Yes let’s touch on the most important topic in hypothesis testing, with some real-life examples and how to implement it in Python
Introduction
In statistics, the term statistical significance refers to the likelihood that a result or relationship observed in a sample data is real or true, and not due to chance. Statistical significance is an important concept in research, as it helps researchers determine whether a relationship or difference observed in their data is likely to be a true phenomenon or simply a random occurrence.
In most research, we begin by formulating a hypothesis, which is an educated guess or prediction about the relationship between different variables or factors. To test this hypothesis, we collect data from a sample of individuals or objects and use statistical tests to determine the probability that the results we observe could have occurred by chance.
If the probability of obtaining the observed results by chance is low, the results are considered statistically significant, and the hypothesis is considered supported by the data. On the other hand, if the probability of obtaining the observed results by chance is high, the results are not considered statistically significant, and the hypothesis is considered not supported by the data.
To summarize, statistical significance is a critical concept in research, as it helps us determine the reliability and validity of their results, and allows us to draw meaningful conclusions from the data.
How to determine statistical significance
To determine statistical significance, we can use statistical tests to evaluate the probability that the results we observed in our data could have occurred by chance.
A p-value is a numerical measure of the probability that the results observed in a sample data occurred by chance. Typically, we will set a threshold for the p-value, called the alpha level, before conducting the statistical test. If the p-value calculated from the data is less than the alpha level, the results are considered statistically significant.
For example, a commonly used alpha level is 0.05, which means that if the p-value is less than 0.05, the results are considered statistically significant.
In addition to p-values, we may also use confidence intervals to determine statistical significance. A confidence interval is a range of values that is likely to contain the true value of a population parameter, based on the data observed in a sample. If the confidence interval does not include the value of the null hypothesis (the hypothesis that there is no relationship or difference between the variables being studied), the results are considered statistically significant.
The determination of statistical significance also depends on the sample size and the number of experiments performed. In general, the larger the sample size, the more likely it is that the results observed in the data are statistically significant. Similarly, the more experiments that are performed, the more likely it is that the results observed are statistically significant. Overall, the determination of statistical significance involves a combination of statistical tests, the use of p-values and confidence intervals, and considerations of sample size and the number of experiments performed. These factors all play a role in helping researchers determine the likelihood that the results observed in their data are real and not due to chance
Real-life example
Let’s see an example of how statistical significance is determined in a real-life scenario
Suppose we are interested in studying the effectiveness of a new drug for treating depression. We recruit 100 participants with depression, and randomly assign half of them to receive the new drug and the other half to receive a placebo. After 8 weeks of treatment, we measure the participants’ depression symptoms using a standard depression scale.
To determine whether the new drug is statistically significantly more effective at reducing depression symptoms than the placebo, We calculate the p-value for the data. The p-value is the probability of obtaining the observed results (or more extreme results) by chance, assuming that the null hypothesis (the hypothesis that the new drug is no more effective than the placebo) is true.
Suppose the p-value calculated from the data is 0.01. This means that there is only a 1% probability that the observed results occurred by chance, assuming that the null hypothesis is true. Since this p-value is less than the commonly used alpha level of 0.05, We can conclude that the new drug is statistically significantly more effective at reducing depression symptoms than the placebo.
In this example, the determination of statistical significance involved calculating the p-value for the observed data and comparing it to the predetermined alpha level. The low p-value and the fact that it was less than the alpha level indicated that the observed results were unlikely to have occurred by chance, and thus were considered statistically significant.
The limitations of statistical significance
While statistical significance is a valuable tool for evaluating the likelihood that observed results in a sample data are real and not due to chance, it does have some limitations.
One limitation of statistical significance is the potential for false positives, also known as Type I errors. A false positive occurs when a statistical test indicates that a result is statistically significant, when in fact it is not. This can happen when the sample size is small, or when the alpha level is set too low.
For example, suppose we conduct a study with a sample size of only 10 participants. If we set the alpha level at 0.05, there is a high likelihood that the observed results will be considered statistically significant, even if they are not. This is because the small sample size increases the likelihood of random fluctuations in the data, and the low alpha level makes it more likely that the results will be considered statistically significant.
Another limitation of statistical significance is the potential for false negatives, also known as Type II errors. A false negative occurs when a statistical test indicates that a result is not statistically significant, when in fact it is. This can happen when the sample size is small, or when the alpha level is set too high.
