THE EXAMPLE BELOW WILL DEMONSTRATE
HOW:

Simple
"arithmetic
mean" pregnancy rates of two embryologists and of
two catheters who were involved
in embryo transfers
(as examples)
are NOT their "TRUE" pregnancy rates.
As a result,
statistical differences in
"arithmetic mean" pregnancy rates
between the two embryologists as well as between the two
catheters (even if detected by simple statistical
tests) are NOT "TRUE" statistical
differences.

HOWEVER,
analyzing the entire data in an
advanced SINGLE STATISTICAL MODEL /
ANALYSIS / TEST, by correcting (adjusting)
those
"arithmetic mean" pregnancy rates for the effect of
the other factor (i.e., by correcting / adjusting the
"arithmetic mean" pregnancy rate of each embryologist for the
catheter effect as well as by correcting / adjusting the
"arithmetic mean" pregnancy rate of each catheter for the
embryologist effect)
dramatically changes those pregnancy rates, and
DETECTS "TRUE" pregnancy rates of embryologists and
catheters.
As a result, the advanced
SINGLE STATISTICAL MODEL /
ANALYSIS / TEST dramatically
changes the degree of statistical differences
in pregnancy rates between the two
embryologists as well as between the two catheters, and
DETECTS the "TRUE"
statistical differences among those "true"
(corrected / adjusted) pregnancy rates.

AND, even further
correcting (adjusting)
those
pregnancy rates
for the effect of
patient's age
dramatically changes those pregnancy rates even further and
DETECTS further "TRUE" pregnancy rates of embryologists and
catheters.
Before going through the example on
this page step by step, let's have a look at the
following table, which summarizes pregnancy
rates of two embryologists and of two catheters who were involved
in embryo transfers, showing (side by
side) simple arithmetic means of pregnancy rates
as well as corrected (adjusted) means of the same
pregnancy rates, accompanied by their respective
statistical differences (P values) (see below for the actual data and the statistical
tests that generated the results in that table).
Embryologist (E)
OR Catheter (C) 
Pregnancy (%)
(arithmetic mean) 
Pregnancy (%)
(corrected for E/C) 
Embryologist 1 (E1) 
65 
P=0.03 
68 
P=0.006 
Embryologist 2 (E2) 
27 
18 
Catheter 1 (C1) 
36 
P=0.26 
25 
P=0.04 
Catheter 2 (C2) 
56 
61 
Test Type 
Simple Tests
(ChiSquare or TTest) 
TwoWay
Analysis of Variance
(TwoWay ANOVA) 
The second column
in the above table shows:

Simple
"arithmetic
mean" pregnancy rates of two embryologists and
of two catheters (who were involved in embryo
transfers) as well as

The degrees of statistical differences
(P values) between those two embryologists
and between those two catheters,
calculated by simple statistical tests.
The third column in the above table shows:

Corrected
(adjusted) "true" pregnancy rates of the same two embryologists and
of the same two catheters, calculated
by the more advanced / comprehensive
TwoWay Analysis of Variance Test (TwoWay ANOVA),
as well as

The corrected (adjusted) "true" degrees of
statistical differences (P values)
between the two embryologists and
between the two catheters, calculated
by the more advanced / comprehensive
TwoWay Analysis of Variance Test (TwoWay ANOVA).
Note that, in the third
column, the "arithmetic
mean" pregnancy rates of the embryologists
have been corrected (adjusted) for the effect of the
catheter involved. Likewise, the "arithmetic
mean" pregnancy rates of the catheters have
been corrected (adjusted) for the effect of the embryologist involved.
As we see in the above table,
after
correcting (adjusting) the "arithmetic mean" pregnancy rates for the effect of the
embryologist / catheter,
each embryologist�s and catheter's �true� pregnancy rate has changed.
AS A RESULT:

The percentagepoint
difference between the two embryologists� �true�
pregnancy rates INCREASED FROM 38 (the difference between
65% & 27%) when using simple statistical tests
TO 50
(the difference between 68% & 18%) when using the more advanced /
comprehensive TwoWay Analysis of Variance Test (TwoWay ANOVA).
As a result, the more
advanced / comprehensive TwoWay ANOVA Test detected a more
significant difference between the two
embryologists� �true� pregnancy rates compared to the simple
statistical tests (P=0.006 vs.
P=0.03).

