1.4 Hypothesis testing

Hypothesis testing is a crucial statistical method in causal inference. It helps researchers make decisions about population parameters based on sample data, using null and alternative hypotheses to assess the significance of treatment effects and compare groups in experiments.

The process involves formulating hypotheses, selecting appropriate tests, and interpreting results. Key concepts include significance levels, p-values, and different types of errors. Researchers must consider limitations and practical significance when drawing conclusions about causal relationships.

Hypothesis testing overview

• Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data
• It involves formulating a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$), and using probability to determine whether to reject or fail to reject the null hypothesis
• Hypothesis testing is a crucial tool in causal inference for assessing the significance of treatment effects and comparing groups in experiments

Null and alternative hypotheses

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• The null hypothesis ($H_0$) states that there is no significant difference or relationship between variables, or that a parameter equals a specific value
• The alternative hypothesis ($H_a$) is the opposite of the null hypothesis and represents the claim being tested, such as the existence of a significant difference or relationship
• Examples:
  • $H_0$: The mean weight of a population is 150 lbs; $H_a$: The mean weight is not 150 lbs
  • $H_0$: There is no association between smoking and lung cancer; $H_a$: There is an association between smoking and lung cancer

Significance level and p-values

• The significance level ($\alpha$) is the probability threshold for rejecting the null hypothesis, typically set at 0.05 or 0.01
• The p-value is the probability of observing a test statistic as extreme as or more extreme than the one calculated from the sample data, assuming the null hypothesis is true
• If the p-value is less than the significance level, the null hypothesis is rejected; otherwise, we fail to reject the null hypothesis
• A smaller p-value provides stronger evidence against the null hypothesis

One-tailed vs two-tailed tests

• A one-tailed test is used when the alternative hypothesis specifies a direction (greater than or less than), focusing on one tail of the distribution
• A two-tailed test is used when the alternative hypothesis does not specify a direction (not equal to), considering both tails of the distribution
• The choice between a one-tailed or two-tailed test depends on the research question and prior knowledge about the direction of the effect

Type I and Type II errors

• A Type I error (false positive) occurs when the null hypothesis is rejected when it is actually true
  • The probability of a Type I error is equal to the significance level ($\alpha$)
• A Type II error (false negative) occurs when the null hypothesis is not rejected when it is actually false
  • The probability of a Type II error is denoted by $\beta$
• The goal is to minimize both types of errors, but there is a trade-off between them

Power of a test

• The power of a test is the probability of rejecting the null hypothesis when it is actually false (i.e., correctly detecting a significant effect)
• Power is calculated as $1 - \beta$, where $\beta$ is the probability of a Type II error
• Factors that influence power include sample size, effect size, significance level, and test type
• Higher power reduces the risk of Type II errors and increases the likelihood of detecting true effects

Common hypothesis tests

• Various hypothesis tests are used depending on the type of data, distribution, and research question
• Some common tests include the z-test, t-test, chi-square test, and F-test
• Each test has specific assumptions and is suitable for different scenarios

Z-test for population mean

• The z-test is used to test hypotheses about a population mean when the population standard deviation is known and the sample size is large (n > 30) or the population is normally distributed
• The test statistic ($z$) is calculated as: $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size
• The z-test assumes that the data are independently and identically distributed (i.i.d.) and that the population is normally distributed

T-test for sample mean

• The t-test is used to test hypotheses about a population mean when the population standard deviation is unknown and the sample size is small (n < 30), or when comparing the means of two independent or paired samples
• The test statistic ($t$) is calculated as: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size
• The t-test assumes that the data are i.i.d. and that the population is normally distributed or that the sample size is large enough for the Central Limit Theorem to apply

Chi-square test for independence

• The chi-square test is used to test for the independence of two categorical variables
• It compares the observed frequencies in a contingency table to the expected frequencies under the null hypothesis of independence
• The test statistic ($\chi^2$) is calculated as: $\chi^2 = \sum \frac{(O - E)^2}{E}$, where $O$ is the observed frequency and $E$ is the expected frequency for each cell
• The chi-square test assumes that the expected frequencies are not too small (usually at least 5) and that the observations are independent

F-test for equality of variances

• The F-test is used to test for the equality of variances between two populations
• It compares the ratio of the sample variances to determine if they are significantly different
• The test statistic ($F$) is calculated as: $F = \frac{s_1^2}{s_2^2}$, where $s_1^2$ and $s_2^2$ are the sample variances of the two populations
• The F-test assumes that the data are i.i.d., the populations are normally distributed, and the samples are independent

