Week 7 Discussion
Kevin Clifford
3/17/2021
Using the “swiss” dataset, conduct a regression of two variables of interest. Interpret the assumptions and results. Post your solutions and R code here.
library(stats)
library(datasets)
library(psych)
library(kableExtra)
require(graphics)
##Swiss Data
swiss## Fertility Agriculture Examination Education Catholic
## Courtelary 80.2 17.0 15 12 9.96
## Delemont 83.1 45.1 6 9 84.84
## Franches-Mnt 92.5 39.7 5 5 93.40
## Moutier 85.8 36.5 12 7 33.77
## Neuveville 76.9 43.5 17 15 5.16
## Porrentruy 76.1 35.3 9 7 90.57
## Broye 83.8 70.2 16 7 92.85
## Glane 92.4 67.8 14 8 97.16
## Gruyere 82.4 53.3 12 7 97.67
## Sarine 82.9 45.2 16 13 91.38
## Veveyse 87.1 64.5 14 6 98.61
## Aigle 64.1 62.0 21 12 8.52
## Aubonne 66.9 67.5 14 7 2.27
## Avenches 68.9 60.7 19 12 4.43
## Cossonay 61.7 69.3 22 5 2.82
## Echallens 68.3 72.6 18 2 24.20
## Grandson 71.7 34.0 17 8 3.30
## Lausanne 55.7 19.4 26 28 12.11
## La Vallee 54.3 15.2 31 20 2.15
## Lavaux 65.1 73.0 19 9 2.84
## Morges 65.5 59.8 22 10 5.23
## Moudon 65.0 55.1 14 3 4.52
## Nyone 56.6 50.9 22 12 15.14
## Orbe 57.4 54.1 20 6 4.20
## Oron 72.5 71.2 12 1 2.40
## Payerne 74.2 58.1 14 8 5.23
## Paysd'enhaut 72.0 63.5 6 3 2.56
## Rolle 60.5 60.8 16 10 7.72
## Vevey 58.3 26.8 25 19 18.46
## Yverdon 65.4 49.5 15 8 6.10
## Conthey 75.5 85.9 3 2 99.71
## Entremont 69.3 84.9 7 6 99.68
## Herens 77.3 89.7 5 2 100.00
## Martigwy 70.5 78.2 12 6 98.96
## Monthey 79.4 64.9 7 3 98.22
## St Maurice 65.0 75.9 9 9 99.06
## Sierre 92.2 84.6 3 3 99.46
## Sion 79.3 63.1 13 13 96.83
## Boudry 70.4 38.4 26 12 5.62
## La Chauxdfnd 65.7 7.7 29 11 13.79
## Le Locle 72.7 16.7 22 13 11.22
## Neuchatel 64.4 17.6 35 32 16.92
## Val de Ruz 77.6 37.6 15 7 4.97
## ValdeTravers 67.6 18.7 25 7 8.65
## V. De Geneve 35.0 1.2 37 53 42.34
## Rive Droite 44.7 46.6 16 29 50.43
## Rive Gauche 42.8 27.7 22 29 58.33
## Infant.Mortality
## Courtelary 22.2
## Delemont 22.2
## Franches-Mnt 20.2
## Moutier 20.3
## Neuveville 20.6
## Porrentruy 26.6
## Broye 23.6
## Glane 24.9
## Gruyere 21.0
## Sarine 24.4
## Veveyse 24.5
## Aigle 16.5
## Aubonne 19.1
## Avenches 22.7
## Cossonay 18.7
## Echallens 21.2
## Grandson 20.0
## Lausanne 20.2
## La Vallee 10.8
## Lavaux 20.0
## Morges 18.0
## Moudon 22.4
## Nyone 16.7
## Orbe 15.3
## Oron 21.0
## Payerne 23.8
## Paysd'enhaut 18.0
## Rolle 16.3
## Vevey 20.9
## Yverdon 22.5
## Conthey 15.1
## Entremont 19.8
## Herens 18.3
## Martigwy 19.4
## Monthey 20.2
## St Maurice 17.8
## Sierre 16.3
## Sion 18.1
## Boudry 20.3
## La Chauxdfnd 20.5
## Le Locle 18.9
## Neuchatel 23.0
## Val de Ruz 20.0
## ValdeTravers 19.5
## V. De Geneve 18.0
## Rive Droite 18.2
## Rive Gauche 19.3str(swiss)## 'data.frame': 47 obs. of 6 variables:
## $ Fertility : num 80.2 83.1 92.5 85.8 76.9 76.1 83.8 92.4 82.4 82.9 ...
## $ Agriculture : num 17 45.1 39.7 36.5 43.5 35.3 70.2 67.8 53.3 45.2 ...
## $ Examination : int 15 6 5 12 17 9 16 14 12 16 ...
## $ Education : int 12 9 5 7 15 7 7 8 7 13 ...
## $ Catholic : num 9.96 84.84 93.4 33.77 5.16 ...
## $ Infant.Mortality: num 22.2 22.2 20.2 20.3 20.6 26.6 23.6 24.9 21 24.4 ...mydescribe <-round(describe(swiss),3)
mydescribe%>%kbl()%>%kable_classic(html_font = "Courier New")| vars | n | mean | sd | median | trimmed | mad | min | max | range | skew | kurtosis | se | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Fertility | 1 | 47 | 70.143 | 12.492 | 70.40 | 70.659 | 10.230 | 35.00 | 92.5 | 57.50 | -0.456 | 0.260 | 1.822 |
