Chapter 13 Notes on Inferential Statistics

Inferential Statistics

• Definition: Involves making inferences about a population based on sample data.

Descriptive Statistics

• Purpose: Describes and summarizes data without making inferences.

• Example: Average words spoken in conversation.

  • Male participants: 54 words.

  • Female participants: 75 words.

Inferential Statistics

• Purpose: Draws conclusions about a population based on sample data.

• Example: Inferring that the average words spoken by women is greater than that by men.

  • Conclusion is uncertain and indicates support but not absolute truth about population values.

Real vs. Chance Differences

• Key Concept: Inferential stats help determine if observed differences are statistically significant and not due to random chance.

  • Descriptive statistics provide numerical differences, but inferential statistics appraise their meaning.

Equivalent Groups Importance

• Control and Random Assignment: Essential for quality in research to isolate effects of the independent variable (IV) alone.

Hypotheses

• Null Hypothesis (H0): No differences exist in means (any observed difference is due to random error).

• Research Hypothesis (H1): Population means are statistically different due to independent variables (IV).

Statistical Significance

• Alpha Level: Typically set at 0.05 (5% chance of Type I error).

• Misconceptions about significance:

  • 5% chance of a false positive does not imply a 95% chance the difference is real.

  • Does not confirm hypothesis is true and often ignores practical significance (effect size and confidence intervals).

Understanding p-values

• A p-value < 0.05 indicates low probability of observing this difference assuming null hypothesis is true (less than 5% chance).

Jury Analogy

• Null Hypothesis as Death by Natural Causes: A low p-value indicates less likelihood that differences observed are due to natural causes, not proving causation by the independent variable.

Types of Errors

• Type I Error: Incorrectly rejecting null hypothesis when no effect exists (false positive).

  • Analogy: A false pregnancy test.

• Type II Error: Failing to reject null hypothesis when an effect is present (false negative).

  • Ability to reject null when it is false reflects the study's power; small sample sizes may lead to this error.

Publishing Issues

• Many non-significant results go unpublished, even when they provide insights.

• Reasons for non-publication may include ceiling effects, floor effects, or poor validity of measures.

Sample Size Considerations

• Larger sample sizes increase reliability of intercepting effect sizes.

  • Small sample with large differences can be misleading.

• Example: Significant results may be seen in large samples even if differences are small, while small samples may show noise masking true differences.

Practical Importance

• Effect Size Considerations:

  • Importance of the findings in real-world context (e.g., IQ changes from investments).

Power Analysis

• Factors influencing power include size of effect, sample size, and type of test used.

• Higher alpha levels increase the likelihood of rejecting the null hypothesis but trade-off must be considered.

Parametric Tests

• More commonly utilized but depend on certain assumptions:

  • Measurement scales being interval or ratio.

  • Normality of sampling distributions.

  • Homogeneity of variance.

Central Limit Theorem

• The theory states that with a sufficiently large sample size (typically n > 30), the sampling distribution of the sample mean approximates a normal distribution regardless of population distribution.

Identifying Outliers

• Use Z-scores or boxplots to identify outliers (values that fall outside the whiskers of a boxplot).

t-tests

Types of t-tests

• Independent Samples t-test: Used to compare means of two unrelated groups.

• Paired Samples t-test: Used when samples are connected (e.g., before and after measurements).

Common Assumptions

• Normality of distribution and independence between samples are key assumptions, and these need to be tested.

F Tests and ANOVA

• F Test: Evaluates differences among three or more groups, assessing systematic versus error variance.

• One-Way ANOVA: Assesses means across several groups; post hoc tests can determine specific group differences.

Model Assumptions

• ANOVA assumes normality, equal variances, and independence of observations.

• If assumptions are violated, alternative methods (e.g., non-parametric tests) may be needed.

Effect Size in ANOVA

• Effect size can indicate the proportion of variance explained by the independent variable.

Replication Importance

• Multiple replications across different settings enhance understanding and establish reliability of findings.

Summary

• The comprehension of inferential statistics, including hypothesis testing, errors, and effect sizes, is crucial for analyzing data effectively and making appropriate conclusions in research contexts.

One-Tailed vs. Two-Tailed Tests

• One-Tailed Test: Tests for a specific direction of an effect (e.g., whether the mean of one group is greater than the mean of another).

  • Example: Testing if a new drug leads to an increase in recovery rates compared to a placebo.

• Two-Tailed Test: Tests for any difference without specifying a direction (e.g., whether the means of two groups are different).

  • Example: Testing whether a new educational program leads to a change in test scores, either higher or lower.

Role of Variance on t and F Tests

• Variance: Measures how data points differ from the mean. The role of variance in statistical tests is crucial:

  • t-Tests: Variance is used to estimate the standard error of the mean. High variance can lead to a larger standard error, making it harder to detect a significant difference.

  • F-Tests/ANOVA: Variance is assessed to determine if systematic variance (variance between group means) is significantly greater than error variance (variance within groups). A significant F-statistic indicates that at least one group mean differs from others.

Post Hoc Tests

• Post Hoc Tests: Conducted after a significant ANOVA to determine which specific group means are different from each other.

  • Common tests include Tukey's HSD, Bonferroni, and Scheffé's Test. These tests control for Type I error across multiple comparisons, ensuring reliable conclusions about specific differences among groups.

Error Bars

• Error Bars: Graphical representations of the variability of data and indicate the uncertainty around sample estimates.

  • Typically represent the standard error or confidence intervals of the mean.

  • Helpful for visualizing the overlap of datasets; less overlap suggests greater differences in means across groups.