Story 15 — Analysis of variance One-way ANOVA

ANOVA SUMMARY TABLE

One-way ANOVA

The analysis showed that we have

a significant effect. The differences

between the means are

significant, i.e., reliable. The p

value is less than 1 in 10000. This 

means that the probability that the

difference we report is a chance

event (and not the result of our

treatment of giving rats vitamin C)

is less than 1 in 10000.

How did you compute the degrees

of freedom, df, you ask.

You know this if you know the

concepts of Between, Within, and

Total.

We said, in order to compute the

variance between, we line up the

means of all the groups and treat

them as scores, and calculate the

variance. How many means we

have here? We have three means.

We really consider them as scores

here. In order to calculate the

variance of three numbers, we

must first calculate the mean. By

calculating the mean, we lose 1

degree of freedom for every mean,

remember? So, our df for the

between term is 3-1=2. 

We calculate variance within as

we said above. We calculate the

mean of the first group and then

the variance of this group. Then

we do the same for the second

group, and then the third group.

We add these 3 variances and this

gives us the variance within. Since

we compute 3 means in the

process of calculating the

variances, we lose 3 degrees of

freedom. How many scores went

into the calculations of variance

within? All the sores, that is 15.

Our degrees of freedom then are

df=15-3=12.

Easy, no formulas needed,

because we understand the

concept of df and also variance

within.

Lastly, we compute the df of Total

as follows:

In order to compute the Total

variance, we said, we take all of

the scores of all the groups 

disregarding what group each

score comes from. In order to

calculate this variance we must

first calculate the mean, therefore

we lose 1 degree of freedom. How

many scores went into the

calculation of the Total variance?

All of the scores, here 15. Our

degrees of freedom for Total then

is 15-1=14.

Note that adding up the df for

Between and Within we find the df

for Total. 

That is what we meant by

partitioning. We said we partition

variance. 

Here we partition the Total

 variance into variance

between and variance within.

Indeed, verify that SS between

plus SS within equals SS Total.

That is,

1259 + 373.2 = 1632.2
 

How did we get the p value? you

ask.

I will be very practical here. As

was the case with the t-distribution,

here too, there is a table with the F

values (see Appendix).

These are in a way the recalculated 

1.96, that is the point on the curve

 beyond which 5% of the curve lies.

In the case of the t curve we

entered the table with the degrees

of freedom and found the required

t at the 5% level. In the present

case, we enter the F table with the

degrees of freedom for Between

(in the present experiment df=2)

and the degrees of freedom

for Within (in the present

experiment df=12). We locate the

F on the table. This is the required

F.

Then we compare the obtained F

(the one we calculated, look at he

summary table) to the required F.

As in the case of the t-test, if our

obtained F is greater than the

required F, we have significance.

We then say p<0.05.

I understand how we calculate df

without formulas, however I do not

see why we need it.

I see why you are confused. As I

said this is what happens every

time we try to teach in a

mechanistic, compulsive, step by

step way. The general practice of

working with formulas blindly, and

using the ANOVA summary table,

often prevents the student from

seeing what is going on.

As I said at the beginning, the

ANOVA summary table is not

needed. What you need to do is

simply calculate the variance, and

compare the variances, that is the

F ratio. df is simply the n in the

variance formula. 

 I understand variance Between,

variance Within, and df.

However, I do not see why the F

ratio can detect significance, you

say.

A very important question, if

indeed, we are sincere when we

say we want to understand the

concepts and the logic of statistics.

Drama

Master of the waves

It is a beautiful, cool, calm day in

Puerto Rico. You are sitting in a San

Juan small café, nested on the rocks

overlooking the magnificent Atlantic

ocean. You are happy, sipping your

coffee, Bacardi on the side, and slowly

nibbling on a sinfully sweet piece of PR

cake. It is quiet, the only thing you

hear is the rhythmic sound of waves

gently breaking on the foundations of

the cafe. Suddenly you hear voices; it

sounds like people are arguing. Soon

their voices become loud enough, you

can clearly hear what they are saying.

You don’t believe me? Look again.

See? I caused that wave.

There is much laughter.

Buddy, you are nuts, that’s what I

say. The only waves you cause is in

your brain. You go and see a shrink!

