Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether or not the means of several groups are all equal, and therefore generalizes t-test to more than two groups. Doing multiple two-sample t-tests would result in an increased chance of committing a type I error. For this reason, ANOVAs are useful in comparing two, three, or more means.




POSTED BY: VON ERJUN SABUSAP

Difference Between Means

Hypothesis Testing of the Difference Between Two Means
Do employees perform better at work with music playing.  The music was turned on during the working hours of a business with 45 employees.  There productivity level averaged 5.2 with a standard deviation of 2.4.  On a different day the music was turned off and there were 40 workers.  The workers' productivity level averaged 4.8 with a standard deviation of 1.2.  What can we conclude at the .05 level?
Solution
We first develop the hypotheses
        H₀:  m₁ - m₂  =  0       
        H₁:  m₁ - m₂  >  0
Next we need to find the standard deviation.  Recall from before, we had that the mean of the difference is 
        m_x  =  m₁ - m₂ 
and the standard deviation is 

 s_x  =      

We can substitute the sample means and sample standard deviations for a point estimate of the population means and standard deviations.  We have

        
and 
Now we can calculate the t-score.  We have
                    0.4
        t  =                       =  0.988
                   0.405
To calculate the degrees of freedom, we can take the smaller of the two numbers n₁ - 1 and n₂ - 1.  So in this example we use 39 degrees of freedom.  The t-table gives a value of 1.690 for the t_.95 value.  Notice that 0.988 is still smaller than 1.690 and the result is the same.  Since the t-score is smaller than 1.690, we fail to reject the null hypothesis and state that there is insufficient evidence to make a conclusion about employees performing better at work with music playing. 




POSTED BY: VON ERJUN SABUSAP

                                       ANOVA for simple linear regression

• Total sum of squared deviations is divided into model (regression) and error
(residual) sums of squares

• Their ratio is the coefficient of determination R2

• These are each divided by their degrees of freedom to obtain the mean SS

• Their ratio is distributed as F and can be tested for significance



J.Santillan

                                         Analysis of Variance (ANOVA)

• Partition the total variance in a population into the model and residual

• If the model has more than one term, also partition the model variance into
components due to each term

• Can be applied to any linear additive design specified by a model

• Each component can be tested for signficance vs. the null hypothesis that it
does not contribute to the model fit



J.Santillan

STATISTICAL FACTS ABOUT TEENAGERS

Every 2 hours a youth is murdered.

In the next twenty four hours, 1,439 Teens will attempt suicide. 

A 1995 study by Children Now and Kaiser Permanente found 40% of teen women know someone who was in an abusive relationship. 

In the next twenty four hours, 2,795 Teenage girls will become pregnant.

Every 4 minutes a youth is arrested for an alcohol related crime.

In the next twenty four hours, 3,506 Teens will run away.

Every 7 minutes a youth is arrested for a drug crime.

G.K. Elio

Analysis of Variance

   The z and t tests should not be used when three orb more means are compared, instead, F-test can be used to compare three or more means. This technique is called analysis of variance or ANOVA .For three groups, the F-test can only show whether or not a difference exists among the three means. It cannot reveal where the difference lies. If the F test indicates that there is a difference among the means, other statistical test are used to find where the difference exists. The most commonly used tests are the Scheffe test and the Tukey test.

One-Way Analysis of Variance
- Analysis of variance used to compare three or more means which contains only one variable.

Two-Way Analysis of Variance
-ANOVA that involves two variables.

Reasons why the t test should not be used on three or more populations:
1. When one is comparing two means at a time, the rest of the means under study are ignored. With the f test, all the means are compared simultaneously.
2. When one is comparing two means at a time, the probability of rejecting the null hypothesis when it is true increased, since more t test are conducted, the greater is the likelihood of getting significant differences by chance alone.
3. The more means there are to compare, 3 the more t tests are needed.

Assumptions for the F test for comparing Three or more means
1. The populations from which the samples were obtained must be normally or approximately normally distributed.
2. The samples must be independent for each.
3. The variances of the population must be equal.


J.Santillan

Procedure in Finding F test Value for the Analysis of Variance

Step 1: Find the mean and variance for each sample.

Step 2: Find the grand mean.

Step 3: Find between- group variance.

Step 4: Find within- group variance.

Step 5: Find the F test value

The degrees of freedom are:
       d.f. N= k-1   where k is the number of groups, and
       d.f D= N-k   where N is the sum of the sample sizes of the groups



J.Santillan