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In general, what the the standard hypothesis tests in all types of regression? We get some coefficients and then test the hypotheses that these coefficients are significant (i.e not equal to $0$)? For example, in logistic regression the p-value indicates that the regression coefficients are significant or not? Likewise with linear regression?

Question: Suppose coefficients $\beta_0$ and $\beta_1$ are significant in linear regression. Can we deduce anything about the significance of $\beta_0+\beta_1$? What about $\beta_0= \beta_1$?

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  • $\begingroup$ Indeed, softwares return the $p$-value for $H_0:\{\theta=0\}$ for each parameter $\theta$ of the model. $\endgroup$ Commented Jan 18, 2013 at 20:00
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    $\begingroup$ @StéphaneLaurent: Is there a way to test for example $H_0: \theta_1+\theta_2 = 2$? $\endgroup$
    – proton
    Commented Jan 18, 2013 at 20:02
  • $\begingroup$ Is this homework? $\endgroup$ Commented Jan 18, 2013 at 20:14
  • $\begingroup$ @proton For a Gaussian linear regression model it is easy to test about any linear combination of the parameters. $\endgroup$ Commented Jan 18, 2013 at 20:16
  • $\begingroup$ @StéphaneLaurent: How about for other types of regression. $\endgroup$
    – proton
    Commented Jan 18, 2013 at 20:21

2 Answers 2

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You can create a contrast matrix $L$ and a vector of parameters $\beta$. Then test any kind of linear combination of the parameters using the null hypothesis, $H_0:L\beta=d$. You can use a Wald test or a Likelihood Ratio Test (LRT) for testing your $H_0$.

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  • $\begingroup$ What if you want to test $H_0: \frac{\beta_0}{\beta_1} = 1$ vs $H_a: \frac{\beta_0}{\beta1} \neq 1$? $\endgroup$
    – proton
    Commented Jan 21, 2013 at 14:56
  • $\begingroup$ It is all the same as $H_0:\beta_0-\beta_1=0$, isn't it? More over you could also test $H_0:\frac{\beta_0}{d\beta_1}=1$ by constructing a contrast matrix $L$ such that $H_0:\beta_0-d\beta_1=0$. $\endgroup$
    – Blain Waan
    Commented Jan 21, 2013 at 18:36
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The standard tests regarding coefficients are (1) Wald test, similar to t and F test in the OLS case (2) score test (3) likelihood ratio test. Recently the gradient test has received more attention. See a summary at Likelihood ratio vs. score vs. Wald test: Different p values, which to use?

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