I would like to integrate T-statistics values from multiple, independent tests.   How can this be done?   Since my degrees of freedom are many ($n&gt;1000$) I was thinking to use the Stouffer's method, which is really for Z-statistics score. However, with a high $df$ value, the T distribution and the Gaussian distribution should be identical.  Can I use Stouffer's method to integrate T values at $df&gt;1000$?
121019
I have $k$ pairs of samples from different distributions. For each pair of samples I want to check if the samples are taken from a normal distribution with the same mean and variance. I assume the normal distributions for each pair is different and independent of other pairs. I do not know the mean and variance of any of the distributions.  I am doing a two sample t-test for each of the pairs. I want to combine the $k$ resulting p-values.   I could use Fisher's method but I am wondering about the distribution of the average t-value. If these were z-values rather than t-values I would use Stouffer's z-test. I could also approximate the t-distribution with a z-distribution and than use Stouffer's z-test, but I am not sure if that is a good approximation.  Is there a test like Stouffer's z-test for t-values?  Is there a better way to combine the t-values?
50279
How would you test or check that sampling is IID (Independent and Identically Distributed)? Note that I do not mean Gaussian and Identically Distributed, just IID.  And idea that comes to my mind is to repeatedly split the sample in two sub-samples of equal size, perform the Kolmogorov-Smirnov test and check that the distribution of the p-values is uniform.  Any comment on that approach, and any suggestion is welcome.  Clarification after starting bounty: I am looking for a general test that can be applied to non time series data.
83631
I have the following problem: two groups A and B.  The proportions of married and single are ($p_1$) and ($p_2$) respectively, and the standard deviations ($s_1$) and ($s_2$). The sample consists of  ($n_1$) A and  ($n_2$) B observations.  Now I want to test whether the proportions are equal. I use the following formula to compute the z-statistic:  z = $abs(p_1 - p_2) / \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }$  and test the hypothesis: h0: $p_1 \neq p_2$  An alternative approach I tried is to estimate the t-value based on the standard deviations (hence just comparing means), that is by the formula (assuming unequal variances):  t = $abs(p_1-p_2)/\sqrt{ s_1^2/n_1 + s_2^2/n_2 }$  My question are:    do you agree with my approach or; should it be two-sided or one-sided?   Thank you.
56566
I'm interested in why dependent observations are a problem in statistics. Let's say you want to know if there is a difference in mean exam scores between two schools. You collect 50 observations in each school. These 50 observations are derived from 5 different classrooms in each school and there is dependence within classrooms. In this instance, how would the results of the t-test be affected and how might they lead to inaccurate conclusions?
53680
I have a fundamental question: is it reasonable to use a statistical test to find out if 2 datasets are similar or not? Some comments have called out the T-test as being unable to answer that question. Why is the T-test insufficient? Furthermore, what other tests are there to determine differences in distributions for the two sample problem.
87725
Is it even possible to use a statistical test in this case? Specifically, I'm analysing cities in two different regions. The aim is to determine if there is a statistically significant difference between the amount of cities that have UK names in each region.   The data look like this:                   region1    region2 UK_named_cities  553        3255 total_cities     9155       55327   The percentages of cities with UK names:  region 1: 6.07% region 2: 5.88%   The difference is 0.19%.   My first thought was an $χ2$ test, which yielded this result:  Test Statistic (X-squared): 0.2840672425979418 p-value: 0.5940477093071286 Degrees of Freedom: 1   I'm not sure this is right though. I don't have expected and observed counts as such. And a p-value of 0.59 doesn't seem that intuitive when there is a difference (albeit a very small one). I can't see how a t-test or z-test or reporting CI's would apply either.  Fundamentally, I'm confused. Can I only report that there is indeed a difference, but not test it for significance?
85974
Suppose that $T_1,…,T_{10}$ are iid $Exp(λ)$ and the goal is to test if $H_0:λ≤1$ versus $H_a:λ≥2$. Suppose that the test statistic is $S=\sum_{i=1}^{10}T_i$, and rejection of the null occurs when $S≤7$.  I realize that $T_1...T_{10}$ is a gamma distribution  of $S =Gamma(10, \lambda$), but I am not sure how to use this information to find the p-value for the type 1 error.
