Understanding P-values

A null hypothesis ($H_0$) in statistical inference often represents a 'no effect' or 'status quo' scenario. Conversely, an alternative hypothesis ($H_1$) represents what researchers aim to find evidence for. When faced with deciding between these two hypotheses using observed sample data, the p-value provides a primary method for making such decisions.

Think of the p-value as a measure of surprise. It answers this specific question:

If the null hypothesis ($H_0$) were actually true, what is the probability of observing sample data that is at least as extreme as what we actually observed?

Let's break that down:

1. "If the null hypothesis ($H_0$) were actually true...": We start by temporarily assuming the null hypothesis is correct. For example, if $H_0$ is "this new drug has no effect on recovery time", we calculate the probability assuming the drug truly has zero effect.
2. "...observing sample data that is at least as extreme as what we actually observed?": We look at our collected sample data (e.g., the average recovery time for patients taking the drug). How likely is it to get a result this far away (or even further away) from what $H_0$ predicted, purely by random chance? "Extreme" means results that provide evidence against $H_0$ and in favor of $H_1$.

Interpreting the P-value

The p-value is a probability, so it ranges between 0 and 1.

• A small p-value (typically ≤ 0.05): This indicates that our observed sample data is quite surprising or unlikely if the null hypothesis were true. It's like saying, "Wow, if nothing special was going on (H0 is true), getting results like these would be really rare." This low probability suggests that our initial assumption (that $H_0$ is true) might be incorrect. Therefore, a small p-value provides evidence against the null hypothesis and supports the alternative hypothesis ($H_1$).
• A large p-value (typically > 0.05): This indicates that our observed sample data is not particularly surprising if the null hypothesis were true. It's like saying, "Well, even if nothing special was going on (H0 is true), results like these could plausibly happen just due to random variation." A large p-value means we don't have strong evidence against the null hypothesis.

The Significance Level: Alpha ($\alpha$)

Okay, but how "small" does a p-value need to be for us to decide it's small enough to reject $H_0$? We need a predefined cutoff point. This cutoff is called the significance level, denoted by the Greek letter alpha, $\alpha$.

The most common significance level used in many fields is $\alpha = 0.05$ (or 5%). Other values like 0.01 (1%) or 0.10 (10%) are sometimes used depending on the context and how cautious you need to be. You choose $\alpha$ before you conduct your test.

The decision rule is simple:

• If $p \le \alpha$: Reject the null hypothesis ($H_0$). We conclude that there is statistically significant evidence in favor of the alternative hypothesis ($H_1$).
• If $p > \alpha$: Fail to reject the null hypothesis ($H_0$). We conclude that there is not enough statistically significant evidence to support the alternative hypothesis ($H_1$).

A flowchart illustrating the decision process using a p-value and significance level ($\alpha$).

An Example

Let's revisit the website design A/B test example:

• $H_0$: The new design does not increase the conversion rate (rate is $\le$ old rate).
• $H_1$: The new design does increase the conversion rate (rate is > old rate).
• We set our significance level $\alpha = 0.05$.
• We run the experiment, collect data on conversions for both designs, and perform a statistical test which yields a p-value.

Scenario 1: The test gives a p-value = 0.02.

• Interpretation: If the new design truly had no positive effect ($H_0$ is true), there's only a 2% chance of seeing an increase in conversion rate as large (or larger) as what we observed in our sample, just due to random luck.
• Decision: Since $p = 0.02$ is less than or equal to $\alpha = 0.05$, we reject $H_0$.
• Conclusion: We have statistically significant evidence to conclude that the new design increases the conversion rate.

Scenario 2: The test gives a p-value = 0.31.

• Interpretation: If the new design truly had no positive effect ($H_0$ is true), there's a 31% chance of seeing an increase in conversion rate as large (or larger) as what we observed in our sample, just due to random luck. This is not very surprising.
• Decision: Since $p = 0.31$ is greater than $\alpha = 0.05$, we fail to reject $H_0$.
• Conclusion: We do not have statistically significant evidence to conclude that the new design increases the conversion rate. It might, but our experiment didn't provide strong enough proof.

Important Clarifications

It's essential to understand what a p-value is and is not:

• A p-value is NOT the probability that the null hypothesis is true. It's calculated assuming $H_0$ is true.
• A p-value is NOT the probability that the alternative hypothesis is true.
• "Failing to reject $H_0$" does NOT mean $H_0$ is true. It simply means our sample didn't provide enough evidence to convince us to abandon $H_0$ at our chosen significance level. Think of it like a "not guilty" verdict in court – it doesn't necessarily mean the person is innocent, just that there wasn't enough proof for a "guilty" verdict.
• Statistical significance (small p-value) does not automatically imply practical significance. With very large datasets, even tiny, unimportant effects can become statistically significant. Always consider the context and the magnitude of the effect alongside the p-value.

Understanding p-values is fundamental for interpreting the results of many statistical tests used in data analysis and machine learning evaluation. They provide a standardized way to assess the strength of evidence against a null hypothesis based on sample data.