# Tag Info

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Your third item is the one I have seen the most often used as rigorous definition. The others are interesting too (+1). In particular the first is appealing, with the difficulty that the sample size not being (yet) defined, it is harder to define the "from" set. To me, the fundamental intuition of the likelihood is that it is a function of the model + its ...

6

If you are concerned about sample size and significance, good concepts to start out with include effect size and power (while on the topic of CIs, you might want to include accuracy as well) As noted previously, 95% CI refers not to probability but to confidence; it is not the likelihood that the current CI contains the population parameter, but that out of ...

6

I think I would call it something different. Likelihood is the probability density for the observed x given the value of the parameter $θ$ expressed as a function of $θ$ for the given $x$. I don't share the view about the proportionality constant. I think that only comes into play because maximizing any monotonic function of the likelihood gives the same ...

6

Fisher uses p-values as a continuous measure of evidence against a null hypothesis? Perhaps. What convinces you of this? So a p-value of 0.06 would indicate that there is no difference and the null hypothesis is true? Not at all. How did you go from 'continuous measure of evidence against' to 'there is no difference'? In particular, Fisher would ...

5

In 1946, geophysicist and Bayesian statistician Harold Jeffreys introduced what we today call the Kullback-Leibler divergence, and discovered that for two distributions that are "infinitely close" (let's hope that Math SE guys don't see this ;-) we can write their Kullback-Leibler divergence as a quadratic form whose coefficients are given by the elements of ...

5

You are correct to be suspicious and you are correct that problems arise from some of the low cell counts in this case. However, there is nothing wrong with Fisher's test itself. We just need to be careful in interpreting its results. Let's review the data: 0 1 Total Site 1 7 2 | 9 Site 2 95 9 | 104 Site 3 0 1 | 1 ...

5

A $p$-value is the probability of someone getting a test statistic as far from 0 as it is observed or farther, given the sample size at hand. It is not related with type I errors. To use MacKay's definition in his book Information Theory, Inference, and Learning Algorithms: p-value is the probability, given a null hypothesis for the probability ...

5

You can perfectly use the mean $p$-value. Fisher’s method set sets a threshold $s_\alpha$ on $-2 \sum_{i=1}^n \log p_i$, such that if the null hypothesis $H_0$ : all $p$-values are $\sim U(0,1)$ holds, then $-2 \sum_i \log p_i$ exceeds $s_\alpha$ with probability $\alpha$. $H_0$ is rejected when this happens. Usually one takes $\alpha = 0.05$ and ...

4

The question asks for ways to display bivariate discrete data using Excel. Although that software is notoriously limited in graphical capabilities, it is still able to generate many different useful graphics. Let's use the example to illustrate. Here are the data, laid out as they might be in a spreadsheet: X, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 Y, ...

3

i) First, a recommendation: Use pchisq( -2*sum(log(p-values)), df, lower.tail=FALSE) instead of 1- ... - you're likely to end up with more accuracy for small p-values. To see that they're sometimes going to give different results, try this: x=70;c(1-pchisq(x,1),pchisq(x,1,lower.tail=FALSE)) ii) Yes, it's one-sided. Small values of the chi-square ...