13801
Using the student t-distribution with $k &gt; 0$ degrees of freedom, location parameter $l$ and scale parameter $s$ having density  $$\frac{\Gamma \left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\sqrt{k \pi s^2}\right)} \left\{ 1 + k^{-1}\left( \frac{x-l}{s}\right)\right\}^{-(k+1)/2},$$  how to show that the Student $t$-distribution can be written as a mixture of Gaussian distributions by letting $X\sim N(\mu,\sigma^2)$, $\tau = 1/\sigma^2\sim\Gamma(\alpha,\beta)$, and integrating the joint density $f(x,\tau|\mu)$ to get the marginal density $f(x|\mu)$? What are the parameters of the resulting $t$-distribution, as functions of $\mu,\alpha,\beta$?  I got lost in calculus by integrating the joint conditional density with the Gamma distribution.
98418
Q  How to evaluate a multivariate integral with a Gaussian weight function?  $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm f}\left(x_{1},x_{2},\ldots,x_{n}\right)\, {\rm d}x_{1}\,{\rm d}x_{2}\ldots{\rm d}x_{n} $$ Where $\displaystyle{% {\rm f}(x_{1},x_{2},\ldots,x_{n}) =\prod_{j} {1 \over \sqrt{1 + {\rm i}\,b\left(x_{j}^2-x_{j+1}^2\right)^2}}}$. I need a hint to solve this integral and this is how I proceeded: $$ \mathcal{Z_{n}}= \int_{-\infty}^{\infty}\prod_{j=1}^{n}{\rm d}x_{j}\, \exp\left(% -\,{a \over 2}\sum_{j = 1}^{n}x_{j}^{2} - {1 \over 2} \log\left(1 + {\rm i}\,b\left[x_{j}^{2} - x_{j + 1}^{2}\right]^{2}\right)\right) $$ Now the integral is of the form of the canonical partition function integrated over the configuration space. Hence the integral can be identified as an $n-$particle partition of the canonical ensemble, which is given by $$ \mathcal{Z}_{n}= \int_{-\infty}^{\infty}\prod_{j = 1}^{n}{\rm d}x_{j}\,  {\rm e}^{-\beta{\cal H}}, $$ Where $$\mathcal{H}=\Bigg(-\frac{a}{2}\sum_{j=1}^{n}x_{j}^2-\frac{1}{2}\log{\Big(1+i b(x_{j}^2-x_{j+1}^2)^{2}\Big)\Bigg)}.$$ Then I got stuck. How to proceed? Thanks.
102815
I have a variable that has 57 kurtosis, so I decided to transform it to log. However, I have multicolleanirity problem due to interacting this variable and others with another variable so I am using z scores to reduce VIF value. So ...           Is it ok to transform the log variable to z score       thank you so much
87886
I am trying to predict the outcome of a complex system using neural networks (ANN's). The outcome (dependent) values range between 0 and 10,000. The different input variables have different ranges. All the variables have roughly normal distributions.   I consider different options to scale the data before training. One option is to scale the input (independent) and output (dependent) variables to [0, 1] by  using the mean and standard deviation values of each variable, independently. The problem with this method is that if I use the sigmoid activation function at the output, I will very likely miss extreme data, especially those not seen in the training set  Another option is to use a z-score. In that case I don't have the extreme data problem; however, I'm limited to a linear activation function at the output.   What are other accepted normalization techniques that are in use with ANN's? I tried to look for reviews on this topic, but failed to find anything useful.
83058
So, I have this question:    And, what I have tried so far is I let $z_1 = \frac{X + Y}{2}$ and let $z_2 = \frac{X - Y}{2}$ to use change of variables for this question. However, when I work this out and plug it into my joint distribution of $X$ and $Y$ above, I end up with the $z_2$ disappearing and only having $z_1$ left in my joint distribution of $z_1$ and $z_2$. So, then when I go to find the marginal for just $z_1$ (which is $\frac{X + Y}{2}$ &lt;- what I am looking for), there is no $z_2$ therefore, integrating wrt to $z_2$, I get $z_2$ just come up (integral of 1). This is an issue because the bounds are from 0 to $\inf$. Obviously, I can't have infinite in my marginal density so what am I doing wrong?   Also, for question 5, I work it out similarly except I let $z_2 = X$ but I just end up with the exact same density as $f_{x,y}$ as seen in question 4 - is that normal? Please help, I been doing this question for a long time trying various dummy transformations.  Also, for question 4, I found the marginals of $x$ and $y$ and realized that the independence consequence (of the product being equal to the joint) is satisfied but using convolution, I get the same issue of my integral coming out to infinite! Thank you for all the help!   EDIT: Am I doing my bounds wrong? the bound for z_2 is from 0 to inf or 0 to z_1?
