# What is the probability that the true effect is greater than z?

Say you run a regression (Y = Beta_0 + Beta_1 + GDP + ... ), and obtain a coefficient estimate for "GDP" of -6 with a standard error = 4. Assume sample size is 250.

How do you answer the question:

What is the probability that the true (population) coefficient for GDP is greater than +1?

Normally, we set up a null hypothesis that the true coefficient is equal to zero and reject/fail to reject -- the "significance test" -- however, this approach doesn't directly answer the question I posed above. All p-values from a significance test are only relevant when you first assume that the null hypothesis is true, which I do not want to do.

Any/all code welcome.

• Note that hypothesis tests cannot answer this sort of questions. It may be the case that you are interested on estimating (pointwise or by interval) this probability rather than testing a hypothesis. – user10525 Jul 31 '12 at 15:50
• I am indeed interested in estimating this probability. I know a hypothesis test cannot answer it, hence my question. – baha-kev Jul 31 '12 at 16:11
• Do you have a model for this variable? If not, according to the regression model you are presenting, calibration models might be of interest. Note also that from a classical point of view, you can obtain interval estimates of this coefficient, but a probability in terms of GDP as a variable can only be calculated from a Bayesian perspective. – user10525 Jul 31 '12 at 16:16

To expand a bit on @Procrastinators comment:

If you are doing classical/frequentist statistics then your question is fairly meaningless. In frequentist statistics the true (population) coefficient is either greater than 1 or it is not, therefore the probability is either 100% or 0% (we just don't know which). Since it is a fixed value (though unkown) it is meaningless to talk about its probability of being in certain ranges.

If you are doing a Bayesian analysis then your question is completely reasonable and easily answered, just fit the regression using Bayesian techniques and compute/estimate the posterior distribution of the coefficient and see what proportion of the posterior is greater than 1. You will need to choose a prior for the coefficients, but don't need to specify hypotheses.

You need to decide which approach to take, it does not work well to try both. Trying to eat the Bayesian omlette without breaking the Bayesian eggs usually leads to you having egg on your face and still being hungry.

You can also change the question to something more frequentist, basically doing a one sided test of hypothesis of $H_0: \beta \le 1$ vs. $H_a: \beta > 1$. Just do the standard frequentist analysis and compute the t-value as $\frac{\hat\beta - 1}{se_{\hat\beta}}$ where $se_{\hat\beta}$ is the standard error of the slope that most statistics packages will compute for you. Look this value up on the t-table (or computer program) using $n-2$ degrees of freedom.

Other ways to do this are to include an offset (R lets you do this directly) or to use $Y - GDP$ as the response variable (but still include GDP as an explanitory variable and use the test of it being different from 0).

• Based on how I posed my question, I implied that the population parameter is a constant rather than a random variable, so this answer is good. – baha-kev Aug 1 '12 at 19:55