💻Intro to Programming in R Unit 15 – Basic Statistical Tests

What's the Deal with Statistical Tests?

• Statistical tests assess whether observed differences between groups or relationships between variables are likely due to chance or represent real effects
• Help researchers make inferences about populations based on sample data
• Provide a systematic way to test hypotheses and draw conclusions from data
• Different tests are used depending on the type of data, number of groups, and research question
• Significance level (usually 0.05) is set to determine if results are statistically significant
• P-values indicate the probability of observing the results if the null hypothesis is true
  • P-values less than the significance level suggest rejecting the null hypothesis
• Statistical power is the ability to detect an effect when one exists and is influenced by sample size, effect size, and significance level

Types of Data: Know Your Numbers

• Nominal data consists of categories without any order or hierarchy (colors, gender)
• Ordinal data has categories with a specific order but no consistent scale (survey responses: strongly agree to strongly disagree)
• Interval data has ordered categories with consistent intervals but no true zero point (temperature in Celsius)
• Ratio data has ordered categories, consistent intervals, and a true zero point (height, weight)
• Discrete data can only take on specific values, often integers (number of siblings)
• Continuous data can take on any value within a range (time, length)
• Knowing the data type is crucial for selecting the appropriate statistical test and interpreting results accurately

T-Tests: Comparing Two Groups

• T-tests compare means between two groups to determine if they are significantly different
• Independent samples t-test is used when the two groups are independent and unrelated
  • Example: comparing exam scores between students who attended a review session and those who did not
• Paired samples t-test is used when the two groups are related or matched
  • Example: comparing blood pressure before and after a medication in the same patients
• Assumptions include normal distribution of data, homogeneity of variances, and independence of observations
• The test statistic (t) is calculated based on the difference between means, variability, and sample size
• Degrees of freedom (df) are determined by sample size and used to find the critical value for the test
• If the calculated t-value exceeds the critical value, the null hypothesis is rejected, indicating a significant difference between the groups

ANOVA: When Two's Not Enough

• Analysis of Variance (ANOVA) is used to compare means across three or more groups
• One-way ANOVA examines the effect of one independent variable (factor) on a dependent variable
  • Example: comparing job satisfaction scores among employees from different departments
• Two-way ANOVA examines the effects of two independent variables and their interaction on a dependent variable
  • Example: comparing plant growth based on both fertilizer type and watering frequency
• F-statistic is calculated to determine if there are significant differences among the group means
• If the F-statistic exceeds the critical value, the null hypothesis is rejected, indicating significant differences among the groups
• Post-hoc tests (Tukey, Bonferroni) are used to determine which specific groups differ significantly from each other
• Assumptions include normal distribution, homogeneity of variances, and independence of observations

Correlation: Relationships Between Variables

• Correlation measures the strength and direction of the linear relationship between two continuous variables
• Pearson's correlation coefficient (r) ranges from -1 to +1
  • r = -1 indicates a perfect negative linear relationship
  • r = 0 indicates no linear relationship
  • r = +1 indicates a perfect positive linear relationship
• Scatterplots are used to visualize the relationship between the variables
• Correlation does not imply causation; other factors may influence the relationship
• Spearman's rank correlation is used for ordinal data or when assumptions of Pearson's correlation are violated
• Correlation can help identify variables that may be used in predictive models (regression analysis)

Chi-Square: Categorical Data's Best Friend

• Chi-square tests are used to analyze relationships between categorical variables
• Chi-square goodness of fit test compares observed frequencies to expected frequencies for a single categorical variable
  • Example: testing if the colors of M&Ms in a bag match the expected distribution
• Chi-square test of independence examines the relationship between two categorical variables in a contingency table
  • Example: testing if there is a significant association between gender and preferred ice cream flavor
• The test statistic ($\chi^2$) is calculated based on the differences between observed and expected frequencies
• Degrees of freedom are determined by the number of categories in the variables
• If the calculated $\chi^2$ value exceeds the critical value, the null hypothesis is rejected, indicating a significant relationship between the variables
• Assumptions include independence of observations, adequate sample size, and expected frequencies greater than 5 in each cell

Implementing Tests in R: Code Time!

• R provides built-in functions for conducting various statistical tests

• t.test()

  function is used for t-tests
  • Specify the formula, data, and type of t-test (paired or independent)

• aov()

  function is used for ANOVA
  • Specify the formula and data
  • Use

    summary()

    to view the ANOVA table and test results

• cor()

  function is used for correlation
  • Specify the variables and method (Pearson or Spearman)

• chisq.test()

  function is used for chi-square tests
  • Specify the observed frequencies or contingency table
• Set the significance level using the

  conf.level

  argument (default is 0.95)
• Extract p-values, test statistics, and other relevant information from the test objects
• Use

  ggplot2

  package to create visualizations (scatterplots, boxplots, bar charts) to support test results

Interpreting Results: What Does It All Mean?

• Statistical tests provide evidence for or against the null hypothesis
• A significant result (p < 0.05) suggests that the observed differences or relationships are unlikely due to chance alone
  • Reject the null hypothesis and conclude that there is a significant effect or association
• A non-significant result (p > 0.05) suggests that the observed differences or relationships could be due to chance
  • Fail to reject the null hypothesis and conclude that there is insufficient evidence for a significant effect or association
• Effect sizes (Cohen's d, eta-squared, r-squared) quantify the magnitude of the difference or relationship
  • Interpret effect sizes in the context of the research question and field of study
• Confidence intervals provide a range of plausible values for the population parameter
  • Narrower intervals indicate greater precision in the estimate
• Consider the practical significance of the results in addition to statistical significance
• Be cautious of type I (false positive) and type II (false negative) errors
  • Adjust significance levels or use multiple comparison corrections when conducting multiple tests
• Interpret results in the context of the study design, limitations, and previous research findings