# Does the t statistic have uses unrelated to hypothesis testing?

Cross Validated Asked on January 1, 2022

Are there non-hypothesis testing uses for Student’s t statistic?

When you ask about "the t-statistic", I think about the concrete quantity $$frac {{bar {X}}-mu }{S/{sqrt {n}}}$$

To actually calculate this quantity, we have to specify $$mu$$. This is typically chosen in reference to some given null hypothesis. So to me it seems awkward to try to disentangle "the statistic" from the null hypothesis to which it is implicitly linked by $$mu$$. Setting $$mu$$ to 0, for example, which you're implicitly doing when you type t.test(rnorm(10))$statistic into R, is implicitly related to the hypothesis test $$H_0: mu = 0$$. Where I think of Student's t-distribution as useful is as a parametric form for fitting to data. At the end of the day, it's just another symmetric, bell-shaped distribution. It just has fatter tails than a Gaussian. So it can be used to model things for which you'd like to preserve that symmetry and bell-shape, but give the extreme outcomes more probability mass than a Gaussian does. I know it's used in finance to model asset-returns (link 1, link 2) for example, but I can't speak to how successful or useful these kinds of models are because I don't use them myself. I'd suspect them to be of particular use to hierarchical modelers who have some prior knowledge that points to fat tails. Gelman briefly discusses using the t instead of the the Gaussian in fat-tail situations in section 17.2 of Bayesian Data Analysis. Answered by klumbard on January 1, 2022 A "hypothesis test" in the strictest sense always results in a binary outcome of either rejecting or failing to reject a null hypothesis. T-statistics are generally turned into p-values, which are then compared against some pre-defined threshold to make that binary determination. It is possible to use the t-statistic itself, however, as a general measure of "deviation from the null", without ever having to take the final step of testing whether the null hypothesis should be rejected or not. Using a t-statistic in this way is still derived from a hypothesis testing framework, but does not actually result in a test of whether the null should be rejected or not, so I'd argue this is not strictly a "hypothesis test". As an example, the t-statistic can be used as a means of ranking features by significance, while accounting for the directionality of the differences. Gene set enrichment analysis, for example, searches for sets of consistently up- or down-regulated genes, so the directionality of differences is important for this method. Ranking features by their p-value will draw no distinction between up- and down-regulated genes, and simply put the most significant genes at the top of the list. Ranking by the t-statistic, on the other hand, will put the most significant up-regulated genes at one end of the list, and the most significant down-regulated genes at the other end. Although the magnitude of the t-statistic is directly related to the p-value, the sign of the t-statistic is lost when calculating a p-value for a hypothesis test. Ranking genes in this way respects the directionality and how incompatible with the null hypothesis each gene is, but does not actually make any determination if any gene is "significantly dysregulated" or not. Answered by Nuclear Hoagie on January 1, 2022 ## Add your own answers! ## Related Questions ### How to determine relationship categorical and numerical data 1 Asked on January 9, 2021 by onhalu ### Multiple Poisson regression (?) in R 2 Asked on January 9, 2021 by jonas8 ### Propose a model for this time series 1 Asked on January 8, 2021 by le-anh-dung ### Would a 3D CNN require less training samples than a corresponding 2D CNN? 0 Asked on January 8, 2021 by alexander-soare ### Can regression to the mean be corrected by linear mixed effects? 0 Asked on January 8, 2021 by lili ### T value vs T-stat 1 Asked on January 8, 2021 by student010101 ### How can I perform a two-sample multivariate t-test where one group is a subset of the other? 0 Asked on January 7, 2021 by grint ### Minimize the limit of K-L (Kullback Leibler) divergence for a given conditional probability$p(y|x)\$ distribution?

0  Asked on January 7, 2021

### Can I use coefficients of one set of regressions as dependent variable in a new regression?

1  Asked on January 7, 2021 by jeremy

### What’s a word meaning “drawn from the same distribution”?

0  Asked on January 6, 2021 by gkhagb

### What Statistical principles are being violated by comparing specific Trainer Fatality Rates to Race Track Fatality rates?

0  Asked on January 6, 2021 by pseudoego

### How to automatically choose the number of components for PCA?

1  Asked on January 6, 2021 by foobar

### Cosine Similarity Intuition

3  Asked on January 6, 2021 by ccb

### Is there a way to get the optimal cutoff points based on probability of topic models and the outcomes?

1  Asked on January 6, 2021 by kuni

### How can I use the box plot to explain the Empirical Rule for a normal distribution?

1  Asked on January 6, 2021 by storymay

### PCA: Dimension Reduction

0  Asked on January 5, 2021 by shank

### How to choose a good operation point from precision recall curves?

4  Asked on January 5, 2021 by amelio-vazquez-reina

### How to develop a likelihood based prediction model to predict chance of rain in a particular hour of a year?

0  Asked on January 5, 2021 by nahid

### How well does GAN (generative adversarial network) perform for small samples?

1  Asked on January 4, 2021

### Using the Hotelling package in R

1  Asked on January 4, 2021 by pitchounet