The argument grows in intensity and

the guy with the claims to

supernatural powers, keeps throwing

small stones into the sea. You decide

to join the noisy group and get the

argument straight.

Guys, I have the answer to your

argument. I will show you who is

right. For now, let the sea rest and

calm down, just in case this guy has

disturbed it. Come sit and have a cup

of coffee.

Ten minutes later, you take the lot to the edge of the rocks.

First, we will measure the heights of the next 40 waves and record these data, you say.

When forty waves have been recorded, you

turn to the guy with the supernatural claims

and say:

Ok, this is your show now.

The guy, his confidence somewhat deflated,

picks up a stone and hurls it into the sea. All

eyes are fixed on the base of the rock, waiting for the next wave. The wave comes and is recorded.

The moment of truth, you say.

The height of the wave is read out loud. It is

not taller than any of the 40 waves previously recorded. The miracle worker receives a truckload of cosmetic epithets and soon the café slips back into the beatific serenity. Sleep hovers over your eyelids. You dream of conquistadors and 

fierce Carib Indians, of rituals and dances that humans created in an attempt to understand their world. 

Cute story, but I still do not

understand why the F ratio can

indeed measure that we have

significance, you say.

The forty waves provided us with

what I call the “endogenous”

variance, baseline, the variation in

the heights of the waves that is

present when no obvious cause

can be seen. The wave after the

action of the ambitious miracle

worker was the presumed effect of

his manipulation (in statistics we

call this treatment). Comparing the

baseline with the claimed effect of

his manipulation can give us

support, or lack thereof, for a

connection between what he did

and the result we observed.

If the result which he claims that

he caused by his manipulation is

bigger than the natural,

(endogenous or spontaneous

variance), then we may say that

he caused the effect by his

manipulation.

A brief parenthesis at this point to make

sure we understand what we mean by

saying ratio. Alas, mechanistic methods

of teaching arithmetic without

development of concepts, often prevent

the pupil from understanding that in

division, what we do is compare two

numbers: the numerator to the

denominator. If you have 20 dollars, and

I have 5 dollars, in dividing 20 by 5, I

compare 20 to 5. You have 4 times

more money than I have.

Our treatment causes variance

between to increase? you say.

Yes, let’s see it in an example.

A pharmacologist is testing a new

drug (tentatively named Coolx)

that is suspected to lower body

temperature. He randomly

selects 10 male college students

and randomly assigns them to

two groups. Group 1 receives

Coolx, Group 2 receives a

placebo (an inert substance that

has no effect on physiology).

Here is the layout of the

experiment.

Before the experiment proper, the

pharmacologist records the

temperature of the subjects, in

order to have the baseline

temperature, the temperature that

is present without any

manipulation on the part of 

the experimenter.

Here is the baseline temperature

(Celsius)

We will now calculate variance

between.

Remember, in order to calculate

variance between we line up the

means and treat them as scores.

We then proceed and calculate the

variance of these scores.

Here we have two means

36.740   36.720

The variance of these two scores

is 0.053. This is variance between.

The next table shows temperature

after the administration of drug

Coolx to Group 1

Note that the mean in Group 1

decreased. It was 36.740 before

giving the drug, it is 35.94 now.

Also note that the variance of this

group did not change.

Now the big moment has arrived.

Has the variance between

changed? If yes, we will be

convinced that variance between 

is sensitive to our manipulation,

i.e., that it senses the effect of the

drug.

As usual, in order to calculate

variance between, we line up the

means and treat them as scores.

We then calculate the variance of

these scores.

The means here are:

35.94 36.720

The variance is 0.3042.

This is variance between.

Let’s compare this to variance

between before our manipulation

of giving the drug:

We saw above that variance was

0.053 .

Voila! After giving the drug, that is

after our treatment, variance

between changed. Variance within

did not change.

Conclusion: The F ratio is

sensitive to our treatment. It does

so, because variance Between

changes because of our

manipulation, while variance

Within does not change.

Why? you say.

Remember that variance

measures the distance of scores

from the mean. The mean can

increase or decrease but the

distance of scores from the mean

does not change. The scores

move up or down with the mean.