Section 1. Pearson & Spearman Correlation

# We start our example by importing the data frist
library(haven)
asgusam5 <- read_sav("asgusam52.sav")
# Take a look of the dataset
head(asgusam5)
## # A tibble: 6 x 58
##      id  gender    month    year language    book home_computer home_desk
##   <dbl> <dbl+l> <dbl+lb> <dbl+l> <dbl+lb> <dbl+l>     <dbl+lbl> <dbl+lbl>
## 1     1 1 [GIR…  1 [JAN… 5 [200… 1 [ALWA… 4 [TWO…       1 [YES]   0 [NO] 
## 2     2 0 [BOY]  9 [SEP… 4 [200… 1 [ALWA… 3 [ONE…       1 [YES]   1 [YES]
## 3     3 0 [BOY] 10 [OCT… 4 [200… 1 [ALWA… 4 [TWO…       1 [YES]   1 [YES]
## 4     4 1 [GIR…  8 [AUG… 4 [200… 1 [ALWA… 3 [ONE…       1 [YES]   1 [YES]
## 5     5 0 [BOY]  8 [AUG… 4 [200… 1 [ALWA… 5 [THR…       1 [YES]   1 [YES]
## 6     6 0 [BOY] 11 [NOV… 4 [200… 1 [ALWA… 3 [ONE…       1 [YES]   1 [YES]
## # … with 50 more variables: home_book <dbl+lbl>, home_room <dbl+lbl>,
## #   home_internet <dbl+lbl>, computer_home <dbl+lbl>,
## #   computer_school <dbl+lbl>, computer_some <dbl+lbl>,
## #   parentsupport1 <dbl+lbl>, parentsupport2 <dbl+lbl>,
## #   parentsupport3 <dbl+lbl>, parentsupport4 <dbl+lbl>, school1 <dbl+lbl>,
## #   school2 <dbl+lbl>, school3 <dbl+lbl>, studentbullied1 <dbl+lbl>,
## #   studentbullied2 <dbl+lbl>, studentbullied3 <dbl+lbl>,
## #   studentbullied4 <dbl+lbl>, studentbullied5 <dbl+lbl>,
## #   studentbullied6 <dbl+lbl>, learning1 <dbl+lbl>, learning2 <dbl+lbl>,
## #   learning3 <dbl+lbl>, learning4 <dbl+lbl>, learning5 <dbl+lbl>,
## #   learning6 <dbl+lbl>, learning7 <dbl+lbl>, engagement1 <dbl+lbl>,
## #   engagement2 <dbl+lbl>, engagement3 <dbl+lbl>, engagement4 <dbl+lbl>,
## #   engagement5 <dbl+lbl>, confidence1 <dbl+lbl>, confidence2 <dbl+lbl>,
## #   confidence3 <dbl+lbl>, confidence4 <dbl+lbl>, confidence5 <dbl+lbl>,
## #   confidence6 <dbl+lbl>, score1 <dbl>, score2 <dbl>, score3 <dbl>,
## #   score4 <dbl>, score5 <dbl>, ParentSupport <dbl>, Home <dbl>,
## #   school <dbl>, StudentBullied <dbl>, learning <dbl>, engagement <dbl>,
## #   confidence <dbl>, ScienceScore <dbl>
# Make a test copy of data
class_example <- asgusam5

The rcorr( ) function in the Hmisc package produces correlations/covariances and significance levels for pearson and spearman correlations. However, input must be a matrix and pairwise deletion is used.

# Usually, we build a scatterplot to precheck whether there is a high correlation between two variables
# We need specify a data set for us to run the scatterplot
confi_VS_sciscore <- cbind(class_example$confidence,class_example$ScienceScore)
plot(confi_VS_sciscore, main="The Scatterplot of ScineceScore VS Confidence",xlab="Sum of confidence",ylab="Science Score")
# Load the package. Don't forget to install the package before you use it. 
# For install the package, type in: install.packages("Hmisc")
library(Hmisc)
## Loading required package: lattice
## Loading required package: survival
## Loading required package: Formula
## Loading required package: ggplot2
## Registered S3 methods overwritten by 'ggplot2':
##   method         from 
##   [.quosures     rlang
##   c.quosures     rlang
##   print.quosures rlang
## 
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:base':
## 
##     format.pval, units

# This time we want to find out what is the correlation between parents support and home stuff.
# First, we specify the data set first.
ParentSupport_VS_HomeStuff <- cbind(class_example$ParentSupport,class_example$Home)
# Calculate the Spearman correlation
rcorr(ParentSupport_VS_HomeStuff, type="pearson")
##       [,1]  [,2]
## [1,]  1.00 -0.11
## [2,] -0.11  1.00
## 
## n
##       [,1]  [,2]
## [1,] 12908 12821
## [2,] 12821 12948
## 
## P
##      [,1] [,2]
## [1,]       0  
## [2,]  0
# Calculate the Pearson correlation
rcorr(ParentSupport_VS_HomeStuff, type="spearman")
##       [,1]  [,2]
## [1,]  1.00 -0.12
## [2,] -0.12  1.00
## 
## n
##       [,1]  [,2]
## [1,] 12908 12821
## [2,] 12821 12948
## 
## P
##      [,1] [,2]
## [1,]       0  
## [2,]  0
# From the outcome table (the correlation matrix), we can see that the more parent support, the more home stuffs (r=-0.11,p=.00). It is because the higher values of sum of parent support indicate less support (see the TISS questionnaire).

