Chi-square test

Chi-square test.

The chi-square test is a non-parametric test because it depends only on the set of observed and expected frequencies and degree of freedom. Since the χ2 test is a does not make any assumption about population parameter, it is also called a distribution-free test. χ2the test is a test which describes the magnitudes of difference between observed frequencies and expected (theoretical frequencies under certain assumptions.

In other words, it describes the magnitudes of discrepancy between theory and observations. It is defined as.

$$Where\, O= observed\, frequencies.$$

$$E=expected\, frequencies.$$

Chi-square test distribution typically looks like a normal distribution which is skewed to the right. It is a continuous distribution which assumes only positive value.

The condition of a chi-square test.

1. The number of observation n must be sufficient large I,e () 30.
2. No theoretical frequency should be very small I,e not less than 5.
3. The expected frequency of any item or cell should not be less than 5, then frequencies taking from the adjacent items or cells must be pooled together in order to make it 5 or more than 5 and adjust the degree of freedom accordingly.

Uses of Chi-square test.

1. To test independent of attributes.
2. To test of goodness of fit.
3. To test of population variance.

Test of association / Independence of attributes.

$$H_0= There \,is \,no \,significance\, association\, between\, two\, factors.$$

$$H_1= There\, is\, the\ ,significance\, association\, between\, two\, factors.$$

Note.

In correlation analysis , the degree of relationship is studied two variable, which can be numerically measured such as age and blood pressure, an amount of rainfall and production of rice etc. But sometimes the variable may not measure such as honesty, beauty, hairstyle, gender, a day of the week or it would be in the form of attributes good/bad , day/night etc. and instead of words variable we use word factors.

Test statistic

$$Where\, O= observed\, frequencies.$$

$$E=expected\, frequencies.$$

$$expected\, frequencies.E\,for\,any\,cell\,is\,determined\,by\,E=\frac{R.T×C.T}{N}$$

$$Where R.T=Row\,total$$

$$C.T=Column\,total$$

Degree of freedom

$$It\,is\,given\,by\,(r-1)(c-1)$$

$$Where\,r=number of rows$$

$$c=number\,of\,columns\,of\,the\,contigency\,table$$

Contingency table.

When two-category variable (factors) are recorded, the data can be summarised by counting the observed number of units that all into each of the various intersection of categories levels. The resulting counts are displayed in an array called a contingency table.

Remark.

1. Under the null hypothesis that two attributes are independent, a 2x2 contingency table with 4 cells frequency is shown as follows.

$$Value \,of\,Χ^2=\frac{N(ad-bc)^2}{(a+b)(c+d)(b+d)}$$

2. Yates correction for continuity for 2x2 table. In a 2x2 table, the number of d.f is (2-1) (2-1)=1. If any cell frequency is less than 5, the use of pooling method will result in d.f=0. Since 1 d.f is lost in pooling) which means that less (χ2) must have at least 1 df. In such conditions, we apply correlation given by F. Yates which is usually known as “yates correlation for continuity consisting in adding 0.5 the cell frequency which is less than 5 and then adjusting the remaining the cell frequency according. The chi-square test for a 2x2 contingency table is given as .

$$Value \,of\,Χ^2=\frac{N(ad-bc)^2}{(a+b)(c+d)(b+d)}$$

The formula after correlation is.

$$Value \,of\,Χ^2=\frac{N\,[ad-bc-\frac{N}{2}]^2}{(a+b)(a+c)(c+d)(b+d)}$$

Reference

Kerlinger, F.N. Foundation of Behavioural Research. New Delhi: Surjeet Publication, 2000.

Kothari, C.R. Research Methodology. India: Vishwa Prakashan, 1990.

Singh, M.L. and J.M Singh. Understanding Research Methodology. 1998.

Singh, Mrigendra Lal. Understanding Research Methodology. Nepal: Natiocentreok centre, 2013.