Homework 3

Question 1

First, let’s load the relevant libraries and set all the graph themes to minimal.

Code

library(readxl)
library(tidyverse)

── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
✔ ggplot2 3.3.6      ✔ purrr   0.3.4 
✔ tibble  3.1.8      ✔ dplyr   1.0.10
✔ tidyr   1.2.1      ✔ stringr 1.4.1 
✔ readr   2.1.3      ✔ forcats 0.5.2 
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()

Code

library(dplyr)
library(alr4)

Loading required package: car
Loading required package: carData

Attaching package: 'car'

The following object is masked from 'package:dplyr':

    recode

The following object is masked from 'package:purrr':

    some

Loading required package: effects
lattice theme set by effectsTheme()
See ?effectsTheme for details.

Code

library(smss)
theme_minimal()

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1.1.1

For Question 1, let’s load in the UN11 data-set and view the first few variables to verify we have loaded in the data.

Code

data(UN11)
head(UN11)

                region  group fertility   ppgdp lifeExpF pctUrban
Afghanistan       Asia  other     5.968   499.0    49.49       23
Albania         Europe  other     1.525  3677.2    80.40       53
Algeria         Africa africa     2.142  4473.0    75.00       67
Angola          Africa africa     5.135  4321.9    53.17       59
Anguilla     Caribbean  other     2.000 13750.1    81.10      100
Argentina   Latin Amer  other     2.172  9162.1    79.89       93

For the sake of the following two subsections, the predictor is “ppgdp” and the response is “fertility.” We are attempting to predict fertility but a country’s PPGDP.

1.1.2

Code

ggplot(UN11, aes(ppgdp, fertility)) +
  geom_point()

A straight-line mean function does seem possible if you remove ppgdp that is less than $5,000. Otherwise, there is a sharp concentration of points at this range that might distort a straight-line mean.

1.1.3

Code

ggplot(UN11, aes(log(ppgdp), log(fertility))) +
  geom_point() + 
  stat_smooth(method = "lm",
              formula = y ~ x,
              se = FALSE)

The simple linear regression would be much more plausible with this graph. We now see a negative association between fertility and ppgdp.The higher the fertility, the lower the ppgdp, and vice-versa.

Question 2

A

Code

ggplot(UN11, aes(log(ppgdp*1.33), log(fertility))) +
  geom_point() +
  stat_smooth(method = "lm",
              formula = y ~ x,
              se = FALSE)

I don’t think the slope has changed, but we can call a quick lm to see.

Code

lm(log(ppgdp) ~ log(fertility), UN11)

Call:
lm(formula = log(ppgdp) ~ log(fertility), data = UN11)

Coefficients:
   (Intercept)  log(fertility)  
        10.780          -2.539  

Code

lm(log(ppgdp*1.33) ~ log(fertility), UN11)

Call:
lm(formula = log(ppgdp * 1.33) ~ log(fertility), data = UN11)

Coefficients:
   (Intercept)  log(fertility)  
        11.065          -2.539  

A little for the intercept, but not when mapped onto the outcome variable (ppgdp does not change its effect on fertility when currency changes).

B

Therefore, the correlation does not change when adjusted from US dollars to British pounds.

Question 3

Let’s load in the water data-set.

Code

data(water)
head(water)

  Year APMAM APSAB APSLAKE OPBPC  OPRC OPSLAKE  BSAAM
1 1948  9.13  3.58    3.91  4.10  7.43    6.47  54235
2 1949  5.28  4.82    5.20  7.55 11.11   10.26  67567
3 1950  4.20  3.77    3.67  9.52 12.20   11.35  66161
4 1951  4.60  4.46    3.93 11.14 15.15   11.13  68094
5 1952  7.15  4.99    4.88 16.34 20.05   22.81 107080
6 1953  9.70  5.65    4.91  8.88  8.15    7.41  67594

Code

pairs(water)

As one can see on the last column, BSAAM, there are three highly positive correlations: OPBPC, OPRC, and OPSLAKE. The other four columns have a positive relationship, but nowhere near as correlated as these three. Indeed, from APSLAKE back, each variable becomes less and less correlated to BSAAM when moving from right to left.