| Agriculture | 2 | 47 | 50.660 | 22.711 | 54.10 | 51.156 | 23.870 | 1.20 | 89.7 | 88.50 | -0.320 | -0.886 | 3.313 |
| Examination | 3 | 47 | 16.489 | 7.978 | 16.00 | 16.077 | 7.413 | 3.00 | 37.0 | 34.00 | 0.446 | -0.137 | 1.164 |
| Education | 4 | 47 | 10.979 | 9.615 | 8.00 | 9.385 | 5.930 | 1.00 | 53.0 | 52.00 | 2.268 | 6.140 | 1.403 |
| Catholic | 5 | 47 | 41.144 | 41.705 | 15.14 | 39.116 | 18.651 | 2.15 | 100.0 | 97.85 | 0.479 | -1.665 | 6.083 |
| Infant.Mortality | 6 | 47 | 19.943 | 2.913 | 20.00 | 19.985 | 2.817 | 10.80 | 26.6 | 15.80 | -0.331 | 0.777 | 0.425 |
##Correlation
cor(swiss)## Fertility Agriculture Examination Education Catholic
## Fertility 1.0000000 0.35307918 -0.6458827 -0.66378886 0.4636847
## Agriculture 0.3530792 1.00000000 -0.6865422 -0.63952252 0.4010951
## Examination -0.6458827 -0.68654221 1.0000000 0.69841530 -0.5727418
## Education -0.6637889 -0.63952252 0.6984153 1.00000000 -0.1538589
## Catholic 0.4636847 0.40109505 -0.5727418 -0.15385892 1.0000000
## Infant.Mortality 0.4165560 -0.06085861 -0.1140216 -0.09932185 0.1754959
## Infant.Mortality
## Fertility 0.41655603
## Agriculture -0.06085861
## Examination -0.11402160
## Education -0.09932185
## Catholic 0.17549591
## Infant.Mortality 1.00000000##Picking Fertility and Education
cor(swiss$Fertility, swiss$Education)## [1] -0.6637889#Based on this there is a strong negative correlation between the two variables.
hist(swiss$Fertility)
#Normally distributed.
hist(swiss$Education)
#Skewed to the right.
##Regression
myreg <- lm(swiss$Fertility~swiss$Education)
summary(myreg)##
## Call:
## lm(formula = swiss$Fertility ~ swiss$Education)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.036 -6.711 -1.011 9.526 19.689
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 79.6101 2.1041 37.836 < 2e-16 ***
## swiss$Education -0.8624 0.1448 -5.954 3.66e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.446 on 45 degrees of freedom
## Multiple R-squared: 0.4406, Adjusted R-squared: 0.4282
## F-statistic: 35.45 on 1 and 45 DF, p-value: 3.659e-07#The intercept is 79.6101 which indicates that when the percent of those who have education beyond primary school for draftees is equal to zero, the fertility equals 79.6101. The coefficient for Education is -0.8624 which is the slope for the regression line and that means when the education beyond primary school for draftees increases by one percentage point, Fertility decreases by 0.8624. The R-squared being 0.4406 shows us that 44.06% of Fertility's variation can be explained by Education.
##ANOVA check
aov <- anova(myreg)
aov## Analysis of Variance Table
##
## Response: swiss$Fertility
## Df Sum Sq Mean Sq F value Pr(>F)
## swiss$Education 1 3162.7 3162.7 35.446 3.659e-07 ***
## Residuals 45 4015.2 89.2
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1sqrt(89.2)## [1] 9.444575##Plots
#Residuals
plot(myreg)
hist(myreg$residuals)
mean(myreg$residuals)## [1] -7.176588e-16#Mean indicates that they are centered at close to zero. The plot seems to show an outlier that may skew the results a bit. The histogram shows that the residuals are normally distributed, but definitely not perfectly.
shapiro.test(myreg$residuals)##
## Shapiro-Wilk normality test
##
## data: myreg$residuals
## W = 0.95346, p-value = 0.05922#We cannot reject the null hypothesis that the residuals are normally distributed although it is close.
#Model
plot(swiss$Fertility, swiss$Education)
abline(myreg)
#The plot shows the negative correlation between the two variables.