# Hypothesis Testing and Interpretation: Repeated Measures ANOVA on Film Brightness Across Different Time Points

In this comprehensive analysis, we conducted a One-Way Repeated Measures ANOVA to examine the impact of time on film brightness. Despite violating the normality assumption, the results, supported by Mauchly's test, revealed a significant difference in mean scores across various time points. Detailed pairwise comparisons further elucidate the nuanced temporal dynamics, contributing valuable insights into the evolution of film brightness over the study period.

## Problem Description:

The statistical analysis assignment aimed to investigate the response time differences between two drugs (Drug A and Drug B) while accounting for the length of time patients were ill. Statistical analyses were performed to assess the relationship between response time and predictors (length of time and drug), considering the potential impact of the length of illness.

### Statistical Analysis Overview:

1. Linear Regression Analysis:

• Conducted to understand the relationship between response time and predictors.
• Results showed limited explanatory power (R-squared = 0.054, adjusted R-squared = 0.008).
• The model's overall significance was not established (p = 0.322).
• Coefficients for drugs and length of time were not statistically significant.

2. Conclusion:

• No significant difference in response time between Drug A and Drug B was found when controlling for the length of time.
• The length of time variable did not significantly impact response time.
• The model explained only a small proportion of the variance in response time (low R-squared).

3. Assumption Checks:

• Normality:Shapiro-Wilk test indicated a deviation from normality for both Drug A (p = 0.008) and Drug B (p = 0.108).
• Homogeneity of Variance:Brown-Forsythe test showed no significant difference in variances between Drug A and Drug B (p = 0.327).

4. Descriptive Statistics:

• Provided information on the length of time and response time for both drugs.

5. JASP Output and Input:

• Included detailed output of statistical analyses such as linear regression, ANOVA, and assumption checks.
• Coefficients, residuals, and graphical representations were presented.

Implications:

While the analysis indicated no significant differences between the drugs or the length of time variable, the model's low explanatory power suggests that other factors might contribute to response time. Further exploration or consideration of additional predictors could enhance the model's predictive capability.

JASP Output: Two-Way ANOVA for Film Brightness

Hypotheses:

1. Development Technique:

• H0: There is no significant effect of the development technique on the brightness of the films produced.

2. Film Manufacturer:

• H0: There is no significant effect of the film manufacturer on the brightness of the films produced.

3. Interaction Effect:

• H0: There is no significant interaction effect between the development technique and film manufacturer on the brightness of the films produced.

### ANOVA Results:

Main Effects:

Film Manufacturers:F(2, 99) = 0.015, p = 0.985 (Non-significant)

Development Techniques:F(2, 99) = 0.998, p = 0.372 (Non-significant)

Interaction Effect:

Film Manufacturers ✻ Development Techniques:

• F(4, 99) = 0.959, p = 0.434 (Non-significant)

Descriptive Statistics:

• The table includes means, standard deviations, standard errors, and coefficients of variation for different combinations of film manufacturers and development techniques.

Assumption Checks:

• Equality of Variances (Levene's Test): F = 0.569, p = 0.801 (Non-significant)
• Q-Q Plot: Graphical representation of normality checks.

Marginal Means:

• Marginal means and 95% confidence intervals for differences in means between film manufacturers and development techniques.

JASP Output: One-Way Repeated Measures ANOVA

Hypothesis Testing:

Step 1: State the hypotheses:

• Null Hypothesis (H0): There is no significant difference in the mean scores across the different time points.
• Alternative Hypothesis (Ha): There is a significant difference in the mean scores across the different time points.

Step 2: Check Assumptions:

• Sphericity assumption: Mauchly's test (p = 0.118) suggests that the assumption of sphericity is met. b) Normality assumption: Violated based on Kolmogorov-Smirnov (p < 0.001) and Shapiro-Wilk (p < 0.001) tests.

Step 3: Calculate the ANOVA:

• Repeated Measures ANOVA was performed using the given data.

Step 4: Interpret the Results:

• ANOVA Results: F(4, 76) = 2.768, p = 0.033 (significant).
• Effect sizes (η², ω²) indicate a small effect.
• Pairwise comparisons show no significant differences between any of the time points.

Interpretation of Tables:

Mauchly's test:

• Suggests sphericity is met (p = 0.118), indicating approximately equal variances of differences between time points.

Test for normality:

• Both KolmBoth Kolmogorov-Smirnov and Shapiro-Wilk tests indicate violations of the normality assumption.

Pairwise comparisons:

• None of the pairwise comparisons are statistically significant after the Bonferroni adjustment.

ANCOVA Suitability:

Would an ANCOVA test be more appropriate for this research? Explain.

• ANCOVA is not more appropriate since there are no covariates involved. ANCOVA is used when covariates need to be controlled for, which is not the case in this study.

JASP Output:

• Repeated Measures ANOVA results are provided, including within-subjects effects, between-subjects effects, and descriptives.
• Descriptive statistics and plots for each time point are presented.