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Module - 7 Hypothesis Testing

Lesson - 4 Analysis of Variance (ANOVA)

**Overview**

Analysis of Variance (ANOVA) could be a measurable strategy utilized to compare the implies of two or more groups to decide whether there's a statistically noteworthy contrast between them. It may be a capable apparatus for analyzing information and can be utilized in a wide range of areas, counting science, medication, social sciences, and building. There are three main types of ANOVA: one-way ANOVA, two-way ANOVA, and rehashed measures ANOVA. In this lesson, we are going center on one-way ANOVA, which is the foremost common sort of ANOVA.

**One-way ANOVA**

One-way ANOVA is used when there is one independent variable (also known as a factor) with three or more levels (also known as groups or treatments) and a continuous dependent variable (also known as an outcome or response variable). The independent variable can be categorical (e.g., sex, race, treatment) or continuous (e.g., age, weight).

**Assumptions**

Before conducting ANOVA, we need to check the following assumptions:

**Normality:**The data should be normally distributed within each group. This assumption can be checked using a normal probability plot or a Shapiro-Wilk test.**Homogeneity of variance:**The variance of the dependent variable should be equal across all groups. This assumption can be checked using a Levene's test or a Bartlett's test.**Independence:**The observations should be independent of each other.

If these assumptions are not met, the results of ANOVA may not be reliable.

**Hypotheses**

The null hypothesis (H0) in ANOVA is that there is no difference in the means of the groups. The alternative hypothesis (HA) is that at least one group's mean is different from the others.

The hypotheses can be written as:

```
H0: μ1 = μ2 = μ3 = ... = μk
HA: At least one μi is different from the others
```

where μi is the mean of the ith group and k is the number of groups.

**Test statistic**

The test statistic in ANOVA is the F-statistic, which is calculated as the ratio of the variance between groups to the variance within groups. The formula for the F-statistic is:

```
F = (SSB / dfB) / (SSW / dfW)
```

where SSB is the sum of squares between groups, dfB is the degrees of freedom for the between-groups variation, SSW is the sum of squares within groups, and dfW is the degrees of freedom for the within-groups variation.

If the F-statistic is large and the p-value is small (less than the chosen significance level, typically 0.05), we reject the null hypothesis and conclude that there is a statistically significant difference between at least two group means.

**Post-hoc tests**

If ANOVA reveals a significant difference between the groups, we need to perform post-hoc tests to determine which groups are significantly different from each other. There are several post-hoc tests available, including Tukey's test, Dunnett's test, and Scheffe's test.

**Example 1:**

**One-way ANOVA with categorical independent variable**

A researcher wants to test whether there is a significant difference in the weight gain of three different groups of rats fed with different diets. The independent variable is the type of diet (A, B, or C), and the dependent variable is the weight gain in grams.

The researcher collected data on 10 rats in each group and checked the assumptions of normality and homogeneity of variance. The data met these assumptions, so the researcher conducted one-way ANOVA using a significance level of 0.05.

The results showed a significant difference in weight gain between the groups (F(2, 27) = 4.89, p = 0.016). The post-hoc test (Tukey's test) revealed that group A had a significantly higher weight gain than group B (p = 0.031), but there was no significant difference between groups A and C or between groups B and C.

**Two Way ANOVA**

Two-Way ANOVA is a statistical method used to analyze the differences between two independent variables (also called factors) and one dependent variable. It is used to test whether there is a significant interaction between the two independent variables and whether there are any main effects of each independent variable on the dependent variable.

The two independent variables can be either categorical or continuous. The dependent variable must be continuous, normally distributed, and have equal variances across all groups.

The two-way ANOVA model is represented as:

```
Yij = μ + αi + βj + (αβ)ij + εij
```

Where:

- Yij represents the dependent variable for the ith level of the first independent variable and the jth level of the second independent variable.
- μ is the overall mean of the dependent variable.
- αi is the effect of the ith level of the first independent variable.
- βj is the effect of the jth level of the second independent variable.
- (αβ)ij is the interaction effect between the ith level of the first independent variable and the jth level of the second independent variable.
- εij is the random error term.

