ANOVA and Test of Independence Video Lecture | What is Data Analytics? - Data & Analytics

hello and welcome to our lecture on
ANOVA
also known as analysis of variance and
the chi-square test of Independence the
toi is test of Independence the
chi-square test of Independence we
present this is the last lecture in the
series on inferential statistics but
it's only fair to say that while some
set of techniques are presented as a
part of inferential statistics most of
the techniques in in statistics use
statistical inference in some way or the
other so for instance after this lecture
we will be we will be talking about
regression and so on and there is a
significant component there which is
associated with statistical inference
but in any case this lecture we're going
to talk about the analysis of variance
in the test of independence okay so so
far statistical inference was confined
to input variables that could take up
two possible values and that was the two
sample cases right the two sample tests
where you would compare two different
samples and the variable of interest
would be for instance either male or
female so we came up with an example
where we said the heights of 10 standard
boys in public schools versus the
heights of 10 standard girls in public
school so if you were interested in
seeing if the average height of a boy in
10 standard in public schools in India
is equal to the high average height of a
girl in 10th standard in public schools
in India here the output essentially can
think of there are two variables that
are involved and you know that you can
think of the output variable as one of
height and your input variable you can
think of as gender so you have two
different sets of data and you're
comparing them now there are also these
cases where it doesn't make sense to
think of these as input and output
variables that's just confusing right so
these are the single sample test cases
where you just have one variable height
and you've defined all the other
parameters around it
and so it could be something like I've
already defined that I am interested in
studying the average height of boys in
10th standard in public schools in India
and I'm interested in seeing if that
average height is less than a hundred
and you know 20 centimeters or something
some some such number there there's no
concept of it you know there's just one
variable which is height and that's it
and you're comparing it to some number
that you had in your head so that's
that's essentially summarizing the
two-sample case in the single sample
case for the first time with something
like the anova we take on a case where
this variable which is essentially like
this input variable can take on two or
more states so essentially you'll see it
takes on three or more states and that's
that's the real differentiator about
ANOVA so just to kind of give you a
little bit more color on this a standard
standard application of ANOVA would be
something like the following where
you're still dealing with an output
variable that's typically quantitative
and continuous like for instance like
height right but your input variable
need not have just two states it can
have three or more states and you're
still probably interested in comparing
their averages so extending the example
that we saw in the two-sample t-tests
case if you wanted to ask a question
such as is the mean height of ten
standard boys in is the average height
of ten standard boys in tamilnadu
equal to the average height of ten
standard boys in public schools and
maharashtra is it equal to the average
height in karnataka so you were not just
interested in comparing two sets of
samples but you had multiple sets of
samples and typically you're the what
you are comparing the the output
essentially variable is still like a
continuous quantitative variable like
height it can be anything else but
essentially it's a quantitative variable
and you have multiple discrete settings
of your input variables so you're not
just in
student comparing boys versus girls or
you know method a versus method B but
you're now going from in the t-test you
were comparing method a versus method B
you can now say is method a equal to
method be equal to method C equal to
method D or are they different ins or
some subset of them different from the
others okay so that's that's essentially
an application of ANOVA now the
chi-square test of Independence is one
that can be used when you want to
compare again multiple proportions right
when you're interested in two variables
and both of these variables are
categorical variables so it could be the
same thing like Tamil Nadu Karnataka so
the different states of India would
could be one of the variables and the
other variable could could be the number
of people at different age groups so
people between number of people between
0 to 20 years between 20 to 40 years and
40 to 60 years so there's one variable
which is the state which is a
categorical variable
and here's another variable age which I
have in some sense made categorical but
essentially I'm just interested in these
two variables which are categorical and
I'm just interested in looking to see if
there's any relationship between them
another way to think of the chi-square
test of Independence you see it as some
form of an extension of your proportions
II test right in a proportion Z test you