Question 4

Code

data(Rateprof)
head(Rateprof)

  gender numYears numRaters numCourses pepper discipline              dept
1   male        7        11          5     no        Hum           English
2   male        6        11          5     no        Hum Religious Studies
3   male       10        43          2     no        Hum               Art
4   male       11        24          5     no        Hum           English
5   male       11        19          7     no        Hum           Spanish
6   male       10        15          9     no        Hum           Spanish
   quality helpfulness  clarity easiness raterInterest sdQuality sdHelpfulness
1 4.636364    4.636364 4.636364 4.818182      3.545455 0.5518564     0.6741999
2 4.318182    4.545455 4.090909 4.363636      4.000000 0.9020179     0.9341987
3 4.790698    4.720930 4.860465 4.604651      3.432432 0.4529343     0.6663898
4 4.250000    4.458333 4.041667 2.791667      3.181818 0.9325048     0.9315329
5 4.684211    4.684211 4.684211 4.473684      4.214286 0.6500112     0.8200699
6 4.233333    4.266667 4.200000 4.533333      3.916667 0.8632717     1.0327956
  sdClarity sdEasiness sdRaterInterest
1 0.5045250  0.4045199       1.1281521
2 0.9438798  0.5045250       1.0744356
3 0.4129681  0.5407021       1.2369438
4 0.9990938  0.5882300       1.3322506
5 0.5823927  0.6117753       0.9749613
6 0.7745967  0.6399405       0.6685579

Code

Rateprof_new <- Rateprof %>%
  select(quality,
         helpfulness,
         clarity,
         easiness,
         raterInterest)
pairs(Rateprof_new)

Quality is highly correlated with helpfulness and clarity. Less so but still positive with easiness and raterInterest.

Question 5

A

Code

data(student.survey)
head(student.survey)

  subj ge ag  hi  co   dh   dr tv sp ne ah    ve pa           pi           re
1    1  m 32 2.2 3.5    0  5.0  3  5  0  0 FALSE  r conservative   most weeks
2    2  f 23 2.1 3.5 1200  0.3 15  7  5  6 FALSE  d      liberal occasionally
3    3  f 27 3.3 3.0 1300  1.5  0  4  3  0 FALSE  d      liberal   most weeks
4    4  f 35 3.5 3.2 1500  8.0  5  5  6  3 FALSE  i     moderate occasionally
5    5  m 23 3.1 3.5 1600 10.0  6  6  3  0 FALSE  i very liberal        never
6    6  m 39 3.5 3.5  350  3.0  4  5  7  0 FALSE  d      liberal occasionally
     ab    aa    ld
1 FALSE FALSE FALSE
2 FALSE FALSE    NA
3 FALSE FALSE    NA
4 FALSE FALSE FALSE
5 FALSE FALSE FALSE
6 FALSE FALSE    NA

Code

ggplot(student.survey, aes(re, pi)) +
  geom_point() +
  stat_smooth(method="lm", se=FALSE)

`geom_smooth()` using formula 'y ~ x'

Code

ggplot(student.survey, aes(tv, hi)) +
  geom_point() +
  stat_smooth(method="lm", se=FALSE)

`geom_smooth()` using formula 'y ~ x'

Due to the categorical nature for (i), it is very difficult to draw conclusions from this graph on how the explanatory variable relates to the outcome variable. There is a hint of a positive relationship, but that is just how the graph is coded. If political ideology was flipped (liberal to conservative from top to bottom), we would see an inverse relationship. Conservatives seem more likely to attend every week, while liberals never do.

For (ii), we see that as an individual observes more tv per week, the lower the high school GPA is. If we removed the outliers, we might see a regression line where the explanatory variable explains more for the outcome variable, but that is not within the scope of this homework, nor is it immediately a best practice.

B

Code

lm(re ~ pi, student.survey)

Warning in model.response(mf, "numeric"): using type = "numeric" with a factor
response will be ignored

Warning in Ops.ordered(y, z$residuals): '-' is not meaningful for ordered
factors

Call:
lm(formula = re ~ pi, data = student.survey)

Coefficients:
(Intercept)         pi.L         pi.Q         pi.C         pi^4         pi^5  
     2.6071       2.0552       0.4501       0.1361      -0.2283      -0.3046  
       pi^6  
     0.4565  

Code

lm (tv ~ hi, student.survey)

Call:
lm(formula = tv ~ hi, data = student.survey)

Coefficients:
(Intercept)           hi  
     20.200       -3.909  

The LM models are a little difficult to interpret, but I shall attempt to explain.

For (i), the outcome variable (pi), is explained by the intercept codes. So a change in re (religiosity), accounts for varying differences for pi (political ideology). This can be a massive change, such as 2.05 for L or -0.30 for pi^5. I attempted to switch the variables in the LM model, as I was unsure if this was correct, and the issue persisted due to the categorical nature of the simple linear regression. Either I am still completing this wrong or the model is incorrect for the data provided - better a logistic regression perhaps.

For (ii), one hour of TV on average explains a change of -3.91 in high school GPA. This shows a strong negative association between the two variables.