We have been using many built-in R functions throughout the lessons.

After you start using R more, you will reach the point when you will want to create reusable sections of statements.

Functions are simply a named group of R expressions that are considered an R object of type function().

Input values or function arguments are passed to functions by value. Each argument is either matched with the function definition by position or by name.

We have used built-in functions that are also considered generic, such as the plot() function. Generic functions are designed to accept a variety of different arguments. R functions can accept a variable length of input arguments and they can even accept references to other functions.

When functions are called the actual arguments are assigned to local variables within the function. The statements in the function body are evaluated based on the input data. Control will be returned to the invoking function when the return() statement is reached or when the final expression in the function body is reached.

If there is no explicit data returned to the calling function, the output generated by the last expression will be returned.

Let's take a look at an example of a simple R function.

We want find that we are often writing code to determine the number of students who passed a course so we decide to define a function to simply our code.

The function is defined using the function() expression. The name of this function has been specified as numPassed and the function takes a single argument.

The comments on line 3 through 6 help explain the usage of the function includes the input and output datatypes expected.

The body or content of the function is contained within the brace brackets ({}). We decided to use the vector enabled sum() function with a conditional expression to calculate the number of students with marks of 50 or higher. The pass object on line 8 is actually a vector of boolean values based on the criteria defined in line 7.

If you want to examine the source code of a function you can type the name of the function, without specifying any arguments, in the R Console.

This function will accepts a single dataframe or a list and it will perform a conditional expression tests for each element of the data structure. If the student has obtained a mark of 50 or more a logical value of TRUE is generated. These logical values are collected in a local vector called pass.

We then use the sum() function to count the number of TRUE values and return the data to the calling function as an integer.

Finally, the function is being called within the concatenate or cat() function.

One of the most used features of R is the *apply() family of functions.

Here we are using the lapply() or List Apply function. The function that we are invoking is the mean or average function within the base package.

The mean function will be executed across each column of the data.frame called m.

The lapply() function always returns a list as its ouput. In our scenario, the list with the mean or average by subject is provided.

In the box of the right side we see how the sapply() or simplified apply function can be used. sapply() basically performs the same task as lapply(), but it coerces the output into a vector or a matrix. Notice how sapply generated a vector of double values where each element has a name associated with it. The final expression demonstrates how to retrieve the element named "Math".

Here we are using the same approach of using lapply() to invoke a function across a dataset.

In the previous example we used the built-in function mean and in this example we are using the function numPassedCourse, that we created, to find the number of students who passed the course.

These examples of using the apply() function are quite simple, but you can probably see how a single line of R code can actually be performing a complex operation including iteration.

Now that we have learned about flow of control and functions, let's examine data relationships across multiple variables.

In our case study we want to determine if there is a relationship between the height of a person and their shoe size.

We will use R to simulate data sets for our analysis and we will use quantitative measures and visualizations to perform our data analysis.

There are a few different quantitative measurements that can be used to measure the degree of varying between multiple variables.

A positive covariance would indicate that there is a positive linear relationship between the two variables, and a negative covariance would indicate the opposite.

Correlation is a similar statistical measurement of the degree of linear relationship between variables, but it has the added benefit that it has a well defined range of values. The coefficient range is from -1 to 1.

For example, a correlation coefficient of 0.8 would indicate that there is a strong linear relationship between the two variables. Therefore, if variable 1 increases there is a similar observed increase in variable 2. If the correlation coefficient is close to zero (0) there is minimal relationship between the changes in values of the variables. Negative coefficients would indicate that the direction of change in value of the 2 variables are opposite one another.

Here we are creating 2 uniform distributions of heights and shoe size for our analysis.

We use the covariance or cov() function to compute the covariance measurement and we use the correlation or cor() function to compute the correlation.

These values are stored in variables as they will be used in our final visualization.

We are using the base graphics package plot() function to create a scatterplot of the generated data. Visually it seems like there is no relationship between these two measurements. The values of correlation and covariance support our visual understanding as their values are close to (0) zero.

On line 22 a linear regression model is created and stored in a variable called lm1.

The linear model function or lm() can be used to perform an analysis of multiple data sets and determine an linear approximation of the relationship between the variables. Notice that the lm() function is provided a formula. In R a formula consists of a tilde (~) character where in this case the variable on the left is considered the response and the variable on the right of the tilde is the term under consideration.

The object lm1 contains quite a large amount of detailed information. In this scenario we use the object to add a line to our graph to visualize the relationship.

We use a math expression on line 26 to adjust the shoe size for our next analysis.

As an aside, the expression uses integer division to introduce a relationship between the variables.

Again we determine the covariance and correlation coefficients of the revised variables.

In our initial simulation of randomly generated uniform shoe sizes and heights there was very little correlation.

Here we see visually that there is a strong linear correlation and the value of the correlation coefficient agrees with our visual analysis.

We have also performed a revised linear regression model and plotted the result on the graph.

Here we see a summary of the before and after of our measurements for heights and shoe sizes.

The strong correlation is quite evident in the revised data for shoe sizes shown on the right.

There are many lab exercises that you can work with following this lesson including: dice simulations, correlations, and recoding of datasets using the apply family of functions in R.