Introduction

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Building high performing machine learning algorithms depends on identifying the relationships between the variables. This helps in feature engineering as well as deciding on the machine learning algorithm. In this guide, you will learn techniques of finding relationships in data with R.

In this guide, we will use a fictitious dataset of loan applicants containing 200 observations and ten variables, as described below:

`Marital_status`

Whether the applicant is married ("Yes") or not ("No")`Is_graduate`

Whether the applicant is a graduate ("Yes") or not ("No")`Income`

Annual Income of the applicant (in USD)`Loan_amount`

Loan amount (in USD) for which the application was submitted`Credit_score`

Whether the applicant's credit score was good ("Good") or not ("Bad").`Approval_status`

Whether the loan application was approved ("Yes") or not ("No").`Investment`

Investments in stocks and mutual funds (in USD), as declared by the applicant`Gender`

Whether the applicant is "Female" or "Male"`Age`

The applicant’s age in years`Work_exp`

The applicant's work experience in years

Let’s start by loading the required libraries and the data.

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`library(plyr) library(readr) library(ggplot2) library(GGally) library(dplyr) library(mlbench) dat <- read_csv("data_test.csv") glimpse(dat)`

{r}

Output:

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`Observations: 200 Variables: 10 $ Marital_status <chr> "Yes", "Yes", "Yes", "Yes", "Yes", "Yes", "Yes", "Yes"... $ Is_graduate <chr> "No", "No", "No", "No", "No", "No", "No", "No", "No", ... $ Income <int> 72000, 64000, 80000, 76000, 72000, 56000, 48000, 72000... $ Loan_amount <int> 70500, 70000, 275000, 100500, 51500, 69000, 147000, 61... $ Credit_score <chr> "Bad", "Bad", "Bad", "Bad", "Bad", "Bad", "Bad", "Bad"... $ approval_status <chr> "Yes", "Yes", "Yes", "Yes", "Yes", "Yes", "Yes", "Yes"... $ Investment <int> 117340, 85340, 147100, 65440, 48000, 136640, 160000, 9... $ gender <chr> "Female", "Female", "Female", "Female", "Female", "Fem... $ age <int> 34, 34, 33, 34, 33, 34, 33, 33, 33, 33, 34, 33, 33, 33... $ work_exp <dbl> 8.10, 7.20, 9.00, 8.55, 8.10, 6.30, 5.40, 8.10, 8.10, ...`

The output shows that the dataset has five numerical (labeled as `int`

, `dbl`

) and five character variables (labelled as `chr`

). We will convert these into `factor`

variables using the line of code below.

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`names <- c(1,2,5,6,8) dat[,names] <- lapply(dat[,names] , factor) glimpse(dat)`

{r}

Output:

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`Observations: 200 Variables: 10 $ Marital_status <fct> Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes,... $ Is_graduate <fct> No, No, No, No, No, No, No, No, No, No, No, No, Yes, Y... $ Income <int> 72000, 64000, 80000, 76000, 72000, 56000, 48000, 72000... $ Loan_amount <int> 70500, 70000, 275000, 100500, 51500, 69000, 147000, 61... $ Credit_score <fct> Bad, Bad, Bad, Bad, Bad, Bad, Bad, Bad, Bad, Bad, Bad,... $ approval_status <fct> Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes,... $ Investment <int> 117340, 85340, 147100, 65440, 48000, 136640, 160000, 9... $ gender <fct> Female, Female, Female, Female, Female, Female, Female... $ age <int> 34, 34, 33, 34, 33, 34, 33, 33, 33, 33, 34, 33, 33, 33... $ work_exp <dbl> 8.10, 7.20, 9.00, 8.55, 8.10, 6.30, 5.40, 8.10, 8.10, ...`

Many machine learning algorithms require that continuous variables should not be correlated with each other, a phenomenon called ‘multicollinearity.’ Establishing relationships between the numerical variables is a common step to detect and treat multicollinearity.

Creating a correlation matrix is a technique to identify multicollinearity among numerical variables. The lines of code below create the matrix.

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`cordata = dat[,c(3,4,7,9,10)] corr <- round(cor(cordata), 1) corr`

{r}

Output:

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`Income Loan_amount Investment age work_exp Income 1.0 0.0 0.1 -0.2 0.9 Loan_amount 0.0 1.0 0.8 0.0 0.0 Investment 0.1 0.8 1.0 0.0 0.1 age -0.2 0.0 0.0 1.0 -0.1 work_exp 0.9 0.0 0.1 -0.1 1.0`

The output above shows the presence of strong linear correlation between the variables `Income`

and `Work_exp`

and between `Investment`

and `Loan_amount`

.

The correlation can also be visualized using a correlation plot, which is implemented using the `ggcorrplot`

package. This library is loaded with the first line of code below.

The second line creates the correlogram plot, where arguments like `colors`

, `outline.color`

, and `show.legend`

are used to control the display of the chart.

