Last updated: 2019-10-01

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Knit directory: STA_463_563_Fall2019/

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CalCOFI data

“The CalCOFI data set represents the longest (1949-present) and most complete (more than 50,000 sampling stations) time series of oceanographic and larval fish data in the world. It includes abundance data on the larvae of over 250 species of fish; larval length frequency data and egg abundance data on key commercial species; and oceanographic and plankton data. The physical, chemical, and biological data collected at regular time and space intervals quickly became valuable for documenting climatic cycles in the California Current and a range of biological responses to them. CalCOFI research drew world attention to the biological response to the dramatic Pacific-warming event in 1957-58 and introduced the term “El Niño” into the scientific literature."

Here, we use only 500 observations to speed up calculation.

You can download orignial data from: https://www.kaggle.com/sohier/calcofi

Load package

library(ggplot2)

Load data

temp: temperature (Celsius)
sal: salinity (amount of salt dissolved in a body of water)
depth: depth (meter)

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
dim(cofi)
[1] 493   3
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

pairwise scatter plot

pairs(cofi)

Version Author Date
e8fc3fd dleelab 2019-09-27 
cor(cofi)
             sal       temp      depth
sal    1.0000000 -0.9229002  0.8363158
temp  -0.9229002  1.0000000 -0.9105742
depth  0.8363158 -0.9105742  1.0000000

scatter plots

# temperature as a function of salinity
ggplot(cofi, aes(x=sal, y=temp, col=depth)) + 
  geom_point() +
  labs(x="Salinity", y="Temperature(C)", col = "Depth(m)")+
  theme_classic()

Version Author Date
e8fc3fd dleelab 2019-09-27 
#salinity as a function of depth
ggplot(cofi, aes(x=depth, y=sal, col=temp)) + 
  geom_point() +
  labs(x="Depth(m)", y="Salinity", col = "Temperature(C)")+
  theme_classic()

Version Author Date
e8fc3fd dleelab 2019-09-27 
#salinity as a function of log10(depth)
ggplot(cofi, aes(x=log10(depth), y=sal, col=temp)) + 
  geom_point() +
  labs(x="log10(Depth(m))", y="Salinity", col = "Temperature(C)")+
  theme_classic()

Version Author Date
e8fc3fd dleelab 2019-09-27

SLR model, temp as a function of sal

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

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

Coefficients:
(Intercept)          sal  
    169.118       -4.796

Add the fitted line to the scatter plot

ggplot(cofi, aes(x=sal, y=temp, col=depth)) + 
  geom_point() +
  geom_smooth(method='lm') +
  labs(x="Salinity", y="Temperature(C)", col = "Depth(m)")+
  theme_classic()

Version Author Date
e8fc3fd dleelab 2019-09-27

Summary of lm() output

summary(cofi.fit)

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

Residuals:
     Min       1Q   Median       3Q      Max 
-2.79153 -0.75022 -0.06611  0.66100  3.04295 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 169.11780    3.03735   55.68   <2e-16 ***
sal          -4.79646    0.09031  -53.11   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.123 on 491 degrees of freedom
Multiple R-squared:  0.8517,    Adjusted R-squared:  0.8514 
F-statistic:  2821 on 1 and 491 DF,  p-value: < 2.2e-16

Questions

Q1. What is sample size?

Q2. What is b0 value?

Q3. What is b1 value?

Q4. What is fitted regression line?

Q5. Test statistic value for a hypothesis test on Beta0 ?

Q6. Test statistic value for a hypothesis test on Beta1?

Q7. What is MSE?

Calculate these manually.

# Residual
# Sum of residuals
# Residual Sum of Squares
# SSE
# DF
# MSE
# Residual Standard Error: sqrt(MSE)
# cor(sal, temp)
# Sxx
# Syy 
# Compute cor(sal, temp) using b1, Sxx and Syy

Confidence Interval of beta0 and beta1

confint(cofi.fit)
                 2.5 %     97.5 %
(Intercept) 163.149993 175.085599
sal          -4.973905  -4.619025

Manually calculate CI for beta1

# b1 +- T(1-alpha/2,n-2)*sqrt(MSE/Sxx)
MSE <- sum(cofi.fit$residuals^2)/cofi.fit$df.residual
Sxx <- sum((cofi$sal-mean(cofi$sal))^2)
tval <- qt(.975, df=cofi.fit$df.residual)

cofi.fit$coefficients[2]-tval*sqrt(MSE/Sxx) #lower bound
      sal 
-4.973905 
cofi.fit$coefficients[2]+tval*sqrt(MSE/Sxx) #upper bound
      sal 
-4.619025

Manually calculate CI for beta0

# b1 +- T(1-alpha/2,n-2)*sqrt(MSE*(1/n+mean(x)^2/Sxx))
MSE <- sum(cofi.fit$residuals^2)/cofi.fit$df.residual
Sxx <- sum((cofi$sal-mean(cofi$sal))^2)
n <- nrow(cofi)
tval <- qt(.975, df=cofi.fit$df.residual)

cofi.fit$coefficients[1]-tval*sqrt(MSE*(1/n+mean(cofi$sal)^2/Sxx)) #lower bound
(Intercept) 
     163.15 
cofi.fit$coefficients[1]+tval*sqrt(MSE*(1/n+mean(cofi$sal)^2/Sxx)) #upper bound
(Intercept) 
   175.0856

Solutions: Calculate these manually.

# Residual
hist(cofi.fit$residuals)

Version Author Date
e8fc3fd dleelab 2019-09-27 
plot(density(cofi.fit$residuals), xlab="Residuals")

# Sum of residuals
sum(cofi.fit$residuals)
[1] 3.298056e-14
# Residual Sum of Squares
RSS <- sum(cofi.fit$residuals^2)
# SSE
SSE <- sum(cofi.fit$residuals^2)
# DF
DF <- cofi.fit$df.residual
# MSE
MSE <- SSE/DF
# Residual Standard Error: sqrt(MSE)
RSE <- sqrt(MSE)
# cor(sal, temp)
cor(cofi$sal, cofi$temp)
[1] -0.9229002
# Sxx
Sxx <- sum((cofi$sal-mean(cofi$sal))^2)
# Syy 
Syy <- sum((cofi$temp-mean(cofi$temp))^2)
# Compute cor(sal, temp) using b1, Sxx and Syy
r <- cofi.fit$coefficients[2]*sqrt(Sxx/Syy)
r
       sal 
-0.9229002 
cor(cofi$sal, cofi$temp)
[1] -0.9229002

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       whisker_0.3-2    magrittr_1.5    
 [5] workflowr_1.4.0  tidyselect_0.2.5 munsell_0.5.0    colorspace_1.4-1
 [9] R6_2.4.0         rlang_0.4.0      dplyr_0.8.3      stringr_1.4.0   
[13] tools_3.6.1      grid_3.6.1       gtable_0.3.0     xfun_0.9        
[17] withr_2.1.2      git2r_0.26.1     htmltools_0.3.6  assertthat_0.2.1
[21] yaml_2.2.0       lazyeval_0.2.2   rprojroot_1.3-2  digest_0.6.20   
[25] tibble_2.1.3     crayon_1.3.4     purrr_0.3.2      fs_1.3.1        
[29] glue_1.3.1       evaluate_0.14    rmarkdown_1.15   labeling_0.3    
[33] stringi_1.4.3    pillar_1.4.2     compiler_3.6.1   scales_1.0.0    
[37] backports_1.1.4  pkgconfig_2.0.2