# 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 ### Different z and Wald values in logistic regression analyses between SPSS and R 0 Asked on February 23, 2021 by afton-nelson ### Creating an index using transactional data 0 Asked on February 23, 2021 by jamzy ### How does a pdf change after a variable transformation with another random variable? 1 Asked on February 23, 2021 by fluctuation ### A tail bound for an unknown distribution via sampling 1 Asked on February 22, 2021 by bashir ### Negative Sampling in Word2Vec – Embedding Vector / Amount of Samples 0 Asked on February 22, 2021 by maxbit ### NxN- fold cross validation 1 Asked on February 21, 2021 by tauling ### Compare regression coefficients within the same model 2 Asked on February 21, 2021 by shenflow ### How to predict an event for different time intervals and compute score? 1 Asked on February 21, 2021 ### What are arguments against using the (log-)likelihood as a loss function? 1 Asked on February 21, 2021 by joel ### How to interpret a ROC curve? 5 Asked on February 20, 2021 by gnal ### Minimum sample size per cluster in a random effect model 2 Asked on February 20, 2021 by roccer ### Outlier detection for a univariate categorical variable? 3 Asked on February 20, 2021 by abdu ### COVID-19: Individual level, cross-country Panel dataset. Which method to use? 0 Asked on February 20, 2021 by user307627 ### Partial dependence plot, GBM multinomial 1 Asked on February 19, 2021 by daniel-l ### Why is it ok to remove variables with low variance from a dataset 0 Asked on February 19, 2021 by amnon ### In a predefined (but realistic, and explaining) example, what$p(x)\$ should be in a variational autoencoder?

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