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Hello again, welcome to the course on Biostatistics
and Design of Experiments. In this class,
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we are going to briefly touch about various
statistical test, that one should understand
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and how to go about using them. So, there
are different types of statistical test, some
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are used for comparing mean of a data and
some are used for comparing variance of data
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and some are used for comparing ratios of
data and so on. So, we will talk about that.
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Before going into that, let me recall again
how you go about doing this hypothesis, you
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have something called the null hypothesis,
which is no difference or state as score.
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So, imagine I am comparing the IQ's of 2 different
classes or class of students, then I would
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say the null hypothesis will be there is no
difference in the IQ of students in class
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a and class b. Then we also have the alternate
hypothesis, if I am going to say yes there
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is possible difference then I would say the
alternate hypothesis will be the, IQ average
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of class a is different from IQ average of
b. Then you can also have another situation
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where instead of just saying different, the
IQ average of class a could be better than
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the IQ average of class b. That is we are
comparing only the better part of it or the
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IQ average of a class a is worst than that
of that class b, then we are comparing only
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the worst part of it. So we have those 3 situations,
no difference is the null hypothesis, there
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is a difference is the alternate hypothesis
and that is called a two-tailed comparison
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that is a different, it could be greater or
worst. Then we are comparing only 1 side of
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it greater side of it or worst side of it,
then that is called one-tailed test.
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So we decide on the hypothesis with the tail
and then we also decide on the p value, that
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is am I going to test against 95 % confidence
or am I going to test against 99 % confidence.
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So p of 0.05 indicates 95 % and p of 0.01
indicates 99 %. Once I decide on all these,
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that means I decide on the type of hypothesis,
I decide on whether it is a single tailed
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or a two-tailed test, then I decide on the
p I want to look at, then I calculate using
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different test something called t, and then
from there I will calculate may be the p value
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and then I will say whether the p value which
I have calculated is less than 0.05 or it
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is more. So if it is less than 0.05 obviously,
there is a difference so obviously, I cannot
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accept null hypothesis so I have to reject
the null hypothesis that means, I have to
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accept the alternate hypothesis. Now if the
p value is greater than 0.05 for a 95 % confidence
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obviously, there is no reason for me to reject
the null hypothesis.
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So, imagine I am testing 2 drugs in the market
which affects this sleeping pattern. So, the
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null hypothesis could be, there is no difference
between the 2 drugs, the alternate there is
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a difference at a 95 % confidence interval.
Now what type of hypothesis equations we will
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put first, the two-tailed under a single tailed.
For a two-tailed your alternate will be mu
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a is not equal to mu b, your null hypothesis
will be mu a equal to mu b, that is the two-tail.
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For single tailed test null hypothesis could
be mu a is equal to mu b, alternate hypothesis
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could be mu a is greater than mu b, that means
drug a has an increased effect than drug b
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or mu a could be less than mu b, that mean
drug a has decreased effect than drug b that
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is a lower tailed and upper tailed. Now there
is something called error, there are 2 types
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of error.
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There is something called type 1 error and
there is something called type 2 error. What
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is type 1 error? We are rejecting null hypothesis
h naught, when h naught is true that means,
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we are rejecting the null hypothesis where
as in reality the null hypothesis is true
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that is called type 1 error. Then you have
the type 2 error, we fail to reject the null
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hypothesis when the alternate hypothesis is
true. Whereas, instead of accepting alternate
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hypothesis we do not accept alternate, we
accept null hypothesis. We have 2 situations,
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one is called the do not reject null hypothesis,
null hypothesis is true null alternate hypothesis
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is true. So here this is the correct, null
hypothesis is true you do not reject null
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hypothesis, similarly alternate hypothesis
need to accept reject null hypothesis so this
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is the correct situation, whereas in some
times although we need to, null hypothesis
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is true we end up rejecting the null hypothesis
that is called the type 1 error or alpha error.
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Whereas in the other situation we do not reject
the null hypothesis whereas, we have H 1 is
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true that is called the type 2 error and that
is called the beta error.
