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Instructor: Hello students and welcome
to Biostat ER example nine, chapter seven.
In this example, we will learn how
to perform hypothesis testing
by using multiple sample independent
continuous outcome data.
And for this purpose we will be using
ANOVA, or analysis of variance.
And for this example, I'm using problem
14 from our textbook
and as always, I'll read the problem first.
Suppose a hypertension trial
is mounted and 18 participants
are randomly assigned
to one of the comparison treatments.
Each participant takes
the assigned medication and his
or her systolic blood pressure
is recorded after six months,
on the assigned treatment.
The data are shown in table 7-58.
Is there a significant difference
in mean systolic blood
pressure among treatments?
Run the appropriate test at alpha equal to 0.05.
Now as a first step,
we are going to set up the hypothesis
and determine level of significance.
We have a continuous outcome
which is systolic blood
pressure measured in millimeter per mercury.
We are comparing the difference
in means between three groups
so we have multiple sample independent
continuous outcome data.
This is an ANOVA test
and the appropriate hypothesis
for null will be all the means are equal
and for the alternative, means are not equal.
And the level of significance here is 0.05.
In step two we will select
the appropriate test statistic.
For our type of data,
the appropriate test statistics is the F statistics.
In step three, we are going to set up the decision.
Our level of significance is 0.05
and our degrees of freedom one is two
because our small K, our number
of groups is three, we deduct one
from that, we get two.
Our degrees of freedom two is going to be 15
because our big N is equal to 18.
And then we deduct three, which is the number
of groups, we get 15.
So from table four in the appendix
we get our F critical value
and hence we will reject
null hypothesis if our F calculated
is greater than or equal to 3.68.
Now in step four
we will compute the test statistic.
The data here are summarized in the following table.
Basically the group means are provided
and the group sample sizes are also listed again.
If we pool N equal to 18,
basically the entire number
of observations, the overall mean
or the grand mean will be 130.
Now given that the sample size is equal
N equal to six for each group,
we can add the group means
and divide it by 3
and calculate the grand mean or overall mean.
And I've shown the calculation here below.
Now, if not equal sample size,
we would add all the values
which is provided
in table 7-58 in our textbook
and divide it by the big N
to calculate the overall mean or grand mean.
Now we will compute the SSB
and the formula for this is given here.
And SSB is basically the sum
of the difference in the group mean
and overall mean for all groups.
So basically the formula here requires us to insert
the sample size from each group
and that is six and I have inserted that here.
Then it requires us to insert X-barj,
which is the mean from each group.
So that is 122.7 for group one,
146.2 for group two and 121.1 for group three.
And I have inserted that here as well.
And then the final thing we have
to insert here is the X bar,
which is the overall mean or grand mean.
And that is 130
and I have inserted that here as well.
So after that, basically all we have
to do is do the algebraic process.
And I have shown you that here
in a step by step fashion
and you should be able to follow it very
easily and I strongly encourage you
to perform the calculation yourself.
It is very important
that you understand the process
and you also want to see
if you're getting the same values
or numbers that I am providing here.
Now in this example, SS total is not
provided to us, hence we must calculate sums
of square within or,
sums of squares error, separately.
And that is what we are going to do next.
For this calculation,
we will use the individual values, the group mean
for each group and then add it for
over all groups.
So the first thing we have to do is go
and set up the table here.
And this is for the first group,
which is the standard
or the group that received this standard treatment.
And here we have inserted the individual values here,
124,
111,
133,
125,
128,
115.
Here in the middle column,
what we are doing is deducting the group mean from each
of the individual values.
And this should come across pretty similar
because we have done this before
when we perform the calculation for standard deviation,
it's a very similar calculation.
So now we are going to do those subtractions
and we get values like 1.3,
-: 11.7,
10.3,
2.3,
5.3,
and negative 7.7.
So now when we add all these values,
the sum should be equal to zero,
but it is not, it is very close,
it is -0.2.
And why is that?
That's because we have decimals here.
Now in the third column
what we are doing is we are taking
the whole square of each
of these values and finally we are adding them up
to get our final value
for this group, which is 337.34.
We are going to do the exact same process
for the placebo group
or the group that received the placebo.
We are going to put the individual values here.
Then we are going to subtract the group mean,
and then we are going to do the summation
to see if we get a zero or not.
Again, we are not going to get an exact zero
because of decimals.
Then we are going to square
root the numbers in the third
column and finally we are going to add them up.
So again, it's an algebraic process,
but I strongly encourage you to do it step by step
to make sure you are getting the values
that I'm providing here.
Now is the last group,
which is the new treatment.
Again, here I have inserted all the values basically
that were provided to us.
And then in the middle column
I have deducted the group mean from each
of the individual values.
So that is again provided here.
And then in the third or the last column,
I have performed the whole square
and then I have added those values
and here I have 124.
Now I do wanna bring it to your attention here that
because in this group we did not have any decimals,
when we did sum up the values for the middle column,
we actually got a perfect zero
because we did not have any decimals.
Now we are going to add all these values up
as you can see here,
and that will give us the final SSW
that we need to move forward with our calculation.
So now we are going to construct the ANOVA table.
Here we have the sums of squares as well
as the degrees of freedom and the mean squares.
And again, I have inserted all these values here
that we obtained so far from our calculation,
2380.2
846.2, degrees of freedom two and 15.
And now we have to do a division.
So 2380.2 will be divided by two
and that will give us 1190.1
and we have to take 846.2 divided by 15
and that will give us 56.4.
So finally we have to divide 1190.1 by 56.4,
and from that, our F value,
our F calculated will be 21.1.
So now as part of our last step, we are going
to draw a conclusion.
So we have calculated
that F is greater than 3.68.
So we have a statistically significant evidence
to reject the null hypothesis
and conclude that there
is a significant difference
in the mean systolic blood pressure
between the treatment groups
with a 5% level of significance.
Now, given that we have rejected
the null hypothesis,
we need to move forward in terms
of determining which groups are different
from each other, okay?
Therefore, we will perform a post-hoc analysis.
And for that purpose, I created a new video
and I will see you in that next video.
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