Title: Correlation matrix
Course: UGQuants (SSPS) > Section: Correlation and Regression

Correlation matrix

Sometimes we are interested in a series of bivariate relationships in our dataset.  We might, for example, have a number of separate indicators of development at the country-wide level: GDP per capita, life expectancy, hospital beds per 1000, and health expenditure.  We might want to know what the bivariate relationship is between any two of these variables but would like a way to examine each of these inter-relationships at the same time.  To do this we can produce a correlation matrix.

A correlation matrix looks like a large table, with variables listed in row headings and again in column headings.  Each cell in the matrix reports the Pearson’s product moment correlation coefficient for the variables listed in the row heading and column heading.  Because all variables are represented twice – once in the rows and once in the columns – we will have an opportunity to examine the correlation between any individual variable and itself.  In the matrix, this produces a diagonal series of cells with a series of 1s for perfect correlations.  The off diagonal cells provide the correlations between two different variables. The matrix is symmetrical, so that the top right portion of the matrix (known as the upper echelon) mirrors the bottom left portion of the matrix (known as the lower echelon). This is because the correlation between variable 1 and variable 2 is the same as the correlation between variable 2 and variable 1.

The correlation matrix for the variables mentioned above appears here:

Data source: WDI 2011.

The table shows that the strongest correlation is between GDP per capita and life expectancy for females (+.583) and the weakest correlation is between health expenditure and hospital beds per 1000 people (+.147). This is a somewhat surprising finding. We might expect health expenditure to have a direct impact on spending in hospitals, which in turn would affect the number of hospital beds.  We can see also the figure we created earlier, namely the +.476 correlation between life expectancy and hospital beds. Clearly, if we wish to understand life expectancy there are other variables that we must consider. The state of the health care system appears to be only part of the answer.

Our interest in variables should be guided by theory but sometimes it is useful for us to understand the bivariate relationships among a series of possibly interrelated variables.  If we are trying to identify different variables that might have an impact on another – if for example we are trying to determine whether age, income, years of education and length of residence in a town make one more likely to participate in local political affairs, we might want to determine whether any of our possible independent variables are related to each other.  If age and length of residence are strongly correlated, for example, it would be difficult for us to identify the independent impact of either variable and so we should likely rely only on the variable for which we have the strongest theoretical explanation, or examine the relationships in isolation.  It is important to remember, of course, that a correlation matrix does not specify whether one variable is independent and the other is dependent; we decide that based on theory prior to looking at the evidence.