Title: Bivariate relationships among two interval / ratio level variables
Course: UGQuants (SSPS) > Section: Correlation and Regression

Bivariate relationships among two interval / ratio level variables

With variables at the interval or ratio level of measurement -  particularly if those variables have a large number of values - then the first step in is to produce a visual representation of the relationship between these variables.  This is best accomplished through a scattergram or scatterplot.  To do this we plot the values of each pair of our variables along the two axes of a graph.  Any point on the graph, referred to as a coordinate, corresponds to a combination of two values, one for each variable.

A scatterplot has two axes, a horizontal axis, called the X axis, and a vertical axis, called the Y axis. Often when we are looking at the relationship between two variables we will have identified one variable – the main variable of interest - as the dependent variable and another variable – that which influences our variable of interest - as the independent variable.  It is customary to put the main variable of interest – the dependent variable – on the Y axis and the variable thought of as a potential influence or cause (the independent, predictor or explanatory variable) on the X axis.

Each case in our dataset (depending on the dataset each case might represent a person, a household or a country) is represented in the scatterplot by the coordinate that reflects its particular value for both the independent and dependent variables.  We might, for example, record the number of leaflets distributed in a constituency during an election campaign and the turnout level in that constituency in the eventual election.  Or we might record the GDP per capita for a particular country and its literacy rate.  When we plot points like this we tend to summarise a particular case by (X, Y), which reflects the scores for the independent and dependent variables, respectively.  A country with a per capita GDP of $41553 and a literacy rate of 97 , for example, would be represented by (41553, 97).  To locate a coordinate, draw a line vertically from its particular value on the X axis and a line horizontally from its particular value on the Y axis. The point where these two lines intersect is where that particular case should be located in the scatterplot.

A video example: http://www.youtube.com/watch?v=G6Edu_RybxA?d=640x440

The scatterplot below plots two variables from the WDI 2011 dataset: the number of hospital beds (per 1000 people) for each country in the dataset, and female life expectancy in each country.  Here we are treating the number of hospital beds as the independent variable because we believe that a well resourced health care system will improve life expectancy, which we are treating here as the dependent variable.

The results show that only in a few countries are there more than 7.5 hospital beds per 1000 people and in only one country are there more than 12.5 beds per 1000 people.  The figures for life expectancy seem to cluster between 50 and 85.

You should now produce the above scatterplot for yourself, opening the WDI Dataset and using the variables 'HospBed' and 'LE_Female')

To produce scatterplots in SPSS select Graphs from the top menu and then select Chart Builder

Click on the gallery tab and in the list of options select Scatter and drag this label to the blank main box.

To select your axes, identify the variables of interest from the list on the left. Click on the independent variable you have selected and drag this to the X (horizontal) axis.  Next click on the dependent variable you have selected and drag this to the Y (vertical) axis.

Click on the Titles/Footnotes tabs and select the features you wish to appear in the scatterplot. You will likely wish to give this a title, so select Title 1. You will likely also wish to have a footnote in which you describe data sources and the number of cases in your study. To do this select Footnote 1.  To populate these fields, in other words to type out the information that will go in your Title and Footnote, select Element Properties from the right side of the main box.

By clicking on the item that you wish to describe you can then insert text in the appropriate box. If you wish to change the description of the variables in the table – if for example you want to use something other than the SPSS labels as the axis labels you can also change these in the same way.  You might also wish to change the scale of your scatterplot. SPSS tends to display information in a way that makes it most meaningful. If there are no countries with more than 15 hospital beds per 1000 people SPSS will select an end point for the graph that is 15 or fewer. Likewise, if none of the countries in our dataset have a life expectancy of less than 42 years, then SPSS will shorten the y axis so that it begins at, say 40.  You might be happy with this. or you might wish to change the scale of the axis so that it begins at 0. To do this, when you are in ‘Element Properties’ select y axis and uncheck the box next to ‘minimum and under ‘Custom’ identify the desired starting point for your y axis. In our plot, for example, we have manually set it to 0. After each change select ‘Apply’ and then when you are done select ‘Close’. You are now ready to build your scatterplot. Select ‘OK’ and SPSS will generate your plot.