What is the relationship between levels of confidence and statistical significance?

As a statistical expert with a strong background in data analysis and interpretation, I often encounter questions regarding the relationship between levels of confidence and statistical significance. These concepts are fundamental to hypothesis testing and inference in statistics, and understanding their interplay is crucial for drawing valid conclusions from data.
Confidence Level refers to the probability that the true value of a parameter lies within a specified range, known as the confidence interval. For instance, a 95% confidence level means that if we were to repeat our sampling process an infinite number of times, 95% of the confidence intervals we construct would contain the true parameter value. It's important to note that confidence levels are not measures of the probability that the parameter is in the interval; rather, they are related to the method's reliability over many samples.
Statistical Significance, on the other hand, is a concept used in hypothesis testing to determine if the observed data provide enough evidence to reject the null hypothesis. The significance level (denoted as alpha, α) is a threshold that we set before conducting a test to decide when we consider our results to be statistically significant. If the P-value (the probability of observing the data given that the null hypothesis is true) is less than the significance level, we reject the null hypothesis and conclude that the results are statistically significant.
Now, let's delve into the relationship between these two concepts:

1. Setting the Significance Level: When we set a significance level, we are determining the threshold for when we will reject the null hypothesis. Commonly used significance levels are 0.05, 0.01, and 0.001, which correspond to 95%, 99%, and 99.9% confidence levels, respectively. This is where the misconception often arises that a 0.05 significance level directly translates to a 95% confidence level. While it's true that a lower P-value corresponds to a higher confidence level, the relationship is not direct because the significance level is about the decision rule for hypothesis testing, not the confidence in the estimate itself.
2. **Confidence Intervals and Hypothesis Testing**: In hypothesis testing, if the calculated confidence interval does not include the value specified by the null hypothesis, it indicates that there is a statistically significant difference from what was expected under the null hypothesis. This is because the non-overlap of the confidence interval with the null value suggests that the observed effect is unlikely to have occurred by chance alone, assuming the null hypothesis were true.

3. Interpreting Results: It's crucial to interpret the results correctly. A statistically significant result (P-value < α) means that the data provide strong evidence against the null hypothesis, but it does not necessarily mean that the effect is large or practically significant. Similarly, a confidence interval that does not include the null value indicates a statistically significant result, but the magnitude of the effect and its practical implications must also be considered.

4. Power of a Test: The power of a statistical test is the probability that it will correctly reject a false null hypothesis. The power of a test is influenced by the significance level, the size of the effect being detected, the sample size, and the variability in the data. A higher power increases the likelihood of detecting a true effect but does not affect the confidence level or significance level directly.
5. **Practical Significance vs. Statistical Significance**: It's important to distinguish between statistical significance and practical significance. Statistical significance tells us that our results are unlikely to be due to chance, but practical significance involves the relevance and magnitude of the findings in a real-world context. A result can be statistically significant but have a negligible effect size, making it of little practical importance.
In conclusion, while confidence levels and statistical significance are related, they serve different purposes in statistical analysis. Confidence levels provide an estimate of the reliability of our parameter estimates, while statistical significance helps us make decisions about hypotheses. It's essential to use these tools appropriately and interpret their results in the context of the research question and the data at hand.

So, if your significance level is 0.05, the corresponding confidence level is 95%. If the P value is less than your significance (alpha) level, the hypothesis test is statistically significant. If the confidence interval does not contain the null hypothesis value, the results are statistically significant.Apr 2, 2015