In this story we will develop the concept of mixed designs and give a practice example.

Elegant research avoids complex

designs also called split-plot designs 

or mixed designs. However, you may not

 be spared of these monsters in your

student or research life.

Let’s get a whiff of these monsters.

A psychiatrist wanted to see, if two

new drugs improve the condition

of depressive and schizophrenic

patients.

He randomly assigned 4

depressive patients to Drug1 and

Drug2 conditions. That is, each of

the depressive patients will be

serving as a subject in both the

Drug conditions. This is a repeated

measures design.

He did the same with the

schizophrenic patients. He

randomly assign 4 schizophrenic

patients to Drug1 and Drug2

conditions. That is, each of the 

schizophrenic patients will be

serving as a subject in both the

Drug conditions. This is a repeated

measures design.

As you see, here we have two

independent groups (depressive

patients, and schizophrenic

patients) but each patient is given

two treatments, that is he is tested

repeatedly, i.e., in both drug

conditions. We have a hybrid

situation, you would say. Both

independence and non-

independence in the same

experiment.

Here is the layout; X stands for scores.

The analysis of data in complex

designs like the above, is, as

always, an operation involving the

calculation of variance. The

interpretation of the results of such

an analysis is like the interpretations

we considered in this book so far.

ANOVA mixed split plot - formula and practice example

What is ANOVA mixed split plot design

ANOVA mixed split plot designs are complex designs that employ both independent and repeated measures. The best way to explain this is to present the layout of these experimental designs.

TABLE SHOWING THE LAYOUT

 OF MIXED SPLIT-PLOT DESIGNS

Observe that there are two independent groups, depressive, and schizophrenic. Also observe that each subject of the depressive and schizophrenic groups is repeatedly tested, once with Drug 1, and later with Drug 2. This is a repeated measures arrangement So here we have a design in which independent and repeated measures are mixed. The name split plot comes from the fact that this design is extensively used in agricultural research.

ANOVA mixed split plot designs formula

As in all ANOVA, the formula for these designs is: 

We read this as follows: Mean square between over mean square within. What is mean square, you ask? It is the mean of squares. What is squares, you ask. Squares is the statistical term for squared deviations (of squared differences) of each score X from the mean. What are the squared differences, you ask. Remember the formula for variance?

Look at the numerator

These are the squared differences summed. To complete our reasoning, we go back to where we started, the F formula, or F ratio, the formula for ANOVA. Why mean sums of squares? Simple because like all averages, we divide by the number of scores. If you are observant, you will notice that the F formula is a modified t formula.

FORMAT OF ANOVA MIXED SPIT PLOT SUMMARY TABLE

HOW TO CALCULATE df OF ANOVA MIXED SPIT PLOT SUMMARY TABLE

ANOVA mixed split plot- practice examples

ANOVA mixed split plot- practice example 1

An experimenter wanted to test drugs (factor A), Drug 1 (A1) and Drug 2 (A2) for their effect on serotonin level in the blood of patients (factor B) suffering from depression (B1) and schizophrenia (B2) . He randomly selected six patients suffering from depression and gave them Drug 1. He waited for one hour and then he measured the level of serotonin in nanograms per liter (ng/lt) of each subject. He recorded the data. One week later he gave these subjects Drug 2. He waited for one hour and measured the level of serotonin of each subject. He also randomly selected six patients suffering from schizophrenia and repeated the same experiment that he performed with the depressive patients. The data are presented in the table below.

ANOVA MIXED SPIT PLOT SUMMARY TABLE

We have already developed the

concept of independence. In those

experiments in which each subject

is used only in one group or

condition, we say that the groups

are independent. So far in this

book we have considered only

independent-groups statistical

designs and experiments.

In designs in which the groups are

not independent, a subject is used

in more than one group or treatment.

That is, each subject experiences

more than one treatment.

For example, John may first be

given behavioral therapy, and

later, several months later, he may

also be given psychoanalytic

therapy. The effects of the two

therapies are then compared.

