# Why can the null hypothesis be something other than equality?

For example, some textbooks will say:

$$H_o:\mu≤0$$ $$H_a:\mu>0$$

If the null hypothesis claims that there's no effect/change, how could the null hypothesis be anything other than $H_o=μ$?

• It's not clear what the difficulty is here. Are you concerned about calculating the distribution of the test statistic under the composite null or about some other issue? Oct 31, 2015 at 9:35
• @Glen_b I think so. Basically I know the null hypothesis is no effect. But I don't understand how we can say $H_o < μ$ also, because wouldn't that be implying there's an effect? Oct 31, 2015 at 16:54
• Your notation's wrong. You are testing two hypotheses which, to avoid having to write them out in full each time, we denote by $H_0$ (null) and $H_1$ (alternative). When we write, say, $H_0 : \mu \le 0$, we are merely defining the null hypothesis (here, that $\mu\le 0$). When we write $H_1 : \mu > 0$, we are defining the alternative. Then we go away and do some test on our data to quantify the weight of evidence in favour of each. $H_0$ and $H_1$ are like pleas to the jury in a court of law; they are not numerical quantities and you must not write them that way. Oct 31, 2015 at 21:01
• @Creosote yes, you are absolutely right, I made a mistake when formulating the question. I fixed that now. Oct 31, 2015 at 22:41

I think the issue is over acceptance of a rule of thumb. That's the problem with thumbs making rules.

My guess is that you've heard something like "the null hypothesis is that there's no effect". But it would be much more accurate to say something along the lines of "often, the null hypothesis is that there's no effect". If $\mu$ is the effect of interest, then certainly $H_o: \mu = 0$ can be correctly interpreted as the null hypothesis being that there's no effect (to be pedantic and circular, this is assuming $\mu=0$ implies no effect. This would not be the case if $\mu$ was a multiplicative factor, for example).

But hypothesis testing is much more general than that special case. You are really testing one set of potential parameters, whether it be $\mu = 0$ or $\mu \leq 0$ against another set of parameters. In theory, there's nothing that limits the structure of the two sets of hypothesis that you choice to construct.

That's theory. Let's talk practice. Recall the fact that we never accept the null hypothesis, but rather at best fail to reject it. On the other hand, through hypothesis testing, we can observe enough evidence to conclude that the parameter belongs to the alternative hypothesis set. Because of this, it makes sense to set up your hypotheses such that from a scientific standpoint, it is very interesting to know that the parameter is in the alternative hypothesis set. So when you design your hypothesis test, your alternative hypothesis should be what your interested in showing, i.e. your drug has a positive effect. As such, your null hypothesis should be the "uninteresting" results, i.e. your drug does not have a positive result.

• This is on the right track to my question, thanks Cliff! So what throws me off is when we "assume" the null hypothesis. When we assume the null hypothesis that $H_o:\mu=0$ in let's say a drug trial, it makes sense because we're assuming that there's no effect. What confuses me is if you assume the null hypothesis $H_o:\mu\le0$, then you're assuming it has an effect or has no effect? I don't see how it would be possible to assume anything other than it having absolutely no effect. Oct 31, 2015 at 22:44
• @rb612: ah, that gets technical. If $H_o: \mu \leq 0$ and $\hat \mu < 0$, we can brush over that situation: clearly the null hypothesis will not get rejected. If $\hat \mu >0$, then the likelihood function, restricted to the values defined in the null hypothesis, will (in most cases) be maximized at $\hat \mu_o = 0$. So the reference distribution is typically defined under the case that $\mu$ is on the boundary of the null hypothesis (i.e. $\mu = 0$), even though the null hypothesis includes other values. Oct 31, 2015 at 22:53
• oh! So even though the null hypothesis includes other values ($H_o:\mu\leq0$), we are assuming it has no effect ($H_o:\mu=0$) regardless? If this is true, is it a matter of simplicity or is there a deep explanation to it? Oct 31, 2015 at 22:57
• @rb612: I think saying that we're assuming $\mu = 0$ is slightly misleading. At one point in our procedure, we are maximizing the likelihood function over all values of $\mu$ in the set defined by $H_o$. If $\bar x > 0$ and $H_o: \mu \leq 0$, then $\mu = 0$ is the value in the null hypothesis set that maximizes this likelihood. Oct 31, 2015 at 23:10
• In a z/t test, we come up with the distribution for how the data would have looked if the null hypothesis was true (i.e. the reference distribution that we compare our outcome to). Again, if the null hypothesis is more than a single value, we want to pick the value in the null set that makes what we observed the most likely. So if $H_o: \mu \leq 0$ and $\bar x > 0$, then the null value that makes $\bar x$ most likely is $\mu =0$. So our reference distribution will use $\mu = 0$. So even though you have many possible $\mu$'s in $H_o$, you still use $\mu =0$ for your reference distribution. Nov 1, 2015 at 1:50

The null hypothesis can be anything at all. It has nothing to do with "nullness" of effect. Usual tests are built to have an arbitrarily low type I error (ie. $\alpha$), so that we don't mistake the null hypothesis being false when in fact it is true.

For example, in a correlations test we set $H_0: \rho = 0$, so that:

• If $\rho$ is indeed 0, it will be very unlikely that we reject the null hypotesis.
• If $\rho$ is not zero, the likelihood of us accepting $H_0$ depends on the power of the particular test.

In conclusion: the null hypothesis should be the one you care about the most not rejecting when it is true.

You talk about one and two tailed tests ,the one-tailed tests allow for the possibility of an effect in just one direction , where with two-tailed test you are testing for the possibility of an effect in two directions , so when you reject the null hypothesis in one tailed test you reject that there is no effect and you reject that there is an effect in the direction which you do not care about it.So in your case $$H_o:\mu≤0$$ when you reject $H0$ you reject both "There is no effect " and "There is a negative effect ".Here there is a nice discussion about One-Tailed vs Two-Tailed Tests? and how a two-tailed test splits your significance level .