3

Here's an attempt at a rigorous mathematical definition: Let $X: \Omega \to \mathbb R^n$ be a random vector which admits a density $f(x | \theta_0)$ with respect to some measure $\nu$ on $\mathbb R^n$, where for $\theta \in \Theta$, $\{f(x|\theta): \theta \in \Theta\}$ is a family of densities on $\mathbb R^n$ with respect to $\nu$. Then, for any $x \in ... 3 Generally speaking, the likelihood ratio and the ordinary Pearson$\chi^2$tests are more accurate than Fisher's "exact" test. But for your situation you need an extremely heavy multiplicity adjustment thrown in, not matter which statistical test is used. Decision trees such as the one you are building require amazingly large datasets for their structure ... 3 For many reasons, classification is not a good goal for most problems; prediction is. Logistic regression (LR) is a more direct probability model to use for prediction, with fewer assumptions. Linear discriminant analysis (LDA) assumes that X has a multivariate normal distribution given Y. Using Bayes' rule to get Prob(Y|X) you get a logistic model. So ... 3 So if you did three studies of similar sizes and got a p-value of 0.05 on all three occasions, your intuition is that the "true value" should be 0.05? My intuition is different. Multiple similar results would seem to make the significance higher. I fear I may be mixing the NP and Fisherian paradigms here. At any rate under the null hypothesis of no ... 2 Have a look at this work on the Tracklet Descriptor. In this case, they need to compute distances between time series of different lengths, and they use a nice dynamic time warping approach to inflate/deflate the two time series in question to find a minimum distance between them based on an idea of optimally distorting them. Outside of a reasonable set of ... 2 If you want to correlate X and Y, then you could consider point biserial correlation:$R_{pb} = \frac{M_1 - M_0}{s_n}\frac{n_1n_0}{n*2}$where$s_n$is the standard deviation of (in your case) Y and the subscripts refer to the two groups as defined by X, if you are willing to posit that Y is "really" continuous (e.g. it's a marker on some latent scale). If ... 2 Including the ten in the analysis you get a significant result and without the 10 you don't get a significant result from prop.test in R. You can simulate your situation in R in order to see if you trust the approximate result: # number of simulation replicates n <- 10000 # variable to hold the output is10 <- rep(NA, times = n) for(i in 1:n){ # ... 2 There is a generalization of the Fisher Exact test for$2 \times2$Tables to more general$R \times C$. In your case$R=2$and$C=9$. The null hypothesis is that the column proportion is independent of the row. So it tests a null hypothesis that the proportions are the same for each configuration versus the alternative that they differ. So the test is ... 2 See the following paper for a very thourough, modern discussion: Bjørnstad, J. F. (1996). On the Generalization of the Likelihood Function and the Likelihood Principle. Journal of the American Statistical Association 91: 791-806. 2 You can find a similar relationship (for a one-dimensional parameter) in equation (3) of the following paper D. Guo (2009), Relative Entropy and Score Function: New Information–Estimation Relationships through Arbitrary Additive Perturbation, in Proc. IEEE International Symposium on Information Theory, 814–818. (stable link). The ... 2 Logistic regression may be the easiest way to do it, if you want the idea of the full 3 dimensional table then include teh interaction term between your predictor variables. There is also the loglin function that will analyze multidimensional contingency tables (though it is more like the chi-squared test than Fisher's exact test). Fisher's Exact test ... 2 Here is a short tale about Linear Discriminant Analysis (LDA) as a reply to the question. When we have one variable and$k$groups to discriminate by it, this is ANOVA. The discrimination power of the variable is$SS_{between groups} / SS_{within groups}$, or$B/W$. When we have$p$variables, this is MANOVA. If the variables are uncorrelated neither in ... 2 'N' or number of samples is usually the number of cases, this can be the number of subject (assuming you have one measurement of each feature per subject) or the number of measurements per feature. If you have multiple measurements per feature, per subject you will need to account for this. Generally, EEG channels should be considered separately, as ... 2 If the null hypothesis is true then the p-values of a test should follow a continuous standard uniform distribution. Think of it this way: Say we a priori decided a significance level of .05 (5%). That would mean that if we were to repeat our experiment many times and our null hypothesis is true we want to (incorrectly) reject our true null hypothesis in ... 2 The issue here is that you need to be clearer on the definitions of these terms, and what those definitions imply. Taking the p-value as a continuous measure of evidence against the null hypothesis means that there is no 'bright line' between "no difference" and "difference". As a result of this,$p=.04$is essentially identical to$p=.06$,$p=.06$is ... 2 I will try for a more intuitive explanation: Consider the case where you are asked to compare 2 distances to see which is further, but one of the distances is given to you in miles and the other in kilometers. You cannot compare the 2 numbers directly, but first need to convert at least one of them. But, it does not matter if you convert the kilometers to ... 1 The multivariate normal central moment generating function is$\exp(t'St/2)$, from which it follows that the mixed fourth central moments are (in two different notations):$m_{22} = m_{20} m_{02} + 2 m_{11}^2 \quad\quad = s_{1122} = s_{11} s_{22} + 2 s_{12}^2 = s_{1}^2 s_{2}^2 (1 + 2 r_{12}^2)m_{31} = 3 m_{20} m_{11} \quad\quad = s_{1112} = 3 ...

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Very short "answer": we've discussed in your other question the roots of the idea of geometrization of statistical models by Jeffreys, how they become Riemannian manifolds with metric $g_{ij}$, etc. Much later, classical statisticians such as Efron, Amari, etc, got interested in the idea of the curvature of these manifolds, because there is a link with the ...

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Whether to use 1 sided or 2 sided tests does not depend on whether the independent variable is categorical or not. It depends on your hypotheses. One sided hypotheses (e.g. $H_0$ industrial companies will be lower or equal to nonindustrial companies on the dependent variable) get 1 sided tests. Two sided hypotheses (e.g. $H_0$ industrial companies will not ...

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The Fisher combination test is intended to combine information from separate tests done on independent data sets in order to obtain power when the individual tests may not have sufficient power. The idea is that if the $k$ null hypotheses are all correct the $p$-value will be uniformly distributed on $[0,1]$ indpendently of each other. This means that \$-2 ∑ ...

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