Untangle the World of ANOVA : A Beginner's Guide to Understanding Analysis of Variance

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Have you ever wondered how researchers determine if a new medical treatment is more effective than the current standard? The answer lies in ANOVA or Analysis of Variance. In this blog post, we will explore

Contents :

  1. What exactly is an ANOVA?
  2. Definition of ANOVA :
  3. Assumptions :
  4. When to use :
  5. Interpretation:
  6. Different types of ANOVA:
  7. Example of Different types of ANOVA:
  8. Visualzation Example of Different types of ANOVA:
  9. Limitations of ANOVA:
  10. Advantages of ANOVA:
  11. Other tests similar to ANOVA:
  12. Common Terms inANOVA:

 

What exactly is an ANOVA?

ANOVA is like when you're feeling sick and you go to the doctor. The doctor gives you two different types of medicine, one is green and one is yellow. The green medicine is for you and the yellow medicine is for your friend.

You both take the medicine for a week and then you tell the doctor how you feel. If green medicine made you feel better than yellow medicine made your friend feel, then green medicine is the best medicine.

ANOVA is used to compare the effectiveness of different medicines and see which one is the best. 

ANOVA stands for Analysis of Variance. It is a statistical method used to test for significant differences between the means of two or more groups. It is used to determine if there is a significant difference between the means of three or more groups, whereas a t-test is used to determine if there is a significant difference between the means of two groups.

Definition of ANOVA :

ANOVA is a statistical method used to test the null hypothesis that the means of two or more groups are equal. It compares the variability between the groups to the variability within the groups.

Assumptions :

There are several assumptions that must be met for ANOVA to be an appropriate statistical test. These assumptions include:

  1. Normality: The data should be approximately normally distributed within each group.
  2. Independence: The observations should be independent of one another.
  3. Equal variances: The variances of the groups should be approximately equal.
  4. Additivity: The effects of the factors should be additive.

It is important to check these assumptions before running an ANOVA, as a violation of these assumptions can lead to invalid results. If the assumptions are not met, non-parametric alternatives such as the Kruskal-Wallis test may be used instead.

When to use :

Scenarios where you want to compare the means of three or more groups, such as:

  1. Comparing the effectiveness of different treatments or interventions on a particular outcome.
  2. Investigating the relationship between a continuous dependent variable and one or more categorical independent variables.
  3. Comparing the means of groups based on one or more factors.
  4. Assessing the impact of different levels of a categorical variable on a continuous outcome.
  5. Investigating the relationship between a continuous dependent variable and a continuous independent variable.

It is also commonly used in experimental designs, such as in randomized controlled trials, to determine if there is a significant difference between the means of the treatment groups and the control group.

Interpretation:

ANOVA results include an F-value and a p-value. If the p-value is less than the significance level (usually 0.05), the null hypothesis is rejected and it can be concluded that at least one of the group means is significantly different from the others.

Different types of ANOVA:

  1. One-way ANOVA: Compare means of three or more groups on a single factor
  2. Two-way ANOVA: Compare means of three or more groups on two factors
  3. Repeated Measures ANOVA: Compare means of groups that are measured multiple times
Example of Different types of ANOVA:

  1. One-way ANOVA: Comparing the effectiveness of three different advertising campaigns on sales.
  2. Two-way ANOVA: Comparing the effectiveness of different advertising campaigns and different demographics on sales.
  3. Repeated Measures ANOVA: Comparing the effectiveness of a treatment for anxiety before and after therapy
Visualization Example of Different types of ANOVA:

  1. One-way ANOVA
import matplotlib.pyplot as plt
import scipy.stats as stats

group1 = [1,2,3,4,5]
group2 = [2,3,4,5,6]
group3 = [3,4,5,6,7]

stats.f_oneway(group1, group2, group3)
  1. Two-way ANOVA
import statsmodels.api as sm
from statsmodels.formula.api import ols

data = {'Advertising':['TV', 'Radio', 'Newspaper', 'TV', 'Radio', 'Newspaper'],
       'Demographics':['Young', 'Young', 'Young', 'Old', 'Old', 'Old'],
       'Sales':[100,150,200,50,75,100]}

model = ols('Sales ~ Advertising + Demographics', data=data).fit()
aov_table = sm.stats.anova_lm(model)
  1. Repeated Measures ANOVA
import pandas as pd
from statsmodels.stats.anova import AnovaRM

data = {'Subject':[1,1,1,2,2,2],
       'Measurement':['Pre','Post','Follow-up','Pre','Post','Follow-up'],
       'Score':[5,7,6,8,10,9]}

df = pd.DataFrame(data)
aov
Limitations of ANOVA:

