1 Chi-square Test Dr. T. T. Kachwala

Using the Chi-Square Test 2 The following are the two Applications: 1. Chi square as a test of Independence 2.Chi square as a test of Goodness of Fit  2 statistics measures difference between f o and f e is always positive if f e is small, it is over estimated (limitation of  2 statistics )

Chi-square as a Test of Independence 3 Objective If we classify a population into two attributes with several classes, we can use Chi-square test to determine whether or not the two attributes are independent of each other.

Chi-square as a test of Independence 4 Contingency Table: It is a table of classification of two attributes into a number of classes. One of the attributes is classified along the rows, while the second attribute is classified along the columns.

Chi-square as a test of Independence 5 Observed and Expected Frequency 1.Observed frequency is based on actual observation (Study or Survey) 2.Expected frequency is based on theoretical calculations. The expected or theoretical frequency can be obtained for any cell as follows: f e =expected frequency of cell RT =Row total corresponding to that cell CT =Column total corresponding to that cell N =Total number of observation

Degree of Freedom (for Contingency table of size m * n) 6 Degree of Freedom is the number of independent observations that can be arbitrarily assigned without violating the restrictions of the problem. For example: Table of size 2*2; = 1 Degree of Freedom for table of size m x n, is given by the following formula: = (m-1)  (n-1){where m is the number of rows & n is the number of columns}. For example: Table of size 2*2 : m = 2, n = 2; = 1*1 = 1 Table of size 3*2 : m = 3, n = 2; = 2*1 = 2

The Chi-Square Distribution 7  2 Critical 1.The sampling distribution of the statistic  2 can be closely approximated by a continuous curve known as Chi-square distribution. 2.As in case of t distribution, there is different  2 distribution for each different number of degrees of freedom. However in practical research work, only a few values of  are popular (0.05, 0.01, 0.1). 3.Chi-square distribution is a skewed distribution & is defined by level of significance  & degree of freedom as indicated below (for  = 0.05) : 0.05

Summarized procedure for Chi-Square Test of Independence 8 Step (i)H 0 : The two attributes are independent H 1 : One attribute depends on the other attribute Step (ii)Assuming α = 0.05,  2 distribution & = (m-1) * (n-1)  2 Critical = {from table} Step (iii) Calculate  2 statistic

Summarized procedure for Chi Square as a Test of Independence 9 Step (iv & v) : Decision Rule & Conclusion  2 Critical If  2 statistic is in acceptance area Accept H 0 : the two attributes are independent If  2 statistic is in rejection area Reject H 0 : the two attributes are independent Accept H 1 : One attribute depends on the other

Chi-square as a Test of Goodness of Fit 10 Objective To assess whether or not there is a significant difference between observed frequency f o & expected frequency f e The term Goodness of Fit signifies how well the theoretical distribution like Binomial, Poisson or Normal distribution fits or represents the observed frequency distribution

Chi-square as a Test of Goodness of Fit 11 The following is the summarized procedure: Step (i)H 0 : f o = f e (Theoretical distribution is a good fit) H 1 : Not all f o are equal to f e (Theoretical distribution is not a good fit) Step (ii) Assuming α =0.05,  2 critical = {from table depending on } ( depends on the Probability Distribution) Step (iii) Calculate  2 statistic

Chi-square as a Test of Goodness of Fit 12 Step (iv & v) : Decision Rule & Conclusion  2 Critical If  2 statistic lies in acceptance area Accept H 0 : f o = f e i..e Theoretical distribution is a Good Fit If  2 statistic lies in rejection area Reject H 0 : f o = f e Accept H 1 : Not all f o are equal to f e i.e. Theoretical distribution is not a Good Fit for the given f o

Precaution about using Chi-square Test 13 When the expected frequencies f e are too small, the value of  2 will be overestimated 1.To avoid making incorrect inferences from  2 test follow the general rule that an expected frequency of less than five in one cell of the contingency table is too small to use. 2.One of the suggested adjustments for small value of f e is combining frequency of the cells in the table. i.e. if the table contains more than one cell with an expected frequency of less than 5, we can combine them in order to get an expected frequency of 5 or more.

14 Thanks and Good Luck Dr. T. T. Kachwala