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So type 1 error, is generally the probability
95 % or 99 % or 90 % which we make use of
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in our statistical calculation. That is where
if you take a larger alpha then obviously
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you are very sure that you will not reject
the null hypothesis, where when H naught is
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true but then your significance level also
gets affected by that. So this table is very
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important when H naught is true, we reject
the H naught that is called the type 1 error.
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When H 1 is true, we fail to reject the null
hypothesis that is called the type 2 error
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or beta error. So type 1 error is called the
alpha error and type 2 is called the beta
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error. So we need to always, sort of balance
between the alpha and the beta error and generally
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we give more importance to the alpha error
actually.
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As you know we have a continuous data and
alternatively we have the discrete data. Discrete
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data we use equations like Binomial or we
use Poisson and so on. So when we have the
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continuous data there are many situations,
we can have one sample that means I know the
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population details, I take a small sample
and I am comparing with the population that
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is called a one sample. For example, I know
the 12th standard average from a particular
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school is 95 %, so I take 10 students in that
school and calculate their 12th standard average.
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I may get some x bar, now I want to know whether
this x bar is related to the 95 % school average,
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which is more like a population or is it very
far away. So here we are collecting only 1
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set of samples and then comparing with the
population that is called a one sample test
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or one sample t-test. Here we are comparing
the mean of the sample with the population
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mean. Now what is Two samples, suppose I am
comparing performance of drug A and performance
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of drug B and trying to tell there is no statistically
significant difference between their performance
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or there is significance difference that means,
I am comparing the means of 2 samples that
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is why it is called the two sample t-test.
Now again you can have Multiple samples, I
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may be comparing drug A, B, C, D so, I could
be having lots of different means that is
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called a Multiple samples actually.
So there are many, many ways by which one
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could analyze these data one is called the
Study Stable or Run Charts, then we can look
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at the Shape we can create a histogram and
see whether it looks normally distributor,
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then we can look at the Data whether it is
and so on actually. And then, there is something
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called Chi Squared Test and then there is
something called t test, t-tests are generally
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meant for comparing means, chi square test
are generally meant for comparing ratios.
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And there is something called F test, which
is generally meant for comparing variances
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or spreads actually.
In the t test we have 3 types of t test, one
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sample t-test that means, I take only 1 sample
and then compare it with the population or
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I can have a two sample t-test, where I am
comparing 2 sets of samples or I may be comparing
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paired t-test that means, there is a relationship
between the sample items with a and sample
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items in b. So when I am comparing variances
there is something called F test or if I am
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having multiple samples and I am comparing
variances then this is something called ANOVA
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analysis of variance test and if I am comparing
the spread there is something called homogeneity
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of variance. There are different types of
test that are possible and we are going to
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spend lot of time on each one of them actually.
So we can have only one sample collected,
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comparing it with the population or we could
be having two samples collected, comparing
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with the population or we could have multiple
samples collected, comparing with the population.
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Then if I am comparing means then there is
something called t test, the one sample t-test,
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the two sample t-test, paired t-test. If I
am comparing variances there is something
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called f test, if I am comparing a variances
of a large number of data sets or samples
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this analysis of variance. If I am looking
at spread of the data I can use something
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called Homogeneity of variance and so on.
So large number of tests are possible we will
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talk about each one of them in detail and
we are going to spend with some examples also,
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so do not worry about it.
Refer Slide Time:
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.
So if you are comparing means, there is something
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called t test. We can get average of a sample
one and then I get a mean of sample two and
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then I am going to find out whether both the
means come from the same population or each
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of the mean come from 2 different population
that is called a t-test, t-tests are quite
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robust even for non normal data. Generally
we can say the standard deviations have to
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be similar, but there can be some difference
in the standard deviation also but still,
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t test is good. In t-test we have 3 types
1 sample t-test, 2 sample t-test and paired
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t test.
So one sample t-test you are taking a sample
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out and then you are getting the mean and
the variance of that sample and you are comparing
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it with the mean of the population like I
gave you some examples actually like, I take
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10 students from a class and then get their
mean average, class average, remarks average
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then I compare it with the school average
and try to tell whether these averages are
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far away from the school average or they fall
into the same population, I can do that sort
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of study. I can collect IQ of 10 students
in a university and then try to say whether
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the mean IQ falls within the university average
IQ or it falls outside that, so that is one
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sample t-test here we are taking only 1 sample.