  1. Assumptions: ANOVA assumes that the data is normally distributed and that the variances of the groups are equal. If these assumptions are not met, the results may be inaccurate.

  2. Limited power: ANOVA has limited power when the sample size is small or the number of groups is large.

  3. Multiple comparisons: ANOVA does not control for multiple comparisons, which means that the chances of finding a significant difference by chance are increased.

  4. Non-parametric data: ANOVA is not suitable for non-parametric data and cannot be used to compare ordinal or categorical data.

  5. Only test for equality of means: ANOVA only tests for equality of means, it cannot be used to test for other relationships such as equality of medians or variances.

Advantages of ANOVA:

  1. Multiple groups: ANOVA can be used to compare multiple groups simultaneously.

  2. Continuous data: ANOVA is suitable for continuous data and can be used to test for differences in means between groups.

  3. Flexibility: ANOVA can be extended to multiple factors and multiple levels, allowing for a wide range of comparisons.

  4. Easy interpretation: The F-statistic and p-value provided by ANOVA are easy to interpret and can be used to determine if the means of the groups are significantly different.

  5. Robustness: ANOVA is robust to the violation of the assumptions of normality and equal variances, as long as the sample size is large enough.

  6. ANOVA can be used as a screening tool to identify which factors have a significant effect on a response variable.

Other tests similar to ANOVA:

There are several other statistical tests that can be used in place of ANOVA, depending on the specific research question and the characteristics of the data. Some of these tests include:

  1. Kruskal-Wallis Test: This is a non-parametric version of ANOVA that can be used to compare multiple groups when the assumptions of normality and equal variances are not met.

  2. chi-square test: This test is used to compare categorical data, and it can be used to compare the distribution of a categorical variable between multiple groups.

  3. Wilcoxon-Mann-Whitney Test: This is a non-parametric test that can be used to compare two groups when the assumptions of normality and equal variances are not met.

  4. Fisher's Exact Test: This is a test used to compare two categorical variables.

  5. Student t-test: This test is used to compare two groups and assumes that the data is normally distributed.

  6. Repeated Measures ANOVA: This test is used when the same subjects are measured under different conditions.

  7. Mixed-design ANOVA: This test is used when both within-subjects and between-subjects factors are present in the data.

It's worth noting that the appropriate test will depend on the specific research question, the type of data, and the assumptions that can be made about the data.

Common Terms in ANOVA:

There are several statistical terms commonly used in ANOVA, including:

  1. Factor: A factor is a variable that is manipulated or controlled in an experiment. In ANOVA, factors are used to divide the data into groups for comparison.

  2. Level: A level is a specific value of a factor. For example, if a factor is "treatment," the levels might be "control" and "experimental."

  3. Treatment: A treatment is a specific combination of levels of the factors in an experiment.

  4. Experimental unit: The experimental unit is the unit of observation in an experiment, such as a person, an animal, or a plot of land.

  5. Response variable: The response variable is the variable of interest in an experiment, also called dependent variable.

  6. Factor means: The mean of the response variable for each level of the factor is called factor mean.

  7. Sum of squares (SS): The sum of the squared differences between the observed values and the overall mean is called sum of squares.

  8. Degrees of freedom (df): Degrees of freedom refers to the number of values in a set of data that are free to vary.

  9. F-statistic: The F-statistic is a ratio of the variability between groups to the variability within groups.

  10. P-value: The p-value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, under the assumption that the null hypothesis is true.

  11. Multiple comparisons: refers to the problem of comparing more than two groups or factors and the risk of committing a type I error.

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