Homework 4

hw4

Template of course blog qmd file

Author

Xiaoyan

Published

April 17, 2023

Code

library(tidyr)
library(dplyr)


Attaching package: 'dplyr'

The following objects are masked from 'package:stats':

    filter, lag

The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union

Code

library(readxl)
library(ggplot2)
library(alr4)

Loading required package: car

Loading required package: carData


Attaching package: 'car'

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

    recode

Loading required package: effects

lattice theme set by effectsTheme()
See ?effectsTheme for details.

Question 1

For recent data in Jacksonville, Florida, on y = selling price of home (in dollars), x1 = size of home (in square feet), and x2 = lot size (in square feet), the prediction equation is ŷ = −10,536 + 53.8x1 + 2.84x2.

A. A particular home of 1240 square feet on a lot of 18,000 square feet sold for $145,000. Find the predicted selling price and the residual, and interpret.

Code

(-10536) + 53.8*1240 + 2.84*18000

[1] 107296

B. For fixed lot size, how much is the house selling price predicted to increase for each square- foot increase in home size? Why?

53.8

C. According to this prediction equation, for fixed home size, how much would lot size need to increase to have the same impact as a one-square-foot increase in home size?

Code

53.8 / 2.84

[1] 18.94366

Question 2

(Data file: salary in alr4 R package). The data file concerns salary and other characteristics of all faculty in a small Midwestern college collected in the early 1980s for presentation in legal proceedings for which discrimination against women in salary was at issue. All persons in the data hold tenured or tenure track positions; temporary faculty are not included. The variables include degree, a factor with levels PhD and MS; rank, a factor with levels Asst, Assoc, and Prof; sex, a factor with levels Male and Female; Year, years in current rank; ysdeg, years since highest degree, and salary, academic year salary in dollars.

A. Test the hypothesis that the mean salary for men and women is the same, without regard to any other variable but sex. Explain your findings.

Code

head(salary)

   degree rank    sex year ysdeg salary
1 Masters Prof   Male   25    35  36350
2 Masters Prof   Male   13    22  35350
3 Masters Prof   Male   10    23  28200
4 Masters Prof Female    7    27  26775
5     PhD Prof   Male   19    30  33696
6 Masters Prof   Male   16    21  28516

Code

unique(salary$rank)

[1] Prof  Assoc Asst 
Levels: Asst Assoc Prof

Code

summary(lm(salary~sex, data=salary))


Call:
lm(formula = salary ~ sex, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-8602.8 -4296.6  -100.8  3513.1 16687.9 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    24697        938  26.330   <2e-16 ***
sexFemale      -3340       1808  -1.847   0.0706 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5782 on 50 degrees of freedom
Multiple R-squared:  0.0639,    Adjusted R-squared:  0.04518 
F-statistic: 3.413 on 1 and 50 DF,  p-value: 0.0706

B. Run a multiple linear regression with salary as the outcome variable and everything else as predictors, including sex. Assuming no interactions between sex and the other predictors, obtain a 95% confidence interval for the difference in salary between males and females.

Code

lm(salary~degree+rank+sex+year+ysdeg, data=salary)|>
  confint()

                 2.5 %      97.5 %
(Intercept) 14134.4059 17357.68946
degreePhD    -663.2482  3440.47485
rankAssoc    2985.4107  7599.31080
rankProf     8396.1546 13841.37340
sexFemale    -697.8183  3030.56452
year          285.1433   667.47476
ysdeg        -280.6397    31.49105

C. Interpret your finding for each predictor variable; discuss (a) statistical significance,

Code

summary(lm(salary~degree+rank+sex+year+ysdeg, data=salary))


Call:
lm(formula = salary ~ degree + rank + sex + year + ysdeg, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-4045.2 -1094.7  -361.5   813.2  9193.1 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 15746.05     800.18  19.678  < 2e-16 ***
degreePhD    1388.61    1018.75   1.363    0.180    
rankAssoc    5292.36    1145.40   4.621 3.22e-05 ***
rankProf    11118.76    1351.77   8.225 1.62e-10 ***
sexFemale    1166.37     925.57   1.260    0.214    
year          476.31      94.91   5.018 8.65e-06 ***
ysdeg        -124.57      77.49  -1.608    0.115    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2398 on 45 degrees of freedom
Multiple R-squared:  0.855, Adjusted R-squared:  0.8357 
F-statistic: 44.24 on 6 and 45 DF,  p-value: < 2.2e-16

interpretation of the coefficient / slope in relation to the outcome variable and other variables

degree, sex, and ysdeg are not statistically significant at this situation. Associate professor make 5292 more than asisstant professor and professor makes 11118 more than assitant professor. As year is a continues varible, one year increase makes 476 more salary

D. Change the baseline category for the rank variable. Interpret the coefficients related to rank again.

Based on this analysis, the assitant professor makes 11118 less then full professor and associate professor make 5826 less than full professor.

