By A. I. Kostrikin, I. R. Shafarevich

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**Extra resources for Algebra II - Noncommunicative Rings, Identities**

**Example text**

Let Qii, i = 1, ... , I, j = 1, ... i = 0. In balanced analysis of variance J I L:L:Qiimii (8) i=l i=l is a contrast in the interactions. i = 0, the contrast (8) can also be written as I J L 2:: QijUl2(ii) i=l i=l which involves only the interactions. The most interpretable way of obtaining a contrast in the interactions is to define the interaction contrast in terms of contrasts in the main effects. Let ai, i = 1, ... , I be a contrast in the rows (thus a. = 0) and let bi, j = 1, ... , J be a contrast in the columns (so b.

1. = ( A A Pu - P21 ß(1 - P) ( n~ )2 + n~ . (2) ) Prove equation (2). By comparing the nij 's to the m~~) 's we can examine the nature of the differences in the two binomials. , the (nij - m~T)'s. In order to make accurate evaluations of how well m~~) is predicting nij, the residuals need to be rescaled or standardized. Define the Pearson residuals as A fii = i = 1, 2, j = 1, 2. 39 indicates that more females support legalized abortion than would be expected under H 0 . 27 indicate that proportionately more females support legalized abortion than males.

In particular, the ratio of the two sets of odds should be one. 2. odds ratio If rows and columns are independent then the (Pu/P21) (PI2/P22) P11P22 P12P21 equals one. Proof. -)2; thus the estimated odds ratio is always one. A more interesting approach is to estimate the odds ratio without assuming independence and then see how close the estimated odds ratio is to one. With this approach Pij = nij jn .. 0 3 which is very close to one. 395. Again there is no evidence against independence. 2. Give a 95% confidence interval for the odds ratio.

### Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich

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