Chi Square/ Cramer’s V/ PSI
Existence of association between two categorical variables implies that the change of level of one variable is related to the change of value of another variable. Tests or measurements of association reveals the relationship between two variables of interest, for example, whether smoking status effects the occurrence of a certain disease, whether gender is related to obesity, and whether an exposure is associated with a health outcome.
Determining the frequencies of data values and building a frequency table gives a preliminary overview of the relationship between two variables. If there is no association, the distribution of the first variable is the same regardless of the level of the other variable.
Chi-squared test is a statistical hypothesis test for the association between two categorical variables. The null hypothesis of Chi-squared test is that there is no association between the two variables, whereas the alternative hypothesis is that the association exists.
Suppose a researcher is investigating the relationship between gender and survival of passengers in the Titanic crash, through Chi-squared test. The null hypothesis of the test is that gender and survival are not associated, which means that the probability of surviving the crash was the same across gender. The alternative hypothesis is that gender and survival are associated, which means the probability of surviving the Titanic crash was not the same for males and females.
The test statistics is:
, following distribution with degrees of freedom .
Where is the observed frequency, is the expected count, is the number of rows of the frequency table and is the number of columns.
If the corresponding p-value of the test statistic is less than the chosen significance level, then the association between the two variables is statistically significant.
Cramer’s V is a measure of association between two nominal variables. For a frequency table with more than 2 rows and 2 columns, Cramer’s V is always non-negative and between 0 and 1. The value of Cramer’s V indicate the how strongly the association between the two categorical variables is.
Cramer’s V is computed as:
- is the PHI coefficient
- is the test statistic of Chi-squared test
- is the number of observations
- is the number of columns
- is the number of rows
The PHI coefficient measures the association between two binary variables. For a 2*2 frequency table, the value of Pearson correlation coefficient is the same as the value of PHI coefficient.
Suppose we have a following frequency table:
Y = 0
Y = 1
X = 0
X = 1
, where a, b, c, d are the frequencies in corresponding groups.
Then the PHI coefficient is computed as:
DATA PERSONS ; INPUT GROUP $ SUCCESS $ @@;
DRUG NO DRUG NO DRUG NO DRUG YES
DRUG YES DRUG YES DRUG YES DRUG YES
DRUG YES DRUG YES
PLACEBO NO PLACEBO NO PLACEBO YES PLACEBO YES
PLACEBO YES PLACEBO YES PLACEBO YES PLACEBO YES
PLACEBO YES PLACEBO YES
PROC FREQ DATA = PERSONS ;
TABLES GROUP * SUCCESS/ NOPERCENT NOCOL NOROW
CHISQ EXPECTED ;
1. Cramér, Harald. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, page 282 (Chapter 21. The two-dimensional case).
2. Yule, G. U. (1912). On the Methods of Measuring Association Between Two Attributes. Journal of the Royal Statistical Society, 75(6), 579. https://doi.org/10.2307/2340126