In the field of statistics, ANOVA plays an exclusive role and is a more helpful tool for researchers and data analysts to evaluate the difference between the means of data sets from an experiment. It helps the researchers and analysts to determine significant differences between the means of three or more groups by using different types of ANOVA.
This technique is essential for understanding variations between data sets of many fields. ANOVA can handle a large variety of experimental factors such as repeated measures on the same experimental unit. It has many applications in different fields including psychology, medicine, agriculture, and marketing to find the variation of data sets and make the experimental work strong.
Many people worry about finding a difference between the mean of data and the characteristics of the experiment but now you don’t worry about it, you can overcome these problems by reading this guide.
In this guide, we will provide a detailed overview of ANOVA, and explain its types. Also, explain the interpretations and assumptions that are necessary for the experiment calculations. For a better understanding, we will solve the example with its detailed steps.
What is ANOVA?
ANOVA is a statistical method that compares the means of different groups to see if at least one of them differs significantly from the others. ANOVA stands for analysis of variance and is simply known as Fisher’s Analysis due to a new idea by the renowned statistician Ronald A. Fisher.
It provides a new way to find the variance of data and make a partition for variance for the response variable based on one or more explanatory factors. Moreover, it extends the t-test that helps us to compare the means between two groups or more than two groups.
By examining the variance within groups and between groups, ANOVA helps in identifying whether the observed differences are due to random chance or a specific factor. It also helps us to find the numeric value of the response variable and analyze the difference between the grouping factors.
Types of ANOVA
There are different types of ANOVA depending on its dependent or independent variables and the number of factors involved in the groups of a data set in experimenting. Here we discuss some of its types.
1. One-Way ANOVA
It is a very simple and easy experiment to analyze the data. In this way, let the three or more groups compare the means of unrelated groups based on one independent variable. Because the ANOVA applies to more than two groups.
In this type, there is one factor in analyzing the data that makes it easiest and understandable. However, the one factor that cannot give the desired results and not provide more detail about the groups during the experiment.
Example: Comparing the test scores of students from three different schools to see if there is a significant difference in performance.
In this experiment, we take one factor such as a score of the students for their performance in class that did not provide the full detail or did not make the proficient decision for the brilliant students or not.
For the brilliant student, we have to check many other qualities like punctuality, a score of the previous years or many others.
2. Two-Way ANOVA
In this approach, compare the means of three or more groups based on two independent variables, allowing us to examine their interaction. It is a comprehensive experiment to analyze the data. Because Two-Way ANOVA applies to scenarios with more than one factor, it provides a deeper understanding of how different variables influence the outcome.
In this type, the interaction between two factors in data analysis makes it more detailed and insightful for any experiment. However, in this experiment by two factors we do not get all the details that are needed for making precise decisions.
Example: Studying the effect of different diets and exercise programs on weight loss.
In this experiment, consider two factors such as diet and exercise to see their combined effect on weight loss. This approach gives more information than considering just one factor, but it still might not give all relevant aspects like the participant’s metabolism rates or pre-existing health conditions that can influence the results.
3. Repeated Measures ANOVA
In this type, compare the means of three or more groups based on repeated observations applied on the same subjects to examine changes over time under different conditions and control the individual variability.
Repeated ANOVA provides a deeper understanding of within-subject factors and their influence on the outcome. By using this way, generate the correlation between repeated measurements and reduce the error variance or find more precise results.
Example: Testing the effect of a new drug over several time points on the same group of patients.
In this experiment select a group of patients and measure their blood pressure at multiple intervals: before starting the medication, after one month of treatment, after two months, and after three months. Using repeated measures ANOVA, analyze the blood pressure readings taken at these different time points to determine if there are significant changes over time due to the medication. This method allows us to note the natural variability in blood pressure readings within each patient and provides a more precise understanding of the drug’s effect on the patient.
Steps to Conduct ANOVA
Follow the below steps to find the numerical value of any data set or any experiment.
- Formulate Hypotheses: Select the hypothesis for the testing of results.
- Null Hypothesis (H0): In this testing, there is no significant difference between group means.
- Alternative Hypothesis (H1): In this selection, at least one group mean is significantly different from the others.
- Collect and Organize Data: Collect the data from the experiment and set it into the table form to find the calculation.
- Calculate Group Means and Total Mean: Find the mean of all groups individually and also find the mean of the sum of all group’s mean by using the below mean formula.
Mean = sum of data/number of elements = X/n
- Find Sum of Squares: now, calculate the Sum of Squares between groups (SSB) by using the mean of every individual group with the mean of all groups and the Sum of Squares within groups (SSW) by using the mean of every individual group. Also, find the total sum of squares by adding both of them. The formula of both is stated as:
SSB = ∑ i=1k ni (Mi − M)2
Where, “ni” is the number of observations in the “ith-group”, “Mi” is the mean of the group, “k” total number of groups, and “M” total mean of all groups.
SSW = ∑ i=1k ∑ j=1nj(Xij – Mj)2
Where, “Xij” is each observation in group “j”, “Mj” is the mean of the group j, “k” total number of groups, and “nj” is the number of observations in group j.
SST=SSB+SSW
- Calculate the Degrees of Freedom: Find the degrees of freedom between the groups (dfB) and degrees of freedom within the groups (dfW) by using the below respective formulas:
Between groups:
Df (B) = k−1
Where, “k” is the number of groups.
