjusted R-squared: 0.418
## F-statistic: 32.61 on 1 and 43 DF, p-value: 9.637e-07
summary(rateCanada) # get t-value -4.017, p-value 0.0007376
##
## Call:
## lm(formula = Canada ~ Year)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.494e-03 -6.161e-04 -8.312e-05 4.951e-04 1.284e-03
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.338e-01 5.480e-02 13.390 3.98e-11 ***
## Year -1.112e-04 2.768e-05 -4.017 0.000738 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.000768 on 19 degrees of freedom
## (24 observations deleted due to missingness)
## Multiple R-squared: 0.4592, Adjusted R-squared: 0.4307
## F-statistic: 16.13 on 1 and 19 DF, p-value: 0.0007376
summary(rateUSA) # get t-value -5.779, p-value 1.44e-05
##
## Call:
## lm(formula = USA ~ Year)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.343e-04 -1.800e-04 -1.714e-05 2.571e-04 3.743e-04
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.201e-01 1.860e-02 33.340 < 2e-16 ***
## Year -5.429e-05 9.393e-06 -5.779 1.44e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0002607 on 19 degrees of freedom
## (24 observations deleted due to missingness)
## Multiple R-squared: 0.6374, Adjusted R-squared: 0.6183
## F-statistic: 33.4 on 1 and 19 DF, p-value: 1.439e-05
Since a small p-value (typically ≤ 0.05) indicates strong evidence against the ### null hypothesis, so since the p-value of the 4 model are all less than 0.05, we ### can say that there is evidence that the roportion of births that are male is
truly declining over this period.
Question 2
library("UsingR") # importing the data set
## Loading required package: MASS
## Loading required package: HistData
## Loading required package: Hmisc
## Loading required package: grid
## Loading required package: lattice
## Loading required package: survival
## Loading required package: Formula
## Loading required package: ggplot2
##
## Attaching package: 'Hmisc'
##
## The following objects are masked from 'package:base':
##
## format.pval, round.POSIXt, trunc.POSIXt, units
##
##
## Attaching package: 'UsingR'
##
## The following object is masked from 'package:ggplot2':
##
## movies
##
## The following object is masked from 'package:survival':
##
## cancer
library(ggplot2) # library for visualization
heightData <- get("father.son")
(a) Perform an exploratory analysis of the dataset. Describe what you find.
At a mini- mum you should produce statistical summaries of the variables,
a visualization of the relationship of interest in this problem, and
a statistical summary of that relationship.
View(heightData) # look at data frame
## Warning: running command ''/usr/bin/otool' -L '/Library/Frameworks/
## R.framework/Resources/modules/R_de.so'' had status 69
summary(heightData) # look at statistical summary of the variables
## fheight sheight
## Min. :59.01 Min. :58.51
## 1st Qu.:65.79 1st Qu.:66.93
## Median :67.77 Median :68.62
## Mean :67.69 Mean :68.68
## 3rd Qu.:69.60 3rd Qu.:70.47
## Max. :75.43 Max. :78.36
ggplot(heightData, aes(fheight, sheight))+geom_point() # get visualization
It’s interesting to notice from the graph, that as the father’s height goes
up, the son’s height has a higher tendency to go up too.
(b) Use the lm function in R to fit a simple linear regression model to predict
son’s height as a function of father’s height. Write down the model,
yˆsheight = βˆ + βˆ × fheight 0i
filling in estimated coefficient values and interpret the coefficient estimates.
attach(heightData) # attach the data frame
sonHeight <- lm(sheight ~ fheight) # fitting a simple linear regression model
summary(sonHeight) # look at information about this function
##
## Call:
## lm(formula = sheight ~ fheight)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.8772 -1.5144 -0.0079 1.6285 8.9685
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.88660 1.83235 18.49 <2e-16 ***
## fheight 0.51409 0.02705 19.01 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.437 on 1076 degrees of freedom
## Multiple R-squared: 0.2513, Adjusted R-squared: 0.2506
## F-statistic: 361.2 on 1 and 1076 DF, p-value: < 2.2e-16
sonHeight
##
## Call:
## lm(formula = sheight ~ fheight)
##
## Coefficients:
## (Intercept) fheight
## 33.8866 0.5141
According to the statistical summary we could write down the following model:
sheight = 33.8866 + 0.5141*fheight
For this linear regression model, the regression coefficients “ßi” mean change
in the response variable (in this case son’s height) for one unite of change in
the predictor variable(in this case father’s height). So in this case, every
additional inch in father’s height, we can expect 0.5141 inch to increase
in son’s height. But this only make sense within the range of normal human
height. So we would not actually shift up or down the fitted line by a full
meter in this case.
(c) Find the 95% confidence intervals for the estimates. You may find the
confint() command useful.
confint(sonHeight) # Find the 95% confidence intervals
## 2.5 % 97.5 %
## (Intercept) 30.2912126 37.4819961
## fheight 0.4610188 0.5671673
(d) Produce a visualization of the data and the least squares regression line.
plot(fheight,sheight) # visualize relationship between fheight and sheight
abline(sonHeight) # draw the least squares regression line
(e) Produce a visualization of the residuals versus the fitted values.
(You can inspect the elements of the linear model object in R using names()).
