STAT 155 Topic 5 In-class Activity
Load some necessary packages:
library(ggplot2)
library(dplyr)Question 1
A real estate appraiser wants to explore the relationship between sale price of an apartment building and other characteristics of the property. The data available is contained in the file, MNSALES.csv. The variables of interest are:
Price: Sale price of the building (in dollars)
NumApts: Number of apartments in the building
Age: Age of the building in years
LotSize: Size of the lot on which the building is built (in square feet)
Parking: Number of parking spots
Condition: Condition of the building (F=Fair, G=Good, E=Excellent)
Read in the data:
MNSALES = read.csv('https://raw.githubusercontent.com/vittorioaddona/data/main/MNSALES.csv')Use this dataset to answer the following questions.
(a) Find the R-squared value for each of the following models:
Price ~ Age
model=lm(formula=Price~Age,data=MNSALES)ssr=sum((fitted(model)-mean(MNSALES$Price))^2)
ssr## [1] 14076534339sse=sum((fitted(model)-MNSALES$Price)^2)
sse## [1] 1.059793e+12sst=ssr+sse
sst## [1] 1.07387e+12Rsquared=1-(ssr/sst)
Rsquared## [1] 0.9868918Rsquared(Price~Age)=0.9868918 #### Price ~ NumApts
model1=lm(formula=Price~NumApts,data=MNSALES)ssr1=sum((fitted(model1)-mean(MNSALES$Price))^2)
ssr1## [1] 915775843108sse1=sum((fitted(model1)-MNSALES$Price)^2)
sse1## [1] 158094104206sst1=ssr1+sse1
sst1## [1] 1.07387e+12Rsquared1=1-(ssr1/sst1)
Rsquared1## [1] 0.147219Rsquared(Price~NumApts)=0.147219 #### Price ~ LotSize
model2=lm(formula=Price~LotSize,data=MNSALES)
ssr2=sum((fitted(model2)-mean(MNSALES$Price))^2)
ssr2## [1] 590946204278sse2=sum((fitted(model2)-MNSALES$Price)^2)
sse2## [1] 482923743036sst2=ssr2+sse2
sst2## [1] 1.07387e+12Rsquared2=1-(ssr2/sst2)
Rsquared2## [1] 0.4497041Rsquared(Price~LotSize)=0.4497041 #### Price ~ Parking
model3=lm(formula=Price~Parking,data=MNSALES)
ssr3=sum((fitted(model3)-mean(MNSALES$Price))^2)
ssr3## [1] 54310439533sse3=sum((fitted(model3)-MNSALES$Price)^2)
sse3## [1] 1.01956e+12sst3=ssr3+sse3
sst3## [1] 1.07387e+12Rsquared3=1-(ssr3/sst3)
Rsquared3## [1] 0.9494255Rsquared(Price~Parking)=0.9494255 #### Price ~ Condition
model4=lm(formula=Price~Condition,data=MNSALES)
ssr4=sum((fitted(model4)-mean(MNSALES$Price))^2)
ssr4## [1] 89083840373sse4=sum((fitted(model4)-MNSALES$Price)^2)
sse4## [1] 984786106941sst4=ssr4+sse4
sst4## [1] 1.07387e+12Rsquared4=1-(ssr4/sst4)
Rsquared4## [1] 0.9170441Rsquared(Price~Condition)=0.9170441
(b) Using your answers to (a), rank the explanatory variables from best to worst.
1.) Age 2.) Parking 3.) Condition 4.) LotSize 5.) NumApts
(c) Now, find the R-squared values for each of the following models, and rank them from best to worst:
Price ~ NumApts + LotSize
model5=lm(formula=Price~NumApts+LotSize,data=MNSALES)
ssr5=sum((fitted(model5)-mean(MNSALES$Price))^2)
ssr5## [1] 915809281076sse5=sum((fitted(model5)-MNSALES$Price)^2)
sse5## [1] 158060666239sst5=ssr5+sse5
sst5## [1] 1.07387e+12Rsquared5=1-(ssr5/sst5)
Rsquared5## [1] 0.1471879Rsquared(Price~NumApts+LotSize)=0.1471879 #### Price ~ NumApts + Condition
model6=lm(formula=Price~NumApts+Condition,data=MNSALES)
ssr6=sum((fitted(model6)-mean(MNSALES$Price))^2)
ssr6## [1] 986169504463sse6=sum((fitted(model6)-MNSALES$Price)^2)
sse6## [1] 87700442851sst6=ssr6+sse6
sst6## [1] 1.07387e+12Rsquared6=1-(ssr6/sst6)
Rsquared6## [1] 0.08166766Rsquared(Price~NumApts+Condition)=0.08166766 #### Price ~ NumApts + Age
model7=lm(formula=Price~NumApts+Age,data=MNSALES)
ssr7=sum((fitted(model7)-mean(MNSALES$Price))^2)
ssr7## [1] 926823801896sse7=sum((fitted(model7)-MNSALES$Price)^2)
sse7## [1] 147046145418sst7=ssr7+sse7
sst7## [1] 1.07387e+12Rsquared7=1-(ssr7/sst7)
Rsquared7## [1] 0.1369311Rsquared(Price~NumApts+Age)=0.1369311
1.) NumApts+LotSize 2.) NumApts+Age 3.) NumApts+Condition
(d) What might be surprising about your rankings in (c)? And how can you explain these results?
they really don’t match anything from (b) this might be due to the NumApts Variable
Question 2
HumanHeight = read.csv('https://raw.githubusercontent.com/vittorioaddona/data/main/HumanHeight.csv')The Correlation Coefficient: Correlation (r, \(\rho\), R) is a measure of the strength of the linear relationship between two quantitative variables. Correlation is always between -1 (perfect negative relationship) and +1 (perfect positive relationship).
(a) To find the correlation between two variables, use the
cor function in R. For example, using the HumanHeight.csv data,
we can find the correlation between Height and
FatherHeight using the command below (uncomment the command
by deleting the “#”, and run it):
HumanHeight %>% summarize( cor( Height , FatherHeight ) )## cor(Height, FatherHeight)
## 1 0.2753548(b) What if you had typed the following (again, uncomment and run the command):
HumanHeight %>% summarize( cor( FatherHeight , Height ) )## cor(FatherHeight, Height)
## 1 0.2753548(c) Describe in your own words at least 1 implication of this property of correlation.
correlation between two variables doesn’t change based on which variable is explanatory or response.