A variation of this arrangement is

to match each subject with

another subject on the basis of

similarity in some measure. This is

done to eliminate carryover effects

that may, obviously, be present in

giving one subject both treatments.

There are obviously advantages

and disadvantages in choosing

matched groups designs over

independent groups designs.

However, this issue is beyond the

goals of the present book. In

general, independent groups

designs are safer, and should, in

my opinion, be preferred.

The concepts in matched groups

designs are the same as those in

independent groups designs. We

will, therefore, confine ourselves to

giving examples of these designs.

First an example for t-test, and

then an example for ANOVA

repeated measures.

An example of t-test for

matched groups

In comparing two new anti-anxiety

drugs, a pharmaceutical company

selected 5 pairs of patients, each

pair matched on the basis of their

anxiety score.

Here is the layout and data of the

experiment.

  Mean for difference=0.4

The formula for the t-test for

dependent groups is

We read it as follows:

t for paired observations equals

mean of differences divided by the

standard deviation over the square

root of the n. (The standard

deviation divided by the square

root of the n is the standard error

of the mean, SEM, remember?)

You know all of the terms of the 

t-formula. You also recognize that it

is the same old story, our old

friend, the z formula.

t=+0.49 df=4

Entering the t-table with df 10 we

find that the required t=2.132

Our obtained t 0.49 is smaller

than the required, therefore we do

not have significance. We say that

the difference we observed is not

significant (p>0.05).

Study the table below..

It adds to our effort toward integration

and understanding beyond a mechanistic

use of a plethora of formulas. 

Study the table below..

It adds to our effort toward integration

and understanding beyond a mechanistic

use of a plethora of formulas. 

Example of ANOVA Repeated

Measures

Four patients with damage in the

hippocampus were treated with

two new drugs in order to see if

their memory improved.

Here is the layout as well as the

scores of the experiment. High

scores indicate improvement in

memory.

ANOVA SUMMARY TABLE

Repeated Measures

Entering the F table in Appendix

with df 1 and 3, we find an F of

10.12. This is the required F in

order to have significance. Our

obtained F (see ANOVA summary

table above) is 7.71. It is less than

the required F, therefore, we do

not have significance. We say:

There was no significant difference

between the means of the two

conditions (p>0.05). 

P greater than point o five.

I see there are questions.

What is Between Columns? You ask.

It is the usual Between variance

that you know. The variance that

our treatments produce. The

variance of the means.

What is Between Rows? you ask.

If you look at the layout above,

you see that the rows are subjects,

one subject per row. The mean of

each subject is the mean of each

row. The variance of these means

are the variance between the

rows.

Why you did not calculate an F for

the Rows? you ask.

There is no reason that I can think

of, that would justify my wanting to

know whether there is a statistical

significant difference between

subjects. That would be an

absurd statement. 

Once again you see that our

conceptual approach allowed us to

attack this design too, without the

need for new formulas. What is of

course more important is the fact

that we understand the logic of this

design too. We feel in command,

comfortable to handle any issue. 

The layout of a 2x3 factorial ANOVA

In this example of a factorial design, we have a 2x3 (we read this as "a two by three") factorial. Two by three, meaning two factors: A and B. "two" meaning two levels for factor A. "three" meaning three levels for B. In another case of a 3x2 factorial design we have two factors, A and B, factor A three levels, factor B two levels.

FORMAT OF 2x3 FACTORIAL ANOVA SUMMARY TABLE

Interaction -factorial designs

Note the term "Interaction" in the ANOVA summary table of the factorial design What is interaction? The best way to grasp the concept of interaction is to graph it.

ANOVA 2x3 factorial practice example

An experimenter wanted to test the effect of two drugs on the emotionality of male and female teenagers. He randomly selected 15 male and 15 female teenagers and randomly assigned them to 6 groups: Group 1, Group 2, Group 3, Group 4, Group 5, Group 6. five subjects in each group as shown in the following table.