The two-way ANOVA allows us to test the following hypotheses:

**Null hypothesis:**There is no significant difference between the means of the groups for each independent variable and no interaction effect between the two independent variables.**Alternative hypothesis:**There is a significant difference between the means of the groups for at least one independent variable and/or there is an interaction effect between the two independent variables.

If the null hypothesis is rejected, we can conduct post-hoc tests (such as Tukey's HSD or Bonferroni) to determine which groups are significantly different from each other.

**Assumptions of Two-Way ANOVA:**

Two-way ANOVA assumes that the following assumptions are met:

**Normality**: The dependent variable is normally distributed in each group.**Homogeneity of variance**: The variance of the dependent variable is equal across all groups.**Independence**: The observations in each group are independent of each other.**Factorial independence**: The levels of each independent variable are independent of each other.

**Example**

**Two-Way ANOVA:** Suppose a researcher wants to investigate the effect of two factors, temperature and pressure, on the yield of a chemical reaction. The researcher conducted an experiment with three different temperature levels (low, medium, and high) and two different pressure levels (low and high). For each combination of temperature and pressure, the researcher conducted three replicates of the reaction and recorded the yield.

The data can be represented as follows:

Temperature | Pressure | Yield |
---|---|---|

Low | Low | 10 |

Low | Low | 12 |

Low | Low | 11 |

Low | High | 20 |

Low | High | 18 |

Low | High | 21 |

Medium | Low | 15 |

Medium | Low | 14 |

Medium | Low | 13 |

Medium | High | 25 |

Medium | High | 23 |

Medium | High | 24 |

High | Low | 20 |

High | Low | 18 |

High | Low | 19 |

High | High | 30 |

High | High | 28 |

High | High | 29 |

To analyze the data using two-way ANOVA, the researcher would first test for normality and homogeneity of variance. If these assumptions are met, the researcher would then perform a two-way ANOVA to determine whether there is a significant effect of temperature, pressure, and their interaction on the yield of the chemical reaction.

The results of the two-way ANOVA would provide information on whether the main effects of temperature and pressure are significant, whether there is a significant interaction effect between temperature and pressure, and which groups are significantly different from each other.

**Mixed ANOVA**

Mixed ANOVA is a statistical method used to analyze the differences between two or more independent variables (factors), one of which is a within-subjects variable and the others are between-subjects variables. It is used to test whether there is a significant interaction between the within-subjects and between-subjects variables and whether there are any main effects of each independent variable on the dependent variable.

The mixed ANOVA model is represented as:

```
Yijk = μ + Ai + Bj + ABij + eijk
```

Where:

- Yijk is the observation of the dependent variable for the ith level of the within-subjects factor, the jth level of the first between-subjects factor, and the kth level of the second between-subjects factor.
- μ is the overall mean of the dependent variable.
- Ai is the effect of the ith level of the within-subjects factor (also known as the main effect of time).
- Bj is the effect of the jth level of the first between-subjects factor (also known as the main effect of group).
- ABij is the interaction effect between the ith level of the within-subjects factor and the jth level of the first between-subjects factor.
- eijk is the residual error, which represents the deviation of the observed value from the expected value based on the model.

The mixed ANOVA allows us to test the following hypotheses:

**Null hypothesis:**

- H0: There is no significant main effect of group on the dependent variable.
- H0: There is no significant main effect of time on the dependent variable.
- H0: There is no significant interaction effect between group and time on the dependent variable.

**Alternative hypothesis:**

- Ha: There is a significant main effect of group on the dependent variable.
- Ha: There is a significant main effect of time on the dependent variable.
- Ha: There is a significant interaction effect between group and time on the dependent variable.

**Assumptions of Mixed ANOVA:**

Mixed ANOVA assumes that the following assumptions are met:

**Normality**: The dependent variable is normally distributed in each group.**Homogeneity of variance**: The variance of the dependent variable is equal across all groups.**Sphericity**: The variances of the differences between all pairs of levels of the within-subjects variable are equal.**Independence**: The observations in each group are independent of each other.

