hw3
simple regression
non-linear function
p value
ggplot
Author

Felix Betanourt

Published

April 11, 2023

Code 
knitr::opts_chunk$set(echo = TRUE, warning = FALSE)

Homework 3

DACSS 603, Spring 2023

Code 
# Loading packages

suppressPackageStartupMessages(library(dplyr))
suppressPackageStartupMessages(library(tidyverse))
library(formattable)
suppressPackageStartupMessages(library(kableExtra))
library(ggplot2)
suppressPackageStartupMessages(library(alr4))

United Nations (Data file: UN11 in alr4) The data in the file UN11 contains several variables, including ppgdp, the gross national product per person in U.S. dollars, and fertility, the birth rate per 1000 females, both from the year 2009. The data are for 199 localities, mostly UN member countries, but also other areas such as Hong Kong that are not independent countries.The data were collected from the United Nations (2011). We will study the dependence of fertility on ppgdp.

Code 
str(UN11)
'data.frame':   199 obs. of  6 variables:
 $ region   : Factor w/ 8 levels "Africa","Asia",..: 2 4 1 1 3 5 2 3 8 4 ...
 $ group    : Factor w/ 3 levels "oecd","other",..: 2 2 3 3 2 2 2 2 1 1 ...
 $ fertility: num  5.97 1.52 2.14 5.13 2 ...
 $ ppgdp    : num  499 3677 4473 4322 13750 ...
 $ lifeExpF : num  49.5 80.4 75 53.2 81.1 ...
 $ pctUrban : num  23 53 67 59 100 93 64 47 89 68 ...
 - attr(*, "na.action")= 'omit' Named int [1:34] 4 5 8 28 41 67 68 72 79 83 ...
  ..- attr(*, "names")= chr [1:34] "Am Samoa" "Andorra" "Antigua and Barbuda" "Br Virigin Is" ...
  1. Identify the predictor and the response.

Predictor: gross national product per person (ppgdp) Response: fertility

  1. Draw the scatterplot of fertility on the vertical axis versus ppgdp on the horizontal axis and summarize the information in this graph. Does a straight-line mean function seem to be plausible for a summary of this graph?
Code 
ggplot(UN11, aes(x = ppgdp, y = fertility)) +
  geom_point()

Seems that the fertility rate varies a lot when the gross domestic product per person is between 0 and 12,500, but above that level the fertility rate seems more homogeneous. Doesn’t seem possible to have a straight line mean between these to variables, at least without making changes to the two distributions. The relationship seems non-linear.

  1. Draw the scatterplot of log(fertility) versus log(ppgdp) using natural logarithms. Does the simple linear regression model seem plausible for a summary of this graph? If you use a different base of logarithms, the shape of the graph won’t change, but the values on the axes will change.
Code 
plot(UN11$ppgdp, UN11$fertility, log = "xy", pch = 16, col = "blue", main = "Scatterplot with Log Axes for Ferlitity and GDP per person", 
     xlab = "GDP-PP (log scale)", ylab = "Fertility  (log scale)")

Now there is a visible negative relationship between Fertility and GDP per person. Fertility rate seems to decrease when the GDP is higher.

  1. Annual income, in dollars, is an explanatory variable in a regression analysis. For a British version of the report on the analysis, all responses are converted to British pounds sterling (1 pound equals about 1.33 dollars, as of 2016).
  1. How, if at all, does the slope of the prediction equation change?

Yes, the slope of the prediction equation will change. This is because the units of measurement for the explanatory variable have changed, which affects the scale and interpretation of the slope.

  1. How, if at all, does the correlation change?

The correlation should not change. Converting the unit of measurement does not change the strength and direction of the relation between two variables.

  1. Water runoff in the Sierras (Data file: water in alr4) Can Southern California’s water supply in future years be predicted from past data? One factor affecting water availability is stream runoff.

If runoff could be predicted, engineers, planners, and policy makers could do their jobs more efficiently. The data file contains 43 years’ worth of precipitation measurements taken at six sites in the Sierra Nevada mountains (labeled APMAM, APSAB, APSLAKE, OPBPC, OPRC, and OPSLAKE) and stream runoff volume at a site near Bishop, California, labeled BSAAM. Draw the scatterplot matrix for these data and summarize the information available from these plots. (Hint: Use the pairs() function.)