The data are presented on the table below. The scores are the values recorded on a device measuring galvanic skin response, a measure of emotionality. Higher values indicate stronger emotion.

.

2x3 FACTORIAL ANOVA SUMMARY TABLE

After we calculate the F, we go to the F table to find the required F value for A, B, and AxB (interaction). Because,( remember?) the F ratio is A over within, B over within, 

AxB over within, we enter the F table with
df of A and df within , which is 1 and 24.
also B and df within, which is 2 and 24
and lastly AxB within., which is 2 and 24. We first choose the F table at 0.05 level of significance.

Factor A: The F at df 1 and 24 is 4.25. In the summary table we see that the F for factor A (rows) is 62.31. This is greater than 4.25, so we conclude that here we have significance at the 0.05 level of significance; we say p<0.05, p less than 0.05. It has been accepted among scientists that at the 0.05 level we are allowed to say that we have significance, that the finding of our experiment is reliable.

Next we look at factor B. We enter the F table with df 2 and 24 and find F=3.50. This is less than the F of our summary table 502.19, therefore we conclude that we have significance at the 0.05 level. We formally express this as follows: p<0.05.

Next we look at AxB. We enter the F table with df 2 and 24 and find F=3.50.. This is greater than the F at the summary table value of 1.75 so we conclude that here we do not have significance. We formally express this as follows: p>0.05.

Step by step calculation of 2x3 ANOVA factorial

The goal of our calculations in ANOVA is to compute the F ratio, The F ratio is MS between over MS within. Mean Square is the mean of the squared deviations (differences).

of each score from the mean. These are very simple calculations involving high school mathematics. Simple as they are, they are very important concepts in data analysis and beyond, that is science in general. You will never need to perform these calculations. There are many free Statistics calculators online. However, for the purpose of developing the concepts of ANOVA here are the steps:

1. Calculate the mean of each group.
2. Subtract each score from the mean.
3. Square each difference
4. Add these squared differences. font red This is the Sum of Squares, the SS on the ANOVA summary table.)
5. calculate the degrees of freedom df (number of scores that went into the calculation of the mean minus 1)
6 Divide the SS by the df. Voila! this the MS.
7. The last step is to calculate the F. Divide MS by the MS of the error term (which is the MS within but may be something else depending on which ANOVA design you have. )


The F ratio, as all ratios, compares two things. For example the ratio 8/4 compares 8 to 4 and finds that 8 is two times greater than 4.

A pharmacology graduate student

working on his thesis wanted to

find whether a new chemical,

DOP-Y, which has been shown to

elevate dopamine levels in the

brain, may be beneficial to

depressive patients. He was also

interested to see if electroshock

has an effect on these patients

when combined with DOP-Y.

He randomly selected 20

depressive patients, and randomly

assigned them to 4 groups:

electroshock - DOP-Y,

electroshock-no DOP-Y

no-electroshock - DOP-Y,

no-electroshock-no DOP-Y

The layout of the pharmacology experiment

The data he recorded are given in

the next table.

The data of the pharmacology experiment

High numbers indicate improvement.

The data of the pharmacology experiment

THE PHARMACOLOGY EXPERIMENT 

ANOVA SUMMARY TABLE

In this table we see that A, B, and

 AxB are significant.

Significance in A means that

electroshock benefited the

depressive patients.

Significance in B means that drug

benefited the depressive patients.

Significance in AxB means that

there was an interaction between electroshock and drug.

Not clear, you say,

You are correct.

Let us look at the graph of the

interaction.

First, we observe that the two lines,

shock and no shock, are not

parallel. Every time we have an

interaction, the two lines are not

parallel.

How about getting to understand

interaction at the gut level, not just

with words? you say.

Let’s do it. Look at the graph

above (previous page).

First, we will visualize the graph

without the effects of the drug. In

that graph the two lines would be

parallel.

Now visualize the effect of drug as 

a force pushing the lines up.