For example, suppose we conduct a study with a sample size of only 20 participants. If we set the alpha level at 0.10, there is a high likelihood that the observed results will not be considered statistically significant, even if they are. This is because the small sample size makes it more difficult to detect real differences or relationships in the data, and the high alpha level makes it more likely that the results will not be considered statistically significant. Overall, the limitations of statistical significance include the potential for false positives and false negatives, as well as the difficulty in detecting real differences or relationships in small sample sizes. It is important for us to consider these limitations when interpreting the results of their studies, and to consider other factors in addition to statistical significance in evaluating the reliability and validity of their results.
How to Avoid these limitations
To avoid the limitations of statistical significance, we can take a number of steps to ensure that their results are reliable and accurate.
First, we can avoid false positives by ensuring that our sample size is large enough to accurately detect real differences or relationships in the data. As a general rule, the larger the sample size, the less likely it is that the observed results will be due to chance, and the more likely it is that the results will be statistically significant.
Second, we can avoid false negatives by setting the alpha level at an appropriate value. As a general rule, the alpha level should be set at 0.05, which is a commonly used threshold for determining statistical significance. Setting the alpha level too high (e.g. 0.10) increases the likelihood of false negatives while setting it too low (e.g. 0.01) increases the likelihood of false positives.
Third, we can avoid the limitations of statistical significance by considering other factors in addition to statistical significance when evaluating the reliability and validity of their results. For example, we can consider the practical implications of their results, such as whether the observed effects are large enough to be meaningful in a real-world setting. Additionally, we can conduct multiple experiments to increase the reliability of our results.
Overall, there are a number of steps that we can take to avoid the limitations of statistical significance, including ensuring a large sample size, setting the alpha level appropriately, and considering other factors in addition to statistical significance in evaluating their results. By following these steps, we can improve the reliability and accuracy of our findings.
Implementation Of Statistical Significance in Python
Here is an example of how statistical significance is determined using the Python programming language
Suppose we are interested in studying the relationship between education level and income. We collect data on the education level and income of 500 individuals and want to determine whether there is a statistically significant relationship between the two variables.
To do this, we can use the scipy library in Python to conduct a Pearson’s correlation test, which is a statistical test that measures the strength of the linear relationship between two variables. We can then calculate the p-value for the observed data, which is the probability of obtaining the observed results (or more extreme results) by chance, assuming that the null hypothesis (the hypothesis that there is no relationship between education level and income) is true.
Here is an example of how this could be done in Python
# Import the necessary libraries
import numpy as np
from scipy.stats import pearsonr
# Define the variables
education = [15, 16, 17, 18, 19, 20, 21, 22, 23, 24]
income = [25000, 30000, 35000, 40000, 45000, 50000, 55000, 60000, 65000, 70000]
# Calculate the Pearson's correlation coefficient and p-value
r, p = pearsonr(education, income)
#Print the results
print("Pearson's correlation coefficient", r)
print("p-value", p)
#output
Pearson's correlation coefficient 0.9999999999999998
p-value 1.0635035875250972e-62In this example, Pearson’s correlation coefficient (r) measures the strength of the linear relationship between education level and income, while the p-value measures the probability of obtaining the observed results (or more extreme results) by chance, assuming that the null hypothesis is true.
If the p-value calculated from the data is less than the alpha level (e.g. 0.05), we can conclude that there is a statistically significant relationship between education level and income. On the other hand, if the p-value is greater than the alpha level, we can conclude that there is no statistically significant relationship between the two variables.
Conclusion
In conclusion, statistical significance is a critical concept in hypothesis testing, as it helps us determine the likelihood that the results or relationships observed in our data are real and not due to chance. By using statistical tests, p-value, and considering factors such as sample size and the number of experiments performed, we can determine whether our results are statistically significant.
However, it is important to note that statistical significance is not the only factor to consider when evaluating the reliability and validity of study results. There are also limitations to statistical significance, including the potential for false positives and false negatives, as well as the difficulty in detecting real differences or relationships in small sample sizes.
To avoid these limitations, we can take steps such as ensuring a large sample size, setting the alpha level appropriately, and considering other factors in addition to statistical significance when evaluating our results. By doing so, we can improve the reliability and accuracy of our findings and draw more meaningful conclusions from our data.