Last updated: 2019-10-01

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Knit directory: STA_463_563_Fall2019/

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Upload a pdf file or word file in Canvas generated using R markdown. You should clearly label the question number, include the r code, the output and any necessary explanation in your file. The plots should be made using ggplot2 package.

Load CalCOFI data using the following R codes:

cofi <- read.table("https://raw.githubusercontent.com/dleelab/STA463_563_Fall2019/master/data/calcofi_500.csv", header=TRUE, sep = ",")
head(cofi)
     sal  temp depth
1 33.440 10.50     0
2 33.440 10.46     8
3 33.437 10.46    10
4 33.420 10.45    19
5 33.421 10.45    20
6 33.431 10.45    30

Question 1 (1pt) Descriptive statistics for all variables in the data.

summary(cofi)
      sal             temp            depth     
 Min.   :32.63   Min.   : 2.780   Min.   :   0  
 1st Qu.:33.07   1st Qu.: 5.020   1st Qu.:  62  
 Median :33.80   Median : 8.120   Median : 200  
 Mean   :33.63   Mean   : 7.821   Mean   : 345  
 3rd Qu.:34.13   3rd Qu.:10.450   3rd Qu.: 600  
 Max.   :34.45   Max.   :12.660   Max.   :1352

Question 2 (1pt) A density plot for one of the variables.

library(ggplot2)
ggplot(cofi, aes(x=sal)) + geom_density()

library(ggplot2)
ggplot(cofi, aes(x=temp)) + geom_density()

library(ggplot2)
ggplot(cofi, aes(x=depth)) + geom_density()

Question 3 (1pt) A scatter plot of salinity and log10(depth+1), use “salinity” as X-axis.

ggplot(cofi, aes(x=sal, y=log10(depth+1))) + geom_point() + theme_classic()

Question 4 (1pt) A correlation matrix of salinity and log10(depth+1).

cofi$log10depth <- log10(cofi$depth+1)
cor(cofi[,c(1,4)])
                 sal log10depth
sal        1.0000000  0.8630779
log10depth 0.8630779  1.0000000

Question 5 (1pt) Use the lm() function in R, fit a simple linear regression using “salinity” as response and log10(depth+1) as predictor. Report the value of b0 and b1 in fitted simple linear regression equation.

cofi.fit <- lm(sal~log10depth, data=cofi)
summary(cofi.fit)

Call:
lm(formula = sal ~ log10depth, data = cofi)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.67472 -0.16683  0.07827  0.14663  1.31027 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 32.12973    0.04158  772.66   <2e-16 ***
log10depth   0.68744    0.01816   37.87   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2835 on 491 degrees of freedom
Multiple R-squared:  0.7449,    Adjusted R-squared:  0.7444 
F-statistic:  1434 on 1 and 491 DF,  p-value: < 2.2e-16

Question 6 (1pt) Verify the sum of the residuals equals to 0.

sum(cofi.fit$residuals)
[1] 7.691764e-15

Question 7 (1pt) Make a density plot for the residuals.

ggplot(cofi.fit, aes(x=cofi.fit$residuals)) + geom_density()

Question 8 (3pt) Instead of using lm(), use R code to calculate b0 and b1 based on the formula we discussed in class. Verify the values match the results using the lm() function.

Sxx <- sum((cofi$log10depth-mean(cofi$log10depth))^2)
Sxx
[1] 243.8907
Sxy <- sum((cofi$log10depth-mean(cofi$log10depth))*(cofi$sal-mean(cofi$sal)))
Sxy
[1] 167.6612
b1 <- Sxy/Sxx
b1
[1] 0.6874438
b0 <- mean(cofi$sal)-b1*mean(cofi$log10depth)
b0
[1] 32.12973

sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ggplot2_3.2.1

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.2       knitr_1.24       magrittr_1.5     workflowr_1.4.0 
 [5] tidyselect_0.2.5 munsell_0.5.0    colorspace_1.4-1 R6_2.4.0        
 [9] rlang_0.4.0      dplyr_0.8.3      stringr_1.4.0    tools_3.6.1     
[13] grid_3.6.1       gtable_0.3.0     xfun_0.9         withr_2.1.2     
[17] git2r_0.26.1     htmltools_0.3.6  assertthat_0.2.1 yaml_2.2.0      
[21] lazyeval_0.2.2   rprojroot_1.3-2  digest_0.6.20    tibble_2.1.3    
[25] crayon_1.3.4     purrr_0.3.2      fs_1.3.1         glue_1.3.1      
[29] evaluate_0.14    rmarkdown_1.15   labeling_0.3     stringi_1.4.3   
[33] compiler_3.6.1   pillar_1.4.2     scales_1.0.0     backports_1.1.4 
[37] pkgconfig_2.0.2