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`library(ggcorrplot) ggcorrplot(corr, hc.order = TRUE, type = "lower", lab = TRUE, lab_size = 3, method="circle", colors = c("blue", "white", "red"), outline.color = "gray", show.legend = TRUE, show.diag = FALSE, title="Correlogram of loan variables")`

{r}

Output:

Correlation Test is another method to determine the presence and extent of a linear relationship between two quantitative variables. In our case, we would like to statistically test whether there is a correlation between the applicants’ investment and work experience.

The first step is to visualize the relationship with a scatter plot, which is done in the line of code below.

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`plot(dat$Investment,dat$work_exp, main="Correlation between Investment Levels & Work Exp", xlab="Work experience in years", ylab="Investment in USD")`

{r}

Output:

The above plot suggests the absence of linear relationship between the two variables. We can quantify this inference by calculating the correlation coefficient using the line of code below.

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`cor(dat$Investment, dat$work_exp)`

{r}

Output:

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`[1] 0.07653245`

The value of 0.07 shows a positive but weak linear relationship between the two variables. Let’s confirm this with the correlation test, which is done in R with the `cor.test()`

function.

The basic syntax is `cor.test(var1, var2, method = “method”)`

, with the default method being `pearson`

. This is done using the line of code below.

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`cor.test(dat$Investment, dat$work_exp)`

{r}

Output:

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`Pearson's product-moment correlation data: dat$Investment and dat$work_exp t = 1.0801, df = 198, p-value = 0.2814 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: -0.0628762, 0.2130117 sample estimates: cor - 0.07653245`

Since the p-value of 0.2814 is greater than 0.05, we fail to reject the null hypothesis that the relationship between the applicant’s investment and their work experience is not significant.

Let’s consider another example of correlation between `Income`

and `Work_exp`

using the line of code below.

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`cor.test(dat$Income, dat$work_exp)`

{r}

Output:

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`Pearson's product-moment correlation data: dat$Income and dat$work_exp t = 25.869, df = 198, p-value < 2.2e-16 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.8423810; 0.9066903 sample estimates: cor - 0.8784546`

In this case, the p-value is smaller than 0.05, so we reject the null hypothesis that the relationship between the applicant’s income and their work experience is not significant.

In the previous sections, we covered techniques of finding relationships between numerical variables. It is equally important is to understand and estimate the relationship between categorical variables.

Creating a frequency table is a simple but effective way of finding distribution between the two categorical variables. The `table()`

function can be used to create the two way table between two variables.

In the first line of code below, we create a two-way table between the variables `marital_status`

and `approval_status`

. The second line prints the frequency table, while the third line prints the proportion table. The fourth line prints the row proportion table, while the fifth line prints the column proportion table.

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`# 2 - way table two_way = table(dat$Marital_status, dat$approval_status) two_way prop.table(two_way) # cell percentages prop.table(two_way, 1) # row percentages prop.table(two_way, 2) # column percentages`

{r}

Output:

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`#Output - two_way table No Yes Divorced 31 29 No 66 10 Yes 52 12 #Output - cell percentages table No Yes Divorced 0.155 0.145 No 0.330 0.050 Yes 0.260 0.060 #Output - row percentages table No Yes Divorced 0.5166667 0.4833333 No 0.8684211 0.1315789 Yes 0.8125000 0.1875000 #Output - column percentages table No Yes Divorced 0.2080537 0.5686275 No 0.4429530 0.1960784 Yes 0.3489933 0.2352941`

The output from the column percentages table shows that divorced applicants (at 56.8 percent) have a higher probability of getting loan approvals compared to married applicants (at 19.6 percent). To test whether this insight is statistically significant, we use the chi-square test of independence.

The chi-quare test of independence is used to determine whether there is an association between two or more categorical variables. In our case, we would like to test whether the marital status of the applicants has any association with the approval status.

The first step is to create a two-way table between the variables under study, which is done in the lines of code below.

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`mar_approval <-table(dat$Marital_status, dat$approval_status) mar_approval`

{r}

Output:

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`No Yes Divorced 31 29 No 66 10 Yes 52 12`

The next step is to generate the expected counts using the line of code below.

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`chisq.test(mar_approval, correct=FALSE)$expected`

{r}

Output:

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`No Yes Divorced 44.70 15.30 No 56.62 19.38 Yes 47.68 16.32`

We are now ready to run the test of independence using the `chisq.test`

function, as in the line of code below.

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`chisq.test(mar_approval, correct=FALSE)`

{r}

Output:

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`Pearson's Chi-squared test data: mar_approval X-squared = 24.095, df = 2, p-value = 5.859e-06`

Since the p-value is less than 0.05, we reject the null hypothesis that the marital status of the applicants is not associated with the approval status.

In this guide, you have learned techniques of finding relationships in data for both numerical and categorical variables. You also learned how to interpret the results of the tests by statistically validating the relationship between the variables. To learn more about data science using R, please refer to the following guides: 1. Interpreting Data Using Descriptive Statistics with R

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