## Solved – Differences between poisson.test and E-test when testing Poisson parameters

Suppose we have two independent Poisson-distributed variables \$X_1\$ and \$X_2\$. We want to test whether the Poisson parameters are equal, i.e. whether \$lambda_1=lambda_2\$. Now we have 4 distinct statistical exact tests to choose: E-test (see Krishnamoorthy and Thomson, A more powerful test for comparing two Poisson means, Journal of Statistical Planning and Inference 119 (2004) … Read more

## Solved – Taylor series expansion of maximum likelihood estimator, Newton-Raphson, Fisher scoring and distribution of MLE by Delta method

Assume \$ellleft(thetaright)\$ is the log-likelihood of parameter vector \$theta\$ and \$widehat{theta}\$ is the maximum likelihood estimator of \$theta\$ then the Taylor series of \$ellleft(thetaright)\$ about \$widehat{theta}\$ is begin{align*} ellleft(thetaright) & approxeqellleft(widehat{theta}right)+frac{partialellleft(thetaright)}{partialtheta}Bigr|_{theta=widehat{theta}}left(theta-widehat{theta}right)+frac{1}{2}left(theta-widehat{theta}right)^{prime}frac{partial^{2}ellleft(thetaright)}{partialthetapartialtheta^{prime}}Bigr|_{theta=widehat{theta}}left(theta-widehat{theta}right)\ frac{ellleft(thetaright)}{partialtheta} & approxeqmathbf{0}+left(mathbf{1}-mathbf{0}right)frac{partialellleft(thetaright)}{partialtheta}Bigr|_{theta=widehat{theta}}+frac{partial^{2}ellleft(thetaright)}{partialthetapartialtheta^{prime}}Bigr|_{theta=widehat{theta}}left(theta-widehat{theta}right)quadoverset{textrm{set}}{=}quadmathbf{0}\ \ theta-widehat{theta} & =-left[frac{partial^{2}ellleft(thetaright)}{partialthetapartialtheta^{prime}}Bigr|_{theta=widehat{theta}}right]^{-}left[frac{partialellleft(thetaright)}{partialtheta}Bigr|_{theta=widehat{theta}}right]\ widehat{theta}-theta & =left[frac{partial^{2}ellleft(thetaright)}{partialthetapartialtheta^{prime}}Bigr|_{theta=widehat{theta}}right]^{-}left[frac{partialellleft(thetaright)}{partialtheta}Bigr|_{theta=widehat{theta}}right]\ widehat{theta}-theta & =left[mathbb{H}left(thetaright)Bigr|_{theta=widehat{theta}}right]^{-}left[mathbb{S}left(thetaright)Bigr|_{theta=widehat{theta}}right] end{align*} As \$ theta=widehat{theta}-left[mathbb{H}left(thetaright)Bigr|_{theta=widehat{theta}}right]^{-}left[mathbb{S}left(thetaright)Bigr|_{theta=widehat{theta}}right] \$ So begin{align*} theta^{left(m+1right)} & =theta^{left(mright)}-left[mathbb{H}left(theta^{left(mright)}right)right]^{-}mathbb{S}left(theta^{left(mright)}right)quadquadleft({textrm{Newton-Raphson}}right)\ \ … Read more

## Solved – Aside from Durbin-Watson, what hypothesis tests can produce inconclusive results

The Durbin-Watson test statistic can lie in an inconclusive region, where it is not possible either to reject or fail to reject the null hypothesis (in this case, of zero autocorrelation). What other statistical tests can produce "inconclusive" results? Is there a general explanation (hand-waving is fine) for why this set of tests are unable … Read more

## Solved – Aside from Durbin-Watson, what hypothesis tests can produce inconclusive results

The Durbin-Watson test statistic can lie in an inconclusive region, where it is not possible either to reject or fail to reject the null hypothesis (in this case, of zero autocorrelation). What other statistical tests can produce "inconclusive" results? Is there a general explanation (hand-waving is fine) for why this set of tests are unable … Read more

## Solved – Why is it wrong to stop an A/B test before optimal sample size is reached

I am in charge of presenting the results of A/B tests (run on website variations) at my company. We run the test for a month and then check the p-values at regular intervals until we reach significance (or abandon if significance is not reached after running the test for a long time), something I am … Read more

## Solved – Why is it wrong to stop an A/B test before optimal sample size is reached

I am in charge of presenting the results of A/B tests (run on website variations) at my company. We run the test for a month and then check the p-values at regular intervals until we reach significance (or abandon if significance is not reached after running the test for a long time), something I am … Read more

## Solved – Statistical significance of sample variance

Suppose that I have a vector X of 50 numbers between 1 and 10. Then I calculate the variance of X. Let us say it turns out to be 8, that is, VAR(X) is equal to 8. Is it meaningful to ask the statistical significance of this result? If yes, how can I calculate it? … Read more

## Solved – Statistical significance of sample variance

Suppose that I have a vector X of 50 numbers between 1 and 10. Then I calculate the variance of X. Let us say it turns out to be 8, that is, VAR(X) is equal to 8. Is it meaningful to ask the statistical significance of this result? If yes, how can I calculate it? … Read more

## Solved – Can a meta-analysis of studies which are all “not statistically signficant” lead to a “significant” conclusion

A meta-analysis includes a bunch of studies, all of which reported a P value greater than 0.05. Is it possible for the overall meta-analysis to report a P value less than 0.05? Under what circumstances? (I am pretty sure the answer is yes, but I'd like a reference or explanation.) Best Answer In theory, yes… … Read more

## Solved – Analysing data from non-equivalent groups in a quasi-experimental design

I am doing a research on two groups, one experimental and one control, which are not randomly selected. In fact the two groups are intact groups. The purpose is to find the effect of the number of languages that the learners know on their academic achievement. The selected design is Pretest Posttest Nonequivalent Group. Which … Read more