Two sample t-test if I am going to have 2
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sets of samples, I am taking 10 students from
1 university 10 students from another university
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and getting their getting their IQ's and then
comparing their IQ's and trying to say whether
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the IQ's are statistically different or there
is no statistically significant difference
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between these 2 IQ's. So that is called two
sample t-test because I am using 2 sets of
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samples.
Paired t-test, if you are pairing 2 sets of
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data then difference in result should be 0
for example, I take 10 cats and I test a drug
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A on the 10 cats and look at their outcome
then on the same 10 cats I give drug B and
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look at the outcome. So the different should
be 0, if there is no difference between in
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the drug A and drug B. If there is a statistically
significant difference away from 0 then I
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can say yes, drug A is different from drug
B because, I have used the same volunteer
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cats and I am testing drug a seeing some performance
change, then I am testing drug B seeing some
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performance change, if drug A and drug B have
to be same then the performance change we
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observed, the difference in performance change
we observed should be equal to 0 that is called
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the paired t-test. So we are going to look
at examples of the each one of them. So you
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do not worry about that.
So, interestingly all these tests are looking
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at means that means averages. Average or mean
of the samples which you are taking it out,
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whether it is one sample t-test, 2 sample.
Now you may ask the question suppose instead
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of 2 samples if I have many more samples what
will I do of course I can do, take 2 sets
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of samples 1 at a time and do a two sample
t-test, but there is another approach which
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is much faster that is called ANOVA analysis
of variance, it is called the 1 way ANOVA
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we will talk about that later in the course.
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Now, if you are comparing variances that means,
you are comparing these standard error. So
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here we are comparing variances, this is generally
valid both for normal and non normal. So the
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H naught will be sigma 1 square is equal to
sigma 2 square and so on, we cannot reject
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h naught when p is greater than 0.05. The
alternate could be sigma a square is different
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from sigma b square, p is less than 0.05 we
have to reject H naught and accept H a. So
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you see the tests are there, for Comparing
Variances. So in the t-test we are comparing
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means, here we are comparing variances.
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One test that is there is called F test, that
means I am comparing the variation from here
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sample 1 and a sample 2. So, the H naught
could be sigma 1 square is equal to sigma
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2 square that means variances are same or
the alternate could be sigma 1 square is different
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from sigma 2 square. So we calculate the f
ratio which is given by s 1 square by s 2
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square, s 1 and the s 2 are the sample standard
deviation, so s 1 square is the variance.
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There is a table called F table, for a 95
% or 99 % you will get F value and if the
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table F value is greater than the F you calculate,
then you accept H naught and if the table
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F value is less than the F you calculate,
you reject H naught and accept H a. For different
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degrees of freedom, the degrees of freedom
for data set 1 is n 1 minus 1, if you have
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used n data sets and degrees of freedom for
data set 2 is n 2 minus 1, if you have collected
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n 2 samples that is called the F test.
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And then you also have ANOVA, when you are
comparing a large number of data sets in the
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previous F test you have only 2 data sets,
1 and 2. So your saying sigma 1 square is
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equal to sigma 2 square, alternative sigma
1 square is not equal to sigma 2 square. So
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ANOVA is very, very powerful because we can
collect a large number of data sets I am comparing
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the IQ's of university A, university B, university
C, university D and trying to find whether
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there is a statistical significant difference
or not. I am comparing drug A, B, C, D in
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clinical trials, I want to perform analysis
to find out whether there is a statistically
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significant difference, then I use ANOVA here.
The variances of the samples are approximately
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equal, the response within any given level
are normally distributed these are the assumptions.
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So H naught will be all these are same, whereas
H a is at least 1 variance is different then
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you get a p value less than 0.05 ok.