Person & Spearman correlation could be also used on Ordinal scales (ranking data).

# We take the Spearman_slide7.sav data for example
# Import the data first
Sperman_rank_example <- read_sav("Sperman_slide7.sav")
# Check the data set
Sperman_rank_example
## # A tibble: 10 x 4
##        X     Y Rank_X Rank_Y
##    <dbl> <dbl>  <dbl>  <dbl>
##  1    12    36      4    8  
##  2    16    25      7    4.5
##  3    10    35      2    7  
##  4    11    38      3    9  
##  5    15    24      6    3  
##  6     9    40      1   10  
##  7    21    18     10    1  
##  8    14    33      5    6  
##  9    17    25      8    4.5
## 10    19    22      9    2
# Calculate the Pearson correlation
Result_Pearson <- rcorr(as.matrix(Sperman_rank_example),type="pearson")
Result_Pearson
##            X     Y Rank_X Rank_Y
## X       1.00 -0.95   0.99  -0.94
## Y      -0.95  1.00  -0.95   0.98
## Rank_X  0.99 -0.95   1.00  -0.92
## Rank_Y -0.94  0.98  -0.92   1.00
## 
## n= 10 
## 
## 
## P
##        X     Y     Rank_X Rank_Y
## X            0e+00 0e+00  0e+00 
## Y      0e+00       0e+00  0e+00 
## Rank_X 0e+00 0e+00        1e-04 
## Rank_Y 0e+00 0e+00 1e-04
# Calculate the Spearman correlation
Result_Spearman <- rcorr(as.matrix(Sperman_rank_example),type="spearman")
Result_Spearman
##            X     Y Rank_X Rank_Y
## X       1.00 -0.92   1.00  -0.92
## Y      -0.92  1.00  -0.92   1.00
## Rank_X  1.00 -0.92   1.00  -0.92
## Rank_Y -0.92  1.00  -0.92   1.00
## 
## n= 10 
## 
## 
## P
##        X     Y     Rank_X Rank_Y
## X            1e-04 0e+00  1e-04 
## Y      1e-04       1e-04  0e+00 
## Rank_X 0e+00 1e-04        1e-04 
## Rank_Y 1e-04 0e+00 1e-04

Section 2. Point Biserial Correlation & Phi Correlation

The point biserial correlation coefficient is a correlation coefficient used when one variable (e.g. Y) is dichotomous; Y can either be “naturally” dichotomous, like whether a coin lands heads or tails, or an artificially dichotomized variable.

# The point-biserial correlation is equivalent to calculating the Pearson correlation between a continuous and a dichotomous variable. 
# You can just use the standard cor.test function in R, which will output the correlation, a 95% confidence interval, and an independent t-test with associated p-value.
cor.test(class_example$learning,class_example$gender)
## 
##  Pearson's product-moment correlation
## 
## data:  class_example$learning and class_example$gender
## t = 2.5608, df = 12878, p-value = 0.01045
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.005292235 0.039815076
## sample estimates:
##        cor 
## 0.02256038
# The result above indicates that Girls (gender = 1) like to learn science less than boys (gender = 0) (r = .023, p = .01). It is because the higher values of sum of learning shows “they don’t like to learn science (see the TIMSS questionnaire).

The Phi Coefficient is a measure of association between two binary variables (i.e. living/dead, black/white, success/failure).

# The phi coefficient is identical to the Pearson coefficient in the case of a 2 x 2 data set.
# We can still use the cor.test function to calculate the phi coefficient.
cor.test(class_example$gender,class_example$home_book)
## 
##  Pearson's product-moment correlation
## 
## data:  class_example$gender and class_example$home_book
## t = 9.697, df = 12898, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.06791693 0.10218083
## sample estimates:
##        cor 
## 0.08507403

Section 3. Partical and Semi-partial Correlation

Partial correlation is the correlation of two variables while controlling for a third or more other variables.

# We can use the package 'ppcor' to calculate the partial and semi-partial correlation
library(ppcor)
## Loading required package: MASS
# Both the calculation of the partial and semi-partial correlation are not allow the Missing Value, so we need prepare a clean data set without all the missing value.
uncleaned_data <- cbind(class_example$ScienceScore,class_example$engagement,class_example$learning)
cleaned_data <- na.omit(uncleaned_data)
names <- c("ScienceScore","Engagement","Learning")
colnames(cleaned_data) <- names
cleaned_data <- as.data.frame(cleaned_data)
# Calculate the correlation between science score and engagement while controlling the variable learning
pcor.test(cleaned_data$ScienceScore,cleaned_data$Engagement,cleaned_data$Learning,method="pearson")
##    estimate      p.value statistic     n gp  Method
## 1 -0.153554 1.257616e-68 -17.61272 12849  1 pearson
# The outcome table indicates that the partial correlation between sum of engagement and ScienceScore after entirely controlling for the effects of sum of learning = -0.154.  
# Similarly, the semi-partial correlations can be calculated with spcor() function.
spcor.test(cleaned_data$ScienceScore,cleaned_data$Engagement,cleaned_data$Learning,method="pearson")
##     estimate      p.value statistic     n gp  Method
## 1 -0.1533073 2.072572e-68 -17.58374 12849  1 pearson
# The outcome table indicates that the semi-partial correlation between all of ScienceScore and that part of sum of engagement which is independent of sum of learning = -0.153.

Note: If you are confused with the difference between partial and semipartial correlation, the link below provide you an elaborative example. http://faculty.cas.usf.edu/mbrannick/regression/Partial.html