Discuss what you see. Do you have any concerns about the linear model?
names(sonHeight)
## [1] "coefficients" "residuals" "effects" "rank"
## [5] "fitted.values" "assign" "qr" "df.residual"
## [9] "xlevels" "call" "terms" "model"
residuals(sonHeight)
## 1 2 3 4 5
## -7.549320518 -3.189432294 -3.937262188 -4.897126876 -1.035698698
## 6 7 8 9 10
## -2.043843444 -3.410828758 -3.165011904 -3.236827219 -4.334628227
## 11 12 13 14 15
## 1.022303623 -0.888502884 -0.939312002 -1.389889738 -1.848607024
## 16 17 18 19 20
## -2.591991494 -2.989964076 -2.052904112 -3.438394247 -2.717010607
## 21 22 23 24 25
## -3.605699113 -4.121340976 0.546305037 -0.312543404 -0.875422298
## 26 27 28 29 30
## -1.312839748 -1.192957249 -1.230181614 -1.812453255 -1.615901663
## 31 32 33 34 35
## -2.375460072 -2.204042616 -1.619202118 -3.046443067 -2.648375866
## 36 37 38 39 40
## -2.626129125 -2.805758011 -3.212520458 -4.390581934 1.251267433
## 41 42 43 44 45
## 0.423048888 -0.045610592 0.017480915 -0.497696232 -0.474796414
## 46 47 48 49 50
## 1.651610444 -0.093493379 -6.433844280 -1.501642356 -0.690917366
## 51 52 53 54 55
## 0.167196487 -0.296185711 -0.663097904 -0.437084264 0.035412008
## 56 57 58 59 60
## -0.270065897 -0.778626280 -0.794462663 -0.920354365 -1.725711657
## 61 62 63 64 65
## -1.230643968 -0.620670541 -1.364184860 -1.572385589 -1.470870305
## 66 67 68 69 70
## -1.513833405 -1.745407337 -2.069684088 -2.846581324 -3.329580662
## 71 72 73 74 75
## -2.946462676 -3.640587809 1.860225243 2.281611743 1.304137765
## 76 77 78 79 80
## 1.278954303 1.282886041 0.483236276 0.464206224 0.684499731
## 81 82 83 84 85
## 0.541204255 0.635911455 -0.167036297 0.447075292 -0.375922595
## 86 87 88 89 90
## -0.216290319 0.116000909 -0.540760234 -0.514137299 -0.965763441
## 91 92 93 94 95
## -0.408536148 -1.177500783 -0.045185671 -1.231809237 -1.256862315
## 96 97 98 99 100
## -1.283594057 -0.807166782 -1.300360878 -0.583115367 -1.592844342
## 101 102 103 104 105
## -1.694391204 -2.726917798 -1.696968809 -2.808427293 -3.721415968
## 106 107 108 109 110
## 2.481860301 2.991852489 2.191823951 2.316841460 1.663529726
## 111 112 113 114 115
## 1.509323610 1.810568992 1.750030102 1.182320365 1.412419651
## 116 117 118 119 120
## 0.593792097 0.609404304 0.130282046 0.484382391 0.873493002
## 121 122 123 124 125
## 0.234369977 -0.507444670 0.256542544 0.113457481 0.050642153
## 126 127 128 129 130
## -0.456481594 -0.714303903 -1.006836579 -0.642387828 -0.821788545
## 131 132 133 134 135
## -0.904210026 -1.227948506 -1.903205886 -3.391252138 2.614688577
## 136 137 138 139 140
## 2.476442442 2.256267373 2.481986131 2.739712775 1.972221011
## 141 142 143 144 145
## 1.206614827 1.720636825 0.898841785 0.733899166 1.800290795
## 146 147 148 149 150
## 0.561337916 0.719309646 0.393519322 0.726053610 0.182004139
## 151 152 153 154 155
## 0.015308736 0.133562580 -0.366483521 -0.228146526 -0.531668437
## 156 157 158 159 160
## -0.322913060 -0.986682154 -2.575892643 4.288392671 2.811879391
## 161 162 163 164 165
## 3.854350538 2.964102656 3.377198232 2.462113473 2.147711164
## 166 167 168 169 170
## 2.500735498 1.712579696 1.766511791 1.668353743 1.576386544
## 171 172 173 174 175
## 1.530073205 1.281262567 1.039887763 1.046845499 1.162054402
## 176 177 178 179 180
## 0.026946632 -0.515032586 -0.680480615 3.857967348 3.386384510
## 181 182 183 184 185
## 3.613236253 2.639723729 3.303625096 2.120430141 2.598257226
## 186 187 188 189 190
## 2.241349711 1.826296299 0.819806704 1.033570797 4.187133519
## 191 192 193 194 195
## 4.073441414 3.622528817 3.353553128 2.995469909 2.616144304
## 196 197 198 199 200
## 2.324543094 8.003255538 5.442858656 3.865042378 3.616255354
## 201 202 203 204 205
## 2.560981725 6.751380784 7.093602167 5.901547869 -4.818473013
## 206 207 208 209 210
## 5.339813720 3.591276323 3.711810584 -4.456682821 -2.036181060
## 211 212 213 214 215
## -7.357182705 3.280337809 -4.094563645 -7.402425332 -4.546391439
## 216 217 218 219 220
## -3.547572138 -2.996731758 -3.058385489 -4.114599078 -5.963614403
## 221 222 223 224 225
## -1.937790543 -2.947703222 -2.850429872 -4.008435512 -3.586611265
## 226 227 228 229 230
## -4.981214007 -0.455548844 -1.843159299 -1.066749004 -2.356374230
## 231 232 233 234 235
## -2.339795194 -2.037973366 -2.089951647 -2.198188552 -3.484180192
