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Since this is a biased estimate of the variance of the unobserved errors, the bias is removed by dividing the sum of the squared residuals by df = n − p − 1, instead of n, where df is the number of degrees of freedom (n minus the number of parameters (excluding the intercept) p being estimated - 1). This forms an unbiased estimate of the ...
In null-hypothesis significance testing, the p-value [note 1] is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. [2] [3] A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis.
Note that a p-value of 0.01 suggests that 1% of the time a result at least that extreme would be obtained by chance; if hundreds or thousands of hypotheses (with mutually relatively uncorrelated independent variables) are tested, then one is likely to obtain a p-value less than 0.01 for many null hypotheses.
The p-value was introduced by Karl Pearson [6] in the Pearson's chi-squared test, where he defined P (original notation) as the probability that the statistic would be at or above a given level. This is a one-tailed definition, and the chi-squared distribution is asymmetric, only assuming positive or zero values, and has only one tail, the ...
The procedures of Bonferroni and Holm control the FWER under any dependence structure of the p-values (or equivalently the individual test statistics).Essentially, this is achieved by accommodating a `worst-case' dependence structure (which is close to independence for most practical purposes).
The Bonferroni correction can also be applied as a p-value adjustment: Using that approach, instead of adjusting the alpha level, each p-value is multiplied by the number of tests (with adjusted p-values that exceed 1 then being reduced to 1), and the alpha level is left unchanged.
The Lagrange multiplier (LM) test statistic is the product of the R 2 value and sample size: =. This follows a chi-squared distribution, with degrees of freedom equal to P − 1, where P is the number of estimated parameters (in the auxiliary regression). The logic of the test is as follows.
The p-value is not the probability that the observed effects were produced by random chance alone. [2] The p-value is computed under the assumption that a certain model, usually the null hypothesis, is true. This means that the p-value is a statement about the relation of the data to that hypothesis. [2]