Logically we would expect to see

both lines pushed up while

maintaining the distance between

them, i.e., the two lines may move

higher on the graph, but they

should remain parallel. However,

in the present experiment we saw

that the drug has pushed the no

electroshock line

disproportionately higher.

This is the concept of interaction.

Understanding the 2x2 factorial ANOVA summary table

A
Looking at the layout tables above, we see that factor A is gender. Factor B is drug. Our calculations gave a p value <0.05 meaning that factor A, gender, gave a significant difference. In other words, there is a difference in emotionality between male and female

B
Looking at the layout tables above, we see that factor B is Drug. Our calculations gave a p value <0.05, meaning that factor B, drug, gave a significant difference. In other words, there is a difference in emotionality between subjects that received drug 1 as compared to subjects that received drug 2.

AxB
This is the interaction term. Definition of the interaction. What is interaction in factorial designs? Interaction is present if one level of one factor has a disproportionate effect on one level of the other factor.

Example 1 of 2x2 factorial experiment ANOVA

A pharmacology graduate student

working on his thesis wanted to

find whether a new chemical,

srt-X, which has been shown to

block serotonin, may be beneficial

to schizophrenic patients. He was

also interested to see if

electroshock has an effect on

these patients when combined

with srt-X.

He randomly selected 20

schizophrenic patients, and

randomly assigned them to 4

groups:

electroshock - srt-X,

electroshock-no srt-X

no electroshock - srt-X,

no electroshock-no srt-X

The layout of this experiment is:

The layout in abstract form is:

Variable A has two levels, a1 and

a2, and variable B has two levels,

b1 and b2.

The next table shows the data he

recorded in running the

experiment. The numbers

represent scores on a psychiatric

test measuring intensity of

schizophrenic behavior. The

higher the number the worse the

condition of the patient.

THE SEROTONIN BLOCKER

PLUS SHOCK EXPERIMENT

ANOVA SUMMARY TABLE OF

THE SEROTONIN BLOCKER

PLUS SHOCK EXPERIMENT

* Interaction

I will first discuss the table in terms

of the calculations we did.

First and most important, the

degrees of freedom.

If you tell me the degrees of

freedom in any ANOVA

experiment, but without the use of

formulas (I do also mean resorting

to memory for the recollection of

formulas - ban formulas!), I know

you know what you are talking

about. Calculation of the F is easy,

high school arithmetic.

If you

Why df for Between A is 1?

Because in order to calculate

variance Between we line up the

means, consider them scores, and

calculate the variance using the

one and only formula for variance

(all the other formulas for variance

that you may see around are

derived from this formula.

Statisticians get their kicks by

producing equivalent formulas, of

considerable complexity and

ornamental value!). Now you and I

know that in order to calculate 

variance, we must first calculate

the mean. Every time we calculate

the mean, we lose 1 degree of

freedom. Because, in the present

example we have 2 scores (never

mind that they are means), we are

left with 1 df. That is 2-1=1.

I do not understand why you say

we have 2 means for A, you ask.

Good question. A has two levels

here, a1 and a2. That is shock and

no shock. You see, when we deal

with variable A, we ignore variable

B. In other words we reduce this

part of the analysis to a one-way,

single-factor ANOVA.

Why df for Between B is 1? you

say.

For the same reasons as in the

previous paragraph, B has two

levels, b1 and b2, drug and no

drug. There are two means

(scores). In order to calculate the

variance of these two scores, we

must first compute the mean. We

therefore lose 1 df. So the df for B

is 2-1=1.

Why df for AxB interaction is 1?

This is easy. Since df for A is 1, and

df for B is also 1, the df for AxB is

1x1=1.

Why is the df for within 16? 