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We have the t test, different types of t tests
which are very powerful for comparing mean,
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then we have the test for comparing variances
like F test and ANOVA and then for ratios
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we have something called chi squared test
which we will talk about later. So if there
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is something called the power of the test
or if H 1 is true, so that the distribution
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of X is specified by H 1 then the probability
of rejecting the H0 is the power of the test
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for distribution. H 1 is true, so you have
2 situations you do not reject H0 that is
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beta error where as you reject H0. If H1 is
true and you reject H0 that is called the
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power of the test for that distribution. So
you have 2 types of error, alpha error where
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you should not reject H0 but you end up rejecting
H0. Whereas the beta error you have to reject
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H0 but you do not reject H0 and that is also
called the power of the test, the beta error
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is also called the power of the test. So these
2 terms are very, very important when you
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are deciding on the alpha error and the beta
error.
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Let us get into problems, the first problem
is called the 1 sample t-test one-sided. The
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average size of barnacle shells is 25 mm,
you know what is barnacle right? It is a marine
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organism, it is got a shell, it gets attached
using glue to hard surfaces and then it feeds
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on it. So it is got a shell and average size
is 25 mm. Now we have collected 10 barnacles
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in South India and we got their sizes, these
are the sizes. Now are the South Indian barnacles
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of smaller size? That is the question or they
are of greater size? So we take 95 % confidence.
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So this is the South Indian barnacles, the
average comes out to be 24.4 but the population
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average is 25 mm, this is the statement. Now
I want to know whether the this 24.4 is statistically
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significantly smaller or it comes from the
same population at a p value of 0.05 or a
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95 % confidence. So how do we that?
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Simple, H0 is equal to mu equal to mu naught,
that is mu naught is your original mean of
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the population, now they are same but alternate
is mu is less than mu naught that means, you
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want to know whether the average size of the
barnacle shell from South India is less than
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this 25. So what do you do, you calculate
t, if you remember this equation we had this
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mu is equal to x bar plus or minus t into
s by square root of n. We rearrange that to
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get this t value, x bar minus mu naught divided
by s square root of n. It is called degrees
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of freedom n minus 1, if t which we calculate
is less than the table t. So there is a table
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t for different degrees of freedom accept
H naught, if t is greater than t then the
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table t then reject H naught this is called
the, of course we are using 1 tail test.
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So, what do we do? We get the average which
is 24.4; we get the standard deviation which
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is 2.59. So the t calculated we can use here
24.4 minus 25 divided by s divided by square
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root of n. So we get the t calculated and
minus 0.73 for 9 degrees of freedom t table
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there is a t table here.
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I want you to look here for 95 % this is a
two-tail test and the top 1 is the single
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tail test, in your problem we are talking
about single tail test because we want to
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know whether this South Indian barnacles are
of smaller size. So for 9 degrees of freedom
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go like this, you go like this and read out
0.05, here you get for 9 degrees of freedom
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1.833. So t table is 1.833, the t calculated
is minus 0.737. So t calculated is less than
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the t table so there is no reason for you
to reject h naught at this condition. So what
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you can say is the South Indian barnacle,
there is no statistical reason for saying,
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the South Indian barnacles are of smaller
size. So South Indian barnacles come from
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the same population of 25 mm. In order to
get your confidence limit on the mean which
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we calculated from the sample, as you know
this equation mu is equal to x bar plus or
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minus t df s by square root of n the s by
square root of n is called the standard error,
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right? I talked about this long time back.
Now you need to know that degree t value,
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now for these you have 10 data sets obviously
the degrees of freedom is 9 you get t as 2.26
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and the mean of this sample is 24.4. So 24.4
plus or minus 2.26 the standard deviation
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is 2.59 divided by square root of 10, which
is equal to 24.4 plus or minus 1.85. So the
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confidence limit 95 % confidence limit for
the this mean is 22.24 to 26.25. One important
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point you need to remember is the t which
you calculated will be for two-tailed both
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the sides because we are talking about plus
or minus that is why we get 2.26. 4 mm as
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the mean of the sample with the standard deviation
of 2.59 and you want to know whether the mu
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which you calculate is equal to mu 0 or the
alternate mu is less than mu 0, so for mu
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is equal to mu 0. So what do you do, we calculate
the t you know t from this equation if you
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adjust it is x bar minus mu divided by s square
root of n. So you get minus 0.7377 for 9 degrees
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of freedom we have from the table, I showed
you t table that top one is for single tailed,
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the bottom one is for two-tailed so for 5
and p of 0.05, 9 degrees of freedom you get
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1.833. So obviously, the t which you have
calculated is much less so there is no reason
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for you to reject the null hypothesis.
