1) A trial was conducted to compare the effects of three drugs (losartan, alasartan, and blasartan) on systolic blood pressure in hypertensive patients. 2) The mean reductions in systolic BP were 4.95, 7.74, and 7.44 mmHg for the losartan, alasartan, and blasartan groups respectively. 3) One-way ANOVA indicated a statistically significant difference between the groups, so pairwise t-tests were used to determine which groups differed. The alasartan and blasartan groups had significantly greater reductions than the losartan group.

1. ANOVA and its Applications
P Guruprasad
SR, Pharmacology,AIIMS
March 20, 2015
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 1 / 18

2. Trial of 2 drugs for management of Hypertension
You are doing an investigator initiated trial to compare two recently
introduced drugs, alasartan and blasartan vs losartan (add on) for
management of hypertension. After ethics committee approval and taking
informed consent, you enroll 150 hypertensive subjects. Baseline Systolic
BP is measured, and after randomization, medicines are precribed such
that the subjects are divided equally among the 3 groups. 3 months later,
Systolic BP is measured again.
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 2 / 18

3. Data of trial on Hypertension
The following table displays the summary of our primary
outcome(difference in Systolic BP) across the groups.
Losartan Grp Alasartan Grp Blasartan Grp
Mean 4.958713 7.7409001 7.4445712
SD 1.0484026 1.0708933 1.0665367
Median 4.7990277 7.6193942 7.4501771
Minimum 2.9642749 5.3501633 4.834001
Maximum 7.9946055 9.865391 9.8347861
Interquartile range 1.4067942 1.5954365 1.5214803
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 3 / 18

4. Which drug is superior?
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 4 / 18

5. Which drug is superior?
Which group has the maximum mean reduction?
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 4 / 18

6. Which drug is superior?
Which group has the maximum mean reduction?
The mean reduction in Systolic BP in the losartan, alasartin and blasartin
groups were 4.958713, 7.7409001 and 7.4445712 millimeters of mercury
respectively.
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 4 / 18

7. Which drug is superior?
Which group has the maximum mean reduction?
The mean reduction in Systolic BP in the losartan, alasartin and blasartin
groups were 4.958713, 7.7409001 and 7.4445712 millimeters of mercury
respectively.
Which test should we use?
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 4 / 18

8. Which drug is superior?
Which group has the maximum mean reduction?
The mean reduction in Systolic BP in the losartan, alasartin and blasartin
groups were 4.958713, 7.7409001 and 7.4445712 millimeters of mercury
respectively.
Which test should we use?
What are the options?
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 4 / 18

9. Which drug is superior?
Which group has the maximum mean reduction?
The mean reduction in Systolic BP in the losartan, alasartin and blasartin
groups were 4.958713, 7.7409001 and 7.4445712 millimeters of mercury
respectively.
Which test should we use?
What are the options?
Parametric – ANOVA
Non-parametric – Kruskal Wallis
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 4 / 18

10. Which test to use
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 5 / 18

11. Which test to use
Lets look at the attributes of the data
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 5 / 18

12. Which test to use
Lets look at the attributes of the data
Independence
Is the reading of each subject independent of others?
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 5 / 18

13. Which test to use
Lets look at the attributes of the data
Independence
Is the reading of each subject independent of others?
Normal or not
ANOVA can be done only if the data is normally distributed.
How can we say if it is normally distributed or not? The following tools
can be used
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 5 / 18

14. Which test to use
Lets look at the attributes of the data
Independence
Is the reading of each subject independent of others?
Normal or not
ANOVA can be done only if the data is normally distributed.
How can we say if it is normally distributed or not? The following tools
can be used
1 Visually – boxplot, histogram, Q-Q plot
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 5 / 18

15. Which test to use
Lets look at the attributes of the data
Independence
Is the reading of each subject independent of others?
Normal or not
ANOVA can be done only if the data is normally distributed.
How can we say if it is normally distributed or not? The following tools
can be used
1 Visually – boxplot, histogram, Q-Q plot
2 Tests - Kolmogorov-Smirnov, Shapiro-Wilk test and others
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 5 / 18

16. Attributes of the data contd.
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 6 / 18

17. Attributes of the data contd.
Equal variance(Homoscedasticity)
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 6 / 18

18. Attributes of the data contd.
Equal variance(Homoscedasticity)
1 Visually – Residuals plot
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 6 / 18

19. Attributes of the data contd.
Equal variance(Homoscedasticity)
1 Visually – Residuals plot
2 Tests – Levene’s test and Barlett’s test.
But ANOVA is resistant/robust against inequality of variances.
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 6 / 18