(d) Consider the following two models for the average height of an individual:
Model 1: Height = m1*FatherHeight + b1
Model 2: FatherHeight = m2*Height + b2
Is m1 = m2?
yes ### Is b1 = b2? yes
(e) The notation often used for correlation is R (not the software!). Why do you think this is? Compare the R-squared values of the two models presented in (d). What do you notice?
model8=lm(formula=Height~FatherHeight,data=HumanHeight)model8=lm(formula=Height~FatherHeight,data=HumanHeight)
ssr8=sum((fitted(model8)-mean(HumanHeight$Height))^2)
ssr8## [1] 873.0753sse8=sum((fitted(model8)-HumanHeight$Height)^2)
sse8## [1] 10641.99sst8=ssr8+sse8
sst8## [1] 11515.06Rsquared8=1-(ssr8/sst8)
Rsquared8## [1] 0.9241797Model1R2=0.9241797
model9=lm(formula=FatherHeight~Height,data=HumanHeight)
ssr9=sum((fitted(model9)-mean(HumanHeight$FatherHeight))^2)
ssr9## [1] 415.013sse9=sum((fitted(model9)-HumanHeight$FatherHeight)^2)
sse9## [1] 5058.628sst9=ssr9+sse9
sst9## [1] 5473.641Rsquared9=1-(ssr9/sst9)
Rsquared9## [1] 0.9241797Model2R2=0.9241797 the R2 for both models is the same
(f) Limitations of correlation: Think about the following two scenarios:
1. Suppose we wanted to find the correlation between
Height and Sex. Try it. What happens and
why?
"HumanHeight%>%
summarize(cor(Height,Sex))"## [1] "HumanHeight%>%\n summarize(cor(Height,Sex))"we get an error message because Sex is a categorical variable
2. Suppose we wanted to model Height by
FatherHeight, MotherHeight, and
Sex all in the same model. Can we find a correlation in
this situation? Can we find an R-squared value?
"HumanHeight%>%
summarize(cor(Height,FatherHeight,MotherHeight,Sex))"## [1] "HumanHeight%>%\n summarize(cor(Height,FatherHeight,MotherHeight,Sex))"model10=lm(formula=Height~FatherHeight+MotherHeight+Sex,data=HumanHeight)
model10##
## Call:
## lm(formula = Height ~ FatherHeight + MotherHeight + Sex, data = HumanHeight)
##
## Coefficients:
## (Intercept) FatherHeight MotherHeight SexM
## 15.3448 0.4060 0.3215 5.2260model10=lm(formula=Height~FatherHeight+MotherHeight+Sex,data=HumanHeight)
ssr10=sum((fitted(model10)-mean(HumanHeight$Height))^2)
ssr10## [1] 7365.9sse10=sum((fitted(model10)-HumanHeight$Height)^2)
sse10## [1] 4149.162sst10=ssr10+sse10
sst10## [1] 11515.06Rsquared10=1-(ssr10/sst10)
Rsquared10## [1] 0.3603248we cannot find a correlation for this, but we can find the Rsquared
value, which is 0.3603248
(g) Why do you think we will prefer R-squared to correlation in our class?
because it works regardless of whether a variable is quantitative or qualitative
Question 3
Answer the following TRUE/FALSE questions regarding R-squared, AND provide a justification for each:
(a) TRUE or FALSE: Consider the following two models:
M1: y ~ x1
M2: y ~ x2 + x3 + x4
where x1, x2, x3, and x4, are explanatory variables. M2 will necessarily have a higher R-squared value than M1.
False, because the addition of additional explanatory variables does not mean a bigger R-squared value necessarily
(b) TRUE or FALSE: Consider the following four models and their R-squared values:
M1: y ~ x1 \(~\rightarrow~\) R-squared = 0.70
M2: y ~ x2 \(~\rightarrow~\) R-squared = 0.60
M3: y ~ x3 \(~\rightarrow~\) R-squared = 0.50
M4: y ~ x4 \(~\rightarrow~\) R-sqaured = 0.40
Form these two models: M5: y ~ x1 + x2 and M6: y ~ x3 + x4. M5 will necessarily have a higher R-squared value than M6.
False, the fact that M5 consists of the two higher Rsquared values does not mean anything, there is no real correlation between the R2 value of 2, single explanatory variable models, and the R2 value of a Model that combines those two explanatory variables.
Question 4
This question reviews the concept of nested models.
(a) Which of the following models are nested in: A ~ B*C + D?
Model 1:\(~\) A ~ B
Model 2:\(~\) A ~ B + D
Model 3:\(~\) B ~ C
Model 4:\(~\) A ~ B + C + D
Model 5:\(~\) A ~ B*D + C
Model 6:\(~\) A ~ D
(b)
Consider the following models involving variables A, B, C, and D.
Model 1:\(~\) A ~ B
Model 2:\(~\) A ~ B + C
Model 3:\(~\) A ~ B + C + B:C
Model 4:\(~\) A ~ C + D
Model 5:\(~\) A ~ B*C
Model 6:\(~\) B ~ A
Model 7:\(~\) B ~ A*C
Answer TRUE or FALSE for each of the following:
(i) Model 1 is nested in Model 2.
(ii) Model 1 is nested in Model 3.
(iii) Model 1 is nested in Model 4.
(iv) Model 2 is nested in Model 3.
(v) Model 3 is nested in Model 2.
(vi) Model 2 is identical to Model 5.
(vii) Model 3 is identical to Model 5.
(viii) Model 1 is identical to Model 6.
(ix) Model 1 has the same R-squared as Model 6. Typically, one shouldn’t compare the R-squared of models with different response variables, but this is a special case.
(x) Model 2 is nested in Model 7.
