Now with the one-way ANOVA in the previous problem, we were only able to identify that one of the groups had a significantly different mean blood pressure reduction. How do we find out which drug (A or B or C) had significant effect?
We perform a post-hoc confirmatory test such as Tukey’s HSD which is mostly used.
8. Perform Tukey’s HSD (Honestly Significant Difference) test
tukey_result <- TukeyHSD(anova1way)
print(tukey_result)
Interpretation of the output:
diff: The difference in the means between the two groups being compared.
lwr: The lower bound of the 95% confidence interval for the difference in means.
upr: The upper bound of the 95% confidence interval for the difference in means.
p adj: The adjusted p-value (for family-wise error rate), indicating whether the difference between the means is statistically significant.
Interpretations of each comparison:
B – A:
Difference: The mean reduction in blood pressure (bp) for Drug B is -11.2 mmHg less than for Drug A. This means Drug B reduces blood pressure by 11.2 mmHg more than Drug A.
Confidence Interval: The 95% confidence interval for the difference is between -19.48 mmHg and -2.92 mmHg. This interval does not include zero, meaning we can be confident that Drug B and Drug A have significantly different effects on blood pressure.
p-value: The p-value is 0.009, which is less than 0.05, indicating that the difference is statistically significant. In other words, Drug A and Drug B reduce blood pressure to significantly different extents.
C – A:
Difference: The mean reduction for Drug C is -18.0 mmHg less than for Drug A. This means Drug C reduces blood pressure by 18.0 mmHg more than Drug A.
Confidence Interval: The 95% confidence interval is between -26.28 mmHg and -9.72 mmHg, which also does not include zero. This indicates that Drug C and Drug A have significantly different effects.
p-value: The p-value is 0.00023, much smaller than 0.05, indicating that Drug C has a significantly different (greater) reduction in blood pressure compared to Drug A.
C – B:
Difference: The mean reduction for Drug C is -6.8 mmHg less than for Drug B. This means Drug C reduces blood pressure by 6.8 mmHg more than Drug B.
Confidence Interval: The 95% confidence interval is between -15.08 mmHg and 1.48 mmHg. Since this interval includes zero, it suggests that the difference in effects between Drug C and Drug B might not be statistically significant.
p-value: The p-value is 0.113, which is greater than 0.05, confirming that there is no significant difference between Drug B and Drug C in terms of their effects on blood pressure.
Summary:
Drug A has the least reduction in blood pressure since both Drug B (-11.2) and Drug C (-18) show significantly more reduction compared to Drug A.
Drug C shows the largest reduction in blood pressure, as it has the largest (and significant) difference from Drug A (-18.0), and even though it’s larger than Drug B (-6.8), this difference is not statistically significant.
Drug B is in between Drug A and Drug C. It significantly reduces blood pressure more than Drug A (-11.2), but the reduction compared to Drug C (-6.8) is not statistically significant.
Conclusion:
The conclusion that Drug A has the smallest reduction (least effective) in blood pressure is supported by the fact that both Drugs B and C significantly outperform it (with negative mean differences and p-values < 0.05). Drug C has the greatest reduction because it shows the largest mean difference from Drug A (-18.0), and although it is larger than Drug B (-6.8), the difference is not statistically significant.