ANOVA (Analysis of Variance) is applied to compare more than two independent sample means.

Assumptions
1. Dependent variable is continuous and normally distributed in all the groups.
2. Samples are drawn using random sampling technique.
3. The observations are independent. There are no observations, which are in more than one group.
4. The variance in all the groups is same. (homogeneity of variance / homoscedasticity). It can be tested by Levene's test of homogeneity of variance.
5. There are no outliers.
6. There is one independent variable, which is nominal or ordinal, with more than two possible levels / categories.

Steps:

1. Null hypothesis and Alternate Hypothesis

Null Hypothesis

A. All the sample means are equal. (μ_i = μ_j)

 Alternate Hypothesis

A. There are at least two means (at least a pair of means), which are different than each other. (μ_i ≠ μ_j)

Where μ_i and μ_j can be any two sample means.

2. Calculate Sum of Squares for Treatment (SS Between)

where n_i is the sample size of group i,
x̄ = Grand mean (Mean of all the observations)
x̄_i = Mean of group i

3. Calculate Sum of Squares Within (SS Error)

x̄ = Grand mean (Mean of all the observations)
x_ij = j^th observation in group i.

4. Calculate Mean Sum of Squares for Treatment and Mean Sum of Squares for Error

df_N = Degrees of freedom for Numerator = k - 1
df_D = Degrees of freedom for Denominator = N - k
N = Grand Sample Size, k = Number of groups

5. Calculate F value

6. Calculate p value, using F table

7. Interpret

If, p < = alpha, then reject the Null hypothesis
If, p > alpha, then accept null hypthesis

8. Post hoc tests

If, p < = alpha, alternate hypothesis is accepted (at least two means are significantly different)
Which pair (or pairs) of means are significantly different is now identified by post hoc tests.
There are multiple post hoc tests available, out of which Benferroni and Tukey HSD are most commonly used.

8. A. Benferroni

Multiple tests can increase the type I error rate. For example if n simulteneous t tests are used for an alpha of 0.05 for each test, then total alpha will be 1 - (1 - alpha)ⁿ.
So, if there are three means to be compared in ANOVA, we need to apply three different t tests. So total resultant alpha will be 1 -(1-0.05)³ = 0.142625, much more than intended 0.05.
Hence, an adjustment is done by applying t test for each pair with adjusted alpha of alpha / (total no. of tests applied).
{Alternately, the p value can be adjusted (Bonferroni adjusted p) by multiplying the p value got during each t test with total no. of tests to be applied.}
In Benferroni post hoc test, for each pair of means t statistics is calculated as follows.

The resultant p value is to be multiplied by no. of tests (No. of pairs), to get Bonferroni adjusted p value. Please that the degrees of freedom, while calculating p value, is equal to the df_D in applied ANOVA. If Bonferroni adjusted p value is less than the inended alpha, then the means in the pairs are significantly different.

If there are n groups, then total no. of tests (No. of pairs) will be n * (n - 1) / 2.

Confidence intervals for Bonferroni post hoc test

Critical t value is calculated using df = df_D and adjusted alpha (original alpha / no. of tests)

LB of CI = (x̄_i - x̄_j) - (critical t) * SE

UB of CI = (x̄_i - x̄_j) + (critical t) * SE

 

Here, SE = SQRT (MS_Between * (1/n_i + 1/n_j)

Please note that, SE is exactly equal to the denominator used for calculating t in above equation.

Example 1:

Random samples of 25 each from rural, semi-urban and urban students have shown mean IQ of 111.68 ± 5.4522, 102.48 ± 12.7381 and 102.40 ± 13.6626, respectively. Apply an approriate statistical test to test whether there is residence-wise significantly difference in IQ, at alpha level of 5%.

As there are more than two means to be compared, appropriate statistical test is ANOVA.
Here,
x̄ ₁ = 111.68, SD₁ = 5.4522, N₁= 25
x̄ ₂ = 102.48, SD₂ = 12.7381, N₂= 25
x̄ ₃ = 102.40, SD₃ = 13.6626, N₃= 25
alpha = 5%
Putting above values at the respective places, we get the output as follows:

As p = 0.0053 in ANOVA, p is significant. We can say that there are at least two means which are statistically significantly different than each other.
To find the pair (pairs), post hoc tests are applied.
Post hoc tests have shown that Mean 1 (rural) and Mean 2 (semi-urban) are significantly different.
Similarly, Mean 1 (rural) and Mean 3 (urban) are also significantly different.
However, Mean 2 (semi-urban) and Mean 3(urban) are not significantly different.

---

 

---

@ Sachin Mumbare

---