The percentagepoint
difference between the two catheter's �true�
pregnancy rates INCREASED FROM 20 (the difference between
36% & 56%) when using simple statistical tests
TO 36
(the difference between 25% & 61%) when using the more advanced /
comprehensive TwoWay Analysis of Variance Test (TwoWay ANOVA).
As a result, the more
advanced / comprehensive TwoWay ANOVA Test detected a significant difference between the two
catheters' �true� pregnancy
rates, which was NOT detected by the simple statistical tests (P=0.04
vs. P=0.26).
NOW LET'S ANALYZE THE ACTUAL DATA, step by step, THAT GENERATED the
results in the above TABLE.
ANALYZING your ENTIRE data in a SINGLE statistical model / analysis /
test (as opposed to analyzing for factors
separately)
DETECTS "true means"
and "true differences"
among those "true means," BECAUSE:

It takes
into account ALL the factors;

It
corrects (adjusts)
data for ALL the factors;

It
removes the effects of the other
factors from the effect of the factor investigated.
Clinical outcomes
(such as pregnancy rate, fertilization rate, etc.) in ART / IVF programs
are the results of NOT ONLY ONE factor but the results of
MULTIPLE factors that are at work simultaneously, i.e.,
those outcomes are simultaneously affected by multiple factors.
Therefore, it is NOT sufficient to
visually compare, or
even statistically analyze, the
arithmetic means of factors separately, i.e., by taking into account the effect of
ONLY ONE
factor at a time.
The examples below with a sample pregnancy
data will illustrate why it is important to analyze your clinical
outcomes (such as pregnancy rate, fertilization rate, etc.)
in a SINGLE statistical model / analysis / test, by
correcting (adjusting) those
arithmetic means for the
effects of ALL the factors involved, i.e., by
taking into
account the effects of ALL the factors involved.
This can only be achieved with
advanced / comprehensive statistical knowledge, expertise and experience
as well as with the use of advanced / comprehensive tests of
statistical analyses, which is what the STATS OF THE ART
is here to help you with.
Below is an example with a sample pregnancy data of 32 patients, with each embryo transfer
procedure involving one of the
two embryologists and one
of the two catheters.
In other words, there were
TWO FACTORS (embryologist and catheter), and
TWO LEVELS within each factor (Embryologists 1 & 2; and
Catheters 1 & 2,
respectively).
These factors (and levels within) are only
examples, so you can include as many important factors / levels as you wish.
In this example, you want to know the
pregnancy rate of each embryologist, and the
pregnancy
rate of each catheter.
However, when you look at the
"arithmetic mean" pregnancy rate of an embryologist, we know that, in each embryo
transfer procedure performed by that embryologist, there was
also
one of the two catheters involved. So the
"arithmetic mean" pregnancy rate
of that embryologist is NOT the �true� pregnancy rate of that
embryologist in its entirety. In other words, the
"arithmetic mean" pregnancy rate
of that embryologist is NOT entirely independent of the
catheter effect.
Likewise, when you look at the
"arithmetic mean" pregnancy rate of a
catheter, we know that, in each embryo
transfer procedure involving that catheter, there was
also
one of the two embryologists involved. So the
"arithmetic mean" pregnancy
rate of that catheter is NOT the �true� pregnancy rate of that
catheter in its entirety. In other words,
the "arithmetic mean" the pregnancy rate of
that catheter is NOT entirely independent of the embryologist
effect.
So because of this, you should not just
want to know the "arithmetic mean" pregnancy rate of each embryologist and of each
catheter, but you should want to know the �true� pregnancy rate
of each embryologist independent of the catheter effect, and
the �true� pregnancy rate of each catheter independent of the
embryologist effect.
So in order to know the
�true�
pregnancy rate of each embryologist independent of the catheter effect,
you need to correct (adjust) the "arithmetic
mean" pregnancy rate of each
embryologist for the catheter effect. In other words, you need
to remove the catheter effect from the
"arithmetic mean" pregnancy rate of
each embryologist. In other words, you need to
take into account
the catheter effect.
Likewise, in order to know the
�true� pregnancy rate of each catheter independent of the embryologist
effect, you need to correct (adjust) the
"arithmetic mean" pregnancy rate of
each catheter for the embryologist effect. In other words, you
need to remove the embryologist effect from the
"arithmetic mean" pregnancy
rate of each catheter. In other words, you need to
take into
account the embryologist effect.
In the example below:

First, we will
statistically analyze the "arithmetic mean" pregnancy
rate within each factor at a time (within
embryologists and within catheters) (i.e., running multiple
statistical analyses / tests separately within factors, i.e.,
NOT correcting / adjusting for the effects of the other
factors, i.e., WITHOUT taking into account the effects of
the other factors, and so which will NOT yield the �true� effects
of the factors).

