**Analysis of Variance (1/2)**

**In this lesson, you’re expected to:**

– understand what the ANOVA test is used for

– learn about the One-Way ANOVA test

– discover how to conduct an F-test and analyze p-values

*There is a wide range of scenarios where we are interested in comparing the means of two or more populations.*

The manager of *Inditex* (https://en.wikipedia.org/wiki/Inditex) may want to know whether T-shirt’s with graphics are more demanded than those without graphics. She might also be interested in whether pants displayed in the first row of their online store generate more sales than those displayed at the bottom.

In all these scenarios, we want to assess * whether there is a significant difference between the means of different groups*.

**What is ANOVA?**

The * ANOVA, or analysis of variance*, is a methodology used to directly

*.*

**compare the means of different groups**Hence, when we have * categorical data*, it is used to test if there is a difference in the mean of numerical variables among the categories.

**What is ANOVA used for?**

The ANOVA methodology allows to study the variation of a random numerical variable with the values taken by other variables called * factors*.

Factors must take * discrete values**. However, the original variable may not be categorical.

*** Discrete**data can only take particular values. It can be numeric — like numbers of apples — but it can also be categorical — like red or blue, male or female, good or bad etc.

*, and can take any positive value. However, since we are interested in the age range, we can*

**continuous***.*

**turn the variable into factors, by computing the age range**Thus, we will transform the exact numerical age into the corresponding age ranges: 0-10, 10-20, 20-35, 35-45, 45-65, 65+.

The ANOVA test is closely related to the fact that * equality in means does not imply equality in medians. **

**Analysis of Variance (ANOVA)**

https://www.youtube.com/watch?v=ITf4vHhyGpc

*of an observation belonging to group i as a combination of:*

**expected value of Y**• the mean for all groups

• the deviation from the mean for group i

• the random noise

**One-Way ANOVA**

In one-way ANOVA, the analysis is limited to evaluating * how the expected value of the dependent variable is conditioned by a single factor*.

For example, the sales of pants might be affected not only by the color but also the type (cargo, chinos, jeans etc.). Thus, with one-way ANOVA, we are limited to analyzing each factor separately.

We could use one-way ANOVA to evaluate how the color affects the number of sales. In this case, the factor would be the color and the groups: yellow, black, white, and beige.

**One-Way ANOVA**

https://statistics.laerd.com/statistical-guides/one-way-anova-statistical-guide.php

**Comparing with the Null Hypothesis**

The * null hypothesis*, usually denoted by H0, represents the hypothesis that sample observations result purely from chance.

By contrast, the * alternative hypothesis*, denoted by H1, is the hypothesis that sample observations are influenced by some non-random cause.

Thus, in our case, the null hypothesis is that there is no significant difference in means among the groups and the population means for the groups are the same.

*For example, in the case of the relation between the sales of pants and the color of the pants:*

The * null hypothesis* is that the means of the number of sales for the different colors is the same.

The * alternative hypothesis*, H1, would be that at least one of the means of the four colors is different. Note that three of the means could be the same, and if just one significantly differs, the alternative hypothesis would become true and we have to reject the null hypothesis.

**Type of Errors and Significance Levels**

When we are testing a hypothesis, we can make two types of errors:

* Type I Error:* Reject the null hypothesis when it is true.

This involves asserting a difference that does not exist and is called a

*False Positive*.

* Type II Error: *Accept the null hypothesis when it is false.

In this case, we are failing to assert a difference that is really present in the data. This is called a

*False Negative*.

When we test a hypothesis, we need to choose a level of significance. The * level of significance*, denoted as α, represents the probability of rejecting the null hypothesis when it is actually true.

**The F-Test & P-Value**

**Rejecting the Null Hypothesis**

ANOVA uses the F statistic to evaluate the hypotheses in what is called an F test.

In addition, a p-value is computed from the F statistic using an F distribution.

This F-test is used for comparing the factors of the total deviation. For example, in one-way, or ANOVA, statistical significance is tested by comparing the F test statistic.

The greater the value of the test, the more unlikely that the null hypothesis is true, as the numerator increases proportionally to the between-group variability, and the denominator represents the within group variability.

Thus, a sufficiently large value of this test statistic results in accepting our alternative hypothesis and asserting difference among the groups.

**MSG**represents a measure of the between-group variability

**MSE**measures the variability within each of the groups

**SSG**= the sum of squares between groups

**SSE**= the sum of squared errors

**n**=

**the number of observations**

**k**= the number of groups

Because larger values of F represent stronger evidence against the null hypothesis, we use the upper tail of the distribution to compute a p-value.

**Disadvantages of using the F-test**

The main problem with the F-test value is that it cannot be interpreted immediately. Once computed, to know whether we can reject the null hypothesis or not, we need to go to the F-tables.

To accept the hypothesis or not, depends not only on the value of F but also on the sample size and the number of groups.

*Let’s go back to the example of the number of sales and the number of pants.*

We had four groups (the categorical variable color had 4 different values). Imagine that we get a result of F=5 for the F-test.

That result would mean very different things for a sample of 10 and for a sample of 100. * In the case of 100 samples*, we will probably reject the null hypothesis and state a difference in means.

In contrast, * for a sample of 10*, we would have to accept the null hypothesis. Note that for k=4, the threshold of F-test value for a significance level of 0.05 is approximately 10 for a sample of size 10 and approximately 4 for a sample of size 100.

**The p-value**

The p-value represents the probability of having the observed values under the null hypothesis. Hence, the probability that we should accept the null hypothesis (H0) is equal to the p-value. The probability that we should not reject H0 is equal to the p-value.

*So what does a very small p-value mean? *

It indicates that differences in means between groups are significant and that we can reject the null hypothesis.

The p-value quantifies the probability of making a Type 1 error. For research and many business applications, the level of significance is chosen as 0.05 (5%). However, other frequent choices include 0.001 (0.1%), 0.01 (1%), and 0.10 (10%).