I’m utterly confused about the interpretation of p value. Should I think of it as :
A critical value or a significance level?
BB. Q 3G and q.10
Both describe the answer as the p value being less than 0.05 (significance level) , hence reject the null and the coefficient is significant.
If it is a significance level, should we not bother about the test statistic vis a vis critical value in the question, and just directly consider the difference between two significance levels (one mentioned and p value)?
P-value is the smallest level of significance that you can reject H0. Generally when you’re looking at a regression results you’d want the p-value of the coefficients to be small so you can reject H0 and accept Ha (the coefficient is significantly different from 0). Now the smaller the p-value, the better chances of rejecting H0 at different alpha (level of significant). Eg. p-value of 0.08 then you can reject H0 at alpha = 10% or above (because 0.08<0.1), but not at 5% (because 0.08>0.05) Using p-value you can be flexible at choosing your own alpha and see if the coefficient is significant at that particular alpha. When you’re running actual regressions (eg. in research papers) it’s really useful to interpret the results.
Its more like if you construct a confidence interval, what is the minimum level of confidence you can have until your coefficients are insignificant. It does take the model into consideration. So if your test stat falls outside the interval suggested by the p value, you can say that your stat is significant. If it fails to fall outside the interval implied by the p value ie your df < p value df, your stat will be insignificant. I am sure someone can explain this better but the essence is that the p value is a reference point to construct confidence intervals and significance tests and the rule of thumb is that if P value < Your degrees of freedom (df) = Your co-efficient is significant.
I’m afraid you’re pretty far off point. It doesn’t make any sense to compare the p-value to the degrees of freedom. The p-value is compared with alpha, the pre selected significance level.
The p-value is also not the reference point to construct a CI-- the p-value is the probability of obtaining a test statistic at least as extreme as the observed one, assuming the null hypothesis is true. In layman’s terms, it’s a summary measure of how much the observed data disagree with the null hypothesis. The smaller the p-value, the more the data “disagree” with the null hypothesis.
You also have the idea of test statistics and critical values mixed up. A critical value is determined based on the preselected alpha level. The test statistic (function of data observed, not in your control, related to p-value) is then compared to the critical value (function of alpha level you choose, not data).