20. Attributes of the data contd.
Equal variance(Homoscedasticity)
1 Visually – Residuals plot
2 Tests – Levene’s test and Barlett’s test.
But ANOVA is resistant/robust against inequality of variances.
Linearity of means
Will be explained later.
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 6 / 18

21. Normality of our data-Histograms
Histogram of Losartan group
Systolic BP reduction
Frequency
2 3 4 5 6 7 8
0510
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 7 / 18

22. Normality of our data-Histograms contd.
Histogram of Alasartan group
Systolic BP reduction
Frequency
5 6 7 8 9 10
048
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 8 / 18

23. Normality of our data-Histograms contd.
Histogram of Blasartan group
Systolic BP reduction
Frequency
4 5 6 7 8 9
051015
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 9 / 18

24. Normality of our data-Boxplot
1 2 3
3579
Boxplots of all 3 groups
Systolic BP reduction
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 10 / 18

25. Normality of our data-QQ Plot
−2 −1 0 1 2
357
QQ Plot of losartan group
Theoretical Quantiles
SampleQuantiles
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 11 / 18

26. Normality of our data-QQ Plot contd.
−2 −1 0 1 2
6789
QQ Plot of Alasartan group
Theoretical Quantiles
SampleQuantiles
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 12 / 18

27. Normality of our data-QQ Plot contd.
−2 −1 0 1 2
579
QQ Plot of Blasartan group
Theoretical Quantiles
SampleQuantiles
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 13 / 18

28. Assumptions of ANOVA
To emphasize, the following are the assumptions of ANOVA which have to
be satistfied.
1 Independence
2 Variables in each groups should have a normal distribution.
3 All groups should have similar variance
4 Linearity of means
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 14 / 18

29. Assumptions of ANOVA
To emphasize, the following are the assumptions of ANOVA which have to
be satistfied.
1 Independence
2 Variables in each groups should have a normal distribution.
3 All groups should have similar variance
4 Linearity of means
What about our data?
Did our data meet the requirements of these assumptions?
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 14 / 18

30. Assumptions of ANOVA
To emphasize, the following are the assumptions of ANOVA which have to
be satistfied.
1 Independence
2 Variables in each groups should have a normal distribution.
3 All groups should have similar variance
4 Linearity of means
What about our data?
Did our data meet the requirements of these assumptions?
YES
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 14 / 18

31. ANOVA Test
Lets go ahead and perform ANOVA.
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 15 / 18

32. ANOVA Test
Lets go ahead and perform ANOVA.
The following table displays the ANOVA table for the comparison of
differences in systolic BP across the three groups.
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 15 / 18

33. ANOVA Test
Lets go ahead and perform ANOVA.
The following table displays the ANOVA table for the comparison of
differences in systolic BP across the three groups.
Df Sum Sq Mean Sq F value Pr(>F)
group 2 233.46 116.73 103.50 0.0000
Residuals 147 165.79 1.13
Table: ANOVA Table of Differences across the 3 groups
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 15 / 18

34. ANOVA Test
Lets go ahead and perform ANOVA.
The following table displays the ANOVA table for the comparison of
differences in systolic BP across the three groups.
Df Sum Sq Mean Sq F value Pr(>F)
group 2 233.46 116.73 103.50 0.0000
Residuals 147 165.79 1.13
Table: ANOVA Table of Differences across the 3 groups
Significance!!
is there a significant difference between the groups?
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 15 / 18

35. Next step, Which group(s) is/are superior?
We shall perform pairwise ”t-tests” between group to find out which
group(s) is superior if there indeed is any.
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 16 / 18

36. Next step, Which group(s) is/are superior?
We shall perform pairwise ”t-tests” between group to find out which
group(s) is superior if there indeed is any.
Indeed we find that the Alasartan and Blasartan groups have a higher
Systolic BP reduction when compared with the Losartan group.
This we did by the ”Bonferroni Correction”
Hence we can conclude that both Alasartan and Blasartan are
superior to losartan in reducing Systolic BP.
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 16 / 18

37. Recommended reads
1 Probability and Statistics (4th Edition) Hardcover - January 6, 2011
by Morris H. DeGroot (Author), Mark J. Schervish (Author)
2 Interpretation and Uses of Medical Statistics Paperback - Import, 4
Jul 2000 by Leslie Daly (Author), Geoffrey J Bourke (Author)
3 Does your data violate one-way ANOVA assumptions?
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 17 / 18

38. This presentation was possible due to!!
P Guruprasad SR, Pharmacology,AIIMS ANOVA and its Applications March 20, 2015 18 / 18