For 95 % confidence we say 24.4 plus or minus
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1.96 s is equal to given here 2.59 divided
by square root of n. So you get the confidence
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limit for the mean a 22.82 to 26 so obviously
your 25 falls within that that is why you
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are not able to reject the null hypothesis
at this condition. So this table is very important
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as we can see this table gives you for different
degrees of freedom the top 1 for single tail,
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the bottom 1 is for two-tail, So if I am interested
in a single tail I will use 95 % this column,
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if I am interested in two-tail 95 % I use
this column. As you go down, down, down as
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you can see for infinite degrees of freedom
we get 1.96. So, as I said two-tailed 0.05
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means a single tailed 0.025 because when you
say 95 % two-tail the tails are divided half
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on both the sides that is why you get 0.05
by 2 which is 0.025 here. Do you understand?
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Do you understand the logic of this particular
table, this table is very important when you
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are calculating t test, when you are calculating
t based on means, whether it is 1 sample t-test
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the example which we saw or later on we are
going to look at 2 sample t-test, paired t
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tests and so on. So this table is very important.
So that you calculate t from the equation
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and then you compare with the t in the table
for a different degrees of freedom, and then
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you say whether the t calculated is greater
than t table, if it is greater than you have
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you can reject the null hypothesis, but if
it is less we calculated less than the t table
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we cannot reject the null hypothesis.
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Now we can also use the particular software
which I mentioned about the GraphPad software,
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which can also calculate the t value, given
the probability value you can see for a 1
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tail it is 1.833. So you compare with the
t which you calculated, which is minus 0.737.
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So there is no reason for you to reject the
null hypothesis, but if I take t value the
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minus 0.737. And then the same software can
be used to calculate the p value, the p comes
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out to be 0.4799. So obviously, you can say
it is not statistically significant at all.
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It should have been 0.05 or less than only
we can call it a statistically significant
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difference. So it is very useful for calculating
this.
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00:29:38,029 --> 00:29:48,229
Or we can also use 1 sample t-test results,
again the GraphPad software can do this and
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again the results as you can see it gives
you a p value of this the difference is considered
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to be not statistically significant. The actual
mean of the sample is 24.4, the hypothetical
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mean you want to reach is 25.0 and so on actually.
So we will see how to do this, it is quite
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simple. We have this data set; I will show
you how to do this.
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So here we have the continuous data here we
need to use this particular thing, here you
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have the one sample t-test as you can see
here one sample t-test you can say continue.
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So we can even enter data like this or we
can copy paste like this. So we can enter
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the data like that also. So I am comparing
it with respect to 25, 25 is my population.
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Now I want to know the sample which is equal
to 22 and so on. So I can copy this, copy
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this and I go to my I paste it here
or if it does not get pasted. So obviously
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we can write, so we can write 22, 23, 22,
25, 28. So we say it 22, then we say 23, then
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22, 25, 28, 25, 28, 27, then again we put
28, then again we put 25, then we put 23.
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So we have 10 data points we can put this
here and then we can say you calculate now.
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So here we put 25 is the global average which
you are interested in so we can say you calculate
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now. So by conventional the difference is
considered to be not statistically significance,
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because the p value is coming out to be 0.4826.
If we get p value less than, because the t
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value is 0.732 where as you want 1.833. So
obviously it is not statistically significant
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different. So you can use the GraphPad software
also to calculate, we can put in your data
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and you can use the GraphPad software also
to perform this type of calculation.
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It is quite useful software and this problem
is quite simple. So you have the population
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mean and you have the sample. So from the
sample you calculate the mean, from this sample
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you calculate the standard deviation and then
you calculate the standard error which is
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given is by s by square root of n, and then
you know you can get the t value and then
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for 9 degrees of freedom you make use of this
particular table. And for a single tailed
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test you use this, you go like this for 9
degrees of freedom you get 1.833, whereas
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t calculated is 0.7377. So obviously, we have
no reason for rejecting the null hypothesis
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that is what it is. So I showed you how to
calculate from the GraphPad software also
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actually. We will continue more on this one
sample t-test as we go along.
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Thank you very much for your time.