would typically handle problems of the
following nature you'd say if you took a
question like are men more likely to
purchase a certain product than women
that would be a proportion Z test right
because you have two categories men and
women it would be um so as to be precise
it would be a two sample proportion Z
test right because you would have two
categories men it would be one category
women would be the other category and
their likelihood of purchasing a certain
product would be represented by a binary
right so upstream of ones and zeros so
30 percent women purchased this product
out of a hundred
samples and 20% men purchased this
product out of 25 samples right but you
take that same problem where you had
just two categories which is purchased
or not purchased men and women and you
extend that to multiple categories on
both dimensions so you know that's what
we did when we said state could be
multiple different states and we said H
could be multiple different ages so if
you're able to take these two variables
and extend them to multiple categories
you can use a test of Independence in a
sense yeah
so let's first jump into the anova and
try and explain the core concept and the
math behind it
the idea behind ANOVA again would be to
test a hypothesis of this nature right
which is mu a equal mu B equals mu C
equals mu D a typical t-test for
instance would have just looked at
something like is mu a equal to MU B
right so I think an a very natural
question that quite often comes up is
what's so special about two versus three
why is it that for two you can use a
t-test three and more it's at the ANOVA
is built for this kind of multiple
comparisons but it works just as well
even if you just want to compare two
samples so you can you can essentially
replace your t-test with a two sample
t-test with an ANOVA and you should be
doing something mathematically identical
okay okay so before we go any further
we've identified the hypothesis now
let's get some nomenclature ready so
these are the different samples so this
these this is the sample corresponding
to a okay and it could be anything it
could be fertilizer a versus fertilizer
B versus fertilizer see a could
represent the state
Karnataka B could represent Tamil Nadu C
could represent Maharashtra whatever so
it depends on the question let's take a
fairly consistent example so let's take
the example of height of boys in public
schools in 10th standard for four
different states so these are the four
in states and this is the data set so
this is data point one so this is data
point one for state one
so that's represented as y1 kebab one
now
data point two first state one is
represented as Y 1 comma 2 and it goes
on till Y 1 comma N and similarly it
goes in this direction Y 2 comma 1 and
ultimately you have Y 4 comma and in the
bottom corner ok
now having defined this I'm going to
throw you in some sense in the deep end
with the formulas of how an ANOVA is
actually computed ok so this is the idea
and so let's just go through this step
by step the idea here is to compute an F
statistic right and what if we learnt
about if there's one thing we've learned
about hypothesis testing it's that you
come up with a hypothesis which we did
which hypothesis was that mu a equals mu
B equals mu C and the alternate
hypothesis by the way out there is that
not all Meuse are equal right because
there are many ways in which mu a that
you can violate the statement that mu a
equals mu B equals mu C equals mu D you
can violate it by saying new a equals mu
B mu B but is not equal to MU C equals
mu D or you know you can put that not
equal to anywhere in that equation so
quite simply the alternate hypothesis is
that not all the Meuse are equal right
ok so we took care of the first step
which is to create a null and alternate
hypothesis the second step is to do some
kind of computation to come up with the
test statistic we're going to talk a
little bit about how first we're going
to talk about the overall structure
which is the math behind this table
leads to an F statistic with a minus 1
degrees of freedom and n minus 1 if you
remember we discussed about how the F
statistic has a numerator degrees of
freedom and a denominator degrees of
freedom and you will calculate the ref
statistic and the rest of the hypothesis
testing is the same you will then
calculate a p-value and then choose to
reject or not reject okay now let's take
one step back we say the F statistic
that we've computed here is nothing but
something called MSB
/ MSE we're going to talk a little bit
further about these terms but for now
you can you can take this again go one
step back to these two values and you
technically don't need to know the mean
square total or any of these terms so
this this entire part you don't need for
actual calculation of the test statistic
but it's there to give you a bigger
picture okay so again we say this MSB is
nothing but SSB okay divided by
something called the degrees of freedom
and that's fairly obvious in terms of
what math needs to happen there and
again the same concept with with MSE
okay so at least we know how things flow
from one side to the next
now let's get to the core of the formula
which is you know calculating the sum of
squares between and the sum of squares
error okay
the sum of squares between is
essentially Y bar I dot minus y bar bar
