Welcome to our explanation of p-values, a fundamental concept in statistical hypothesis testing. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from your sample data, assuming that the null hypothesis is true.
In hypothesis testing, we start with two competing hypotheses. The null hypothesis, denoted as H₀, typically represents no effect or no difference. The alternative hypothesis, denoted as H₁, represents the presence of an effect or difference.
This graph shows a standard normal distribution, which represents the distribution of a test statistic under the null hypothesis. The red shaded areas in the tails represent the p-value - the probability of observing a test statistic this extreme or more extreme if the null hypothesis were true.
The p-value is then compared to a pre-determined significance level, typically 0.05. If the p-value is less than or equal to this threshold, we reject the null hypothesis. If it's greater, we fail to reject the null hypothesis.
Let's see what happens when our test statistic changes. As the test statistic becomes more extreme, the p-value decreases, making us more likely to reject the null hypothesis.
Now, let's discuss how to correctly interpret p-values, as they are often misunderstood. There are two common misinterpretations we should avoid.
First, the p-value is NOT the probability that the null hypothesis is true. It assumes the null hypothesis is true and calculates the probability of obtaining your results or more extreme results.
Second, the p-value is NOT the probability that your results occurred by chance. Again, it's a conditional probability based on the assumption that the null hypothesis is true.
The correct interpretation is that the p-value is the probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true. Let's visualize this with our normal distribution.
Here we have a p-value of 0.15, which is greater than our significance level of 0.05. Therefore, we fail to reject the null hypothesis. This doesn't mean the null hypothesis is true; it just means we don't have enough evidence to reject it.
Now, let's see what happens when our p-value decreases to 0.03. Since this is less than our significance level of 0.05, we would reject the null hypothesis.
And if our p-value is very small, like 0.001, we have very strong evidence against the null hypothesis. The smaller the p-value, the stronger the evidence against the null hypothesis.
Remember, the p-value does not tell us the size or importance of an effect. A small p-value only indicates that the observed data is unlikely under the null hypothesis.
Now, let's explore the key factors that influence p-values. Understanding these factors is crucial for correctly interpreting statistical results.
The first important factor is sample size. Larger samples tend to produce smaller p-values, even when the effect sizes are small. This is because larger samples provide more statistical power to detect effects.
The second factor is effect size. Larger effects, or stronger signals, tend to produce smaller p-values. This makes intuitive sense: the larger the difference from what we'd expect under the null hypothesis, the less likely that difference occurred by chance.
The third factor is variability in the data. Less variability tends to produce smaller p-values. When data points are more tightly clustered, it's easier to detect a genuine effect.
Let's see how these factors interact. If we have a small effect size but a large sample, we can still get a statistically significant result with a p-value less than 0.05.
Conversely, with a large effect size, even a smaller sample can yield a significant result.
This is why it's important to consider not just the p-value, but also the effect size and sample size when interpreting statistical results. A tiny p-value with a large sample might represent an effect that's statistically significant but practically meaningless.
Now, let's look at how p-values are used in common statistical tests. Different tests have different distributions and critical values, but they all use p-values in a similar way.
The t-test compares means between groups or against a reference value. It's commonly used for normally distributed data with unknown variance. Here, we have a t-distribution with 10 degrees of freedom. The red areas represent the critical regions where we would reject the null hypothesis.
ANOVA, or Analysis of Variance, extends the t-test to compare means across three or more groups. It uses the F-distribution, which is always right-skewed. Our test statistic F is in the critical region, giving us a p-value of 0.033, which is less than 0.05, so we reject the null hypothesis.
The Chi-Square test is used for categorical data to test association between variables or goodness-of-fit to an expected distribution. It also has a right-skewed distribution. Our chi-square statistic of 11 gives a p-value of 0.027, which is less than 0.05, so we reject the null hypothesis.
Correlation tests examine relationships between continuous variables. Tests like Pearson's r can be transformed to follow a normal distribution. Our test statistic z of 2.3 gives a p-value of 0.021, which is less than 0.05, so we reject the null hypothesis of no correlation.
Despite the different distributions and test statistics, all these tests use p-values in the same way: to determine whether to reject the null hypothesis. If the p-value is less than our significance level, typically 0.05, we reject the null hypothesis.