Question 5
One measure of fitness is percent body fat. Unlike body mass index (BMI), percent body fat is not easy to calculate. BodyFat.csv has estimates of percent body fat (determined by underwater weighing) and various body circumference measurements for 252 men. The variables are:
BF: percent body fat from Siri’s equation (percentage, 0-100)
Age: individual’s age in years
Weight: individual’s weight in pounds (lbs)
Height: individual’s height in inches
Neck: individual’s neck circumference in cm
Chest: individual’s chest circumference in cm
Abdomen: individual’s abdomen circumference in cm
Hip: individual’s hip circumference in cm
Thigh: individual’s thigh circumference in cm
Knee: individual’s knee circumference in cm
Ankle: individual’s ankle circumference in cm
Biceps: individual’s bicep circumference in cm
Forearm: individual’s forearm circumference in cm
Wrist: individual’s wrist circumference in cm
Read in the data:
BodyFat = read.csv('https://raw.githubusercontent.com/vittorioaddona/data/main/BodyFat.csv')Use this dataset to answer the following questions.
(a) Find the R-squared of the model: BF ~ Weight.
model1A=lm(formula=BF~Weight,data=BodyFat)
ssr1A=sum((fitted(model1A)-mean(BodyFat$BF))^2)
ssr1A## [1] 6593.016sse1A=sum((fitted(model1A)-BodyFat$BF)^2)
sse1A## [1] 10985.97sst1A=ssr1A+sse1A
sst1A## [1] 17578.99Rsquared1A=1-(ssr1A/sst1A)
Rsquared1A## [1] 0.6249491R2(BF~Weight)=0.6249491
(b) Find the R-squared of the model: BF ~ Height.
model2A=lm(formula=BF~Height,data=BodyFat)
ssr2A=sum((fitted(model2A)-mean(BodyFat$BF))^2)
ssr2A## [1] 140.7976sse2A=sum((fitted(model2A)-BodyFat$BF)^2)
sse2A## [1] 17438.19sst2A=ssr2A+sse2A
sst2A## [1] 17578.99Rsquared2A=1-(ssr2A/sst2A)
Rsquared2A## [1] 0.9919906R2(BF~Height)=0.9919906
(c) Based on your answers to (a) and (b), what do you think are the smallest and largest possible R-squared values for the model: BF ~ Weight + Height.
the smallest is probably around 0.5 and the highest is probably about 1
(d) Find the R-squared value of the model: BF ~ Weight + Height.
model3A=lm(formula=BF~Weight+Height,data=BodyFat)
ssr3A=sum((fitted(model3A)-mean(BodyFat$BF))^2)
ssr3A## [1] 8097.391sse3A=sum((fitted(model3A)-BodyFat$BF)^2)
sse3A## [1] 9481.599sst3A=ssr3A+sse3A
sst3A## [1] 17578.99Rsquared3A=1-(ssr3A/sst3A)
Rsquared3A## [1] 0.5393711R2(BF~Weight+Height)=0.5393711
(e) By comparing the values found in (a), (b), and (d), explain in your own words what you have learned about R-squared.
that the addition of additional variables decreases how accurate a model is.
Question 6
Answer the following short answer questions in your own words:
(a) How does R-squared summarize the quality of a model (that is, describe what R-squared is actually measuring)?
(b) What does it mean for one model to be nested inside another model?
(c) State at least 2 ways that the correlation coefficient differs from R-squared?
(d) Consider models of the form: response ~ 1. Briefly explain in your own words why R-squared = 0 for such models.
(e) Generally, in comparing two models with the same response variable, the model with the larger R-sqaured is better. Add some nuance to this rule of thumb.
Question 7
Condsider the following four scatterplots with response variables y and four different explanatory variables x. If we were to fit a simple linear regression model y ~ x to each of these sets of data, order the R-squared values from largest to smallest for the four models.
Question 8
Our primary research question for the FEV data is: what is the impact of smoking on lung function in children? Forced expiratory volume (FEV) is a measure of lung capacity. Higher values indicate higher lung capacity - the ability to blow out more air in one second - and therefore better lung function. Data are available for 654 children, ages 3-19. Each row in the data corresponds to a child’s visit to a doctor. All children attended the same pediatric clinic. The date of the study is unknown, but it was prior to 2005. Variables collected include:
age: at time of measurement (years)
fev: forced expiratory volume fev
(liters per second)
height: at time of measurement (inches)
sex: (male/female)
smoke: indicator of smoking habits (smoker/nonsmoker)
Load the FEV data:
fevdata = read.csv('https://raw.githubusercontent.com/vittorioaddona/data/main/fev.csv')(a) We’ll be evaluating and comparing 4 models:
Model 1:\(~\) fev ~ age
Model 2:\(~\) fev ~ smoke
Model 3:\(~\) fev ~ smoke + age
Model 4:\(~\) fev ~ smoke + age + smoke:age
Uncomment (delete the ‘#’), and complete the code chunk below to fit
these 4 models and save them as objects called mod1,
mod2, mod3, and mod4.
mod1=lm(formula=fev~age,data=fevdata)
mod2=lm(formula=fev~smoke,data=fevdata)
mod3=lm(formula=fev~smoke+age,data=fevdata)
mod4=lm(formula=fev~smoke+age+smoke:age,data=fevdata)(b) Most of our model evaluation techniques focus on the model residuals. Answer the following 2 brief review questions to help remind ourselves of some key terms:
What is a fitted value?
the predicted response based on an explanatory value
What is a residual?
the difference between the observed value and the fitted value
(c) Using Model 1, calculate the fitted value and residual for an eight-year-old with an FEV of 1.724 liters/second.