Code 
pairs(water[,1:8])

The water stream runoff in Bishop seems to have a positive correlation with the volume of precipitation in certain sites of the sierra nevada, specifically with OPSLAKE, OPRC and OPBPC. This means that while more precipitation in those sites, the water stream runoff in Bishop is higher.

  1. Professor ratings (Data file: Rateprof in alr4) In the website and online forum RateMyProfessors.com, students rate and comment on their instructors. Launched in 1999, the site includes millions of ratings on thousands of instructors. The data file includes the summaries of the ratings of 364 instructors at a large campus in the Midwest (Bleske-Rechek and Fritsch, 2011).

Each instructor included in the data had at least 10 ratings over a several year period. Students provided ratings of 1–5 on quality, helpfulness, clarity, easiness of instructor’s courses, and raterInterest in the subject matter covered in the instructor’s courses. The data file provides the averages of these five ratings. Create a scatterplot matrix of these five variables. Provide a brief description of the relationships between the five ratings.

Code 
str(Rateprof)
'data.frame':   366 obs. of  17 variables:
 $ gender         : Factor w/ 2 levels "female","male": 2 2 2 2 2 2 2 2 2 2 ...
 $ numYears       : int  7 6 10 11 11 10 7 11 11 7 ...
 $ numRaters      : int  11 11 43 24 19 15 17 16 12 18 ...
 $ numCourses     : int  5 5 2 5 7 9 3 3 4 4 ...
 $ pepper         : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...
 $ discipline     : Factor w/ 4 levels "Hum","SocSci",..: 1 1 1 1 1 1 1 1 1 1 ...
 $ dept           : Factor w/ 48 levels "Accounting","Anthropology",..: 17 42 3 17 45 45 45 17 34 17 ...
 $ quality        : num  4.64 4.32 4.79 4.25 4.68 ...
 $ helpfulness    : num  4.64 4.55 4.72 4.46 4.68 ...
 $ clarity        : num  4.64 4.09 4.86 4.04 4.68 ...
 $ easiness       : num  4.82 4.36 4.6 2.79 4.47 ...
 $ raterInterest  : num  3.55 4 3.43 3.18 4.21 ...
 $ sdQuality      : num  0.552 0.902 0.453 0.933 0.65 ...
 $ sdHelpfulness  : num  0.674 0.934 0.666 0.932 0.82 ...
 $ sdClarity      : num  0.505 0.944 0.413 0.999 0.582 ...
 $ sdEasiness     : num  0.405 0.505 0.541 0.588 0.612 ...
 $ sdRaterInterest: num  1.128 1.074 1.237 1.332 0.975 ...
Code 
subset_rate <- Rateprof[, c("quality", "helpfulness", "clarity", "easiness", "raterInterest")]

pairs(subset_rate)

Seems an strong positive relationship among clarity, helpfulness and quality, then easiness seems to have a moderate positive relationship with clarity, helpfulness and quality, and if we see raterInterest as a dependant variable, it shows some patter of low relationship with those 4 qualities.