## 236 237 238 239 240
## -3.311773147 -4.661443598 -0.009016003 -0.379773844 -0.055167765
## 241 242 243 244 245
## -0.966452762 -0.676800413 -1.302263927 -1.650971009 -1.416779793
## 246 247 248 249 250
## -1.019166136 -2.330428929 -1.957194664 -1.842430538 -3.005425792
## 251 252 253 254 255
## -2.799590649 -2.524552853 -3.378735919 -4.134563822 0.902462882
## 256 257 258 259 260
## 0.600731530 0.536295550 0.953143203 -0.014966588 -0.237475177
## 261 262 263 264 265
## 0.265131054 -0.228149267 -5.391354890 -2.849681116 -1.850511525
## 266 267 268 269 270
## -3.589695641 -0.102142686 -1.043803584 -0.666873710 -0.613530122
## 271 272 273 274 275
## -0.782423016 -0.765505802 -0.981290464 -1.070444318 -0.568468247
## 276 277 278 279 280
## -1.610169354 -1.765660462 -1.642620503 -1.486344337 -2.148516404
## 281 282 283 284 285
## -2.177641887 -1.785823380 -2.458158060 -2.459333375 -3.180891716
## 286 287 288 289 290
## -2.358938099 -3.344303343 1.737870221 1.529971642 2.126584122
## 291 292 293 294 295
## 1.837056693 1.339263165 1.180360089 0.674466681 0.463427763
## 296 297 298 299 300
## -0.123244601 0.742131310 -0.293302886 -0.188462701 0.335081797
## 301 302 303 304 305
## 0.145177756 -0.264314883 -0.427049864 0.084303897 -0.927216569
## 306 307 308 309 310
## -0.791968391 -1.187685475 -0.234938274 -1.272797267 -1.642547450
## 311 312 313 314 315
## -0.822138191 -0.732354563 -1.497519308 -1.531592328 -2.319124873
## 316 317 318 319 320
## -1.420630068 -1.632832247 -1.879364631 -3.047175858 -2.731998400
## 321 322 323 324 325
## 3.238853634 2.267149069 2.325138632 1.685806236 1.877961690
## 326 327 328 329 330
## 1.616795594 0.998786670 1.879586855 1.013560311 1.266195138
## 331 332 333 334 335
## 0.417645158 0.461415500 0.255616549 0.157141033 0.998830954
## 336 337 338 339 340
## -0.265556588 0.105104884 -0.294442609 -0.151975195 -0.122545542
## 341 342 343 344 345
## 0.047138344 -0.187101821 -0.329330033 -1.375334272 -1.103788678
## 346 347 348 349 350
## -0.655542638 -1.121097939 -1.017182163 -1.979390319 2.955116189
## 351 352 353 354 355
## 2.662128458 1.985806920 1.903415364 2.163431715 1.717846883
## 356 357 358 359 360
## 2.148443709 0.821629353 1.132662390 1.446649298 1.690406308
## 361 362 363 364 365
## 1.158488113 0.988784179 1.243774578 0.607180922 0.101055811
## 366 367 368 369 370
## 0.542196241 0.177075898 0.517863104 -0.564554608 0.047002321
## 371 372 373 374 375
## -0.653364255 -0.963612058 -1.659336497 4.138422805 3.552991910
## 376 377 378 379 380
## 3.349257552 2.174165598 2.560942560 2.337009107 2.182338515
## 381 382 383 384 385
## 1.917710543 1.828863893 1.772559456 1.904477732 1.488699510
## 386 387 388 389 390
## 1.486941263 1.954373022 1.111390949 0.859528685 1.197521578
## 391 392 393 394 395
## 0.358195739 0.240338195 -0.148113758 4.182020693 3.565863898
## 396 397 398 399 400
## 3.250366427 2.324158514 2.389633338 2.537895513 2.374891897
## 401 402 403 404 405
## 2.617766937 1.520946125 0.864434305 0.729090171 4.699917114
## 406 407 408 409 410
## 4.672887051 3.795810853 3.707780237 3.487485465 2.732503400
## 411 412 413 414 415
## 1.666374075 1.963024008 5.064569422 5.484583810 3.387810643
## 416 417 418 419 420
## 3.637810778 2.024863332 4.629300349 8.151747800 -4.165330595
## 421 422 423 424 425
## -4.397042931 0.092183980 -8.271382039 -1.875112825 0.074072255
## 426 427 428 429 430
## -0.294812949 2.968543040 -1.649808898 -5.637762696 1.390831221
## 431 432 433 434 435
## 0.549083220 -4.497157752 -3.497778992 -4.525255176 -5.520827464
## 436 437 438 439 440
## -1.986475485 -2.387577959 -3.302808121 -3.025704983 -3.618434519
## 441 442 443 444 445
## -4.864695754 0.821873405 -1.462888129 -0.933447502 -2.631248222
## 446 447 448 449 450
## -2.581049069 -2.088575488 -2.392377869 -2.632835083 -2.985479647
## 451 452 453 454 455
## -3.746318585 -4.570049223 0.123878967 0.017222335 -0.636215736
## 456 457 458 459 460
## -0.343478636 -1.308177593 -0.996858657 -1.356835908 -1.482939709
## 461 462 463 464 465
## -1.045068074 -2.380873782 -2.035909345 -2.405678827 -2.577727027
## 466 467 468 469 470
## -2.282927149 -2.269134787 -2.804976822 -3.263977852 1.507905714
## 471 472 473 474 475
## 1.401090723 0.220064646 -0.064974348 0.629756131 -0.350102478
## 476 477 478 479 480
## -0.248751352 -0.711485837 -3.486091137 -4.676935637 -1.760437753
## 481 482 483 484 485
## -2.470469682 -0.068363565 -0.003059954 -0.667482049 -0.834641792
## 486 487 488 489 490
## -1.015693781 -0.740362645 -0.875387261 -1.605599758 -1.349426932
## 491 492 493 494 495
## -0.864106255 -0.919440855 -0.822352004 -1.658792381 -1.976064268
## 496 497 498 499 500
## -1.658146023 -1.891878029 -2.229115654 -1.760182271 -2.455244350
## 501 502 503 504 505
## -3.012410151 -2.582018386 2.218731817 2.286712411 1.032497054
## 506 507 508 509 510
## 1.134678004 0.918782672 1.073320623 0.796418419 -0.170741632
## 511 512 513 514 515
## 0.287647361 -0.118998035 0.014531984 -0.341609600 -0.512497595
## 516 517 518 519 520
## -0.166179153 0.058683229 0.157404465 -0.070758990 -0.434102977
## 521 522 523 524 525
## -0.193019062 -0.485859072 -0.864082612 -1.463250029 -0.804839027
## 526 527 528 529 530
## -1.232652814 -1.401964163 -1.186450226 -1.064415076 -1.499661599
## 531 532 533 534 535
## -1.203505578 -2.081020787 -1.941188075 -2.456274391 -2.494140757
## 536 537 538 539 540
## 3.857451958 3.073832008 2.427127077 1.603469118 2.417727796
## 541 542 543 544 545
## 1.823501510 1.593041702 0.940236448 0.577220003 0.910136172
## 546 547 548 549 550
## 0.515233980 0.703753170 -0.233272042 0.196287715 0.100233396
## 551 552 553 554 555
## 0.476631073 -0.467795173 -0.484116104 -0.394556034 0.026199662
## 556 557 558 559 560
## 0.121192064 -0.615982756 -0.349044921 -0.111379734 -0.520127446
## 561 562 563 564 565
## -0.956518584 -0.900222426 -1.879284146 -1.571540721 3.390140321
## 566 567 568 569 570
## 2.714972594 2.529337171 2.331660578 2.185673182 2.205944965
## 571 572 573 574 575
## 1.617506217 1.841601958 0.994719840 1.925553736 1.311455510
## 576 577 578 579 580
## 0.206828320 1.042816425 0.505300846 0.523866950 0.632331719
## 581 582 583 584 585
## 0.122432249 0.605779235 0.420151602 -0.498952678 -0.565600690
## 586 587 588 589 590
## -0.006776941 -1.034715921 -0.712179528 4.139316630 3.906986317
## 591 592 593 594 595
## 3.293088672 2.426219526 2.224091815 2.731735177 2.764149030
## 596 597 598 599 600
## 1.733662782 2.318791395 2.003560969 1.544138142 1.888508198
## 601 602 603 604 605
## 1.384333399 1.614683392 1.222460389 1.958252376 0.723632483
## 606 607 608 609 610
## 0.246105987 -0.211022623 0.359560060 5.307896020 4.380018759
## 611 612 613 614 615
## 3.032073323 3.081352240 2.559210884 3.158959859 1.998324951
## 616 617 618 619 620
## 2.155955791 2.167794730 1.924072275 0.860063659 1.295236203
## 621 622 623 624 625
## 4.209837674 3.804104926 4.104828835 3.690333035 3.165760595
## 626 627 628 629 630
## 1.968800825 1.632180526 6.933304176 4.754464919 4.110180686
## 631 632 633 634 635
## 2.919973210 2.715892034 6.406283771 4.432464139 8.343688769
## 636 637 638 639 640
## -6.912291784 2.729159120 -2.864912195 -1.004644842 1.402358037
## 641 642 643 644 645
## 0.310794534 4.414340456 0.593138224 -6.197892353 -0.863888343
## 646 647 648 649 650
## -0.960672325 -3.003959566 -7.392165742 -2.912550245 -3.631370065
## 651 652 653 654 655
## -4.822340049 -2.113980471 -2.389505185 -2.863227011 -3.241792826
## 656 657 658 659 660
## -3.660770957 -3.557003416 0.347453597 -1.046381008 -1.210875925
## 661 662 663 664 665
## -1.199720032 -2.198016722 -1.753317097 -2.283677937 -2.250404110
## 666 667 668 669 670
## -3.154485655 -4.073990691 -3.496240977 1.147346305 -0.176037520
## 671 672 673 674 675
## -0.807078631 -0.557475725 -0.869483985 -0.886144096 -1.789389167
## 676 677 678 679 680
## -1.879991423 -2.216156355 -1.873581809 -2.065795216 -2.224427754
## 681 682 683 684 685
## -2.153025312 -2.756852481 -2.821668994 -2.891438726 -3.107287648
## 686 687 688 689 690
## 2.556559092 1.585146235 0.912853044 0.243258742 0.084565160
## 691 692 693 694 695
## -0.176879207 -0.582579831 0.728382928 -2.031688861 -5.673090644
## 696 697 698 699 700
## 0.128901089 -0.848534235 -0.425037232 -0.026043450 -0.461419334
## 701 702 703 704 705
## -0.666059743 -0.583233298 -0.576804909 -1.025709317 -1.727829408
## 706 707 708 709 710
## -0.656146609 -1.173511313 -1.483330169 -1.102655255 -1.984596098
## 711 712 713 714 715
## -1.925230237 -2.075224673 -2.241378133 -2.105931208 -2.162257874
## 716 717 718 719 720
## -2.301354229 -2.591485584 -2.352828690 2.679176774 2.552197385
## 721 722 