## Problem Description

In the ANOVA assignment, we conducted a Univariate Analysis of Variance (ANOVA) to explore the variation in doubling time among three distinct cohorts of patients. Doubling time is an essential parameter in medical research, especially when studying the growth of cells or diseases. Our goal was to determine if there are significant differences in doubling time among these cohorts. Additionally, we tested for heteroskedasticity, a condition where within-group standard deviations differ, which could potentially affect the ANOVA results.

## Summary of Assignment Solution:

Descriptive Statistics: The following tables present descriptive statistics for doubling time across the entire dataset and for each of the three cohorts separately:

Statistics |
Overall Data |
Cohort 1 |
Cohort 2 |
Cohort 3 |
---|---|---|---|---|

N | 18 | 3 | 11 | 4 |

Mean | 2.78 | 10.2 | 1.17 | 1.63 |

Std. Deviation | 3.51 | 1.6 | 0.46 | 1.12 |

Kurtosis | 2.43 | - | - | - |

Skewness | 1.94 | - | - | - |

**Table 1: Descriptive Statistics for Doubling Time.**

For the overall dataset, the mean doubling time for the 18 patients is 2.78, with a standard deviation of 3.51, skewness of 1.94, and kurtosis of 2.43. Cohort 1, comprising 3 participants, has a mean doubling time of 10.2 with a standard deviation of 1.6. Cohort 2, with 11 participants, has a mean doubling time of 1.17 and a standard deviation of 0.46. Cohort 3, with 4 participants, has a mean doubling time of 1.63 and a standard deviation of 1.12.

Heteroskedasticity Testing: Heteroskedasticity is a violation of one of the assumptions of ANOVA, indicating that the variances of doubling time for the three cohorts are different. We employed the Levene Test for Equality of Variances to determine this:

**Null Hypothesis**: The variances are equal across the three cohorts (homoscedasticity).**Alternative Hypothesis**: The variances are different across the three cohorts (heteroscedasticity).

The test resulted in a p-value less than the significance level, leading us to reject the null hypothesis and conclude that heteroskedasticity is present in the dataset. This underscores the importance of testing this assumption, as unequal variances can influence the ANOVA results.

ANOVA and Post Hoc Analysis: To compare the cohorts for differences in means, we performed an ANOVA. The F-test revealed a significant difference in cohort means, as indicated by F(2,15) = 134.54, p = 0.000. This means that at least one cohort's mean doubling time differs significantly from the others.

To determine where these differences lie, we conducted a post hoc analysis using the Games-Howell test due to the presence of unequal variances. The results are as follows:

- Cohort 1 vs. Cohort 2: p = 0.02 (significant difference)
- Cohort 1 vs. Cohort 3: p = 0.009 (significant difference)
- Cohort 2 vs. Cohort 3: p = 0.738 (no significant difference)

This post hoc analysis clarifies that there are significant differences between the mean doubling time of cohort 1 compared to both cohort 2 and cohort 3, but no significant difference between cohort 2 and cohort 3.

In conclusion, the ANOVA and subsequent post hoc analysis allowed us to identify significant differences in doubling time means between certain cohorts, providing valuable insights for our research.