Description
Data on US cancer mortality rates for over 3000 counties are available in the dataset
cancer_reg.csv available on Blackboard. The data were obtained from the Data World website (https:
//data.world/nrippner/ols-regression-challenge). Read the data set into R and use it to answer the questions
that follow. We’ll use the subset of variables listed below:
- incidencerate: Mean per capita (100,000) cancer diagnoses1
- medincome: Median annual income (dollars) per county (2
- povertypercent: Percent of county population in poverty2
- studypercap: Per capita number of cancer-related clinical trials per county1
- medianage: Median age (in years) of county residents2
- pctunemployed16_over: Percent of county residents aged 16 and over that are unemployed2
- pctprivatecoverage: Percent of county residents with private health coverage2
- pctbachdeg25_over: Percent of county residents aged 25 and over with bachelor’s degree as highest
education attained2
- target_deathrate: Response variable. Mean per capita (100,000) cancer mortalities1
1 Years 2010-2016 2 2013 Census Estimates
- Create a new dataset called cancer2 that contains only the subset of variables listed above.
Based on a summary of the variables in the dataset and the plots below, identify any variable or
variables that have obviously incorrect values. For the variables you identify, write and implement code
to fifilter out the incorrect values. Give the number of observations left in the dataset.
100
200
300
250
500
750 1000 1250
Mean cancer diagnoses
per 100,000
100
200
300
250005000075000100000125000
Median income per county
100
200
300
10
20
30
40
Percent of population
in poverty
100
200
300
0
2500 5000 7500 10000
Number of cancer−related
clinical trials per county
0
100
200
300
0
200
400
600
Median age of county
100
200
300
0
10
20
30
% aged 16 and over
who are unemployed
100
200
300
20
40
60
80
% with private
health coverage
100
200
300
10
20
30
40
% aged 25 and over with
Bachelor’s degree as highest qualification
- Some data cleaning is done on cancer2 and a new dataset cancer3.csv (available on
Blackboard) is created. Construct a scatterplot matrix of all variables in the new dataset. List any
key points of note from the scatterplot matrix, including any considerations you might make during a
regression analysis.
2
Mortality
Mortality
Mortality
Mortality
Mortality
Mortality
Mortality
Mortalityc. Fit a linear model to the data in cancer3, including all predictors with no transformations
or interactions. Present a summary of the model in a table. Give an estimate of σ2 , the error variance.
- Suppose two counties diffffer by 1 per 100,000 in mean cancer diagnoses with all else being
equal. Based on the model fifitted in part (c), what is the difffference in expected cancer mortality for
these two counties?
- Does it make practical sense to interpret the intercept for the model in part (c)? Justify
your answer.
- The model fifitted in part (c) is to be used to predict cancer mortality for a county with
the predictor values below. Obtain 95% confifidence and prediction intervals for such a county. Explain
brieflfly why the prediction interval is wider than the confifidence interval.
- incidencerate: 452
- medincome: 23000
- povertypercent: 16
- studypercap: 150
- medianage: 40
- pctunemployed16_over: 8
- pctprivatecoverage: 70
- pctbachdeg25_over: 50
- Assuming all regression assumptions hold, are the intervals you obtained in part (f) likely
to be valid? Explain your answer brieflfly.
- Based on a global usefulness test, is it worth going on to further analyse and interpret a
model of target_deathrate against each of the predictors? Carry out the test, give the conclusion
and justify your answer.
- The plots below are constructed from the cleaned dataset cancer3. Which predictors, if
any, would you consider applying log or polynomial transformations to? Explain your answer brieflfly.
100
200
300
250
500
750 1000 1250
Mean cancer diagnoses
per 100,000
100
200
300
250005000075000100000125000
Median income per county
100
200
300
10
20
30
40
Percent of population
in poverty
100
200
300
0
2500 5000 7500 10000
Number of cancer−related
clinical trials per county
100
200
300
30
40
50
60
Median age of county
100
200
300
0
10
20
30
% aged 16 and over
who are unemployed
100
200
300
20
40
60
80
% with private
health coverage
100
200
300
10
20
30
40
% aged 25 and over with
Bachelor’s degree as highest qualification
3
Mortality
Mortality
Mortality
Mortality
Mortality
Mortality
Mortality
MortalityFrancis Galton’s 1866 dataset (cleaned) lists individual observations on height for 899
children. Galton coined the term “regression” following his study of how children’s heights related to heights
of their parents. The data are available in the fifile galton.csv and contain the following variables:
- familyID: Family ID
- father: Height of father
- mother: Height of mother
- gender: gender of child
- height: Height of child
- kids: Number of childre in family
- midparent: Mid-parent height calculated as (‘father + 1.08*mother)/2
- adltchld: height if gender=M, otherwise 1.08*height if gender= F
All heights are measured in inches.
- Read the data into R and fifit a linear model for height with the variables father, mother,
gender, kids and midparent as predictors. Provide a summary of the fifitted model. You will notice
that estimates for midparent are listed as NA. Why might this be the case and what regression problem
does this point to?
- What action might you take to resolve the problem identifified in part (a)?
- Based on the model fifitted in part (a) give an interpretation of the coeffiffifficient for genderM.
- Determine the number of families in the dataset.
- The problem in part (a) is resolved and a new linear model is fifitted.No observations are
excluded. The plots below are obtained to investigate regression assumptions for this new model. Based
on your answer in part (d) and the plots below, do the data meet all the regression assumptions?
Explain your answer brieflfly.
62
64
66
68
70
72
74
Fitted values
Residuals vs Fitted
479
289
60
−3
−2
−1
0
1
2
3
Theoretical Quantiles
Normal Q−Q
479
289
60
62
64
66
68
70
72
74
Fitted values
Scale−Location
479
60289
0.000
0.005
0.010
0.015
0.020
Leverage
Cook’s distance
Residuals vs Leverage
815
60
126
4
−10 0
10
Residuals
−4
0
4
Standardized residuals
0.0 1.0 2.0
Standardized residuals
−4 0
4
Standardized residuals
