Instructor: Dear students, welcome to Biostat
ER example eight, chapter seven.
In this example, we will learn
how to perform hypothesis testing
by using multi-sample independent continuous outcome data.
And for this purpose, we will be using ANOVA,
or analysis of variance.
For this example, we will use problem 28 from our textbook,
and first, I'm going to read the problem.
Use the data shown in problem 25
and test if there is a significant difference
in mean age among the three groups.
Hint, SStotal is equal to 2,893.
Use a 5% level of significance.
So for your convenience,
I have copied and pasted the table here.
And before I get started,
I want to bring your attention to a couple of items.
So, why are we using ANOVA?
Well, we are using ANOVA because here, we have three groups,
therefore we cannot use T-test or Z-test.
Also here, our outcome age is measured
on a continuous scale, hence we cannot use chi-squared.
Therefore, we will be using ANOVA for this example.
So for step one, we are going to set up the hypotheses
and we are going to determine the level of significance.
The null hypothesis here states
that all the means are equal,
so mu one is equal to mu two is equal to mu three,
and the alternative states that the means are not equal.
The alpha here is provided to us, which is 0.05.
In step two, we are going to select
the appropriate test statistics, and that is F statistics.
And to determine the F statistics, we need to divide
MSB by MSW, and MSB stands for means squared between,
and MSW stands for means squared within.
Now, in step three, we are going to set up the decision.
We have three groups, so the k is equal to three.
Now, our big N is 120
because each group has 40 participants,
so when we add it up, we get 120.
The degrees of freedom one is k minus 1,
and the k here is again three,
so 3 - 1, we get two.
And for degrees of freedom two,
we have to subtract the k from the big N,
and the big N here is 120, and the k is three,
so when we subtract three from 120, we get 117.
Now, the F-table on page 351,
which is available at the back of our book,
does not include 117,
hence, we will use 100 as the DF2 value,
and based on that, our F critical value will be 3.09.
Hence, we will reject the null
if our F calculated is greater than or equal to 3.09.
Now in step four, we have to compute the test statistics.
To organize our computations,
we will complete the ANOVA table,
and for that purpose, first we need
to calculate the overall, our grand mean,
which is X bar, and for that purpose we are going to add
the three group means and divide it by three.
Now for this problem, we can calculate the grand mean,
or overall mean by adding the three means
and dividing by three because we have equal number
of participants in each group, and that is 40.
So now we can start to calculate the SSB,
which is the sums of square between,
and for that, the formula is given here,
which is the summation sign, n sub j,
open parentheses X bar sub j
minus X bar, closed parentheses, and the whole square.
So the X bar sub j here
is the group mean for each group.
And that is 75.2 for group one, 75.6 for group two,
and 74.7 for group three.
And the X bar is 75.2, which is our grand.
So once we insert all the values
and we perform the calculation,
our SSB is equal to 16.4.
So the total, our SStotal,
is equal to 2,893, which is provided to us.
So by subtracting SSB from SStotal,
we can ascertain that SSW,
or SSE, is going to be 2876.6.
Now, we need to insert these values in our ANOVA table
and start calculating the mean squares
by dividing sums of squares by degrees of freedom.
So first, we are going to take 16.4
and divide that by two and we will get 8.2,
then we are going to divide 2876.6 by 117,
and we will get 24.6.
Now, to calculate the final F value,
we are going to divide 8.2 by 24.6,
and that will give us 0.33.
So now in step five, we have to draw our conclusion.
So here, we will fail to reject null
because 0.33 is less than 3.09.
So we do not have statistically significant evidence
at alpha value of 0.05 to show that
there is a difference in mean age among the three groups.
I hope this was helpful,
and I would like to thank you for your time and attention,
and please feel free to contact me
if you have any questions.