anova and its application

ANOVA and its Applications

P GuruprasadSR, Pharmacology,AIIMS

March 20, 2015

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

Trial of 2 drugs for management of Hypertension

You are doing an investigator initiated trial to compare two recentlyintroduced drugs, alasartan and blasartan vs losartan (add on) for

management of hypertension. After ethics committee approval and takinginformed consent, you enroll 150 hypertensive subjects. Baseline SystolicBP 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.

Data of trial on Hypertension

The following table displays the summary of our primaryoutcome(difference in Systolic BP) across the groups.

Losartan Grp Alasartan Grp Blasartan Grp

Mean 4.958713 7.7409001 7.4445712SD 1.0484026 1.0708933 1.0665367Median 4.7990277 7.6193942 7.4501771Minimum 2.9642749 5.3501633 4.834001Maximum 7.9946055 9.865391 9.8347861Interquartile range 1.4067942 1.5954365 1.5214803

Which drug is superior?

Which group has the maximum mean reduction?

The mean reduction in Systolic BP in the losartan, alasartin and blasartingroups were 4.958713, 7.7409001 and 7.4445712 millimeters of mercuryrespectively.

Which test should we use?

What are the options?

Parametric – ANOVANon-parametric – Kruskal Wallis

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 toolscan be used

1 Visually – boxplot, histogram, Q-Q plot

2 Tests - Kolmogorov-Smirnov, Shapiro-Wilk test and others

Which test to use

Independence

Normal or not

Which test to use

Independence

Normal or not

Which test to use

Independence

Normal or not

Which test to use

Independence

Normal or not

Which test to use

Independence

Normal or not

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.

Linearity of means

Normality of our data-Histograms

Histogram of Losartan group

Systolic BP reduction

2 3 4 5 6 7 8

Figure: Histogram of losartan group

Normality of our data-Histograms contd.

Histogram of Alasartan group

5 6 7 8 9 10

Figure: Histogram of Alasartan group

Normality of our data-Histograms contd.

Histogram of Blasartan group

4 5 6 7 8 9

Figure: Histogram of Blasartan group

Normality of our data-Boxplot

Boxplots of all 3 groups

Figure: Boxplot of all three groups

Normality of our data-QQ Plot

−2 −1 0 1 2

QQ Plot of losartan group

Theoretical Quantiles

Figure: QQ Plot of losartan group

Normality of our data-QQ Plot contd.

−2 −1 0 1 2

QQ Plot of Alasartan group

Figure: QQ Plot of Alasartan group

Normality of our data-QQ Plot contd.

−2 −1 0 1 2

QQ Plot of Blasartan group

Figure: QQ Plot of Blasartan group

Assumptions of ANOVA

To emphasize, the following are the assumptions of ANOVA which have tobe 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?

1 Independence

ANOVA Test

Lets go ahead and perform ANOVA.

The following table displays the ANOVA table for the comparison ofdifferences in systolic BP across the three groups.

Df Sum Sq Mean Sq F value Pr(>F)

group 2 233.46 116.73 103.50 0.0000Residuals 147 165.79 1.13

Table: ANOVA Table of Differences across the 3 groups

Significance!!

is there a significant difference between the groups?

ANOVA Test

Lets go ahead and perform ANOVA.The following table displays the ANOVA table for the comparison ofdifferences in systolic BP across the three groups.

group 2 233.46 116.73 103.50 0.0000Residuals 147 165.79 1.13

Significance!!

ANOVA Test

group 2 233.46 116.73 103.50 0.0000Residuals 147 165.79 1.13

Significance!!

ANOVA Test

group 2 233.46 116.73 103.50 0.0000Residuals 147 165.79 1.13

Significance!!

Next step, Which group(s) is/are superior?

We shall perform pairwise ”t-tests” between group to find out whichgroup(s) is superior if there indeed is any.

Indeed we find that the Alasartan and Blasartan groups have a higherSystolic BP reduction when compared with the Losartan group.

This we did by the ”Bonferroni Correction”

Hence we can conclude that both Alasartan and Blasartan aresuperior to losartan in reducing Systolic BP.

Next step, Which group(s) is/are superior?

We shall perform pairwise ”t-tests” between group to find out whichgroup(s) is superior if there indeed is any.

Indeed we find that the Alasartan and Blasartan groups have a higherSystolic BP reduction when compared with the Losartan group.

This we did by the ”Bonferroni Correction”

Hence we can conclude that both Alasartan and Blasartan aresuperior to losartan in reducing Systolic BP.

anova and its application

Health & Medicine

maximum mean reduction

aiims anova

systolic bp

blasartan vs losartan

data of trial

recentlyintroduced drugs

hypertensive subjects

takinginformed consent