Steps in hypothesis testing

• Hypothesis testing follows a systematic procedure to ensure valid and reliable results
• The steps include formulating hypotheses, selecting an appropriate test, calculating the test statistic, determining the critical value, and making a decision based on the p-value

Formulating hypotheses

• Clearly state the null hypothesis ($H_0$) and the alternative hypothesis ($H_a$) based on the research question and available information
• Ensure that the hypotheses are mutually exclusive and exhaustive
• Consider the implications of rejecting or failing to reject the null hypothesis

Selecting appropriate test

• Choose the appropriate hypothesis test based on the type of data (categorical or numerical), the distribution of the data (normal or non-normal), the sample size, and the research question
• Consider the assumptions of each test and whether they are met by the data
• Determine whether a one-tailed or two-tailed test is appropriate based on the alternative hypothesis

Calculating test statistic

• Calculate the test statistic using the appropriate formula for the selected hypothesis test
• Substitute the sample data and hypothesized values into the formula
• Double-check the calculations to ensure accuracy

Determining critical value

• Determine the critical value(s) based on the significance level ($\alpha$) and the type of test (one-tailed or two-tailed)
• Use statistical tables or software to find the critical value(s) for the specific test and degrees of freedom
• The critical value(s) represent the boundary between the rejection and non-rejection regions of the distribution

Making decision to reject or fail to reject

• Compare the calculated test statistic to the critical value(s) or calculate the p-value
• If the test statistic falls in the rejection region or the p-value is less than the significance level, reject the null hypothesis; otherwise, fail to reject the null hypothesis
• Interpret the decision in the context of the research question and the implications for the study

Interpreting results

• After conducting a hypothesis test, it is essential to interpret the results correctly and consider their implications
• Interpretation should include confidence intervals, effect sizes, practical significance, and limitations of the test

Confidence intervals

• A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence (usually 95%)
• Confidence intervals provide more information than a simple hypothesis test by indicating the precision and uncertainty of the estimate
• A narrow confidence interval suggests a more precise estimate, while a wide confidence interval indicates greater uncertainty

Effect size and practical significance

• The effect size measures the magnitude of the difference or relationship between variables, independent of the sample size
• Common effect size measures include Cohen's d, Pearson's r, and odds ratios
• Practical significance refers to the real-world importance or relevance of the effect, beyond statistical significance
• A statistically significant result may not be practically significant if the effect size is small or the consequences are minimal

Limitations of hypothesis testing

• Hypothesis testing has several limitations that should be considered when interpreting results:
  • It does not prove causality, only the existence of a significant relationship or difference
  • It is sensitive to sample size, with large samples more likely to yield significant results even for small effects
  • It can be affected by violations of assumptions, such as non-normality or lack of independence
  • It does not account for multiple testing, which can inflate the Type I error rate
• Researchers should be cautious in drawing conclusions based solely on hypothesis tests and consider other factors, such as study design, data quality, and theoretical plausibility

Applications in causal inference

• Hypothesis testing is a key tool in causal inference for assessing the significance of treatment effects and comparing groups in experiments
• It helps researchers determine whether observed differences or relationships are likely due to chance or to a genuine causal effect

Testing for significant treatment effects

• In randomized controlled trials (RCTs) and other experimental designs, hypothesis testing is used to assess the significance of the difference between treatment and control groups
• A significant result suggests that the treatment has a causal effect on the outcome, while a non-significant result indicates that the observed difference could be due to chance
• Example: Testing whether a new drug significantly reduces blood pressure compared to a placebo

Comparing groups in experiments

• Hypothesis testing is also used to compare multiple groups in experiments, such as different treatment conditions or subpopulations
• Tests like ANOVA (analysis of variance) and post-hoc comparisons help determine which groups differ significantly from each other
• Example: Comparing the effectiveness of three different teaching methods on student performance

Assessing validity of causal claims

• Hypothesis testing can be used to assess the validity of causal claims made in observational studies or quasi-experiments
• By testing for significant associations between variables or differences between groups, researchers can evaluate the strength of the evidence for a causal relationship
• However, hypothesis testing alone cannot establish causality, as other factors like confounding and reverse causation must be considered

Hypothesis testing vs estimation approaches

• While hypothesis testing focuses on making decisions about the existence of effects or differences, estimation approaches aim to quantify the size and uncertainty of effects
• Estimation methods, such as confidence intervals and Bayesian analysis, provide more informative results than simple hypothesis tests
• In causal inference, a combination of hypothesis testing and estimation approaches is often used to assess the significance, magnitude, and precision of causal effects