dot dot what do we mean by that
okay so let me first we'll talk about
the terms here and then I'll come to N
and a and so on okay
essentially what Y I bar dot means is y
a bar dot is nothing but the average of
all these terms Y bar bar dot dot is
nothing but the average of all the
numbers here okay so if I want a row
wise addition I say Y a bar dot and
similarly Y I would say B bar dot for
this row
right so this would be why a bar doc
this could be why B bar dot and so on
and the idea is that if you're going to
change a B and so on and a B and C and D
can be coded as 1 2 3 4 then I might as
well just say why I bar right dot and
put that through a loop essentially so
that's what I'm doing here I'm saying
why I bar dot to say I'm going from I
equals 1 to a where a is the total
number of treatments which in my
particular case is 4 so there are 4
different treatments or there are 4
different sets of samples that I'm
comparing so essentially I am taking i
from 1 to 4 and I'm taking each of those
means that I calculate so I'm
essentially taking that this average
this average this average this average
separately I'm taking each of those and
I'm subtracting that from the grand mean
the overall mean the overall mean is
nothing but every single output variable
this entire table in a sense this entire
table average is y bar Y bar bar dot dot
okay so essentially what I'm doing out
here is I'm doing something if you look
at this formula very similar to standard
deviation but it's essentially the
standard deviation of the means right so
there's an overall mean and you have
individual treatment means so I'm
essentially looking at out here this
term looks it's not the standard
deviation because I've not done the
square root and I'm not divided by the
number n minus 1 in this case it would
be a minus 1 so not divided that so
that's why it's still I'm going to
divide it once I divide it it's kind of
like variance but I'm not divided it yet
so out here it's a sense
like a sum of squares once it gets
divided by that a minus one then it
becomes that MSB right but in concept
this is and we're going to talk about
the multiplication by n in a second but
in concept this is a lot like the
standard the variance of the means right
so it's it's like the variance so if you
were to compute a set of means for each
row right a set of like Y bars I dot for
each row the standard deviation of these
values or rather the variance of these
values is essentially what you're
computing with MSB okay
now what's the concept behind here here
what you're doing is you're looking at
the standard deviation of each data
point all right so I I comma J just
means that I goes from 1 to a which
means I'm going through each row and I'm
going through each column out here J
goes from 1 to N so J is going from 1 to
N okay I is going from 1 to a right so
it's literally like I'm going through
each data point right and you know all
through this matrix and I'm looking at
the deviation of each data point from
its row average right we saw these two
are the same terms that we're using here
the concept here is that we are looking
at essentially the standard deviation
within each row right I'm not looking at
the deviation of Y 1 1 from the grand
mean Y bar bar but I'm looking at the
deviation of Y 1 1 from this rose
essentially this rows average okay
and I'm averaging that across every row
so it's almost like I'm taking a row
wise average of every data point
and averaging that so okay so that's
called the summer squares error and
again you're dividing by the total
number of points that you see it's a set
of N and capital n by the way Archer
this there's a difference between the
small n and the capital n capital n is
essentially nothing but the total number
of data points so small a is the total
number of treatments in this case it's
equal to four and we've not actually
defined out here but essentially it's
the total number of number of columns
right so that is the number of data
points that each combination has and
capital n is the total number of data
points so essentially it's nothing but a
times n so a times n is capital n so
using that normal feature again
calculating MSc also should be obvious
okay so the mechanics of how you do the
anova with the f-test and calculate a
p-value should at this point be fairly
clear to you but let's spend a few
minutes and try and get a little bit
more intuition on this which is why are
we using an f-test
which we know from earlier experience is
used to see the difference between two
variances right we use the two sample F
test for detecting the difference
between two variances now why are we
using that to calculate the difference
between means right and the core idea is
the following that the mean square
between is one way of calculating the
total variance now mean square error is
another way of calculating total
Whelan's
now these three which is mean square
between mean square error and mean
square total which is what is described
here are all going to be equal in a
sense again they're not they're going to
be statistically equal not actually
equal if the null hypothesis is true so
if mu if we go back to the null
hypothesis which is that mu a equals mu
B equals mu C equals mu
then MSB MSE and MST would be
statistically equivalent however if it's
not true then you will find that MSB