Within the groups:
Df (W) = N − k
Where, “N” is the total number of observations, and “k” is the number of groups.
- Determine the Mean Squares: Now, determine the mean square between groups and within groups by dividing the sum of squares with the degree of freedom for respective selection.
Mean Square between Groups:
MSB = SSB/DF (B)
Mean Square within Groups:
MSW = SSW/DF (W)
- Calculate the F-Statistic: The F-statistic is a ratio of the variance between groups to the variance within groups. A higher F-value indicates a greater likelihood that the observed differences are significant. Its formula is stated as:
F = MSB/MSW
- Determine the F-table value: Use an F-distribution table to find its value using the degree of freedom of the groups and compare it with the calculated F-value at a chosen significance level. If the F-value is less than the chosen significance level (usually 0.05) leads to rejecting the null hypothesis.
F (α, df1, df2) = value from table
- Interpreting ANOVA Results:
- SignificantF-Statistic: If F-Test Result > Critical Value (F-table value) then reject the null hypothesis and make a new hypothesis.
- Non-Significant F-Statistic: If F Test Result < Critical Value (F-table value) then accept the null hypothesis.
How to Perform ANOVA Calculation?
Here we solve the mathematical examples for a better understanding of ANOVA calculation.
Example: If the data of the weight loss by using the three different diets (A, B, and C) and randomly assigned to different participants. After a fixed period record the weight loss for each participant and the data is given below. Now find the effect of weight loss By ANOVA comparison when “α = 0.05”.
Solution:
Step 1: First make the hypothesis H0 and H1.
Null Hypothesis: H0= no significant difference in the mean weight loss among the three diets.
Alternative Hypothesis: H1 = At least one diet results in a significantly different mean weight loss.
Step 2: Now, calculate the Mean of every group and the mean of all group means by using the mean formula.
Mean for Diet (A) = M1 = 8+ 9 + 6 + 7 + 5/ 5 = 7
Mean for Diet (B) = M2 = 5 + 6 + 4 + 7 + 5/ 5 = 5.4
Mean for Diet (C) = M3=7 + 8 + 6 + 9 + 7 /5 =7.4
Mean of all groups = M =7 + 5.4 +7.4 / 3 = 6.6
Step 3: Find the sum of squares of all groups individually and collectively by using the above formulas.
SSW = ∑ i=1k ∑ j=1nj(Xij – Mj)2
SSW for Diet A = (8 −7)2+ (9−7)2 + (6−7)2 + (7−7)2 + (5−7)2 = 1 + 4 + 1 + 0 + 4 = 10
SSW for Diet B = (5−5.4)2 + (6−5.4)2 + (4−5.4)2 + (7−5.4)2 + (5−5.4)2
= 0.16 + 0.36 + 1.96 + 2.56 + 0.16 = 5.2
SSW for Diet C = (7−7.4)2 + (8−7.4)2 + (6−7.4)2 + (9−7.4)2 + (7−7.4)2
= 0.16 + 0.36 + 1.96 + 2.56 + 0.16 = 5.2
Adding the value of all diet.
SSW = 10 + 5.2 + 5.2= 20.4
SSB = ∑ i=1k ni (Mi − M)2
SSB of all groups = 5[(7−6.6)2+(5.4−6.6)2+(7.4−6.6)2]
=5[(0.4)2 + (−1.2)2 + (0.8)2]
=5 [0.16 + 1.44 + 0.64] =5 × 2.24 = 11.2
Total Sum of Squares = SST= SSB + SSW
= 11.2 + 20.4 = 31.6
Step 4: Now, calculate the degree of freedom of the individual or all groups.
Df B = k−1 = 3−1 = 2
Df W = N−k = 15−3 = 12
Step 5: Calculate the values of mean squares by putting the values from step 3 & step 4 in its formula.
MS-between Groups:
MSB = SSB/DF (B) = 11.2/2 = 5.6
MS-within Groups:
MSW = SSW/DF (W) = 20.4 / 12 = 1.7
Step 6: Determine the F-statistics value by using the values of step 5.
F-statistics value = MSB/MSW = 5.6/1.7 = 3.29
Step 7: Using the F-distribution table to find the F-table value by its degree of freedom.
F-table value = F (α, df1, df2) = 4.26
Thus, we note that “F-statistics value < F-table value” then we accept the null hypothesis.
To overcome the long calculation method, use the ANOVA Calculator to make your calculation faster and error-free. This tool provides the solution with detailed steps of your data groups in seconds by just entering the values in given input fields and pressing the button.
Assumptions for ANOVA
Some assumptions are essential for valid ANOVA results and they must rely on any experiment data:
- Independence: Observations within each group must be independent of each other.
- Normality: The data within each group should be approximately normally distributed.
- Homogeneity of Variances: The variance among the groups should be roughly equal.
Any violations of the above assumptions can lead to inaccurate results. So check them before proceeding with ANOVA and make sure every experiment follows the assumption rule for exact results.
Conclusion
ANOVA is an essential method for researchers and analysts to determine the difference between multiple groups. By analyzing the variance within and between groups, ANOVA helps us whether observed differences are due to random chance or a specific factor. Understanding its types, assumptions, and interpretation is crucial for accurate data analysis and informed decision-making.
By following this guide anyone can solve their problems and make remarkable results by applying ANOVA in their research and analysis endeavors. Moreover, by understanding the concepts of ANOVA researchers and analysts can draw better meaningful insights from their experiments.