fitted(mod1)## 1 2 3 4 5 6 7 8
## 2.430017 2.207976 1.985935 2.430017 2.430017 2.207976 1.763894 1.763894
## 9 10 11 12 13 14 15 16
## 2.207976 2.430017 1.763894 2.207976 2.207976 2.207976 2.207976 1.985935
## 17 18 19 20 21 22 23 24
## 1.541853 1.763894 2.430017 2.430017 1.541853 1.541853 1.319812 1.985935
## 25 26 27 28 29 30 31 32
## 2.430017 1.097771 2.430017 1.541853 2.207976 2.430017 1.541853 2.430017
## 33 34 35 36 37 38 39 40
## 2.207976 1.985935 1.541853 2.207976 2.430017 2.207976 2.207976 2.207976
## 41 42 43 44 45 46 47 48
## 2.430017 2.207976 1.541853 2.207976 1.541853 2.430017 1.985935 2.207976
## 49 50 51 52 53 54 55 56
## 1.763894 2.207976 1.541853 2.430017 2.430017 2.207976 1.763894 2.430017
## 57 58 59 60 61 62 63 64
## 2.430017 1.985935 1.319812 2.207976 2.207976 2.207976 1.763894 1.319812
## 65 66 67 68 69 70 71 72
## 2.207976 1.763894 2.430017 1.985935 1.541853 2.430017 2.207976 2.207976
## 73 74 75 76 77 78 79 80
## 2.430017 2.430017 2.430017 1.985935 1.541853 1.541853 2.430017 1.763894
## 81 82 83 84 85 86 87 88
## 1.985935 1.763894 2.207976 2.207976 1.985935 2.207976 1.985935 2.430017
## 89 90 91 92 93 94 95 96
## 1.541853 2.430017 2.430017 2.430017 1.985935 2.207976 2.207976 2.430017
## 97 98 99 100 101 102 103 104
## 2.430017 2.430017 1.985935 2.207976 2.207976 1.985935 2.430017 1.319812
## 105 106 107 108 109 110 111 112
## 2.430017 1.763894 2.207976 1.763894 1.985935 1.985935 2.207976 1.985935
## 113 114 115 116 117 118 119 120
## 1.985935 1.985935 1.985935 2.207976 1.985935 1.541853 2.207976 1.985935
## 121 122 123 124 125 126 127 128
## 2.430017 1.985935 1.985935 1.763894 2.207976 2.207976 2.207976 2.430017
## 129 130 131 132 133 134 135 136
## 1.985935 2.207976 2.430017 2.207976 2.207976 2.430017 2.207976 1.763894
## 137 138 139 140 141 142 143 144
## 1.763894 2.207976 2.430017 1.541853 1.985935 2.430017 1.763894 2.430017
## 145 146 147 148 149 150 151 152
## 2.430017 2.430017 1.763894 2.207976 2.430017 2.207976 2.207976 2.430017
## 153 154 155 156 157 158 159 160
## 2.430017 2.430017 1.985935 2.207976 1.763894 2.430017 2.430017 2.430017
## 161 162 163 164 165 166 167 168
## 1.985935 2.207976 1.541853 2.207976 2.430017 1.763894 2.430017 1.763894
## 169 170 171 172 173 174 175 176
## 2.207976 1.541853 1.985935 1.985935 1.319812 2.430017 2.207976 2.430017
## 177 178 179 180 181 182 183 184
## 2.430017 2.430017 1.541853 2.430017 1.985935 1.763894 2.430017 2.430017
## 185 186 187 188 189 190 191 192
## 2.430017 1.985935 1.541853 2.207976 2.430017 1.985935 2.430017 2.207976
## 193 194 195 196 197 198 199 200
## 2.430017 1.763894 1.763894 2.207976 2.430017 1.541853 1.763894 1.763894
## 201 202 203 204 205 206 207 208
## 2.430017 1.985935 2.430017 2.207976 1.541853 1.985935 1.763894 2.430017
## 209 210 211 212 213 214 215 216
## 1.985935 2.430017 2.430017 2.207976 2.430017 1.985935 2.430017 1.319812
## 217 218 219 220 221 222 223 224
## 2.430017 1.541853 2.207976 2.430017 2.207976 1.097771 2.430017 2.207976
## 225 226 227 228 229 230 231 232
## 1.763894 2.430017 2.207976 2.207976 1.985935 1.763894 2.207976 2.430017
## 233 234 235 236 237 238 239 240
## 1.319812 1.985935 2.207976 2.207976 2.430017 1.763894 2.207976 1.763894
## 241 242 243 244 245 246 247 248
## 2.207976 2.430017 2.207976 1.985935 2.430017 2.207976 1.985935 2.430017
## 249 250 251 252 253 254 255 256
## 2.207976 2.430017 1.763894 2.207976 2.430017 2.207976 2.430017 2.430017
## 257 258 259 260 261 262 263 264
## 2.207976 1.985935 1.541853 1.985935 2.207976 2.430017 2.430017 1.763894
## 265 266 267 268 269 270 271 272
## 2.207976 1.985935 2.430017 1.985935 1.985935 1.541853 2.430017 2.430017
## 273 274 275 276 277 278 279 280
## 2.207976 2.207976 2.430017 1.763894 1.985935 1.541853 2.430017 1.541853
## 281 282 283 284 285 286 287 288
## 1.985935 1.763894 2.207976 1.985935 2.207976 1.319812 2.207976 1.541853
## 289 290 291 292 293 294 295 296
## 2.207976 1.985935 1.985935 2.430017 2.430017 2.207976 2.430017 1.763894
## 297 298 299 300 301 302 303 304
## 2.207976 2.430017 1.319812 1.763894 1.985935 2.430017 2.207976 1.763894
## 305 306 307 308 309 310 311 312
## 2.207976 1.985935 1.541853 2.207976 1.985935 2.874099 2.652058 3.540222
## 313 314 315 316 317 318 319 320
## 2.874099 2.874099 3.096140 2.652058 2.874099 2.652058 3.540222 3.318181
## 321 322 323 324 325 326 327 328
## 3.540222 3.096140 3.096140 2.652058 3.318181 2.652058 2.874099 2.652058
## 329 330 331 332 333 334 335 336
## 2.874099 2.652058 3.318181 3.540222 2.874099 2.652058 2.874099 3.318181
## 337 338 339 340 341 342 343 344
## 2.652058 2.652058 3.096140 2.652058 2.652058 2.652058 2.874099 2.874099
## 345 346 347 348 349 350 351 352
## 2.874099 2.652058 2.874099 2.874099 3.318181 3.318181 2.874099 2.874099
## 353 354 355 356 357 358 359 360
## 3.540222 2.874099 