Statistical AnalysisANOVA

Roderick GrahamFashion Institute of Technology

ANOVA ANOVA means Analysis Of Variance.

With ANOVA, we have one variable that is ungrouped, and one categorical, grouped variable.

The question that we try to answer with ANOVA is: “for any collection of groups, is at least one group different from the others?”

This is hypothesis testing. This time your critical statistic will be an F ratio.

The types of procedures we’ve used…

Steps in Conducting ANOVA tests Step 1 – State the null hypothesis and critical

region Step 2 – Identify the critical statistic Step 3 – Compute the test (calculated)

statistic Step 4 – Interpret results

Example: Capital Punishment and Religious Affiliation Suppose we ask 20 people to rank their support

for capital punishment on a scale from 1 to 30. 1 being absolutely no support under no

circumstances 30 being always support under any circumstance

Suppose these 20 people were equally divided into 5 religions (Protestant, Catholic, Jewish, Buddhist, and other)

Now, we have an ungrouped, scale variable (support for capital punishment) and a grouped, categorical variable (religion). This is the perfect situation for ANOVA!

Example: Capital Punishment and Religious Affiliation Now let’s say we got this data:

What do we do?

Protestant

Catholic Jewish Buddhist

Other

8 12 12 15 1012 20 13 16 1813 25 18 23 1217 27 21 28 12

Example: Capital Punishment and Religious AffiliationStep 1 – State the null hypothesis and critical

regionH0: “The population means for each category are the same.”

H1: “At least one of the population means is different.”

Let’s set our critical region at .05 (Meaning we will accept 95% of all findings, and if we get a calculated statistic that falls in the .05 region, we will reject the null hypothesis).

Example: Capital Punishment and Religious AffiliationStep 2 – Identify the critical statistic.

To get the critical statistic, (F critical), we must look for two values.

dfb (degrees of freedom between) – columns for F chart dfb = (k – 1), with k = number of categories

dfw (degrees of freedom within) – rows for F chart dfw = (N – k), with N = number of respondents, and k

= number of categories

Example: Capital Punishment and Religious AffiliationStep 2 – Identify the critical statistic.

For our example we have a k of 5 (five categories) and an N of 20 (20 respondents). Thus, dfb = (5 – 1) = 4

dfw = (20 – 5) = 15, F(critical) = 3.06

Example: Capital Punishment and Religious AffiliationWhat the upcoming symbols mean: SST = sum of squares total SSW = sum of squares within SSB = sum of squares between

MSB = mean squares between MSW = mean squares within

Example: Capital Punishment and Religious AffiliationStep 3 – Compute the test/calculated statistic

requires using these equations. Important…these formulas should be used in this order!!!

MSWMSBratioF )(

knSSWMSW

1

kSSBMSB

22 XNXSST 2)( XXNSSB kk SSBSSTSSW

Example: Capital Punishment and Religious Affiliation Let’s take a closer look at the beginning formulas

Note: SSB will have to be calculated for each category and then summed. In our example, we have five religious groups, so we will compute SSB five times, and then sum.

22 XNXSST 2)( XXNSSB kk

Mean value for the entire sample

Number of cases in EACH category

Total number of cases in a sample

Mean value of EACH category

Example: Capital Punishment and Religious Affiliation Understanding the final formula. It is a ratio of

the differences between groups and the differences within groups

When MSB and MSW are similar, the F is low, and the less chance we will reject the null.

But when these two values differ, the F increases. We then begin to believe that at least one of the populations that these samples represent is different from the other populations.

MSWMSBratioF )(

Example: Capital Punishment and Religious Affiliation This is all we need to being building our table

for ANOVA.

What new information do we need in order to use our starting formulas?

Protestant

Catholic Jewish Buddhist

Other

8 12 12 15 1012 20 13 16 1813 25 18 23 1217 27 21 28 12

22 XNXSST 2)( XXNSSB kk

Example: Capital Punishment and Religious Affiliation Setting up the table for ANOVA…

You also need: The N (sample), = 20, and Nk (no. of cases per category) = 4 The mean of the entire sample, = (50+84+64+82+52)/20 =

16.6

Protestant

Catholic Jewish Buddhist Other

x x2 x x2 x x2 x x2 x x2

8 64 12 144 12 144 15 225 10 10012 144 20 400 13 169 16 256 18 32413 169 25 625 18 324 23 529 12 14417 289 27 729 21 441 28 784 12 144

∑ 50 66 84 1898

64 1078

82 1794

52 712

12.5

21.0

16.0

20.5

13.0

= 16.6

kXX

X

Our F(critical) was….3.06Our F(test) was…2.57Thus, we do not reject the null hypothesis.

Your turn… Are sexually active teens better informed about AIDS than

teenagers who are sexually inactive? A 15 item test of knowledge about sex was administered to teens who were “inactive”, “active with one partner” and “active with more than one partner”. Here are the results. Test at the .05 level.

Give H0, H1, F-critical, F-ratio, Concluding Results

Inactive Active – 1 Partner

Active – More than 1 Partner

10 11 1212 11 128 6 1010 5 48 15 35 10 15

END