Measures of Association Based on Chi-Square in Software Get pdf417 2d barcode in Software Measures of Association Based on Chi-Square

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14.4.1 Measures of Association Based on Chi-Square using barcode development for software control to generate, create pdf 417 image in software applications. console app Some notation rst: Le barcode pdf417 for None t Nij denote the number of events that occur with the rst variable x taking on its i th value and the second variable y taking on its j th value. Let N denote the total number of events, the sum of all the Nij s. Let Ni denote the number of events for which the rst variable x takes on its i th value regardless of the value of y; N j is the number of events with the j th value of y regardless of x.

So we have X X Nij Nj D Nij Ni D N D X. Ni D (14.4.1).

14.4 Contingency Table Analysis of Two Distributions In other words, dot pdf417 for None is a placeholder that means, sum over the missing index . N j and Ni are sometimes called the row and column totals or marginals, but we will use these terms cautiously since we can never keep straight which are the rows and which are the columns! The null hypothesis is that the two variables x and y have no association. In this case, the probability of a particular value of x given a particular value of y should be the same as the probability of that value of x regardless of y.

Therefore, in the null hypothesis, the expected number for any Nij , which we will denote nij , can be calculated from only the row and column totals, Ni nij D Nj N which implies nij D Ni N j N (14.4.2).

Notice that if a colum n or row total is zero, then the expected number for all the entries in that column or row is also zero; in that case, the never-occurring bin of x or y should simply be removed from the analysis. The chi-square statistic is now given by equation (14.3.

1), which, in the present case, is summed over all entries in the table:. X .Nij nij /2 nij (14.4.3).

The number of degrees of freedom is equal to the number of entries in the table (product of its row size and column size) minus the number of constraints that have arisen from our use of the data themselves to determine the nij . Each row total and column total is a constraint, except that this overcounts by one, since the total of the column totals and the total of the row totals both equal N , the total number of data points. Therefore, if the table is of size I by J , the number of degrees of freedom is IJ I J C 1.

Equation (14.4.3), along with the chi-square probability function ( 6.

2), now give the signi cance of an association between the variables x and y. Incidentally, the two-sample chi-square test for equality of distributions, equation (14.3.

3), is a special case of equation (14.4.3) with J D 2 and with the y variable simply a label distinguishing the two samples.

Suppose there is a signi cant association. How do we quantify its strength, so that (e.g.

) we can compare the strength of one association with another The idea here is to nd some reparametrization of 2 that maps it into some convenient interval, like 0 to 1, where the result is not dependent on the quantity of data that we happen to sample, but rather depends only on the underlying population from which the data were drawn. There are several different ways of doing this. Two of the more common are called Cramer s V and the contingency coef cient C.

The formula for Cramer s V is r. N min .I 1; J (14.4.4).

where I and J are agai n the numbers of rows and columns, and N is the total number of events. Cramer s V has the pleasant property that it lies between zero and one inclusive, equals zero when there is no association, and equals one only when the association is perfect: All the events in any row lie in one unique column, and vice.
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