723 724 725
## 1.475345605 1.081885037 1.336060999 1.320918248 -0.163937907
## 726 727 728 729 730
## 0.226689115 0.155647222 0.615597172 0.841266298 0.306471871
## 731 732 733 734 735
## -0.177464407 0.347112755 0.422040392 -0.615294736 -0.348105883
## 736 737 738 739 740
## -0.791977664 -0.518978094 -0.636249443 -0.960427270 -0.107662718
## 741 742 743 744 745
## -1.446767273 -0.847153161 -0.853009530 -1.087654005 -1.037981022
## 746 747 748 749 750
## -1.341329846 -1.545713529 -2.050437241 -1.931290112 -3.112732942
## 751 752 753 754 755
## -2.649142210 -3.309949996 2.785279007 2.263076122 1.531179258
## 756 757 758 759 760
## 1.905201983 1.764683451 1.213794428 1.666860943 1.826353542
## 761 762 763 764 765
## 1.094292047 0.642080206 0.786003233 0.293665575 0.561125983
## 766 767 768 769 770
## 0.183822293 -0.296645354 0.459255598 0.257705551 -0.463951074
## 771 772 773 774 775
## -0.687939067 0.165302297 -0.485975879 -0.702747144 -0.516705767
## 776 777 778 779 780
## -0.798849741 -1.189519749 -1.284811307 -1.137687029 -2.215477427
## 781 782 783 784 785
## 3.981979061 3.257037431 2.441659396 1.865190889 2.184233319
## 786 787 788 789 790
## 2.438748847 1.523302204 1.744129527 1.792821336 0.977441966
## 791 792 793 794 795
## 0.761255989 0.987603262 0.845677064 0.751871029 1.311344445
## 796 797 798 799 800
## 1.121127694 0.939099389 -0.076258444 0.328537243 0.054671476
## 801 802 803 804 805
## 0.280665660 0.367567953 -0.843409666 -1.183476486 4.547311529
## 806 807 808 809 810
## 3.923177995 3.928424900 2.511007480 3.132594693 2.421288524
## 811 812 813 814 815
## 2.210515640 2.619886458 2.615086752 2.144988651 1.925027600
## 816 817 818 819 820
## 2.140899245 0.995929277 1.676481318 0.899878283 1.421671863
## 821 822 823 824 825
## 0.706562463 1.010632168 0.330221258 -0.676492952 5.512939661
## 826 827 828 829 830
## 3.582462901 3.565717639 3.207917457 2.921717005 2.891910573
## 831 832 833 834 835
## 1.970314406 2.583362960 2.434336198 1.514810192 1.805870562
## 836 837 838 839 840
## 1.103459451 4.316685112 4.858847022 3.737191773 2.971788177
## 841 842 843 844 845
## 3.110645457 2.544724923 2.029458434 6.796174721 5.605938444
## 846 847 848 849 850
## 4.054330034 3.889141502 3.209280692 5.693750433 5.221715677
## 851 852 853 854 855
## 8.968478813 -8.743284614 1.540916577 0.396943250 1.471849840
## 856 857 858 859 860
## 2.905875337 -1.061552721 -6.910148818 1.510890334 2.068747073
## 861 862 863 864 865
## -5.731984395 -2.572537507 -4.664720036 -2.422006258 -3.815460108
## 866 867 868 869 870
## -4.266914447 -0.520498395 -2.221230289 -2.771972252 -2.997079420
## 871 872 873 874 875
## -2.925450034 -3.579032884 -6.245864307 -0.992296736 -1.651270933
## 876 877 878 879 880
## -1.731973307 -1.784466334 -2.309649546 -2.035579488 -2.274169372
## 881 882 883 884 885
## -3.139890119 -3.757720851 -3.203750953 -4.590130726 0.172744471
## 886 887 888 889 890
## -0.916724729 -0.730549555 -0.887759564 -0.822638186 -1.364119639
## 891 892 893 894 895
## -1.144799831 -1.514609136 -2.067856275 -1.915276036 -1.980686728
## 896 897 898 899 900
## -1.697328359 -2.642619860 -2.485659262 -3.179389457 -3.698535260
## 901 902 903 904 905
## -4.187737492 1.975566233 1.138536012 0.039401362 0.476744056
## 906 907 908 909 910
## -0.358221327 0.233489628 0.326412660 -0.385024862 -5.470119405
## 911 912 913 914 915
## -0.943585721 -3.858235770 1.721985741 0.411202107 -0.570615118
## 916 917 918 919 920
## -0.584183699 -0.618681331 -1.021739968 0.031520090 -1.290869634
## 921 922 923 924 925
## -1.464087582 -1.339519526 -0.922204781 -1.427366675 -1.125437078
## 926 927 928 929 930
## -1.689962615 -1.566518148 -1.994245576 -2.162774545 -1.965427571
## 931 932 933 934 935
## -1.932382933 -2.225195673 -2.552031012 -2.562924943 1.567945618
## 936 937 938 939 940
## 1.705210854 1.478785397 1.632961910 0.689264831 1.447444579
## 941 942 943 944 945
## 0.386428459 0.627950374 0.422352636 0.435988172 0.030293202
## 946 947 948 949 950
## 0.403164614 0.131251766 0.216064953 0.141792061 -0.575767023
## 951 952 953 954 955
## -0.894873844 -1.086366203 -0.876240723 -0.332976163 -0.722298637
## 956 957 958 959 960
## -0.940250308 -0.954278596 -1.291021065 -1.371546810 -1.005424910
## 961 962 963 964 965
## -1.193308727 -1.825408209 -1.966702622 -2.205402332 -2.280651354
## 966 967 968 969 970
## -2.668852466 -2.801679762 3.350174082 1.944998698 1.834501349
## 971 972 973 974 975
## 1.240292894 1.318071750 1.018360789 1.293638770 0.673796039
## 976 977 978 979 980
## 0.978309682 0.547953795 0.711440073 0.545759462 0.662990964
## 981 982 983 984 985
## 0.273431755 0.180373017 0.208792946 -0.016661794 0.164318768
## 986 987 988 989 990
## 0.179359096 0.199053067 0.037893666 -0.341811880 -1.120156569
## 991 992 993 994 995
## -0.837729072 -1.109617521 -0.933549710 -1.109290052 -0.735894733
## 996 997 998 999 1000
## -1.633203894 3.028308450 2.181361796 3.050111495 2.889867743
## 1001 1002 1003 1004 1005
## 2.195399495 1.535078644 2.191354836 1.104142120 1.014290782
## 1006 1007 1008 1009 1010
## 1.997966028 1.384245909 1.004036442 0.598681741 1.378897978
## 1011 1012 1013 1014 1015
## 1.086436763 0.119568553 0.163910582 0.322355679 0.554619095
## 1016 1017 1018 1019 1020
## -0.414866878 0.171365277 -0.236351870 -0.520593564 -1.634873942
## 1021 1022 1023 1024 1025
## 3.436372382 3.506197296 3.668358411 2.939776083 2.466993091
## 1026 1027 1028 1029 1030
## 2.385146036 2.790695191 2.584203554 1.583647664 1.912701017
## 1031 1032 1033 1034 1035
## 1.586134683 1.580070675 1.332671761 1.946137415 1.254843310
## 1036 1037 1038 1039 1040
## 1.309027961 0.602998826 0.414716046 -0.364875489 -1.278200379
## 1041 1042 1043 1044 1045
## 4.136000253 3.682608593 3.774236726 3.211133930 2.378770554
## 1046 1047 1048 1049 1050
## 2.131856040 2.588984847 2.049679070 1.836506945 1.220304149
## 1051 1052 1053 1054 1055
## 0.962337404 4.331060985 4.028872564 4.195720592 3.619775234
## 1056 1057 1058 1059 1060
## 2.313494450 2.811195962 2.270234930 1.997450895 5.865147140
## 1061 1062 1063 1064 1065
## 4.680763131 3.452156878 2.926232447 1.284000836 4.123250188
## 1066 1067 1068 1069 1070
## 7.400658027 -7.523332626 5.628732981 -4.251839278 -7.593037101
## 1071 1072 1073 1074 1075
## -3.658603782 -6.671128941 -8.877150660 2.423122015 -2.290051307
## 1076 1077 1078
## -1.483926919 -0.950717935 -3.015475796
plot(sonHeight$fitted.values, sonHeight$residuals) # visualization of the residuals
# versus the fitted values
From the plotting we can see that the relationship between residuals versus
the fitted values are very similar to the relation between the father’s height
and the son’s height. My concern would be that this is not the best model to
interpret the dataset.
(f) Using the model you fit in part (b) predict the height was 5 males whose
father are 50, 55, 70, 75, and 90 inches respectively. You may find the predict()
function helpful.
predfheight = data.frame(fheight = c(50, 55, 70, 75, 90)) # set heights
predict(sonHeight, predfheight, interval = "predict") # predict son's heights
## fit lwr upr
## 1 59.59126 54.71685 64.46566
## 2 62.16172 57.33140 66.99204
## 3 69.87312 65.08839 74.65785
## 4 72.44358 67.64470 77.24246
## 5 80.15498 75.22740 85.08255
From the fitted value we could see that the son’s heights are:
59.59126, 62.16172, 69.87312, 72.44358, 80.15498, when the father’s heights are:
50, 55, 70, 75, 90
Question 3
Loading the data set
(a) Describe the data and variables that are part of the Boston dataset.
library(MASS) # load the data set
View(Boston)
## Warning: running command ''/usr/bin/otool' -L '/Library/Frameworks/
## R.framework/Resources/modules/R_de.so'' had status 69
summary(Boston)
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00
str(Boston)
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
This data frame contains the following columns:
crim: per capita crime rate by town.
zn: proportion of residential land zoned for lots over 25,000 sq.ft.
indus: proportion of non-retail business acres per town.
chas: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
nox: nitrogen oxides concentration (parts per 10 million).
rm: average number of rooms per dwelling.
age: proportion of owner-occupied units built prior to 1940.
dis: weighted mean of distances to five Boston employment centres.
rad: index of accessibility to radial highways.
tax: full-value property-tax rate per $10,000.
ptratio: pupil-teacher ratio by town.
black: 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town.
lstat: lower status of the population (percent).
medv: median value of owner-occupied homes in $1000s.
