One-Way ANOVA

ANOVA, or Analysis of Variance, is a statistical method for comparing means among three or more groups, crucial in understanding group differences and relationships in diverse fields. In this article, we’ll focus on One-way ANOVA.

What is ANOVA?

ANOVA, or Analysis of Variance is a parametric statistical technique that helps in finding out if there is a significant difference between the mean of three or more groups. It checks the impact of various factors by comparing groups (samples) based on their respective mean. ANOVA tests the null hypothesis that all group means are equal, against the alternative hypothesis that at least one group mean is different.

Assumptions for ANOVA

1. The dependent variable is approximately normally distributed within each group. This assumption is more critical for smaller sample sizes.
2. The samples are selected at random and should be independent of one another.
3. All groups have equal standard deviations.
4. Each data point should belong to one and only one group. There should be no overlap or sharing of data points between groups.

In the case of two-way ANOVA, there are additional assumptions related to the interaction between the independent variables.

• The effect of one independent variable on the dependent variable should be consistent across all levels of the other independent variable.
• Combined effect of two independent variables is equal to the sum of their individual effects.

Types of ANOVA

There are two main types of ANOVA:

1. One-way ANOVA: This is the most basic form of ANOVA and is used when there is only one independent variable with more than two levels or groups. It assesses whether there are any statistically significant differences among the means of the groups.
2. Two-way ANOVA: Extending the one-way ANOVA, two-way ANOVA involves two independent variables. It allows for the examination of the main effects of each variable as well as the interaction between them. The interaction effect explores whether the effect of one variable on the dependent variable is different depending on the level of the other variable.

How to perform One-Way ANOVA?

It is a type of hypothesis test where only one factor is considered. We use F-statistic to perform a one-way analysis of variance. 

Steps Involved

Step 1 – Define the null and alternative hypothesis.

• H0 -> μ1 = μ2 = μ3 (where μ = mean)
• Ha -> At least one difference among the means.

Step 2 – Find the degree of freedom between and within the groups. [Using Eq-1 and Eq-2]

• n = number of samples in all groups combined
• k = number of groups.

Then, find total degree of freedom. [Using Eq-3]

For the next step, we need to understand what F-statistic is. 

F-value: It is defined as the ratio of the variance between samples to variance within samples. It is obtained while performing ANOVA test.

Eq-4 shows the F-value formula for one-way ANOVA.

Step 3 – Refer the F-Distribution table and find F_table using df_between and df_within.

As per the given F-Distribution table,

[use the given value of α while referring the table.]

Step 4 – Find the mean of all samples in each group.

Then use Eq-5 to find the Grand mean.
)

where,

• = sum of all sample values.
• n = number of samples.

Step 5 – Find the sum of squares total using Eq-6 and sum of squares within using Eq-7.

where,

• x_{i =} i_th sample.
• μ_{i =} mean of i_th group.

Then find sum of squares between using Eq-8.

Step 6 – Find the variance (μ² or S²) between and within samples using Eq-9 and Eq-10.

Step 7 – Find Fcalc using Eq-11.

(Equation-11)

Interpreting the results

• If Fcalc < Ftable :
  • Don’t reject null hypothesis.
  • i.e
• If Fcalc > Ftable :
  • Reject null hypothesis.

One-Way ANOVA Example

Consider the example given below to understand step by step how to perform this test. The marks of 3 subjects (out of 5) for a group of students is recorded. (as given in the table below) 
[Take α = 0.05]

Std/Sub	English (e)	Math (m)	Science (s)
Student 1	2	2	1
Student 2	4	3	2
Student 3	2	4	5

Step 1: Define the hypothesis

• Null hypothesis, (where μ = mean)
• Alternate hypothesis, .

Step 2: Degres of Freedom

As per the table, k = 3, n = 9

Step 3 Find the F-value corresponding to Degree of Freedom and alpha

On referring to the F-Distribution table, using df1 = 2 and df2 = 6 at α = 0.05: we get, Ftable = 5.14

Step 4 Compute the mean

Step 5 Compute the Sum of Squares

Step 6 Find the variance

Step 7 Compute F-statistics

Since, Fcalc < Ftable (0.05 < 5.14)

We cannot reject the null hypothesis.

Thus, we can say that the means of all three subjects is the same.

Conclusion

One-way ANOVA compares three or more than three categorical groups to establish whether there is a difference between them. The fundamental strategy of ANOVA is to systematically examine variability within groups being compared and also examine variability among the groups being compared. For any doubt/query, comment below. 

Frequently Asked Questions (FAQs)

Q. What is ANOVA used for?

It analyzes group variances to test differences in means, commonly applied in comparing group effects.

Q.How to use ANOVA using Python?

In Python, the scipy.stats module provides functions for ANOVA, and libraries like statsmodels offer comprehensive ANOVA capabilities.

Q. What is the role of ANOVA in EDA?

In Exploratory Data Analysis (EDA), ANOVA helps identify significant variations between groups, aiding insights into data patterns.

Q. What is the basic principle of ANOVA?

ANOVA compares means across groups, assessing if observed differences are statistically significant, using variances within and between groups.