The p-value from the T-test is denoted as P(|T|>=t). Yet, observing a t-statistic similair or more extreme then T under H0 based on what I know does not fit with this notation. Often a small letter is used to depict a statistic and a large letter the hypothesis (parameters). I am also confused to the order P(|T|>=t) because hypothesised value T, is "reversed" with the statistic (t) and the assumption on H0 is left out.
I currently expect for a two sided test P(|t|>=x| T) since P(Data>=x|H0), here the small |t| the absolute t-statistic, x=0 and the larger value T would depict the assumed H0. The probability of the t-statistic (t) being similair or more extreme than the 0 (x) under (T). I am (assume to be) wrong, but then I would like to understand why?
Thank you in advance for the clarification.
P.S. I am mixing Baysian and Frequentist notations here.
Usually capital letters are used to denote random variables, and corresponding small letters to denote realizations of these random variables.
T is a random variable, which is t(𝜈)-distributed under H0.
t is your sample statistic, which may, under H0, be considered a realization of T.
If t is a realization of T, the large values of |t| are unlikely.
The notation Pr(|T|≥t) is slightly strange, however. For the two-sided test this should be Pr(|T|≥|t|): the probability that the absolute value random variable T is a value larger than the absolute value of the observed test statistic t. In fact, in R this is usually noted a bit shorter as "Pr(>|t|)". Since T (all under H0) is symmetric with E(T)=0, Pr(|T|≥|t|) = 2*Pr(T≥|t|).
Generally, Pr(X≥x) = 1-Pr(X
Jochen Wilhelm ah, I see my mistake thanks for the correction. I am still a bit confused. I am not really sure if I can explain this confusion properly. I will try nontheless.
I interprete it as observing the random variable T being bigger or more extreme as my test statistic under H0 as if I would obtain my posterior and calculate the T over the posterior P(|T|>=|t| | Data). This only because the |T| is the first premises in the notation P(|T|>=|t|).
Or, is the premise (under H0) sort of explicitly assumed as in: P( |T| | H0 >= |t| ) = the probability of observing the |T| (under H0) being >= |t| . But but this - in bold notated - |T| | H0 assumption is not specifically specified and simply denoted as |T|. And so as you write, T (all under H0)? Or, does this sound like giberish? :)
Thanks in advance for your time btw.
I think your last paragraph is correct. The distribution of T is considered/known under H0. t is just a value - whatever you get from your sample (note that we don't know from which distribution t is a realization or under which hypothesis t is generated).
I further think it is wrong to think of T in terms of some posterior. T is just a random variable. The distribution of T can be derived from the distribution of Y and the sample size. Under H0 we know the distribution of T (often only approximately, though). T is never conditioned on data, so I don't see how it can be called a "posterior".
- Badges
- Science method