2.652058 2.652058 2.652058 3.540222 3.318181 2.652058
## 361 362 363 364 365 366 367 368
## 3.540222 2.652058 2.874099 3.318181 3.096140 3.318181 2.652058 3.318181
## 369 370 371 372 373 374 375 376
## 2.874099 3.540222 2.874099 3.318181 2.874099 2.874099 2.652058 2.874099
## 377 378 379 380 381 382 383 384
## 2.874099 2.652058 2.874099 3.318181 3.096140 2.652058 2.652058 3.540222
## 385 386 387 388 389 390 391 392
## 2.874099 2.652058 2.874099 2.652058 2.874099 3.318181 3.318181 2.652058
## 393 394 395 396 397 398 399 400
## 2.874099 2.874099 3.096140 2.652058 2.652058 2.874099 2.652058 2.874099
## 401 402 403 404 405 406 407 408
## 3.540222 3.318181 3.096140 2.874099 2.874099 2.874099 3.540222 3.096140
## 409 410 411 412 413 414 415 416
## 2.652058 3.096140 2.874099 2.652058 2.874099 3.318181 2.652058 2.652058
## 417 418 419 420 421 422 423 424
## 2.874099 3.318181 2.652058 2.874099 2.652058 3.318181 2.874099 2.652058
## 425 426 427 428 429 430 431 432
## 2.874099 2.874099 3.540222 2.874099 3.318181 2.874099 2.874099 2.652058
## 433 434 435 436 437 438 439 440
## 3.318181 2.652058 3.318181 2.652058 3.096140 2.652058 3.540222 3.096140
## 441 442 443 444 445 446 447 448
## 2.652058 2.874099 3.540222 3.096140 2.652058 2.652058 2.652058 2.652058
## 449 450 451 452 453 454 455 456
## 3.096140 3.318181 2.874099 3.096140 2.874099 3.096140 2.874099 2.874099
## 457 458 459 460 461 462 463 464
## 3.096140 3.096140 3.318181 2.874099 3.096140 2.652058 3.096140 3.318181
## 465 466 467 468 469 470 471 472
## 2.652058 3.096140 2.652058 3.096140 2.652058 2.874099 2.652058 3.096140
## 473 474 475 476 477 478 479 480
## 3.540222 2.652058 2.652058 3.096140 2.652058 2.652058 3.318181 3.096140
## 481 482 483 484 485 486 487 488
## 3.096140 2.874099 3.318181 3.096140 2.652058 2.874099 2.874099 3.318181
## 489 490 491 492 493 494 495 496
## 3.096140 3.318181 3.318181 2.652058 3.096140 3.096140 3.540222 2.874099
## 497 498 499 500 501 502 503 504
## 2.652058 3.318181 2.874099 2.874099 3.318181 3.096140 2.652058 2.652058
## 505 506 507 508 509 510 511 512
## 3.096140 3.318181 2.874099 2.652058 2.874099 2.874099 2.874099 2.874099
## 513 514 515 516 517 518 519 520
## 2.874099 3.540222 3.096140 3.318181 3.318181 2.652058 3.096140 2.652058
## 521 522 523 524 525 526 527 528
## 2.652058 3.096140 2.874099 3.096140 2.874099 2.874099 3.096140 3.096140
## 529 530 531 532 533 534 535 536
## 3.540222 2.874099 2.652058 2.874099 3.096140 3.318181 3.096140 2.874099
## 537 538 539 540 541 542 543 544
## 2.874099 2.874099 3.540222 2.874099 3.318181 3.096140 2.652058 3.096140
## 545 546 547 548 549 550 551 552
## 3.318181 2.652058 2.652058 2.652058 2.652058 3.540222 3.096140 2.874099
## 553 554 555 556 557 558 559 560
## 2.874099 3.096140 3.540222 3.540222 2.652058 2.874099 2.874099 2.652058
## 561 562 563 564 565 566 567 568
## 2.652058 3.096140 3.096140 2.874099 3.096140 2.652058 3.096140 3.318181
## 569 570 571 572 573 574 575 576
## 2.652058 3.096140 2.652058 3.318181 3.096140 2.652058 3.096140 2.652058
## 577 578 579 580 581 582 583 584
## 2.874099 3.096140 2.874099 3.096140 2.652058 3.318181 3.096140 2.874099
## 585 586 587 588 589 590 591 592
## 2.874099 2.874099 2.874099 3.096140 3.540222 2.874099 2.874099 3.096140
## 593 594 595 596 597 598 599 600
## 3.540222 2.874099 3.318181 2.874099 2.652058 3.318181 3.096140 2.874099
## 601 602 603 604 605 606 607 608
## 3.318181 3.540222 2.652058 2.874099 2.874099 3.762263 3.762263 4.428386
## 609 610 611 612 613 614 615 616
## 4.650427 4.650427 3.984304 4.206345 3.762263 3.762263 3.762263 3.762263
## 617 618 619 620 621 622 623 624
## 3.762263 4.650427 4.428386 3.984304 4.206345 3.984304 3.762263 3.762263
## 625 626 627 628 629 630 631 632
## 3.762263 4.428386 4.206345 3.762263 4.206345 4.206345 3.984304 4.206345
## 633 634 635 636 637 638 639 640
## 3.762263 3.762263 3.984304 3.984304 3.762263 4.428386 3.762263 3.984304
## 641 642 643 644 645 646 647 648
## 4.206345 3.984304 3.984304 3.762263 4.428386 3.762263 3.984304 4.206345
## 649 650 651 652 653 654
## 3.984304 3.984304 3.762263 4.428386 3.984304 3.762263resid(mod1)## 1 2 3 4 5 6
## -0.722016889 -0.483975913 -0.265934937 -0.872016889 -0.535016889 0.128024087
## 7 8 9 10 11 12
## 0.155106039 -0.348893961 -0.220975913 -0.488016889 -0.161893961 -0.472975913
## 13 14 15 16 17 18
## -0.014975913 -0.089975913 0.050024087 -0.053934937 -0.069852986 0.114106039
## 19 20 21 22 23 24
## -0.078016889 0.173983111 -0.141852986 -0.285852986 -0.480812010 0.592065063
## 25 26 27 28 29 30
## 0.557983111 0.306228966 -0.082016889 0.213147014 0.772024087 -0.330016889
## 31 32 33 34 35 36
## -0.259852986 0.569983111 0.465024087 0.107065063 0.070147014 -0.032975913
## 37 38 39 40 41 42
## 0.294983111 -0.136975913 -0.660975913 -0.203975913 0.704983111 0.212024087