Then, in order to see the
�true� effects of those factors (embryologists and catheters),
we will statistically reanalyze the same pregnancy data within and
across all factors (i.e., running one SINGLE statistical
model / analysis
/ test, i.e., correcting / adjusting for the
effects of the other factors, i.e., taking into account
the effects of the other factors).
Now let�s look at, and statistically analyze,
the (imaginary) pregnancy data below, involving 32 patients, with
two
embryologists (E1 & E2) and two
catheters (C1 & C2) who were
involved
in embryo transfer procedures of those patients. Again, this is only an
example, and you can include as many important factors and levels as you want in
the statistical analysis of these data. Although the analyzed outcome in
this example is �pregnancy,� the statistical analysis is applicable to
other outcomes as well, such as live birth, fertilization rate, etc.
Patient
ID 
Patient�s
Age 
Embryologist
(E) 
Catheter
(C) 
Pregnancy 
Patient 01 
26 
E1 
C2 
YES 
Patient 02 
27 
E1 
C1 
NO 
Patient 03 
28 
E2 
C2 
YES 
Patient 04 
29 
E1 
C1 
NO 
Patient 05 
29 
E1 
C2 
YES 
Patient 06 
30 
E1 
C1 
YES 
Patient 07 
30 
E1 
C2 
YES 
Patient 08 
30 
E2 
C2 
YES 
Patient 09 
31 
E1 
C1 
YES 
Patient 10 
31 
E2 
C1 
YES 
Patient 11 
31 
E2 
C2 
NO 
Patient 12 
31 
E2 
C2 
NO 
Patient 13 
32 
E1 
C2 
YES 
Patient 14 
33 
E1 
C2 
YES 
Patient 15 
33 
E2 
C2 
YES 
Patient 16 
34 
E2 
C2 
NO 
Patient 17 
35 
E1 
C1 
NO 
Patient 18 
35 
E1 
C1 
NO 
Patient 19 
35 
E2 
C1 
NO 
Patient 20 
36 
E1 
C1 
YES 
Patient 21 
36 
E2 
C1 
NO 
Patient 22 
36 
E2 
C2 
NO 
Patient 23 
37 
E1 
C1 
YES 
Patient 24 
37 
E1 
C1 
NO 
Patient 25 
37 
E1 
C2 
YES 
Patient 26 
37 
E1 
C2 
YES 
Patient 27 
38 
E1 
C1 
NO 
Patient 28 
38 
E2 
C1 
NO 
Patient 29 
38 
E2 
C2 
NO 
Patient 30 
38 
E2 
C2 
NO 
Patient 31 
39 
E2 
P2 
NO 
Patient 32 
40 
E2 
P2 
NO 
So now we have two �factors�
(embryologist and catheter), and then we have
two �levels� within each �factor.� We also have patient's age as �covariate.�
Factors 
Levels of Factors 
Embryologist 
Embryologist 1 (E1)
Embryologist 2 (E2) 
Catheter 
Catheter 1 (C1)
Catheter 2 (C2) 
The following table summarizes
"arithmetic mean" pregnancy
rate (WITHOUT any statistical analyses)
for each embryologist (E1
& E2) and for each catheter (C1 & C2).
Embryologist (E)
OR Catheter (C) 
Pregnancy (%)
(arithmetic mean) 
E1 
65 (11/17) 
E2 
27 (4/15) 
C1 
36 (5/14) 
C2 
56 (10/18) 
More specifically, the
data constitute a "twobytwo factorial design"
as shown below.


FACTOR 2
Catheter 
Pregnancy (%) 


