Statistical Power and t-test Basics

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Questions and Answers

What is the purpose of a p-value in hypothesis testing?

  • To determine the probability of observing the data if the null hypothesis is true (correct)
  • To identify the sample size needed for accurate estimates
  • To calculate the standard error of the sample
  • To measure the effect size of a population

Which of the following correctly defines a Type I error?

  • Incorrectly estimating the standard error in the analysis
  • Rejecting the null hypothesis when it is true (correct)
  • Accepting the alternative hypothesis when it is false
  • Failing to reject the null hypothesis when it is false

How can statistical power be maximized in hypothesis testing?

  • By decreasing the effect size
  • By lowering the confidence level
  • By increasing the sample size (correct)
  • By rejecting the null hypothesis more often

In the context of statistical testing, what does the term 'alpha' (α) refer to?

<p>The threshold for statistical significance (D)</p> Signup and view all the answers

What does a false negative (Type II error) indicate in hypothesis testing?

<p>The null hypothesis is accepted when it should be rejected (D)</p> Signup and view all the answers

What is the definition of statistical power in a hypothesis test?

<p>The ability to reject the null hypothesis when it is false. (A)</p> Signup and view all the answers

Which factor does NOT directly impact the power of a statistical test?

<p>Accuracy of the data collection method (A)</p> Signup and view all the answers

What is the conventional target power level typically aimed for in studies?

<p>80% (B)</p> Signup and view all the answers

Which of the following statements is true regarding one-sided vs. two-sided tests?

<p>One-sided tests have more power than two-sided tests when the effect is in one direction. (D)</p> Signup and view all the answers

What is the relationship between sample size and statistical power?

<p>Larger sample sizes increase the power. (D)</p> Signup and view all the answers

How can increasing the significance threshold (α) affect statistical power?

<p>It increases power by reducing β. (C)</p> Signup and view all the answers

What does a false negative imply in the context of statistical testing?

<p>Failing to detect an effect when one actually exists. (D)</p> Signup and view all the answers

When planning research, what is a common use of power analysis?

<p>To calculate the sample size required for a desired power level. (B)</p> Signup and view all the answers

What is the primary reason for needing a larger sample size with a smaller effect size when aiming for 80% power?

<p>To detect true effects that are otherwise too small (D)</p> Signup and view all the answers

In the context of hypothesis tests, a two-sided test is used when the alternative hypothesis states what?

<p>The group mean is not equal to a given value (D)</p> Signup and view all the answers

What issue arises from conducting studies with low power?

<p>They may overestimate effect sizes (C)</p> Signup and view all the answers

Why is the z-test not commonly used in practice?

<p>It requires knowing the true population SD (D)</p> Signup and view all the answers

When discussing statistical significance, what must be fixed to ensure the reliability of effect sizes?

<p>Power and alpha level (B)</p> Signup and view all the answers

What is one of the consequences of running underpowered studies?

<p>They inflate the perceived effect sizes (A)</p> Signup and view all the answers

In hypothesis testing, what does the null hypothesis (H0) generally assert?

<p>The group mean equals a specified value (C)</p> Signup and view all the answers

What role does publication bias play in the interpretation of research results?

<p>It leads to fewer significant results being published (A)</p> Signup and view all the answers

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Flashcards

Type I Error (False Positive)

A type of statistical error where you reject the null hypothesis (H0) when it is actually true. Essentially, you find a significant effect when there is none.

Alpha (α)

The probability of making a Type I error, often set at 0.05 (5%).

Type II Error (False Negative)

A type of statistical error where you fail to reject the null hypothesis (H0) when it is actually false. You miss a real effect.

Statistical Power (1 - β)

The probability of correctly rejecting the null hypothesis (H0) when it is false. This means you have a good chance of detecting an actual effect.

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Beta (β)

The probability of making a Type II error, often represented by β.