## 43 44 45 46 47 48
## 0.234147014 -0.276975913 -0.198852986 -0.354016889 -0.361934937 -0.863975913
## 49 50 51 52 53 54
## -0.113893961 0.524024087 0.475147014 0.366983111 1.125983111 -0.504975913
## 55 56 57 58 59 60
## -0.129893961 0.139983111 0.585983111 0.433065063 0.249187990 -0.509975913
## 61 62 63 64 65 66
## -0.084975913 0.273024087 -0.282893961 0.257187990 -0.267975913 -0.016893961
## 67 68 69 70 71 72
## -0.361016889 -0.354934937 -0.005852986 0.129983111 -0.245975913 0.323024087
## 73 74 75 76 77 78
## 0.284983111 0.026983111 -0.340016889 -0.196934937 0.316147014 -0.089852986
## 79 80 81 82 83 84
## 1.411983111 -0.044893961 0.125065063 -0.068893961 0.003024087 -0.413975913
## 85 86 87 88 89 90
## -0.068934937 -0.063975913 -0.732934937 0.228983111 0.038147014 -0.304016889
## 91 92 93 94 95 96
## 0.598983111 0.533983111 -0.374934937 0.007024087 0.180024087 -0.234016889
## 97 98 99 100 101 102
## -0.679016889 -0.265016889 -0.303934937 -0.684975913 -0.915975913 -0.336934937
## 103 104 105 106 107 108
## 0.157983111 -0.523812010 0.143983111 0.215106039 0.146024087 -0.045893961
## 109 110 111 112 113 114
## -0.243934937 -0.382934937 0.431024087 -0.156934937 0.098065063 0.234065063
## 115 116 117 118 119 120
## -0.512934937 0.133024087 -0.287934937 -0.345852986 -0.335975913 0.233065063
## 121 122 123 124 125 126
## -0.010016889 -0.158934937 -0.524934937 -0.425893961 -0.117975913 -0.510975913
## 127 128 129 130 131 132
## -0.645975913 -0.390016889 -0.376934937 0.250024087 0.219983111 -0.778975913
## 133 134 135 136 137 138
## -0.532975913 -0.483016889 -0.138975913 -0.191893961 -0.415893961 0.080024087
## 139 140 141 142 143 144
## -0.657016889 -0.750852986 -0.080934937 0.032983111 -0.332893961 0.200983111
## 145 146 147 148 149 150
## 0.683983111 -0.295016889 -0.236893961 0.085024087 0.611983111 0.719024087
## 151 152 153 154 155 156
## 0.457024087 -0.129016889 0.029983111 0.161983111 -0.235934937 -0.448975913
## 157 158 159 160 161 162
## -0.227893961 -0.171016889 -0.382016889 0.140983111 0.060065063 -0.427975913
## 163 164 165 166 167 168
## 0.010147014 -0.254975913 0.462983111 -0.050893961 0.420983111 -0.139893961
## 169 170 171 172 173 174
## 0.423024087 0.277147014 -0.327934937 0.172065063 0.469187990 0.573983111
## 175 176 177 178 179 180
## 0.295024087 -0.497016889 -0.339016889 -0.114016889 0.162147014 -0.824016889
## 181 182 183 184 185 186
## -0.820934937 0.338106039 -0.110016889 -0.200016889 -0.714016889 -0.195934937
## 187 188 189 190 191 192
## -0.395852986 -0.020975913 0.286983111 -0.189934937 -0.477016889 -0.872975913
## 193 194 195 196 197 198
## -0.311016889 -0.097893961 0.062106039 0.501024087 0.440983111 -0.449852986
## 199 200 201 202 203 204
## 0.498106039 0.340106039 -0.264016889 -0.295934937 0.542983111 -0.062975913
## 205 206 207 208 209 210
## 0.429147014 0.109065063 -0.066893961 0.024983111 -0.065934937 -0.266016889
## 211 212 213 214 215 216
## -0.300016889 0.785024087 0.098983111 -0.259934937 0.011983111 -0.217812010
## 217 218 219 220 221 222
## -0.374016889 0.266147014 0.097024087 -0.461016889 -0.651975913 -0.025771034
## 223 224 225 226 227 228
## -0.388016889 -0.695975913 -0.340893961 1.250983111 -0.216975913 -0.310975913
## 229 230 231 232 233 234
## -0.615934937 -0.425893961 -0.191975913 0.208983111 0.069187990 -0.373934937
## 235 236 237 238 239 240
## -0.072975913 0.473024087 0.792983111 0.032106039 -0.197975913 -0.240893961
## 241 242 243 244 245 246
## -0.463975913 0.054983111 0.127024087 -0.570934937 -0.354016889 0.227024087
## 247 248 249 250 251 252
## -0.257934937 0.419983111 -0.363975913 -0.676016889 -0.420893961 0.095024087
## 253 254 255 256 257 258
## -0.184016889 0.268024087 0.808983111 0.026983111 0.174024087 -0.345934937
## 259 260 261 262 263 264
## 0.047147014 0.070065063 0.018024087 -0.544016889 0.402983111 -0.048893961
## 265 266 267 268 269 270
## 0.423024087 0.564065063 -0.518016889 -0.108934937 -0.050934937 -0.002852986
## 271 272 273 274 275 276
## 0.372983111 0.492983111 0.150024087 -0.113975913 -0.575016889 -0.228893961
## 277 278 279 280 281 282
## 0.149065063 0.388147014 -0.248016889 -0.182852986 0.016065063 -0.064893961
## 283 284 285 286 287 288
## 0.292024087 0.380065063 -0.138975913 0.098187990 0.125024087 -0.027852986
## 289 290 291 292 293 294
## -0.449975913 0.549065063 0.578065063 0.056983111 -0.839016889 -0.583975913
## 295 296 297 298 299 300
## 0.367983111 -0.072893961 -0.208975913 -0.561016889 -0.315812010 -0.336893961
## 301 302 303 304 305 306
## -0.159934937 0.257983111 -0.550975913 -0.091893961 -0.192975913 0.385065063
## 307 308 309 310 311 312
## 0.573147014 0.120024087 -0.490934937 0.009901159 -0.324057865 -0.159221769
## 313 314 315 316 317 318
## -0.704098841 0.595901159 -0.038139817 -0.841057865 -0.350098841 -0.010057865