LEVEL 1
Catheter 1 
LEVEL 2
Catheter 2 
FACTOR 1
Embryologist 
LEVEL 1
Embryologist 1 
40% (4 / 10) 
100%
(7 / 7) 
65% (11 / 17) 
LEVEL 2
Embryologist 2 
25% (1 / 4) 
27% (3 / 11) 
27% (4 / 15) 
Pregnancy (%) 
36% (5 / 14) 
56% (10 / 18) 
47% (15 / 32) 
The following is the
list of the steps we will go through
(you can also click on them to go directly to each step):
STEP 1
 ARITHMETIC MEAN (UNTRUE) PREGNANCY RATES COMPARED BY SIMPLE STATISTICAL TESTS
Now we want to know the
�overall�
pregnancy rate for each embryologist (E1 & E2) and
for each catheter (C1 & C2), who were involved in embryo transfers of those patients.
And we also want to know
whether the
pregnancy rates of embryologists are statistically different
from one another. Likewise, we also want to know
whether the
pregnancy rates of catheters are statistically different
from one another.
For this purpose, we will statistically
analyze the pregnancy data within each factor at a time (i.e.,
running multiple statistical analyses / tests separately within
factors, i.e., NOT correcting / adjusting for the effects of the
other factors, i.e., WITHOUT taking into account the effects of
the other factors, and so which will NOT
yield the �true� effects of
the factors).
Accordingly, using a statistics software,
we will analyze the pregnancy data within embryologists, and then
also within catheters, by
ChiSquare Test, or alternatively
by TTest, both of which will generate the same results.
NOTE THAT:
ChiSquare Test and TTest are appropriate
to compare "arithmetic means," when there is ONLY
ONE factor, and ONLY TWO levels within that factor (such as two embryologists or
two
catheters). When there are more than two embryologists and/or
catheters, those tests have to be repeated multiple times, i.e., between
each pair of levels of the factor (i.e., between Embryologists 1 & 2;
between Embryologists 1 & 3; between Embryologists 2 & 3; and so on),
which makes these tests impractical, and so necessitates the use of a
more advanced / comprehensive statistical analysis / test called
OneWay ANALYSIS OF VARIANCE Test (OneWay ANOVA),
although those simple tests still have to be repeated multiple times between
each pair of levels of the factor, which will be illustrated in the
next step. However, when there are more than ONE factor involved
(e.g., two factors as "embryologist" and "catheter"), OneWay
ANOVA has to be repeated for each factor. But even then, OneWay
ANOVA Test still and only compares
"arithmetic means," which are NOT "true
means," as they are NOT corrected (adjusted) for all the
factors. This necessitates the use of
TwoWay ANOVA, ThreeWay ANOVA,
or FourWay ANOVA, and so on, when
there are two, three, or four factors involved, and so on, which will
also be illustrated in the
next step. 
After analysis of the pregnancy data, now
let�s update the above table by adding P values in the table, showing
statistical differences in "arithmetic
means" between the two embryologists as well as between the two
catheters.
See below for the updated table, and let�s
see whether those "arithmetic mean" pregnancy rates are different between the two
embryologists as well as between the two catheters.
Embryologist (E)
OR Catheter (C) 
Pregnancy (%)
(arithmetic mean) 
E1 (n=17) 
65 
P=0.03 
E2 (n=15) 
27 
C1 (n=14) 
36 
P=0.26* 
C2 (n=18) 
56 
*Note: P=0.26 by ChiSquare Test; P=0.28
by TTest.
As we see in the above table:

There is a 38
percentagepoint difference between the two embryologists (65%
vs. 27%), which is statistically significant (P=0.03).

There is a 20
percentagepoint difference between the two catheters (36% vs.
56%), although statistically not significant (P=0.26).
However, the above comparisons
cannot be conclusive, because, when we look at the
"arithmetic mean" pregnancy
rate of, say, Embryologist 1 (65%) who did the embryo transfers of those
patients, we know that Catheter 1 or Catheter 2 was also involved in
those embryo transfers performed by Embryologist 1, and so
the two
catheters (as independent factors)
may have affected the "arithmetic mean" pregnancy rate
of Embryologist 1 differently from one another.
So the 65%
"arithmetic mean" pregnancy rate of
Embryologist 1 is NOT the �true� pregnancy rate of Embryologist 1,
as it is NOT corrected (adjusted) for different effects of
Catheter 1 and Catheter 2, as some of those embryo transfers were
performed by Catheter 1 and some others by Catheter 2, and those two
catheters (as independent factors) may result in different pregnancy
rates. The same is true for the "arithmetic
mean" pregnancy rates of Embryologist 2,
Catheter 1 and Catheter 2, and so their
"arithmetic mean" pregnancy rates
shown in the above table are NOT their �true� pregnancy rates.
The above necessitates the use of a more
advanced / comprehensive statistical analysis / test, which is
discussed
in the next step.
STEP 2
 THE �TRUE� PREGNANCY RATES CALCULATED AND COMPARED BY A SINGLE
ADVANCED STATISTICAL MODEL / ANALYSIS / TEST
That is, as opposed to "arithmetic
means":

The �true� pregnancy rate of each
embryologist corrected (adjusted) for the effect of the catheter;
and

