           Research Design in Occupational Education Copyright 1997. James P. Key. Oklahoma State University Except for those materials which are supplied by different departments of the University (ex. IRB, Thesis Handbook) and references used by permission. MODULE S7 - CHI SQUARE

The "t" test and the F test described in previous modules are called parametric tests. They assume certain conditions about the parameters of the population from which the samples are drawn.

Parametric and nonparametric statistical procedures test hypotheses involving different assumptions. Parametric statistics test hypotheses based on the assumption that the samples come from populations that are normally distributed. Also, parametric statistical tests assume that there is homogeneity of variance (variances within groups are the same). The level of measurement for parametric tests is assumed to be interval or at least ordinal. Nonparametric statistical procedures test hypotheses that do not require normal distribution or variance assumptions about the populations from which the samples were drawn and are designed for ordinal or nominal data.

The main weakness of nonparametric tests is that they are less powerful than parametric tests. They are less likely to reject the null hypothesis when it is false. When the assumptions of parametric tests can be met, parametric tests should be used because they are the most powerful tests available.

There are, however, certain advantages of nonparametric techniques such as Chi Square (X2). For one thing, nonparametric tests are usually much easier to compute. Another unique value of nonparametric procedures is that they can be used to treat data which have been measured on nominal (classificatory) scales. Such data cannot, on any logical basis, be ordered numerically, hence there is no possibility of using parametric statistical tests which require numerical data.

The general pattern of nonparametric procedures is much like that seen with parametric tests, namely, certain sample data are treated by a statistical model which yields a value or statistic. This value is then interpreted for the likelihood of its chance occurrence according to some type of statistical probability distribution. With Chi Square, a value is calculated from the data using Chi Square procedures and then compared to a critical value from a Chi Square table with degrees of freedom corresponding to that of the data. If the calculated value is equal to or greater than the critical value (table value), the null hypothesis is rejected. If the calculated value is less than the critical value, the null hypothesis (Ho) is accepted. This procedure is similar to that used with the "t" test and F test.

Purpose of Chi Square

The Chi Square (X2) test is undoubtedly the most important and most used member of the nonparametric family of statistical tests. Chi Square is employed to test the difference between an actual sample and another hypothetical or previously established distribution such as that which may be expected due to chance or probability. Chi Square can also be used to test differences between two or more actual samples.

Basic Computational Equation Example: A U D Observed responses (Fo) 8 8 14 Expected responses (Fe) (10) (10) (10) Fo - Fe -2 -2 4 (Fo - Fe)2 4 4 16 .4 .4 1.6 2.4

Degrees of freedom - (number of levels - 1) = 2

X2.05 = 5.991 2.4 < 5.991

Therefore, accept null hypothesis.

When there is only one degree of freedom, an adjustment known as Yates correction for continuity must be employed. To use this correction, a value of 0.5 is subtracted from the absolute value (irrespective of algebraic sign) of the numerator contribution of each cell to the above basic computational formula. The basic chi square computational formula then becomes: One-Way Classification

The One-Way Classification (or sometimes referred to as the Single Sample Chi Square Test) is one of the most frequently reported nonparametric tests in journal articles. The test is used when a researcher is interested in the number of responses, objects, or people that fall in two or more categories. This procedure is sometimes called a goodness-of-fit statistic. Goodness-of-fit refers to whether a significant difference exists between an observed number and an expected number of responses, people or objects falling in each category designated by the researcher. The expected number is what the researcher expects by chance or according to some null hypothesis.

Example of a One-Way Classification (with Yates Correction):

Suppose that we flip a coin 20 times and record the frequency of occurrence of heads and tails. We know from the laws of probability that we should expect 10 heads and 10 tails. We also know that because of sampling error we could easily come up with 9 heads and 11 tails or 12 heads and 8 tails.

Let us suppose our coin-flipping experiment yielded 12 heads and 8 tails. We would enter our expected frequencies (10 - 10) and our observed frequencies (12 - 8) in a table.

 Observed Expected (Fo-Fe-0.5) (Fo-Fe-0.5)2 Heads 12 10 1.5 2.25 0.225 Tails 8 10 -1.5 2.25 0.225 20 20 0.450

The calculation of x in a one-way classification (Yates Correction) is very straight forward. The expected frequency in a category ("heads") is subtracted from the observed frequency, and since Yates Correction is being used, 0.5 is subtracted from the absolute value of Fo - Fe, the difference is squared, and the square is divided by its expected frequency. This is repeated for the remaining categories, and as the formula for x2 indicates, these results are summed for all categories.

How does a calculated X2 of 0.450 tell us if our observed results of 12 heads and 8 tails represent a significant deviation from an expected 10-10 split? The shape of the Chi Square sampling distribution depends upon the number of degrees of freedom. The degrees of freedom for a one-way classification X2 is r - 1, where r is the number of levels. In our problem above r = 2, so there would obviously be 1 degree of freedom. From our statistical reference tables, a X2 of 3.84 or greater is needed for X2 to be significant at the .05 level, so we conclude that our X2 of 0.450 in the coin-flipping experiment could have happened by sampling error and the deviations between the observed and expected frequencies are not significant. We would expect any data set yielding a calculated X2 value less than 3.84 with one degree of freedom at least 5% of the time due to chance alone. Therefore, the observed difference is not statistically significant at the .05 level.