  1. For the student.survey data file in the smss package, conduct regression analyses relating (by convention, y denotes the outcome variable, x denotes the explanatory variable)
  1. y = political ideology and x = religiosity, (ii) y = high school GPA and x = hours of TV watching. (You can use student.survey in the R console, after loading the package, to see what each variable means.)**
Code 
suppressPackageStartupMessages(library(smss))
data("student.survey")
student <- student.survey
str(student)
'data.frame':   60 obs. of  18 variables:
 $ subj: int  1 2 3 4 5 6 7 8 9 10 ...
 $ ge  : Factor w/ 2 levels "f","m": 2 1 1 1 2 2 2 1 2 2 ...
 $ ag  : int  32 23 27 35 23 39 24 31 34 28 ...
 $ hi  : num  2.2 2.1 3.3 3.5 3.1 3.5 3.6 3 3 4 ...
 $ co  : num  3.5 3.5 3 3.2 3.5 3.5 3.7 3 3 3.1 ...
 $ dh  : int  0 1200 1300 1500 1600 350 0 5000 5000 900 ...
 $ dr  : num  5 0.3 1.5 8 10 3 0.2 1.5 2 2 ...
 $ tv  : num  3 15 0 5 6 4 5 5 7 1 ...
 $ sp  : int  5 7 4 5 6 5 12 3 5 1 ...
 $ ne  : int  0 5 3 6 3 7 4 3 3 2 ...
 $ ah  : int  0 6 0 3 0 0 2 1 0 1 ...
 $ ve  : logi  FALSE FALSE FALSE FALSE FALSE FALSE ...
 $ pa  : Factor w/ 3 levels "d","i","r": 3 1 1 2 2 1 2 2 2 2 ...
 $ pi  : Ord.factor w/ 7 levels "very liberal"<..: 6 2 2 4 1 2 2 2 1 3 ...
 $ re  : Ord.factor w/ 4 levels "never"<"occasionally"<..: 3 2 3 2 1 2 2 2 2 1 ...
 $ ab  : logi  FALSE FALSE FALSE FALSE FALSE FALSE ...
 $ aa  : logi  FALSE FALSE FALSE FALSE FALSE FALSE ...
 $ ld  : logi  FALSE NA NA FALSE FALSE NA ...
  1. Graphically portray how the explanatory variable relates to the outcome variable in each of the two cases
Code 
ggplot(student, aes(x=re, y=pi)) + geom_point() +
  labs(title="Scatter Plot Religion and Political Ideology",
        x ="Frequency attending religious services", y = "Political ideology")

Seems that there a correlation between frequency of attending relegious services and political ideology. Specifically, seems that being conservative is related to attending more services.

Code 
ggplot(student, aes(x=tv, y=hi)) + geom_point() +
  labs(title="Scatter Plot Hours watching TV and High School GPA",
        x ="Average hours watching TV / week", y = "HS GPA Score")

In the case of GPA score in High school and average hours watching TV per week, it is difficult to see a clear relationshp, but we can say that seems that when the studnet watch TV 10 hours or less per week, there is no correlation with the GPA, but we can say that there is an light trend to get lower GPA when the student watch TV more than 10 hours per week.

  1. Summarize and interpret results of inferential analyses.
Code 
student <- student.survey %>%
 mutate(pi_num = case_when(
         pi == "very liberal" ~ 1,
         pi == "liberal" ~ 2,
         pi == "slightly liberal" ~ 3,
         pi == "moderate" ~ 4,
         pi == "slightly conservative" ~ 5,
         pi == "conservative" ~ 6,
         pi == "very conservative" ~ 7,
         ))

student$pi_num <- as.numeric(student$pi_num)


model1 <- lm(pi_num ~ re, data=student)
summary(model1)

Call:
lm(formula = pi_num ~ re, data = student)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.8889 -0.5172 -0.2667  1.2040  2.7333 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   3.5253     0.1958  18.000  < 2e-16 ***
re.L          2.1864     0.3919   5.579 7.27e-07 ***
re.Q          0.1049     0.3917   0.268    0.790    
re.C         -0.6958     0.3915  -1.777    0.081 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.315 on 56 degrees of freedom
Multiple R-squared:  0.3872,    Adjusted R-squared:  0.3544 
F-statistic:  11.8 on 3 and 56 DF,  p-value: 4.282e-06

When we see the rregression output, we can see that seems there is a significant incidence from frequency to attend religious services and political ideology, for instance, seems that attending to religious services more frequently is associated to being more conservative.

Code 
model2 <- lm(hi ~ tv, data=student)
summary(model2)

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

Residuals:
    Min      1Q  Median      3Q     Max 
-1.2583 -0.2456  0.0417  0.3368  0.7051 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.441353   0.085345  40.323   <2e-16 ***
tv          -0.018305   0.008658  -2.114   0.0388 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4467 on 58 degrees of freedom
Multiple R-squared:  0.07156,   Adjusted R-squared:  0.05555 
F-statistic: 4.471 on 1 and 58 DF,  p-value: 0.03879

Seems that there is a negative relationship between average TV hours per week incidence in the GPA score, where as more TV hours watching TV is related to lower GPA score, specifically for every additional hour of TV per week the score can get lower in 3 points. And even thought the relationship is significant at 0.05, still that incidence from TV hours to score seems weak or low. I would elminate 2-3 outliners (the cases with more than 20 hours per week) and look again at the regression output, but I can anticipate non-significant incidence from hours watching TV.

Also, important to note that R squared is very low, so TV hours explain very little of the GPA score.