(b) Consider this data what is the response variable of interest?
Consider that this dataset contains information about median house value for
506 neighborhoods in Boston, MA. The response variable of interest should be the
last variable “medv”: the median value of the homes.
(c) For each predictor, fit a simple linear regression model to predict the
response. In which of the models is there a statistically significant
association between the predictor and the response? Create some plots to
back up your assertions.
attach(Boston) # attach the data set to work with
crimAndPrice <- lm(medv ~ crim) # fit a linear regression between crim and medv
summary(crimAndPrice)
##
## Call:
## lm(formula = medv ~ crim)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.957 -5.449 -2.007 2.512 29.800
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 24.03311 0.40914 58.74 <2e-16 ***
## crim -0.41519 0.04389 -9.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.484 on 504 degrees of freedom
## Multiple R-squared: 0.1508, Adjusted R-squared: 0.1491
## F-statistic: 89.49 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ crim)
znAndPrice <- lm(medv ~ zn) # fit a linear regression between zn and medv
summary(znAndPrice)
##
## Call:
## lm(formula = medv ~ zn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.918 -5.518 -1.006 2.757 29.082
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.91758 0.42474 49.248 <2e-16 ***
## zn 0.14214 0.01638 8.675 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.587 on 504 degrees of freedom
## Multiple R-squared: 0.1299, Adjusted R-squared: 0.1282
## F-statistic: 75.26 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ zn)
indusAndPrice <- lm(medv ~ indus) # fit a linear regression between indus and medv
summary(indusAndPrice)
##
## Call:
## lm(formula = medv ~ indus)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.017 -4.917 -1.457 3.180 32.943
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.75490 0.68345 43.54 <2e-16 ***
## indus -0.64849 0.05226 -12.41 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.057 on 504 degrees of freedom
## Multiple R-squared: 0.234, Adjusted R-squared: 0.2325
## F-statistic: 154 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ indus)
noxAndPrice <- lm(medv ~ nox) # fit a linear regression between nox and medv
summary(noxAndPrice)
##
## Call:
## lm(formula = medv ~ nox)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.691 -5.121 -2.161 2.959 31.310
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 41.346 1.811 22.83 <2e-16 ***
## nox -33.916 3.196 -10.61 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.323 on 504 degrees of freedom
## Multiple R-squared: 0.1826, Adjusted R-squared: 0.181
## F-statistic: 112.6 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ nox)
rmAndPrice <- lm(medv ~ rm) # fit a linear regression between rm and medv
summary(rmAndPrice)
##
## Call:
## lm(formula = medv ~ rm)
##
## Residuals:
## Min 1Q Median 3Q Max
## -23.346 -2.547 0.090 2.986 39.433
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -34.671 2.650 -13.08 <2e-16 ***
## rm 9.102 0.419 21.72 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.616 on 504 degrees of freedom
## Multiple R-squared: 0.4835, Adjusted R-squared: 0.4825
## F-statistic: 471.8 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ rm)
abline(rmAndPrice)
ageAndPrice <- lm(medv ~ age) # fit a linear regression between age and medv
summary(ageAndPrice)
##
## Call:
## lm(formula = medv ~ age)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.097 -5.138 -1.958 2.397 31.338
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.97868 0.99911 31.006 <2e-16 ***
## age -0.12316 0.01348 -9.137 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.527 on 504 degrees of freedom
## Multiple R-squared: 0.1421, Adjusted R-squared: 0.1404
## F-statistic: 83.48 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ age)
abline(ageAndPrice)
disAndPrice <- lm(medv ~ dis) # fit a linear regression between dis and medv
summary(disAndPrice)
##
## Call:
## lm(formula = medv ~ dis)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.016 -5.556 -1.865 2.288 30.377
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.3901 0.8174 22.499 < 2e-16 ***
## dis 1.0916 0.1884 5.795 1.21e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.914 on 504 degrees of freedom
## Multiple R-squared: 0.06246, Adjusted R-squared: 0.0606
## F-statistic: 33.58 on 1 and 504 DF, p-value: 1.207e-08
plot(medv ~ dis)
radAndPrice <- lm(medv ~ rad) # fit a linear regression between rad and medv
summary(radAndPrice)
##
## Call:
## lm(formula = medv ~ rad)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.770 -5.199 -1.967 3.321 33.292
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.38213 0.56176 46.964 <2e-16 ***
## rad -0.40310 0.04349 -9.269 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.509 on 504 degrees of freedom
## Multiple R-squared: 0.1456, Adjusted R-squared: 0.1439
## F-statistic: 85.91 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ rad)
taxAndPrice <- lm(medv ~ tax) # fit a linear regression between tax and medv
summary(taxAndPrice)
##
## Call:
## lm(formula = medv ~ tax)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.091 -5.173 -2.085 3.158 34.058
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 32.970654 0.948296 34.77 <2e-16 ***
## tax -0.025568 0.002147 -11.91 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.133 on 504 degrees of freedom
## Multiple R-squared: 0.2195, Adjusted R-squared: 0.218
## F-statistic: 141.8 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ tax)
ptrAndPrice <- lm(medv ~ ptratio) # fit a linear regression between ptr and medv
summary(ptrAndPrice)
##
## Call:
## lm(formula = medv ~ ptratio)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.8342 -4.8262 -0.6426 3.1571 31.2303
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 62.345 3.029 20.58 <2e-16 ***
## ptratio -2.157 0.163 -13.23 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.931 on 504 degrees of freedom
## Multiple R-squared: 0.2578, Adjusted R-squared: 0.2564
## F-statistic: 175.1 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ ptratio)
blackAndPrice <- lm(medv ~ black) # fit a linear regression between black and medv
summary(blackAndPrice)
##
## Call:
## lm(formula = medv ~ black)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.884 -4.862 -1.684 2.932 27.763
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.551034 1.557463 6.775 3.49e-11 ***
## black 0.033593 0.004231 7.941 1.32e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.679 on 504 degrees of freedom
## Multiple R-squared: 0.1112, Adjusted R-squared: 0.1094
## F-statistic: 63.05 on 1 and 504 DF, p-value: 1.318e-14
plot(medv ~ black)
lstatAndPrice <- lm(medv ~ lstat) # fit a linear regression between lstat and medv
summary(lstatAndPrice)
##
## Call:
## lm(formula = medv ~ lstat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.168 -3.990 -1.318 2.034 24.500
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.55384 0.56263 61.41 <2e-16 ***
## lstat -0.95005 0.03873 -24.53 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.216 on 504 degrees of freedom
## Multiple R-squared: 0.5441, Adjusted R-squared: 0.5432
## F-statistic: 601.6 on 1 and 504 DF, p-value: < 2.2e-16
plot(medv ~ lstat)
We can see from the plotting that rm and medv fits into a positive linear regression, while age and medv fits into a negative linear regression.
