Let us take a look at an example. In 1985, the USDA commissioned a study of women’s nutrition. Nutrient intake was measured for a random sample of 737 women aged 25-50 years. The following variables were measured:

• Calcium(mg)
• Iron(mg)
• Protein(g)
• Vitamin A(μg)
• Vitamin C(mg)

options ls=78;   /*This sets the max number of lines per page to 78.*/
title "Example: Nutrient Intake Data - Descriptive Statistics";   
/*This sets a title that will appear on each page of the output until it's changed.*/
data nutrient;   /*This defines a data set called 'nutrient'.*/
  infile "D:\Statistics\STAT 505\data\nutrient.csv" firstobs=2 delimiter=',';   /*SAS will look in this path for the nutrient.csv file.*/
  input id calcium iron protein a c;   /*This is where we provide names for the variables in order of the columns in the data set. If any were categorical (not the case here), we would need to put a '$' character after its name.*/
  run;   
proc means;   
  var calcium iron protein a c;   /*If not all variables are of interest, we can specify here the ones we want to work with.*/
  run;   
proc corr pearson cov;   /*The 'pearson' option specifies the pearson correlation to be computed. The 'cov' option requests the sample covariance matrix.*/
  var calcium iron protein a c;   /*If not all variables are of interest, we can specify here the ones we want to work with.*/
  run;

Analysis

Descriptive Statistics

A summary of the descriptive statistics is given here for ease of reference.

Notice that the standard deviations are large relative to their respective means, especially for Vitamin A & C. This would indicate a high variability among women in nutrient intake. However, whether the standard deviations are relatively large or not, will depend on the context of the application. Skill in interpreting the statistical analysis depends very much on the researcher's subject matter knowledge.

The variance-covariance matrix is also copied into the matrix below.

\[S = \left(\begin{array}{RRRRR}157829.4 & 940.1 & 6075.8 & 102411.1 & 6701.6 \\ 940.1 & 35.8 & 114.1 & 2383.2 & 137.7 \\ 6075.8 & 114.1 & 934.9 & 7330.1 & 477.2 \\ 102411.1 & 2383.2 & 7330.1 & 2668452.4 & 22063.3 \\ 6701.6 & 137.7 & 477.2 & 22063.3 & 5416.3 \end{array}\right)\]

Interpretation

Because this covariance is positive, we see that calcium intake tends to increase with increasing iron intake. The strength of this positive association can only be judged by comparing s₁₂ to the product of the sample standard deviations for calcium and iron. This comparison is most readily accomplished by looking at the sample correlation between the two variables.

• The sample variances are given by the diagonal elements of S. For example, the variance of iron intake is \(s_{2}^{2}\). 35. 8 mg².
• The covariances are given by the off-diagonal elements of S. For example, the covariance between calcium and iron intake is \(s_{12}\)= 940. 1.
• Note that, the covariances are all positive, indicating that the daily intake of each nutrient increases with increased intake of the remaining nutrients.

Sample Correlations

The sample correlations are included in the table below.

Here we can see that the correlation between each of the variables and themselves is all equal to one, and the off-diagonal elements give the correlation between each of the pairs of variables.

Generally, we look for the strongest correlations first. The results above suggest that protein, iron, and calcium are all positively associated. Each of these three nutrient increases with increasing values of the remaining two.

The coefficient of determination is another measure of association and is simply equal to the square of the correlation. For example, in this case, the coefficient of determination between protein and iron is \((0.623)^2\) or about 0.388.

\[r^2_{23} = 0.62337^2 = 0.38859\]

This says that about 39% of the variation in iron intake is explained by protein intake. Or, conversely, 39% of the protein intake is explained by the variation in the iron intake. Both interpretations are equivalent.