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When using ANOVA to compare means, I understand the null hypothesis is u1=u2=u3.... as we need to combine all groups into a single group when assuming their means are the same. But for ANOVA for Regression, I don't understand why the null hypothesis is : The model is ineffective(slope is zero).Why can't we have The model is effective(slope is non-zero) as the null hypothesis?

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Linear regression and ANOVA are the same thing except that the former attempts to predict a continuous outcome using one or more continuous predictor variables, whereas the latter uses one or more categorical predictor variables to predict a continuous outcome.

So, if you have a continuous predictor you can test whether a model with slope is different from a model without slope. A model without slope is the simplest model we can fit and that's the mean and also the null hypothesis in a linear regression context. So if your regression coefficient $b_1$ is not significantly different from $b_0$ (the intercept), you might as well use the mean (the intercept) to describe your data, which could also be translated as a model with slope is ineffective.

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  • $\begingroup$ I stii don;t get it. If I have the null hypothesis to be slope is non-zero, we can still use F-distribution, can't we? Or why "slope is 0" enable us to use F-distribution. $\endgroup$ – whoisit Dec 22 '15 at 0:35
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In hypothesis testing, the null hypothesis is never what we want to demonstrate (translated in your case, that the model is effective), but the reverse, for which we know the distribution of the statistic.

Under H0 (slope is zero), we know that MSr divided by MSe will follow a F-distribution.

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  • $\begingroup$ "In hypothesis testing, the null hypothesis is never what we want to demonstrate" How? Why? $\endgroup$ – jona Dec 21 '15 at 14:11
  • $\begingroup$ @jona: see stats.stackexchange.com/questions/163957/… $\endgroup$ – user83346 Dec 22 '15 at 12:01
  • $\begingroup$ @fcop For example, if I set The model is effective(slope is non-zero) as the null hypothesis, I can still use F test to calculate the ratio of mean squre model and mean square error? Is "slope is 0" a necessary requirement to use F test? If so, why? $\endgroup$ – whoisit Dec 23 '15 at 1:46
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    $\begingroup$ @whoisit: it is not only when $\beta_i=0,\forall i$, you could also find an F-distribution (with a slightly different formaula) if e.g. $\beta_i=5,\forall i$, but the hypothesis $\beta_i \ne 0$ means that you may take many values for for $\beta_i$, so which value would you take ? For a textbook that explains this: D.N. Gujarati, "Basic Econometrics". $\endgroup$ – user83346 Dec 24 '15 at 5:15
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    $\begingroup$ @whoisit: I find Gujarati a good introduction, if you want a more technical one you could try W.H. Greene, ''Econometric Analysis''. $\endgroup$ – user83346 Dec 24 '15 at 9:51

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