(d) Fit a multiple regression model to predict the response using all of the
predictors. Describe your results. For which predictors can we reject the null
hypothesis H0 : βj = 0?
# fit multipul linear regression model using function:
# lm(response ~ explanatory_1 + explanatory_2 + ... + explanatory_p)
price = lm(medv ~ crim+zn+indus+chas+nox+rm+age+rad+tax+ptratio+black+lstat)
summary(price)
##
## Call:
## lm(formula = medv ~ crim + zn + indus + chas + nox + rm + age +
## rad + tax + ptratio + black + lstat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.6876 -2.9477 -0.7247 1.7515 27.7518
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 23.024356 5.022590 4.584 5.78e-06 ***
## crim -0.079610 0.034373 -2.316 0.020963 *
## zn 0.005526 0.013232 0.418 0.676389
## indus 0.116597 0.063301 1.842 0.066082 .
## chas 2.787236 0.907203 3.072 0.002241 **
## nox -9.799607 3.859335 -2.539 0.011417 *
## rm 4.222808 0.436167 9.682 < 2e-16 ***
## age 0.029160 0.013308 2.191 0.028901 *
## rad 0.313831 0.069860 4.492 8.79e-06 ***
## tax -0.012876 0.003959 -3.252 0.001224 **
## ptratio -1.043352 0.137167 -7.606 1.44e-13 ***
## black 0.009748 0.002828 3.447 0.000615 ***
## lstat -0.539183 0.053368 -10.103 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.997 on 493 degrees of freedom
## Multiple R-squared: 0.7118, Adjusted R-squared: 0.7048
## F-statistic: 101.5 on 12 and 493 DF, p-value: < 2.2e-16
From the summary of the statistical coefficients, we could see that zn has
a p-value of 0.676389, indus has a p-value of 0.066082, they are both larger
than 0.05, which should be considered to be large. So apart from zn and indus,
we could reject the null hypothesis of all the rest predictors.
(e) How do your results from (c) compare to your results from (d)?
Create a plot displaying the univariate regression coefficients from (c)
on the x-axis and the multiple regression coefficients from part (d)
on the y-axis. Use this visualization to support your response.
It’s interesting to notice and prove what we learned in class:
R2 will always increase when more variables are added to the model.
(f) Is there evidence of a non-linear association between any of the predictors
and the response?
pricelog <- lm(formula = log(medv) ~ crim + chas + nox + rm + dis + ptratio + black + lstat, data = Boston, family = "binomial")
## Warning in lm.fit(x, y, offset = offset, singular.ok = singular.ok, ...):
## extra argument 'family' is disregarded.
summary(pricelog)
##
## Call:
## lm(formula = log(medv) ~ crim + chas + nox + rm + dis + ptratio +
## black + lstat, data = Boston, family = "binomial")
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.70579 -0.09972 -0.01652 0.09180 0.90262
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.7964008 0.1965807 19.312 < 2e-16 ***
## crim -0.0080244 0.0012107 -6.628 8.89e-11 ***
## chas 0.1186049 0.0350353 3.385 0.000767 ***
## nox -0.6880284 0.1304358 -5.275 1.99e-07 ***
## rm 0.1100190 0.0162770 6.759 3.90e-11 ***
## dis -0.0426965 0.0066075 -6.462 2.47e-10 ***
## ptratio -0.0356303 0.0045174 -7.887 1.98e-14 ***
## black 0.0003696 0.0001084 3.408 0.000708 ***
## lstat -0.0286950 0.0019509 -14.709 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.195 on 497 degrees of freedom
## Multiple R-squared: 0.776, Adjusted R-squared: 0.7724
## F-statistic: 215.2 on 8 and 497 DF, p-value: < 2.2e-16
From the statistical summary of the mutiple linear regression, we can see that
we have a higher R-squared value of 0.7724. Since R-squared is a statistical measure of how close the data are to the fitted regression line, we can see this as the evidence of a non-linear association between the predictors and the response.