## 319 320 321 322 323 324
## 0.200778231 1.017819207 1.301778231 1.453860183 -0.255139817 0.513942135
## 325 326 327 328 329 330
## 0.497819207 -0.091057865 0.779901159 -0.171057865 -0.209098841 0.550942135
## 331 332 333 334 335 336
## 0.230819207 -1.304221769 0.347901159 0.458942135 0.615901159 -0.171180793
## 337 338 339 340 341 342
## -0.132057865 -0.360057865 -0.207139817 -0.406057865 -0.715057865 -0.006057865
## 343 344 345 346 347 348
## 0.082901159 1.132901159 -0.488098841 0.598942135 -0.112098841 0.136901159
## 349 350 351 352 353 354
## 0.986819207 0.587819207 0.708901159 0.361901159 -0.104221769 0.183901159
## 355 356 357 358 359 360
## 0.354942135 0.836942135 0.211942135 -0.112221769 -0.499180793 -0.402057865
## 361 362 363 364 365 366
## 1.142778231 -0.300057865 0.233901159 0.675819207 1.296860183 -0.110180793
## 367 368 369 370 371 372
## -0.060057865 -0.125180793 -1.180098841 0.416778231 -0.528098841 1.470819207
## 373 374 375 376 377 378
## 0.640901159 -0.120098841 0.067942135 -0.411098841 -0.241098841 0.395942135
## 379 380 381 382 383 384
## 0.236901159 0.426819207 -0.712139817 -0.558057865 0.530942135 -0.466221769
## 385 386 387 388 389 390
## 1.102901159 0.701942135 0.536901159 -0.265057865 0.296901159 0.568819207
## 391 392 393 394 395 396
## -0.672180793 -0.148057865 0.712901159 0.970901159 -0.125139817 0.238942135
## 397 398 399 400 401 402
## -0.829057865 -0.457098841 -0.477057865 -0.139098841 0.732778231 -0.342180793
## 403 404 405 406 407 408
## 0.738860183 1.190901159 -0.556098841 0.721901159 -0.145221769 -0.345139817
## 409 410 411 412 413 414
## 0.020942135 -0.540139817 -0.332098841 -0.044057865 -0.520098841 -0.719180793
## 415 416 417 418 419 420
## -1.194057865 1.142942135 -0.383098841 -0.258180793 -0.107057865 0.118901159
## 421 422 423 424 425 426
## 0.652942135 1.437819207 0.899901159 0.202942135 0.113901159 -0.376098841
## 427 428 429 430 431 432
## -0.371221769 0.012901159 -0.614180793 0.640901159 0.550901159 -0.365057865
## 433 434 435 436 437 438
## -0.884180793 -0.287057865 -0.232180793 0.043942135 -0.228139817 0.160942135
## 439 440 441 442 443 444
## 0.768778231 0.158860183 0.760942135 1.718901159 0.570778231 -1.180139817
## 445 446 447 448 449 450
## -0.794057865 0.322942135 0.697942135 0.248942135 -0.855139817 0.906819207
## 451 452 453 454 455 456
## 0.348901159 2.127860183 1.198901159 0.983860183 -0.268098841 0.294901159
## 457 458 459 460 461 462
## 1.314860183 0.694860183 -0.229180793 -0.409098841 0.246860183 0.547942135
## 463 464 465 466 467 468
## -0.183139817 1.558819207 -0.294057865 0.182860183 -0.071057865 -0.749139817
## 469 470 471 472 473 474
## 0.038942135 -0.047098841 -0.779057865 0.654860183 -1.002221769 0.105942135
## 475 476 477 478 479 480
## 0.397942135 -0.017139817 -0.451057865 -0.794057865 -1.102180793 0.306860183
## 481 482 483 484 485 486
## 0.404860183 -0.296098841 -0.240180793 0.089860183 -0.987057865 -0.793098841
## 487 488 489 490 491 492
## 0.099901159 -0.021180793 0.976860183 1.129819207 0.665819207 -0.402057865
## 493 494 495 496 497 498
## -0.344139817 -0.792139817 0.139778231 0.227901159 0.209942135 -0.641180793
## 499 500 501 502 503 504
## 0.148901159 0.806901159 -0.063180793 0.595860183 -0.296057865 1.938942135
## 505 506 507 508 509 510
## -0.014139817 -0.021180793 0.383901159 -0.436057865 0.372901159 1.449901159
## 511 512 513 514 515 516
## -0.512098841 -0.311098841 0.331901159 0.044778231 1.623860183 0.012819207
## 517 518 519 520 521 522
## 1.764819207 0.845942135 -0.679139817 -0.288057865 -0.311057865 -0.337139817
## 523 524 525 526 527 528
## 0.078901159 0.134860183 0.203901159 0.494901159 0.432860183 -0.230139817
## 529 530 531 532 533 534
## -0.649221769 0.147901159 0.474942135 -0.008098841 -0.491139817 -0.262180793
## 535 536 537 538 539 540
## -0.527139817 -0.373098841 0.445901159 -0.751098841 0.239778231 0.972901159
## 541 542 543 544 545 546
## 0.466819207 0.827860183 -0.520057865 -0.344139817 -0.869180793 0.803942135
## 547 548 549 550 551 552
## 0.420942135 0.035942135 0.676942135 0.730778231 0.433860183 0.053901159
## 553 554 555 556 557 558
## -0.185098841 -0.764139817 -0.606221769 -1.264221769 0.457942135 0.019901159
## 559 560 561 562 563 564
## 1.762901159 -0.217057865 0.185942135 -0.061139817 1.734860183 -0.062098841
## 565 566 567 568 569 570
## -0.382139817 0.433942135 0.422860183 0.913819207 0.117942135 0.244860183
## 571 572 573 574 575 576
## 0.437942135 -0.787180793 -0.274139817 0.385942135 -0.161139817 -0.084057865
## 577 578 579 580 581 582
## -0.487098841 -0.597139817 1.255901159 -0.095139817 0.479942135 0.258819207
## 583 584 585 586 587 588
## 0.125860183 0.405901159 -0.215098841 -0.052098841 -0.734098841 1.106860183
## 589 590 591 592 593 594
## -0.543221769 0.245901159 -0.312098841 -0.014139817 