The �true� pregnancy rate of each
catheter corrected (adjusted) for the effect of the embryologist.
As discussed in the previous step, when we look at the
"arithmetic mean" pregnancy
rate of, say, Embryologist 1 (65%) who did the embryo transfers of those
patients, we know that Catheter 1 or Catheter 2 was also involved in
those embryo transfers performed by Embryologist 1, and so
the two
catheters (as independent factors)
may have affected the "arithmetic mean" pregnancy rate
of Embryologist 1 differently from one another.
So the 65% pregnancy
"arithmetic mean" rate of
Embryologist 1 is NOT the �true� pregnancy rate of Embryologist 1,
as it is NOT corrected (adjusted) for different effects of
Catheter 1 and Catheter 2, as some of those embryo transfers were
performed using Catheter 1 and some others using Catheter 2, and those two
catheters (as independent factors) may result in different pregnancy
rates. The same is true for the "arithmetic
mean" pregnancy rates of Embryologist 2,
Catheter 1 and Catheter 2, and so their
"arithmetic mean" pregnancy rates
shown in the above table are NOT their �true� pregnancy rates.
So now we want to know
the �true�
pregnancy rate of each embryologist when each embryologist�s
pregnancy rate is corrected (adjusted) for the effect of the
catheter factor (i.e., for the effect of the catheter that was
used) (i.e., when the effect of the catheter factor is
taken
into account). Likewise, we want to know
the �true� pregnancy
rate of each catheter when each catheter's pregnancy rate is
corrected (adjusted) for the effect of the embryologist factor
(i.e., for the effect of the embryologist who was involved) (i.e., when
the effect of the embryologist factor is taken into account.
And we also want to know
whether the
�true� (corrected / adjusted) pregnancy rates of embryologists are
statistically different from one another. Likewise, we also want
to know whether the �true� (corrected / adjusted)
pregnancy rates
of catheters are statistically different from one another.
So, in order to see the �true�
pregnancy rate of each embryologist and of each catheter,
let�s
correct (adjust) each embryologist�s "arithmetic mean" pregnancy rate for the effect of
the catheter that was used; and likewise,
let�s correct
(adjust) each catheter's "arithmetic mean" pregnancy rate for the effect of the
embryologist who was involved.
For this purpose, instead of statistically
analyzing the pregnancy data within each factor at a time (which was
done in the previous step), we will statistically reanalyze the same
pregnancy data within and across all factors (i.e., running
one
SINGLE statistical analysis / test, i.e., correcting / adjusting
for the effects of the other factors, i.e., taking into account
the effects of the other factors).
Accordingly, using
a statistical analysis
software that is more advanced / comprehensive than in the previous
step, we will reanalyze the same pregnancy data, in a SINGLE
statistical model / analysis / test, within and across
embryologists and catheters at the same time, In other words, we will
reanalyze the same pregnancy data by TwoWay ANALYSIS OF VARIANCE Test (TwoWay ANOVA)
in a "twobytwo
factorial design," with "embryologist" and "catheter"
being FACTORS, and with "Embryologist 1", "Embryologist 2", "Catheter 1" and "Catheter 2" being LEVELS within those factors.
NOTE THAT:
As discussed previously here,
you can include in the same
single statistical model / analysis / test any number of factors you wish. Likewise, you can include any number of levels
within each factor (e.g., two embryologists,
three types of embryo transfer catheters, etc.).
For example:

You can
include two factors with
two
levels in each, and analyze them by "TwoWay
Analysis of Variance" in a "twobytwo
factorial design."

You can
include two factors with
two and
three levels, and analyze them by "TwoWay
Analysis of Variance" in a "twobythree
factorial design."

You can
include three factors with
two,
three and four levels, and analyze them by "ThreeWay
Analysis of Variance" in a "twobythreebyfour
factorial design."
For the
sake simplicity, our example here involves only
two factors with
two
levels in each, analyzed by "TwoWay
Analysis of Variance" in a "twobytwo
factorial design."

After reanalysis of the same pregnancy
data, now let�s update the above table by adding a third column,
showing pregnancy rates for each embryologist and for each
catheter corrected (adjusted) for the effect of embryologist / catheter AND
showing P values (statistical differences) between the two embryologists
as well as between the two catheters .
See below for the updated table, and
let�s
see whether those corrected (adjusted) pregnancy rates, and the
corrected (adjusted) P values for the differences
between the two embryologists as well as between the two catheters,
have changed in comparison to the table in the previous statistical
analysis.
Embryologist (E)
OR Catheter (C) 
Pregnancy (%)
(arithmetic mean) 
Pregnancy (%)
(corrected for E/C) 
E1 (n=17) 
65 
P=0.03 
68 
P=0.006 
E2 (n=15) 
27 
18 
C1 (n=14) 
36 
P=0.26* 
25 
P=0.04 
C2 (n=18) 
56 
61 
Test Type 
Simple Tests
(ChiSquare or TTest) 
TwoWay
Analysis of Variance
(TwoWay ANOVA) 
*Note: P=0.26 by ChiSquare Test; P=0.28
by TTest.
As we see in the above table,
after
correcting (adjusting) the pregnancy data for the effect of the
embryologist / catheter (with TwoWay Analysis of Variance Test, or
TwoWay ANOVA),
each embryologist�s and catheter's �true� pregnancy rate has changed.
AS A RESULT:

The percentagepoint
difference between the two embryologists� �true�
pregnancy rates INCREASED FROM 38 (the difference between
65% & 27%) when using the simple ChiSquare or TTest
TO 50
(the difference between 68% & 18%) when using the more advanced /
comprehensive TwoWay ANOVA Test.
As a result, the more
advanced / comprehensive TwoWay ANOVA Test detected a more
significant difference between the two embryologists� �true� pregnancy
rates compared to the simple ChiSquare or TTest
(P=0.006
vs. P=0.03).

The percentagepoint
difference between the two catheters' �true�
pregnancy rates INCREASED FROM 20 (the difference between
36% & 56%) when using the simple ChiSquare or TTest
TO 36
(the difference between 25% & 61%) when using the more advanced /
comprehensive TwoWay ANOVA Test.
As a result, the more
advanced / comprehensive TwoWay ANOVA Test detected a significant difference between the two
catheters' �true� pregnancy
rates, which was NOT detected by the simple ChiSquare or TTest
(P=0.04
vs. P=0.26).
However, the above comparisons still
cannot be conclusive, because, we also know that
patient�s
age is an independent factor that adversely affects pregnancy rate,
and that not all patients are of the same age in this sample data.
As a matter of fact, a separate
statistical analysis of our sample data indicates that
patients' pregnancy in our sample is
inversely and significantly
correlated
to patient's age, with a correlation coefficient of �0.41
with the P value of
0.02.
So the pregnancy rate of each embryologist
and of each catheter in the above table (already corrected / adjusted
for the effect of embryologist / catheter) are still NOT the �true�
pregnancy rate of each embryologist and of each catheter, as they
are NOT corrected (adjusted) for the effect of patient's age.
The above necessitates the use of a more
advanced / comprehensive statistical analysis / test, which is
discussed
in the next step.
STEP 3
 THE �TRUER� PREGNANCY RATES CALCULATED AND COMPARED BY A MORE ADVANCED
STATISTICAL MODEL / ANALYSIS / TEST, TAKING INTO ACCOUNT PATIENT'S AGE
That is:

The �truer� pregnancy rate of each
embryologist corrected (adjusted) for the effect of the catheter and for the effect of the patient's
age; and

The �truer� pregnancy rate of each
catheter corrected (adjusted) for the effect of the embryologist
and for the effect of the patient's age.
We know that patient�s
age is an independent factor that adversely affects pregnancy rate,
and that not all patients are of the same age in this sample data.
As a matter of fact, a separate
statistical analysis of our sample data indicates that
patients' pregnancy in our sample is
inversely and significantly
correlated
to patient's age, with a correlation coefficient of �0.41
with the P value of
0.02.
So the pregnancy rate of each embryologist
and of each catheter in the above table (already corrected / adjusted
for the effect of embryologist / catheter) are still NOT the �true�
pregnancy rate of each embryologist and of each catheter, as they
are NOT corrected (adjusted) for the effect of patient's age.
So, in order to see the �truer�
pregnancy rate of each embryologist and of each catheter,
let�s
further correct (adjust) those (alreadycorrected / adjusted) pregnancy
rates for the effect of patient�s age.
For this purpose, we will repeat the same
statistical analysis as in the previous step (TwoWay ANOVA Test), but now
we will include, in the statistical
model / analysis / test, patient's age as additional factor
(as COVARIATE) (i.e., now also correcting / adjusting for
the effects of patient's age, i.e., taking into account the
effects of patient's age as well).
Accordingly, using
a statistical analysis
software that is more advanced / comprehensive than in the previous
step, we will reanalyze the same pregnancy data within and across
embryologists and catheters, and while correcting / adjusting
for the effects of patient's age as covariate at the same time, in a
SINGLE statistical model / analysis / test, by what is called now
ANALYSIS OF COVARIANCE Test.
After reanalysis of the same pregnancy
data, now let�s update the above table by adding a fourth column,
showing pregnancy rate for each embryologist and
catheter corrected
(adjusted) for the effect of embryologist / catheter as well as
corrected for the effect of patient's age AND showing P values
(statistical differences) between the two embryologists as well as
between the two catheters.
See below for the updated table, and
let�s
see whether those pregnancy rates, and the P values for the differences
between the two embryologists as well as between the two catheters,
have changed even further in comparison to the table in the previous
statistical analysis.
[ Of course, you also have the option to
do these analyses within the age groups; however, correcting /adjusting
for the age factor directly will still be useful. ]
Embryologist (E)
OR Catheter (C) 
Pregnancy (%)
(arithmetic mean) 
Pregnancy (%)
(corrected for E/C) 
Pregnancy (%)
(corrected for E/C & Age) 
E1 (n=17) 
65 
P=0.03 
68 
P=0.006 
64 
P=0.02 
E2 (n=15) 
27 
18 
23 
C1 (n=14) 
36 
P=0.26* 
25 
P=0.04 
28 
P=0.07 
C2 (n=18) 
56 
61 
59 
Test Type 
Simple Tests
(ChiSquare or TTest) 
TwoWay
Analysis of Variance
(TwoWay ANOVA) 
Analysis of Covariance 
*Note: P=0.26 by ChiSquare Test; P=0.28
by TTest.
As we see in the above table,
after
correcting (adjusting) the pregnancy data for the effect of the
embryologist / catheter as well as further for the effect of patient's age
(with Analysis of Covariance Test), each embryologist�s and
catheter's �true� pregnancy rate has FURTHER changed. AS A
RESULT:

The percentagepoint
difference between the two embryologists� �true�
pregnancy rates INCREASED FROM 38 (the difference between
65% & 27%) when using the simple ChiSquare or TTest
TO 41
(the difference between 64% & 23%) when using the more advanced /
comprehensive Analysis of Covariance Test.
As a result, the more
advanced / comprehensive Analysis of Covariance Test detected a
more significant difference between the two embryologists� �true�
pregnancy rates compared to the simple ChiSquare or TTest
(P=0.02
vs. P=0.03).

The percentagepoint
difference between the two catheters' �true�
pregnancy rates INCREASED FROM 20 (the difference between
36% & 56%) when using the simple ChiSquare or TTest
TO 31
(the difference between 28% & 59%) when using the more advanced /
comprehensive Analysis of Covariance Test.
As a result, the more
advanced / comprehensive Analysis of Covariance Test detected a
more significant difference between the two catheters' �true�
pregnancy rates compared to the simple ChiSquare or TTest
(P=0.07
vs. P=0.26).
So far, we have ONLY looked at, and
analyzed, the �overall� pregnancy rate of each embryologist and of
each catheter, corrected (adjusted) for the
effect of embryologist or catheter as well as for the effect of
patient's age.
However, we don�t know yet
what the
pregnancy rates of the four different combinations of
embryologists and catheters are, AND how they are
different from one another. In other words, we don�t know yet
about
the nature of the interaction between embryologists and catheters,
with regard to pregnancy rates.
The above necessitates the use of a more
advanced / comprehensive statistical analysis / test, which is
discussed
in the next step.
STEP 4
 PREGNANCY RATES OF COMBINATIONS OF
EMBRYOLOGISTS AND CATHETERS CALCULATED AND COMPARED
So far, we have ONLY looked at, and
analyzed, the �overall� pregnancy rate of each embryologist and of
each catheter, corrected (adjusted) for the
effect of embryologist or catheter as well as for the effect of
patient's age.
However, we don�t know yet
what the
pregnancy rates of the four different combinations of
embryologists and catheters are, AND how they are
different from one another. In other words, we don�t know yet
about
the nature of the interaction between embryologists and catheters,
with regard to pregnancy rates.
So now, we want to know
the
pregnancy rates of the four different combinations of
�embryologist x catheter.� We also want to know
whether those
pregnancy rates are statistically different from one another.
So we will break the pregnancy rate
of each embryologist down by catheters; and likewise, we will
break the pregnancy rate of each catheter down by embryologists
(i.e., break pregnancy results down by �embryologist x catheter�
combinations).
After we break down the
pregnancy data, we will reanalyze the pregnancy
data:

First, WITHOUT correcting
(adjusting) pregnancy rates for the effect of patient's AGE (analyzing by TwoWay ANOVA Test),
and

Then, by
CORRECTING (adjusting) them for the effect of patient's AGE (analyzing by ANALYSIS OF COVARIANCE Test; with patients� age as
covariate).
After reanalysis of the pregnancy
data by TwoWay ANOVA Test (WITHOUT correcting / adjusting pregnancy rates
for the effect of
patient's AGE), the following table shows pregnancy rate
(arithmetic mean) for
each combination of �embryologist x catheter� AND
shows P values (statistical
differences) among those combinations.
Levels of Factors 

Combinations of Levels
(arithmetic means) 

Comparisons among
Combinations 
Difference
(P Value) 
Embryologist 1 (E1) 