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Statistical Power

The probability of finding a significant effect when there is actually an effect. In other words, how confident are we that we can detect a real difference or relationship if it exists.

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Power Threshold

The minimum likelihood of detecting an effect set by researchers, typically at 80%. Aiming for 80%+ power means a higher chance of correctly identifying a real effect.

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Effect Size (d)

The size of the difference or relationship we are trying to detect. Larger effect sizes are easier to detect and require less power.

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Sample Size (n)

The number of participants or observations in a study. Larger samples provide more data and generally lead to greater power.

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Significance Threshold (α)

The threshold for rejecting the null hypothesis. Typically set at 0.05 (5%), meaning we are willing to accept a 5% chance of making a Type I error.

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Statistical Test

The type of statistical test used to analyze the data, e.g., z-test or t-test. Different tests have varying power based on their assumptions and design.

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True Positive Rate / Sensitivity

The ability to correctly reject the null hypothesis when it is false. In other words, we have a good chance of detecting an actual effect.

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One-Sample Z-Test

A statistical test that compares the mean of a sample to a known or hypothesized population mean. It assumes we know the true population standard deviation.

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Sampling Distribution of Z-Statistic under H0

The distribution of z-scores (obtained from a sample) if the null hypothesis is true. It helps us determine if the sample mean is significantly different from the hypothesized population mean.

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Null Hypothesis (H0)

A statistical hypothesis that states there is no difference between the population mean and a specific value. It's essentially a statement of no effect.

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Alternative Hypothesis (H1)

A statistical hypothesis that states there is a difference between the population mean and a specific value. It's the opposite of the null hypothesis.

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Two-Sided Test

A type of hypothesis test that examines if the population mean is significantly different from a given value, allowing for either direction (greater or less than).

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One-Sided Test

A type of hypothesis test that examines if the population mean is significantly different from a given value in a specific direction (either greater than or less than).

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Study Notes

Statistical Power and the t-test

  • Statistical power is the likelihood of detecting an effect if it truly exists.
  • Low statistical power leads to a higher chance of missing effects, leading to wasted resources.
  • Power should be considered when designing studies.
  • The t-test is a fundamental statistical test for comparing group means.
  • There are different types of t-tests: one-sample, independent groups, and paired.

Assumptions of the t-test

  • One-sample t-test: Assumes the sample's mean is compared to a known value. The data needs to be normally distributed.
  • Independent groups t-test: Assumes that the data is normally distributed and observation within and between groups are independent. Also, the variance between groups must be homogeneous.
  • Paired t-test: Assumes pairs of observations are independent; difference scores are normally distributed.

Cohen's d

  • Cohen's d is a measure of effect size.
  • Values of 0.2, 0.5, and 0.8 are used to describe small, medium, and large effects but are arbitrary and should be contextually interpreted.

Interpretation

  • Statistical tests can sometimes produce incorrect results.
  • False positive (Type I error): null hypothesis incorrectly rejected even though it is true
  • False negative (Type II error): null hypothesis incorrectly accepted despite the alternative hypothesis being true.
  • These values (alpha and beta) should be reported along with effect sizes to give a complete picture of the significance in a statistical assessment.

Statistical Tests

  • t-test: This test is frequently used when the standard deviation is unknown. The sample standard deviation is applied to give an approximate estimate.
  • Z-test: Assumes the population standard deviation is known.

Choosing the Correct Test

  • One-sided vs two-sided tests: Using a one-sided test will have slightly higher power when you are only interested in one direction of results.
  • Student's t-test vs. Welch's t-test: By default, R applies Welch's test. While Welch's is generally preferred, Student's t-test assumes homogeneity of variance (a shared variance for each group).

Additional Considerations

  • Statistical tests and their results should be examined in context to determine significance.
  • The t-test can be considered to be an outcome in a regression model.
  • The results should be reviewed with the correct contextual information.

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Lecture 4 - Stats Power and t Tests PDF

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