#### Contingency tables, Biology tutorial

Introduction:

Contingency tables are generally constructed for the aim of studying the relationship among two or more variables of classification. One might wish for to recognize whether the two variables are independence or there is a relationship between them.  By means of chi square (χ2) test it is possible to test the hypothesis which the two variables are independent (that is, Independence test).

Chi-square Distribution:

Chi-square (χ2) is the general process or method for testing the compatibility based on a measure of the extent to which the observed and estimated frequencies agree. Chi-square is as well, termed to as test for the homogeneity randomness, association, independence and goodness of fit. The suppositions for the chi-square goodness-of-fit test are as follows:

1) The data are acquired from the random sample.

2) The expected frequency for each and every category should be 5 or more.

3) Chi-square can be represented by the formula:

χ2 = i=1Σk [(oi - ei)2/ei]

Where oi and ei symbolize the observed and expected frequencies, correspondingly for the ith cell and k symbolizes the number of cells.

The frequencies we examine are compared to those we wait for on the basis of some null hypotheses. When the differences among the observed and expected frequencies are great and surpass the critical value at suitable degrees of freedom, we are then forced to the refuse the null hypothesis (Ho) and accept the alternative hypothesis (H1).

Properties of chi-square:

1) It is mainly concerned with the series of normally distribution observations, thus approximating the variance.

2) Chi-square values tend to become normal as the sample size (n) rises.  The degree of freedom is (n - 1).

3) Chi-square is non-symmetrical

4) χ2 can take any value from zero to the infinity.

5) χ2 is additive and for all time positive.

6) Chi-square can be:

i) One-way categorization as in testing the goodness of fit of a hypothesis, having two or more classes.

ii) Two-way categorization as in finding out relation or differences among the two different classes or the test might involve more than two classes.

Chi-square Testing:

Chi-square distributions are employed in a procedure which includes the comparison of the differences among the sample frequencies of percentages or occurrences which are in reality observed and the theoretical or hypothetical population frequencies of occurrences or percentages which are expected when the hypothesis is true.

Steps in the general χ2 testing procedure:

a) Create the null and alternative hypotheses.

b) Choose the level of significance to be employed in the specific testing condition.

c) Take arbitrary samples from the populations, and record the observed frequencies which are in reality obtained.

d) Calculate the frequencies of percentages which would be estimated when the null hypothesis is true.

e) Make use of the observed and the estimated frequencies to calculate the χ2.

f) Compare the value of χ2 calculated in step (e) with the χ2 table value at the particular level of significance (step b).

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