E1C1 = 40% 
n=10 

E1C1 vs. E1C2 (40% vs. 100%) 
0.01 
Embryologist 2 (E2) 

E1C2 = 100% 
n= 7 

E1C1 vs. E2C1 (40% vs. 25%) 
0.57 
Catheter 1 (C1) 

E2C1 = 25% 
n= 4 

E1C1 vs. E2C2 (40% vs. 27%) 
0.51 
Catheter 2 (C2) 

E2C2 = 27% 
n=11 

E1C2 vs. E2C1 (100% vs. 25%) 
0.01 




E1C2 vs. E2C2 (100% vs. 27%) 
0.002 




E2C1 vs. E2C2 (25% vs. 27%) 
0.93 
As we see in the above table:
The highest pregnancy rate
resulted from the combination of E1C2 (100%), whereas
the lowest
pregnancy rates resulted from the combinations of E2C1 (25%)
and E2C2 (27%).
Statistically significant
differences existed between combinations of:
E1C1 vs.
E1C2 (40% vs.
100%)
(P=0.01)
E1C2 vs.
E2C1 (100% vs. 25%)
(P=0.01)
E1C2 vs.
E2C2 (100% vs. 27%)
(P=0.002)
Although not statistically
significant:
There was a
15
percentagepoint difference between E1C1
and E2C1 (40% vs. 25%) (P=0.57)
There was a
13
percentagepoint difference between E1C1
and E2C2 (40% vs. 27%) (P=0.51)
After reanalysis of the same pregnancy
data by ANALYSIS OF COVARIANCE Test (by correcting /
adjusting pregnancy rates for the effect of patient's AGE),
the following table shows pregnancy rate for each combination of
�embryologist x catheter� AND shows P values (statistical differences) among
those combinations.
Levels of
Factors 

Combinations of Levels
(corrected for age) 

Comparisons among
Combinations 
Difference
(P Value) 
Embryologist 1 (E1) 

E1C1 = 39% 
n=10 

E1C1 vs. E1C2 (39% vs. 94%) 
0.01 
Embryologist 2 (E2) 

E1C2 = 94% 
n= 7 

E1C1 vs. E2C1 (39% vs. 30%) 
0.71 
Catheter 1 (C1) 

E2C1 = 30% 
n= 4 

E1C1 vs. E2C2 (39% vs. 30%) 
0.61 
Catheter 2 (C2) 

E2C2 = 30% 
n=11 

E1C2 vs. E2C1 (94% vs. 30%) 
0.03 




E1C2 vs. E2C2 (94% vs. 30%) 
0.005 




E2C1 vs. E2C2 (30% vs. 30%) 
1.00 
As we see in the above table:
The highest pregnancy rate
resulted from the combination of E1C2 (94%), whereas the lowest
pregnancy rates resulted from the combinations of E2C1 (30%)
and E2C2 (30%).
Statistically significant
differences existed between combinations of:
E1C1 vs.
E1C2 (39% vs.
94%) (P=0.01)
E1C2 vs.
E2C1 (94% vs. 30%) (P=0.03)
E1C2 vs.
E2C2 (94% vs. 30%)
(P=0.005)
Although not statistically
significant:
There was a
9
percentagepoint difference between E1C1
and E2C1 (39%
vs. 30%) (P=0.71)
There was a
9
percentagepoint difference between E1C1
and E2C2 (39%
vs. 30%) (P=0.61)
CONCLUSION
Clinical outcomes
(such as pregnancy, live birth and fertilization rates) in ART / IVF programs
are the results of NOT ONLY ONE factor but the results of
MULTIPLE factors that are at work simultaneously, i.e.,
those outcomes are simultaneously affected by multiple factors.
Therefore, it is NOT sufficient to
visually compare, or
even statistically analyze by simple
statistical tests, the
arithmetic means of factors separately, i.e., by taking into account the effect of
ONLY ONE
factor at a time.
The above examples with a sample pregnancy
data illustrates why it is important to analyze your clinical
outcomes (such as pregnancy, live birth and fertilization rates)
in a SINGLE statistical model / analysis / test, by
correcting (adjusting) those
arithmetic means for the
effects of ALL the factors involved, i.e., by
taking into
account the effects of ALL the factors involved, AS OPPOSED TO
simply calculating arithmetic means and
simply comparing those arithmetic means by
simple statistical tests.
This can only be
achieved with
advanced / comprehensive statistical knowledge, expertise and experience
as well as with the use of advanced / comprehensive tests of
statistical analyses, which is what the STATS OF THE ART
is here to help you with.