This is simple, too. We said variance

within is variance for the first

group plus variance for the second

group, plus variance for the third

groups and so on. We have four

groups here. In order to calculate

the variance of each group we

must first calculate a mean. The

consequence of this is that we

lose 1 df for every mean we

calculate. How many scores go

into the calculation of variance for

group 1? Five scores. Therefore

df for the first group is 5-1=4. We

calculate the variance of the

remaining 3 groups in a similar

way. Since we have 4 groups

here, the df for Within is 4x4=16.

he 

Note: Checksum. The sum of df

for A, B, AxB, Within, equals df

Total

SS for A, B, AxB, and Within

equals SS Total.

Remember, we said that in ANOVA

we partition variance.

Discussion of the experiment

with the schizophrenic patients.

Look at the ANOVA Table again:

ANOVA SUMMARY TABLE OF

THE SEROTONIN BLOCKER

PLUS SHOCK EXPERIMENT

  * Interaction

The p value (the probability that

the difference or effect we are

reporting may not be reliable or

significant) for A is less than 1 in

ten thousand (p<.0001|). 

Variable A is electroshock in this

experiment. This means that the

two conditions, electroshock and

no electroshock (condition 1: 

electroshock-drug, electroshock-no drug; condition 2: no-

electroshock-drug, no- electroshock-no drug) produced a result, a significant difference. 

In other words those patients who received electroshock ended up different from those patients that did not receive electroshock. 

The p value (the probability that the difference or effect we are reporting may not be reliable or significant) for B is less than 1 in ten thousand (p<.0001|). 

Variable B is drug in this experiment. This means that the two conditions, drug and no drug (condition 1: drug-electroshock, drug-no electroshock, condition 2: no drug-electroshock, no drug-no electroshock) produced a result, a significant difference. In other words, those patients who received the drug were different from those patients that did not receive the drug. 

The p value of AxB, the interaction 

is p>.05, We read this as follows:

p greater than five per cent. This

means that if we were to say that

there was significant interaction

between electroshock and drug,

we would be running the chance

of reporting an effect that is not

reliable, not significant, meaning

that if we or someone else were to

do the same experiment again,

most likely would not find a

difference as we did

As we said earlier the concept of

interaction is a new one for us,

and we need to understand it our

way, at the gut level, as we are

used to.

We will now consider an experiment in which the interaction is significant.

Analysis of Variance

Factorial Designs

Two-Way ANOVA

The ANOVA that we discussed so

far is called 'One-way ANOVA' or

'Single-factor ANOVA'.

Now we will consider two-way

ANOVA or two-factor ANOVA.

The concepts we developed so far

also apply to two-way ANOVA.

What do you mean by one-way,

single-factor, two-way, or

two-factor? you say.

Drama

Beam storm

Rutgers College. May 9, 1999, 9:00 in the

morning. Two sections of Statistics 101 are in class: two adjacent classrooms, C120 and

C121. USS Spaceship Enterprise flew over the two classrooms and locked on the bio readings of the students. Then, classroom C120 was bombarded with a X-Z-LOBX beam for 10 milliseconds. The security cameras recorded an almost imperceptible tilt of the head to the left, while the professor of Statistics, without being aware, wrote the same complex formula for MS 5 times. Two nanoseconds after classroom C120 was bathed in the benevolent X-Z-LOBX beam, classroom C121 was bombarded by the same X-Z-LOBX beam for 100 milliseconds. All students raised the index finger of their right hand and stuck it in their left nostril. The professor started reciting the t-table but stopped short in a deluge of laughter from the students.

The duration of the students’ responses was

recorded by the spaceship and instantly

transmitted to Houston where a robot was

waiting to manually enter the data on the

layout of the experiment. The layout of the

experiment was made public, the data not. 

Discussion of data was forbidden by a

unanimous decision of the Congress.

The layout of the USS Enterprise

experiment

This experiment is a one-way

ANOVA design.

Why? Because each student was

bombarded with one beam.

We also say that this design is a

single-factor ANOVA.

Why?

Because each student was

bombarded with a single beam.

Another way of saying this is, that

each score in this experiment is

the result of one beam, one factor,

or one treatment. You may also

 come across the term 

one-way classification.