Click for printer friendely version of this HowTo P-Values In publications, you will often times see p-values reported as the result of some statistical test. A p-value is the probability of an event (or series of events) taking place that would create a statistic with a more extreme value3.1than the one you derived, assuming your model under the null hypothesis is correct. Regardless of the type of model you are assuming describes the source of the data under the null hypothesis, you can create what are called one-sided tests. With these tests, there are two typical hypotheses that people make about the mean of the underlying model. One type of hypothesis is that the mean is less than some value. For example, you might propose that the mean is less than zero, or H $_0: \mu \le 0$. Alternatively, the hypothesis might be that the mean is greater than zero, or H $_0: \mu \ge 0$. For the first type of one-sided hypothesis, the p-value is defined as: $$\textrm{p-value} = \textrm{Pr}(x \ge \textrm{your statistic})$$ $$= \int_{\textrm{your statistic}}^{\infty} f(x) {\textrm d}x,$$ where $f(x)$ is the probability distribution you are assuming the data came from, and your statistic is some value derived from a function of the data (for example, the mean of the data). This is illustrated in Figure 3.5.1. Since our hypothesis is $H_0: \mu \le 0$, the larger the mean of the data is (and thus, the smaller the p-value), the more likely we will reject the proposed model. Figure: The p-value for a one-sided statistic where we are testing H $_0: \mu \le 0$. The second type of one-sided hypothesis, where we are testing to see if the mean is greater than some value, is very similar. The only difference is that we integrate in the other direction. That is: $$\textrm{p-value} = \textrm{Pr}(x \le \textrm{your statistic})$$ $$= \int^{\textrm{your statistic}}_{-\infty} f(x) {\textrm d}x.$$ This is illustrated in Figure 3.5.2. Figure: The p-value for a one-sided statistic where we are testing H $_0: \mu \ge 0$. If the type of model you are assuming describes the source of the data is symmetric (like the distributions in Figures 3.5.1 and 3.5.2) you can create what are called two-sided tests. In this case your typical null hypothesis is that that the mean is equal to a certain value. For example, you might propose the hypothesis $H_0: \mu = 0$. Thus, if the mean of your data is much larger or much smaller than zero, then you have good reason to reject $H_0$. In this case, the p-value is defined as: $$\textrm{p-value} = \textrm{Pr}(x \ge \vert \textrm{your statistic} \vert)$$ $$\quad + \textrm{Pr}(x \le - \vert \textrm{your statistic} \vert)$$ $$= 2 \times \textrm{Pr}(x \ge \vert \textrm{your statistic} \vert)$$ $$= 2\int_{\vert \textrm{your statistic} \vert}^{\infty} f(x) {\textrm d}x.$$ This is illustrated in Figure 3.5.3. Figure: The p-value for a two-sided statistic where we are testing H $_0: \mu = 0$. Obviously, the smaller the p-value, the less likely an event as rare or rarer will take place. Often times the model proposed by the null hypothesis, H$_0$, is rejected if the p-value is less than $0.05$. That is to say, it is assumed that the proposed model does not explain the data if the p-value is less than $0.05$. Click for printer friendely version of this HowTo

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