**Example of Mixed ANOVA:**

Suppose a researcher wants to investigate the effect of a new medication on the pain levels of patients with a certain medical condition. The researcher conducted an experiment with two groups of patients: one group received the medication (treatment group) and the other group received a placebo (control group). Each patient was tested twice, once before the treatment and once after the treatment, to measure their pain levels.

The data can be represented as follows:

Group | Time | Pain level |
---|---|---|

Treatment | Pre | 10 |

Treatment | Pre | 11 |

Treatment | Pre | 12 |

Treatment | Post | 5 |

Treatment | Post | 6 |

Treatment | Post | 7 |

Control | Pre | 15 |

Control | Pre | 14 |

Control | Pre | 13 |

Control | Post | 12 |

Control | Post | 11 |

Control | Post | 10 |

To analyze the data using mixed ANOVA, the researcher would first test for normality and homogeneity of variance. If these assumptions are met, the researcher would then perform a mixed ANOVA to determine whether there is a significant effect of group, time, and their interaction on the pain levels of the patients.

The results of the mixed ANOVA would provide information on whether the main effects of group and time are significant, whether there is a significant interaction effect between group and time, and which groups and time points are significantly different from each other. For example, the results might show that the treatment group had significantly lower pain levels than the control group after the treatment, but not before the treatment, indicating that the medication had a significant effect on pain levels.

**Conclusion**

In conclusion, ANOVA is a powerful tool for analyzing data with one independent variable and a continuous dependent variable. Before conducting ANOVA, we need to check the assumptions of normality, homogeneity of variance, and independence. If these assumptions are met and the F-statistic is significant, we can conclude that at least two group means are significantly different. Post-hoc tests can be used to determine which groups are significantly different from each other.

**Key Takeaways**

- ANOVA is a statistical method used to compare the means of two or more groups to determine whether there is a statistically significant difference between them.
- One-way ANOVA is used when there is one independent variable with three or more levels and a continuous dependent variable.
- If ANOVA reveals a significant difference between the groups, we need to perform post-hoc tests to determine which groups are significantly different from each other.
- Two-way ANOVA is used to analyze the differences between two independent variables and one dependent variable.
- Mixed ANOVA is a statistical method used to analyze the differences between two or more independent variables (factors), one of which is a within-subjects variable and the others are between-subjects variables.

**Quiz**

**1. What is ANOVA?**

- A statistical method used to compare the means of two or more groups to determine whether there is a statistically significant difference between them.
- A machine learning algorithm used to predict future outcomes.
- A method used to estimate the population parameters from sample statistics.
- A data visualization tool used to create histograms.

**Answer**: a. A statistical method used to compare the means of two or more groups to determine whether there is a statistically significant difference between them.

**2. What is the most common type of ANOVA?**

- One-way ANOVA
- Two-way ANOVA
- Repeated measures ANOVA
- Factorial ANOVA

**Answer**: a. One-way ANOVA

**3. What are the three assumptions that need to be met before conducting ANOVA?**

- Randomness, normality, and independence
- Normality, homogeneity of variance, and independence
- Homogeneity of variance, randomness, and correlation
- Homogeneity of variance, correlation, and normality

**Answer**: b. Normality, homogeneity of variance, and independence

**4. What is the test statistic used in ANOVA?**

- t-statistic
- p-value
- F-statistic
- Chi-square statistic

**Answer**: c. F-statistic

**5. What are post-hoc tests?**

- Tests that are conducted before ANOVA to check assumptions.
- Tests that are conducted during ANOVA to determine which groups are significantly different from each other.
- Tests that are conducted after ANOVA to determine which groups are significantly different from each other.
- Tests that are conducted to determine the sample size needed for a study.

**Answer**: c. Tests that are conducted after ANOVA to determine which groups are significantly different from each other.

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