Chi-Square 2 x 2 Contingency Table

The proportions of two alternatively nominal scaled variables are represented in a contingency table. The Chi-Square test examines whether there is an interrelation between the two variables or not.

Requirements:
Every observation must be assignable unambiguously to exactly one cell.
The expected frequency should not fall below 7 for each cell. If there are expected frequencies below 7 the Fisher’s exact test should be used instead.

Hypothesis:
H0: The two variables are independent
H1: The two variables are dependent

For the assumption that the two variables are independent (H0) the following holds for the probabilities of the four cells:

The expected frequencies under H0 can be obtained by multiplication of the cell probability by the total number of observations.

Probability Table:

 Frequency Table:

For each cell the expected frequency is estimated in the following way:

whereas f_e means expected frequency, f_o means observed frequency.

The following term is a measure for the deviation between the observed and expected frequencies, and it is approximately Chi-Square distributed:

Chi-Square is the sum over all cells of the squared cells’ residual (f_o - f_e ) divided by the cells’ expected frequency f_e.

Degrees of freedom are determined as follows:

• If , , and are known
  df = number of cells - 1 = 3
• If , , and are estimated from the sample
  df = (number of rows - 1)*(number of columns - 1) = 1

Continuity Correction after Yates:

The continuity correction after Yates considers the fact that frequency and Chi-Square values are different, the first being discrete the second being continuous.

The Chi-Square value is corrected the following way:

Interpretation of a significant result:

The interrelation between the two variables is expressed by the deviance of the cells observed percentages  from the row or column percentages  and .

The standardized residual is another measure for the strength of the deviance of the cells observed frequency from its expected frequency. For each cell it is calculated as follows:

For sufficient big sample sizes the standardized residual is comparable to a z-value. As a rule of thumb a standardized residual of –2 or less indicates that the cells observed frequency is significantly lower than its expected frequency and a standard residual of +2 or more indicates that the cells observed frequency is significantly higher than its expected frequency.