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We express the probabilities of transformation in what are called log-odds scores. The scores matrix S is defined as The scores matrix S is defined as S i , j = log p i ⋅ M i , j p i ⋅ p j = log M i , j p j = log observed frequency expected frequency , {\displaystyle S_{i,j}=\log {\frac {p_{i}\cdot M_{i,j}}{p_{i}\cdot p_{j ...
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of event A taking place in the presence of B, and the odds of A in the absence of B. Due to symmetry, odds ratio reciprocally calculates the ratio of the odds of B occurring in the presence of A, and the odds of B in the absence of A.
Log probabilities make some mathematical manipulations easier to perform. Optimization. Since most common probability distributions —notably the exponential family —are only logarithmically concave , [ 2 ] [ 3 ] and concavity of the objective function plays a key role in the maximization of a function such as probability, optimizers work ...
In practice the odds ratio is commonly used for case-control studies, as the relative risk cannot be estimated. [1] In fact, the odds ratio has much more common use in statistics, since logistic regression, often associated with clinical trials, works with the log of the odds ratio, not relative risk. Because the (natural log of the) odds of a ...
The log diagnostic odds ratio can also be used to study the trade-off between sensitivity and specificity [5] [6] by expressing the log diagnostic odds ratio in terms of the logit of the true positive rate (sensitivity) and false positive rate (1 − specificity), and by additionally constructing a measure, :
In this case the odds ratio for allele T is A:B (meaning 'A to B', in standard odds terminology) divided by X:Y, which in mathematical notation is simply (A/B)/(X/Y). When the allele frequency in the case group is much higher than in the control group, the odds ratio is higher than 1, and vice versa for lower allele frequency.
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If p is a probability, then p/(1 − p) is the corresponding odds; the logit of the probability is the logarithm of the odds, i.e.: = = = = (). The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base e is the one most often used.