Basic Statistics

The Chi Square Test of Independence

Chi Square Test of Independence

• A measure of association similar to the correlations we studied earlier.

• Pearson and Spearman are not applicable if the data are at the nominal level of measurement.

• Chi Square is used for nominal data placed in a contingency table.– A contingency table is a two-way table showing the

contingency between two variables where the variables have been classified into mutually exclusive categories and the cell entries are frequencies.

An Example

• Suppose that the state legislature is considering a bill to lower the legal drinking age to 18. A political scientist is interested in whether there is a relationship between party affiliation and attitude toward the bill. A random sample of 150 registered republicans and 200 registered democrats are asked their opinion about the proposed bill. The data are presented on the next slide.

Political Party and Legal Drinking Age Bill

Attitude Toward Bill

For Undecided Against Total

Republican 38 17 95 150

Democrat 92 18 90 200

Total 130 35 185 350

The bold numbers are the observed frequencies (fo)

Determining the Expected Frequencies (fe)

• First, add the columns and the rows to get the totals as shown in the previous slide.

• To obtain the expected frequency within a particular cell, perform the following operation: Multiply the row total and the column total for the cell in question and then divide that product by the Total number of all respondents.

Attitude Toward Bill

For Undecided Against Total

Republican 38 17 95 150

Democrat 92 18 90 200

Total 130 35 185 350

Calculating the Expected Value for a Particular Cell

1. 130*150 = 19500 2. 19500/350 = 55.7

55.7

Attitude Toward Bill

For Undecided Against Total

Republican 38

55.7

17

15

95

79.3

150

Democrat 92

74.3

18

20

90

105.7

200

Total 130 35 185 350

Political Party and Attitude toward Bill

Numbers in Black are obtained (fo), Numbers in Purple are expected (fe)

The Null Hypothesis and the Expected Values

• The Null Hypothesis under investigation in the Chi Square Test of Independence is that the two variables are independent (not related).

• In this example, the Null Hypothesis is that there is NO relationship between political party and attitude toward lowering the legal drinking age.

Understanding the Expected Values

• If the Null is true, then the percentage of those who favor lowering the drinking age would be equal for each political party.

• Notice that the expected values for each opinion are proportional for the number of persons surveyed in each party.

Attitude Toward Bill

For Undecided Against Total

Republican 38

55.7

17

15

95

79.3

150

42.8%

Democrat 92

74.3

18

20

90

105.7

200

57.2%

Total 130 35 185 350

Political Party and Attitude toward Bill

The numbers in Green are the percentage of the total for each Party.

The Expected Values

• The expected values for each cell are also equal to the percentage of each party for the column total.

• For example, Republicans were 42.8% of the total persons surveyed

• If 130 people were in favor of the bill, then 42.8% of them should be Republican (55.7), if there is no relationship between the variables

Calculating the Chi Square Statistic

e

eo

f

ffx

22 )(

The Chi Square statistic is obtained via this formula

The Chi Square statistic is (1) the sum over all cells of (2) the difference between the obtained value and the expected value SQUARED, which is then (3) divided by the expected frequency.

The numbers in Purple on the next slide illustrate this calculation

Calculating the Chi Square Statistic

15

)1517( 23.79

)3.7995( 2

7.55

)7.5538( 2

3.74

)3.7492( 2

20

)2018( 27.105

)7.10590( 2

=5.62 =0.27 =3.11

=4.22 =0.20 =2.33

X2 = 5.62 + 0.27 + 3.11 + 4.22 + 0.20 + 2.33 = 15.75

The calculated value for the chi square statistic is compared to the critical value found in Table H, page 544.

Interpreting the Results2X

Note: The distribution of the Chi Square Statistic is not normal and the critical values are only on one side. If the obtained values are close to the expected value, then the chi square statistic will approach 0. As the obtained value is different from the expected, the value of chi square will increase. This is reflected in the values found in Table H.

The Degrees of Freedom for the Chi Square Test of Independence is the product of the number of rows minus 1 times the number of columns minus 1

Interpreting Our Results

• In our study, we had two rows (Republicans and Democrats) and three columns (For, Undecided, Against).

• Therefore, the degrees of freedom for our study is (2-1)(3-1) = 1(2) = 2.

• Using an of .05, the critical value from Table H would be 5.991

• Since our calculated chi square is 15.75, we conclude that there IS a relationship between political party and opinion on lowering the drinking age, thereby rejecting the Null Hypothesis