Welcome back to the AI Bayeslab Statistics series. Today, let's grasp some basic concepts of the Power function so that we can better understand the effect size, omega squared, and signal detection theory in the next post.
Definition of Today:
Statistical power:
The probability of rejecting the incorrect null hypothesis successfully, denoted as 1-ß.
Power function:
The function relationship is established using potential parameter values suggested by the alternative hypothesis (H₁) as independent variables, alongside the corresponding power values (1-ß) serving as dependent variables.
Through this post, you will grasp the obscure statistical concepts discussed above and learn how to apply them to your business decisions and management in natural language. I'm glad to provide zero-cost consumer information about statistics to the general public, and hope this will significantly impact your daily life.
1.What does statistical power mean？
If we denote the probability of a Type II error as ß, then 1-ß represents the probability of not making a Type II error. In other words, it also signifies that we successfully reject the incorrect null hypothesis. This is known as the statistical power.
First, let's review the context of Type II error (β error) and Type I error (α error). This somewhat relates to the principle that there is no 100% certainty in hypothesis testing. In some cases, the individual we adopt may fall into the area of Type II error (β error), while in others, it may fall into the area of Type I error (α error). All of these things have at their root the concepts of random sampling.
1.1 Calculation for Type I error (α error):
Definition: Rejecting a correct null hypothesis and accepting an incorrect alternative hypothesis is also known as an α error.
The Type I error is easy to confirm when we set the significance level (α); therefore, the Type I error is equivalent to the significance level (α).
1.2 Calculation for Type II error (β error):
Definition: Accepting an incorrect null hypothesis and rejecting a correct alternative hypothesis, also known as a β error.
But how to calculate the Type II error? This is more complex to confirm. Example for the difference of means in a two-tailed t-test: the data from a sample of 30 teenagers from a region were taken for an IQ test, and the sample mean X̅, or μ₁ = 106, with a sample standard deviation of σ₁ = 16.23. Meanwhile, the historical data μ₀ = 108, σ₀ = 16.23.. The question remains whether the average IQ of teenagers in this region can be considered 108, with a significance level of α = 0.05.
Calculate the confidence intervals for the two data sets separately using the formula.
The sample data set:( μ₁ ±Zₐ/₂ xσ²/ √𝑛) → (106±1.96x16.23²/√30) → (100.20, 111.80)
The history data set: (μ₀ ±Zₐ/₂ xσ²/ √𝑛) → (108±1.96x16.23²/√30) → (102.20, 113.80)
So the overlapping area between the two confidence intervals is quite large. Therefore, the larger the overlapping region, the greater the probability that a Type II error may occur.
How to calculate it? Let's check the definition of a Type II error: Accepting an incorrect null hypothesis. So if there were two populations, the area under the curve would represent the probability. If the null hypothesis is incorrect, but we accept it, then it also means we take a confidence interval of another population, right?
We stated that if we have a standard distribution shape, defined by its mean and standard deviation, we can calculate the overlapping area between the mean of A distribution and the confidence interval of B distribution.
The probability of a Type II error will be calculated using the distinctive value:
In the alternative hypothesis, H₁: μ₁ = 106, as the mean,
while the Region for acceptance of the null hypothesis is H₀: μ₀ = 108.
$$P\left\{\frac{102.20 - 106}{\frac{\sigma}{\sqrt{30}}} \leq \frac{\bar{x} - \mu_1}{\frac{\sigma}{\sqrt{n}}} \leq \frac{113.80 - 106}{\frac{16.23}{\sqrt{30}}}\right\}$$={-1.28 ≤ Z ≤ 2.63}= 0.9957 - 0.1003 = 0.8954
1.3 AI Calculator for Type II error (β error)
We can instruct the AI agent to do it for us, allowing us to ignore the complex formula above. One prompt: "Calculate the Type II error." Then, we will obtain the desired result.
2.What is the power function in hypothesis testing?
The definition is:
The function relationship is established using potential parameter values suggested by the alternative hypothesis (H₁) as independent variables, alongside the corresponding power values (1-ß) serving as dependent variables.
You can refer to the table below, which provides calculations of β errors and corresponding power values (1- β) for different alternative hypotheses (μ_1) in the range from 90 to 120, assuming various sample distribution means.
We can observe that when the alternative hypotheses (μ₁) are closer to the null hypothesis (μ₀), the probability of making a Type I error increases. Indeed, there is no possibility for μ₁ = μ₀ in reality.
Thank you for reading this installment today. We discussed what statistical power is and explained the power function. In the next post, let's explore effect size and omega squared in more detail.
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