Multivariate non-graphical EDA - Statistics, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Multivariate non-graphical EDA

Multivariate non-graphical EDA techniques generally show the relationship between two or more variables in the form of either cross-tabulation or statistics.

4.4.1 Cross-tabulation

For categorical data (and quantitative data with only a few different values) an extension of tabulation called cross-tabulation is very useful. For two variables, cross-tabulation is performed by making a two-way table with column headings that match the levels of one variable and row headings that match the levels of the other variable, then filling in the counts of all subjects that share a pair of levels. The two variables might be both explanatory, both outcome, or one of each. Depending on the goals, row percentages (which add to 100% for each row), column percentages (which add to 100% for each column) and/or cell percentages (which add to 100% over all cells) are also useful. 

Here is an example of a cross-tabulation. Consider the data in table 4.1. For each subject we observe sex and age as categorical variables. 

Table 4.2 shows the cross-tabulation.

We can easily see that the total number of young females is 2, and we can calculate, e.g., the corresponding cell percentage is 2/11×100 = 18.2%, the row percentage is 2/5×100 = 40.0%, and the column percentage is 2/7×100 = 28.6%. 

Cross-tabulation can be extended to three (and sometimes more) variables by making separate two-way tables for two variables at each level of a third variable.

Table 4.1: Sample Data for Cross-tabulation
 

Table 4.2: Cross-tabulation of Sample Data

For example, we could make separate age by gender tables for each education level.

4.4.2 Correlation for categorical data

Another statistic that can be calculated for two categorical variables is their correlation. But there are many forms of correlation for categorical variables, and that material is currently beyond the scope of this book.

4.4.3 Univariate statistics by category

For one categorical variable (usually explanatory) and one quantitative variable (usually outcome), it is common to produce some of the standard univariate nongraphical statistics for the quantitative variables separately for each level of the categorical variable, and then compare the statistics across levels of the categorical variable. Comparing the means is an informal version of ANOVA. Comparing medians is a robust informal version of one-way ANOVA. Comparing measures of spread is a good informal test of the assumption of equal variances needed for valid analysis of variance.

4.4.4 Correlation and covariance

For two quantitative variables, the basic statistics of interest are the sample covariance and/or sample correlation, which correspond to and are estimates of the corresponding population parameters from section 3.5. The sample covariance is a measure of how much two variables “co-vary”, i.e., how much (and in what direction) should we expect one variable to change when the other changes.

Positive covariance values suggest that when one measurement is above the mean the other will probably also be above the mean, and vice versa. Negative covariances suggest that when one variable is above its mean, the other is below its mean. And covariances near zero suggest that the two variables vary independently of each other.
 

Covariances tend to be hard to interpret, so we often use correlation instead. The correlation has the nice property that it is always between -1 and +1, with -1 being a “perfect” negative linear correlation, +1 being a perfect positive linear correlation and 0 indicating that X and Y are uncorrelated. The symbol r or rx,y is often used for sample correlations.

Table 4.3: Covariance Sample Data
 

4.4.5 Covariance and correlation matrices

When we have many quantitative variables the most common non-graphical EDA technique is to calculate all of the pairwise covariances and/or correlations and assemble them into a matrix. Note that the covariance of X with X is the variance of X and the correlation of X with X is 1.0. For example the covariance matrix of table 4.5 tells us that the variances of X, Y , and Z are 5, 7, and 4 respectively, the covariance of X and Y is 1.77, the covariance of X and Z is -2.24, and the covariance of Y and Z is 3.17. 

Similarly the correlation matrix in figure 4.6 tells us that the correlation of X and Y is 0.3, the correlation of X and Z is -0.5. and the correlation of Y and Z is 0.6.

Table 4.4: Covariance Calculation

Table 4.5: A Covariance Matrix