0.265778231 0.464901159
## 595 596 597 598 599 600
## -0.166180793 -0.416098841 -0.261057865 -0.177180793 -0.517139817 0.229901159
## 601 602 603 604 605 606
## 0.726819207 1.222778231 -0.552057865 0.194901159 -0.089098841 0.521737255
## 607 608 609 610 611 612
## 0.743737255 -1.522385673 0.451573351 -1.131426649 -0.296303721 0.222655303
## 613 614 615 616 617 618
## 0.516737255 0.737737255 -1.127262745 -1.083262745 -1.564262745 -1.305426649
## 619 620 621 622 623 624
## -1.346385673 -0.597303721 -1.124344697 -1.081303721 -0.758262745 2.030737255
## 625 626 627 628 629 630
## 0.222737255 -0.208385673 0.517655303 -0.031262745 -0.800344697 -0.706344697
## 631 632 633 634 635 636
## -0.310303721 1.426655303 -0.640262745 -0.432262745 -1.376303721 -0.339303721
## 637 638 639 640 641 642
## 0.036737255 -0.342385673 -0.875262745 0.085696279 -0.246344697 0.314696279
## 643 644 645 646 647 648
## -1.003303721 -1.498262745 -0.024385673 -1.484262745 0.519696279 1.431655303
## 649 650 651 652 653 654
## 0.887696279 0.285696279 -0.035262745 -1.575385673 -1.189303721 -0.551262745fitted value=2.207976 residual=-0.483975913
(d) What does it tell us if we have a residual of exactly zero? What if our residual is positive? Negative?
if our residual is exactly 0, that means that the fitted value and the observed value are the same, if the residual is positive, that means that the observed value is greater than the fitted value, if the residual is negative, then the observed value is less than the fitted value.
(e) Uncomment the commands in the chunk below, and complete the
lines of code to store the fitted values and the residuals (according to
mod1) for all of the observed FEV data:
residuals=resid(mod1)
fitted=fitted(mod1)(f) Look at the second value of the residuals and fitted values you found in (e). These should match your answers to (c). Why?
because the 2nd line of the table is the 8 year old kid with the 1.724 fev.
(g) An alternative to R-squared is obtained by calculating the
standard deviation of the residuals (i.e., the residual standard
error). This provides a measure of the typical size of a
residual (a summary of how well a model can predict the response
variable). Compute the standard deviation of the residuals for
mod1.
sd(resid(mod1))## [1] 0.5670923SD=0.5670923
(h) Now compute the R-squared value for mod1, and write
down an interpretation of it.
ssr1B=sum((fitted(mod1)-mean(fevdata$fev))^2)
ssr1B## [1] 280.9192sse1B=sum((fitted(mod1)-fevdata$fev)^2)
sse1B## [1] 210.0007sst1B=ssr1B+sse1B
sst1B## [1] 490.9198Rsquared1B=1-(ssr1B/sst1B)
Rsquared1B## [1] 0.4277698R2=0.4277698 this value is pretty low, meaning that the variance in the explanatory does not really explain the variance in the response.
(i) Now find the R-squared and the residual standard error for
mod2, mod3, and mod4. Which model
has the best residual standard error? The best R-squared?
"Rfunction(x)=function(){ssr=sum((fitted(x)-mean(fevdata$fev))^2)
sse=sum((fitted(x)-fevdata$fev)^2)
sst=ssr+sse
1-(ssr1B/sst1B)}"## [1] "Rfunction(x)=function(){ssr=sum((fitted(x)-mean(fevdata$fev))^2)\nsse=sum((fitted(x)-fevdata$fev)^2)\nsst=ssr+sse\n1-(ssr1B/sst1B)}"ssr1C=sum((fitted(mod2)-mean(fevdata$fev))^2)
ssr1C## [1] 29.56968sse1C=sum((fitted(mod2)-fevdata$fev)^2)
sse1C## [1] 461.3502sst1C=ssr1C+sse1C
sst1C## [1] 490.9198Rsquared1C=1-(ssr1C/sst1C)
Rsquared1C## [1] 0.9397668"print(Rfunction(mod2))
print(Rfunction(mod3))
print(Rfunction(mod4))"## [1] "print(Rfunction(mod2))\nprint(Rfunction(mod3))\nprint(Rfunction(mod4))"R2(mod2)=0.9397668 R2(mod3)=0.4234125 R2(mod4)=0.4059151
Question 9
The high_peaks data includes information on hiking
trails in the 46 “high peaks” in the Adirondack mountains of northern
New York state. Our goal will be to understand the variability in the
time in hours that it takes to complete each hike. In doing
so, we’ll separately consider four possible predictors:
ascent: a hike’s vertical ascent (feet)
elevation: highest elevation (feet)
length: length (miles)
rating: difficulty rating (easy / moderate / difficult)
Read in the data:
peaks <- read.csv('https://raw.githubusercontent.com/vittorioaddona/data/main/high_peaks.csv')(a) Construct a separate model of time by each predictor.
(b) Graph the data and the model for each of the 3 quantitative predictors.
(c) Look at the model coefficients for the categorical explanatory variable. Interpret each coefficient, and use this model to predict the time it takes to complete an easy hike. Then, graph the data for this categorical variable.
(d) Which of the 4 relationships appears to be the strongest? Or, which variable seems to be the best predictor of the time required to complete the hike?
(e) Compute the R-squared values for each of the 4 models. Does this agree with your intuition from (d)?