30185
Here are some grades. (They aren't test scores. They aren't a sample.) I want to test whether a normal distribution derived from the mean and variance of the grades provides a good fit for the grades. What other distribution might fit is, for now, an orthogonal issue. The question is whether the normal distribution provides a valid model.  &gt; grades   [1] NA 75 70 70 80 75 85 85 75 85 75 80 85 80 75 70 70 85 90 90 NA 85 NA 75 80  [26] 80 85 80 85 85 75 NA 75 70 80 65 80 75 70 80 85 65 85 75 75 85 75 60 85 80  [51] 85 85 70 70 80 75 80 80 NA 80 75 85 80 80 55 75 60 90 80 75 80 70 85 75 80  [76] 75 80 75 75 70 85 85 80 80 75 70 NA 75 70 75 70 65 85 70 70 80 80 85 85 70 [101] 75 70 75 75 65 70 80 75 65 65 &gt; t &lt;- table(grades) &gt; t grades 55 60 65 70 75 80 85 90  1  2  6 18 27 25 22  3   The grades come from a discrete set of possible values from five to 100 in increments of five. Because of that, a Chi-squared test might be appropriate. First, the normal distribution probabilities that any of the discrete values will occur:  &gt; p &lt;- dnorm(seq(55,90,5), mean(grades, na.rm=TRUE), sd(grades, na.rm=TRUE)) &gt; p [1] 0.0004433536 0.0031786310 0.0136901824 0.0354207364 0.0550535301 [6] 0.0514034173 0.0288322176 0.0097150119   Why are the values of p so small? Why do they not add-up to one?  &gt; tf &lt;- as.data.frame(t) &gt; tf   grades Freq 1     55    1 2     60    2 3     65    6 4     70   18 5     75   27 6     80   25 7     85   22 8     90    3 &gt; chisq.test(tf$Freq, p=p, rescale.p=TRUE)      Chi-squared test for given probabilities  data:  tf$Freq X-squared = 7.0452, df = 7, p-value = 0.4242  Warning message: In chisq.test(tf$Freq, p = p, rescale.p = TRUE) :   Chi-squared approximation may be incorrect   Why does the chisq.test function produce the warning?  I need numbers, because I have a lot of these data sets and want fit results for a large sample of them. Visual inspection isn't proof. The closest relevant question I've found is  ("Testing Whether a Binomial Distribution Fits Data")
15072
There is a very confusing question in my mind. I have data, and would like to compare numeric scores between men and women.  There is a big difference in those two groups: the number of men is 34, while the number of women is 310, and variances are not equal.  As far as I know, when the variances are not equal, I can use the Welch–Satterthwaite equation (independent t-test assuming unequal variances). My question is: can I still use this equation although there is a really big difference in sample size between my two samples? Or is there a certain limit for the difference in sample size between two samples?
68510
Why can't we  separately test each variable(using t-test), whether or not it is equal to 0? Why should we use the F-test for checking if all coefficients combined are equal to 0?
99953
Situation: Comparing mean scores of level of satisfaction across multiple institutions and overall.   Let's say there are 50 institutions, all with different sample sizes.   Level of satisfaction is rated 1 to 5.  I was asked to perform a t-Test to determine outliers and significance of the score differences, with a control range for 1 SD and 2 SD.   Eg. Institute = Mean score  Institute A = 3.8 Institute B = 3.9 Institute C = 4.1 . . . Overall = 4.0  This is so that we can rank the institutions based on their mean score.  So here's my question:   How do I make a fair comparison of the mean scores? It is possible? How does a t-Test determine statistical significance in this scenario? Do I look at P-value or overlap in distribution to assess usefulness of the test?  I conducted multiple t-Tests for 2 groups (eg. Institute A &amp; Overall, Institute B &amp; Overall, Institute C &amp; Overall, ...) but I'm aware that it is not appropriate to conduct multiple t-Tests comparing 2 groups at a time. What should I do?   Please kindly help. I am quite lost and confused.
103675
I've been going back over my notes from Stats class and came across the Probability Integral Transform.  From my limited understanding, the basic idea is that from a cdf in terms of one variable, can be transformed into another cdf in terms of different variable:    i.e. from $F_x(x)$ to --> $F_y(y)$   Is this understanding correct?  What is the purpose behind this?  Finally, is there a general procedure in performing the transformation?