Two-Way Classification

The two-way Chi Square is a convenient technique for determining the significance of the difference between the frequencies of occurrence in two or more categories with two or more groups. For example, we may see if there is any difference in the number of freshmen, sophomores, juniors, or seniors in regards to their preference for spectator sports (football, basketball, or baseball). This is called a two-way classification since we would need two bits of information from the students in the sample, their class and their sports preference.

Example of a Two-Way Classification

Suppose an investigator wishes to see if 20 boys and girls respond differently to an attitudinal question regarding the educational value of extracurricular activities and observed the following (A = very valuable, U = uncertain, and D = little value).

Boys A = 60 U = 20 D = 20

Girls A = 40 U = 0 D = 60

Expected frequencies (Fe) for each cell are determined by the following formula. Example - For the cell "Boys - A", the corresponding row subtotal = 100, the corresponding column subtotal = 100, and the total number of observations = 200. NOTE: Row subtotals and column subtotals must have equal sums, and total expected frequencies must equal total observed frequencies. A U D Row Subtotals Boys 60 (50) 20 (10) 20 (40) 100 Girls 40 (50) 0 (10) 60 (40) 100 Column Subtotals 100 20 80 200 Degrees of Freedom = (Rows - 1)(Columns - 1) = (2 - 1)(3 - 1) = 2

Table value of X2.05 with 2 degrees of freedom = 5.991

Therefore, reject null hypothesis.

Degrees of Freedom

A value of X2 cannot be evaluated unless the number of degrees of freedom associated with it is known. The number of degrees of freedom associated with any X2 may be easily computed.

If there is one independent variable, df = r - 1 where r is the number of levels of the independent variable.

If there are two independent variables, df = (r - l) (s - l) where r and s are the number of levels of the first and second independent variables, respectively.

If there are three independent variables, df = (r - l) (s - 1) (t - 1) where r, s, and t are the number of levels of the first, second, and third independent variables, respectively.

Assumptions

Even though a nonparametric statistic does not require a normally distributed population, there still are some restrictions regarding its use.

1. Representative sample (Random)

2. The data must be in frequency form (nominal data) or greater.

3. The individual observations must be independent of each other.

4. Sample size must be adequate. In a 2 x 2 table, Chi Square should not be used if n is less than 20. In a larger table, no expected value should be less than 1, and not more than 20% of the variables can have expected values of less than 5.

5. Distribution basis must be decided on before the data is collected.

6. The sum of the observed frequencies must equal the sum of the expected frequencies.

SELF ASSESSMENT

1. Explain the purpose and importance of Chi Square as a nonparametric statistic.

2. Write the basic computational equation for Chi Square.

3. Explain the difference between a one-way classification and a two-way classification.

4. How do you compute the degrees of freedom for the following:

X2 with one independent variable
X2 with two independent variables
X2 with three independent variables

6. What purpose does Yates correction for continuity serve?

7. What are some of the major differences between parametric and nonparametric statistics?

8. What level of measurement is required for Chi Square?

9. A public opinion polling team in a small town was interested in the type of sporting events that adults in the age bracket of 20-50 years prefer to watch on TV. A random sample of 120 was selected and asked, "Given your preference, would you prefer to watch baseball (P1), basketball (P2), or football (P3) on TV?" Of the respondent, 39 indicated a preference for baseball, 25 selected basketball, and 56 selected football.

Null Hypothesis

General - In the population being sampled, the proportions of people in each category are equal. Ho: P1 = P2 = P3

Specific - In the population being sampled, equal proportions of people prefer baseball, basketball, and football.

 P1 P2 P3 Observed responses (Fo) Expected responses (Fe) Fo - Fe (Fo - Fe)2  Degrees of freedom - (number of levels - 1) =
X2.05 =
Ho = Accept or Reject?

10. In a school with a merit system for pay raises a random sample of the faculty were asked if they wished that system to be continued. Of the 10 faculty members responding, 7 wanted to continue and 3 did not want to continue. Use the one-sample case technique with Yates correction and determine if the difference in proportions are statistically significant at the 0.05 level.

Observed Expected (Fo-Fe-0.5)
 (Fo-Fe-0.5)2 Continue
Discontinue

Degrees of freedom - (number of levels - 1) =
X2.05 =
Ho = Accept or Reject?

11. A representative of a major university was interested in how undergraduate males and females responded differently to a question regarding a proposed athletic fee. Of the 100 males and 100 females who responded, 20 males and 60 females agreed, 70 males and 20 females disagreed, and 10 males and 20 females were undecided. A U D Row Subtotals Males Females Column Subtotals Degrees of Freedom = (Rows - 1)(Columns - 1) =
X2.05 =
Ho = Accept or Reject?           