Code

salary$rank<-relevel(salary$rank, ref = "Prof")
summary(lm(salary~., data = salary))


Call:
lm(formula = salary ~ ., data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-4045.2 -1094.7  -361.5   813.2  9193.1 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  26864.81    1375.29  19.534  < 2e-16 ***
degreePhD     1388.61    1018.75   1.363    0.180    
rankAsst    -11118.76    1351.77  -8.225 1.62e-10 ***
rankAssoc    -5826.40    1012.93  -5.752 7.28e-07 ***
sexFemale     1166.37     925.57   1.260    0.214    
year           476.31      94.91   5.018 8.65e-06 ***
ysdeg         -124.57      77.49  -1.608    0.115    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2398 on 45 degrees of freedom
Multiple R-squared:  0.855, Adjusted R-squared:  0.8357 
F-statistic: 44.24 on 6 and 45 DF,  p-value: < 2.2e-16

E. Finkelstein (1980), in a discussion of the use of regression in discrimination cases, wrote, “[a] variable may reflect a position or status bestowed by the employer, in which case if there is discrimination in the award of the position or status, the variable may be ‘tainted.’” Thus, for example, if discrimination is at work in promotion of faculty to higher ranks, using rank to adjust salaries before comparing the sexes may not be acceptable to the courts. Exclude the variable rank, refit, and summarize how your findings changed, if they did.

Code

summary(lm(salary~degree+sex+year+ysdeg, data=salary))


Call:
lm(formula = salary ~ degree + sex + year + ysdeg, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-8146.9 -2186.9  -491.5  2279.1 11186.6 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 17183.57    1147.94  14.969  < 2e-16 ***
degreePhD   -3299.35    1302.52  -2.533 0.014704 *  
sexFemale   -1286.54    1313.09  -0.980 0.332209    
year          351.97     142.48   2.470 0.017185 *  
ysdeg         339.40      80.62   4.210 0.000114 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3744 on 47 degrees of freedom
Multiple R-squared:  0.6312,    Adjusted R-squared:  0.5998 
F-statistic: 20.11 on 4 and 47 DF,  p-value: 1.048e-09

F. Everyone in this dataset was hired the year they earned their highest degree. It is also known that a new Dean was appointed 15 years ago, and everyone in the dataset who earned their highest degree 15 years ago or less than that has been hired by the new Dean. Some people have argued that the new Dean has been making offers that are a lot more generous to newly hired faculty than the previous one and that this might explain some of the variation in Salary. Create a new variable that would allow you to test this hypothesis and run another multiple regression model to test this. Select variables carefully to make sure there is no multicollinearity. Explain why multicollinearity would be a concern in this case and how you avoided it. Do you find support for the hypothesis that the people hired by the new Dean are making higher than those that were not?

Code

salary$newyear<-ifelse(salary$year <= 20,10,0)
cor.test(salary$newyear, salary$year)


    Pearson's product-moment correlation

data:  salary$newyear and salary$year
t = -3.5608, df = 50, p-value = 0.0008223
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.6436643 -0.2016149
sample estimates:
       cor 
-0.4497686

Code

summary(lm(salary~degree+sex+newyear+ysdeg, data=salary))


Call:
lm(formula = salary ~ degree + sex + newyear + ysdeg, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-8248.6 -2611.6  -786.1  2910.8 11147.9 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 19996.62    4605.45   4.342 7.47e-05 ***
degreePhD   -4087.81    1375.85  -2.971  0.00466 ** 
sexFemale   -2693.61    1251.85  -2.152  0.03659 *  
newyear      -161.96     432.94  -0.374  0.71001    
ysdeg         467.24      66.54   7.022 7.57e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3973 on 47 degrees of freedom
Multiple R-squared:  0.5845,    Adjusted R-squared:  0.5491 
F-statistic: 16.53 on 4 and 47 DF,  p-value: 1.602e-08

Code

summary(lm(salary~degree+sex+newyear+ysdeg+year, data=salary))


Call:
lm(formula = salary ~ degree + sex + newyear + ysdeg + year, 
    data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-8314.2 -2146.3  -222.6  2240.8 11044.4 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 15012.81    4821.50   3.114 0.003175 ** 
degreePhD   -3411.96    1335.79  -2.554 0.014017 *  
sexFemale   -1233.85    1329.06  -0.928 0.358065    
newyear       202.79     437.23   0.464 0.644984    
ysdeg         341.08      81.38   4.191 0.000125 ***
year          375.97     152.72   2.462 0.017633 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3775 on 46 degrees of freedom
Multiple R-squared:  0.6329,    Adjusted R-squared:  0.593 
F-statistic: 15.86 on 5 and 46 DF,  p-value: 4.589e-09

Question 3

(Data file: house.selling.price in smss R package)

A. Using the house.selling.price data, run and report regression results modeling y = selling price (in dollars) in terms of size of home (in square feet) and whether the home is new (1 = yes; 0 = no). In particular, for each variable; discuss statistical significance and interpret the meaning of the coefficient.

B. Report and interpret the prediction equation, and form separate equations relating selling price to size for new and for not new homes.

C. Find the predicted selling price for a home of 3000 square feet that is (i) new, (ii) not new.

D. Fit another model, this time with an interaction term allowing interaction between size and new, and report the regression results

E. Report the lines relating the predicted selling price to the size for homes that are (i) new, (ii) not new.

F. Find the predicted selling price for a home of 3000 square feet that is (i) new, (ii) not new.

G. Find the predicted selling price for a home of 1500 square feet that is (i) new, (ii) not new. Comparing to (F), explain how the difference in predicted selling prices changes as the size of home increases.

H. Do you think the model with interaction or the one without it represents the relationship of size and new to the outcome price? What makes you prefer one model over another?