will be greater than MST will be greater
than MSE so you've got two different
ways of computing variance and what
you're doing is you are doing an f-test
of these two different variance
calculations right so you're doing an F
test on these two different variance
calculations and you will you will not
reject the null hypothesis if the null
hypothesis is true which is mu equals mu
B equals mu C equals mu D now if that if
those means are not equal then these two
computations of variance right don't
really represent the total variance and
this computation of variance becomes an
over estimate and this computation of
variance becomes an underestimate of the
overall variance and therefore a test of
whether these two are different
variances will show up to be true
essentially you'll wind up rejecting the
null hypothesis that these two variances
are equal and thereby commenting on the
fact that the null hypothesis of means
was probably not true okay so let's get
a little bit more intuition on why this
becomes if mu a if the null hypothesis
is not true and here the null hypothesis
is that mu a equals mu B equals mu C dot
dot dot if this null hypothesis is not
true let's take a look at why this
computation becomes an overestimate and
this becomes an underestimate okay so
here's the first case where the null
hypothesis is true okay so the null
hypothesis is true that means mu A
equals mu B equals mu C equals mu D
which means the mean of MU is so this is
the distribution of a this is the
distribution of B and this is it so this
is just a sample case so there's it has
ABC and here are the distributions and
Hey look the means of this you know the
same and here I've gone further and even
made the distributions look identical
right so if you were to take samples
from a samples from B samples from C and
you're just going to put them all in one
bucket called
total distribution why this is how it
would look right because essentially
since these distributions are identical
and their means are identical it looks
this this is also identical so how do
you compute mean squared error the way
you compute mean squared error is
essentially look at the variance of each
distribution all right
you basically take each data point and
look at the standard deviation of
variance of the data point with respect
to its mean right
so you're essentially computing the
variance of each of these distributions
and you're just averaging them and
because the variance of these
distributions are going to be the same
right the that's going to match the
variance of the total distribution of Y
right and we just explained why okay
now let's step to a mean square between
mean square between is so here's the
total distribution of Y and you're
interested in capturing the variance of
the total distribution of Y and the way
you're going to do that is you're going
to take the mean of distribution a
you're going to take a sample mean of
distribution E and as you've known from
the earlier part of this course if you
take five data points of six data points
on a and you compute a sample mean that
sample mean need not exactly fall on the
population mean right so you might get
another value here for this distribution
B now as long as the null hypothesis is
true which is the distribution a equals
distribution B equals distribution C
blah blah blah then you are going to get
some sampling distribution if you take
the means of each of these distributions
now if you take the means of these each
of these distributions then if all of
these are equal then you're going to get
a new distribution called the
distribution of the means of
BC and what is it that we've discussed
about the variance of this distribution
we've discussed that as long as you're
sampling from essentially the same
distribution right you're sampling from
the same distribution because a now
looks identical to B looks identical to
see they all have the same means as long
as you're doing that we know that this
standard deviation right is equal to the
overall standard deviation that is the
standard deviation of the total
distribution or you can think of it as a
standard deviation of any of these
distributions because they're all
identical right divided by square root
of n all right or you can think of it
you can think of it in variance terms
and say it's Sigma square divided by n
okay so as long as I can compute this
variance and then move this and multiply
it by n right I could cancel this out
and I can get an estimate of this
variance so that's what I'm trying to do
what I'm trying to do out here is I'm
calculating means from each of these
distributions and I'm taking the
standard deviation of those means I'm
taking the variance of those means I'm
taking the variance of those means and
multiplying it my n with the belief that
if the null hypothesis is true
that should be a great estimate of the
variance of the total distribution which
is fine that's going to work great as
long as the null hypothesis is true so
as long as the null hypothesis is true
this approach to calculating the
variance of this distribution right
variance of this distribution and this
approach to calculating the variance of
this distribution so the msie approach
to calculating variance of the total
distribution and msb approach to
calculating the variance of the total
distribution both should work perfectly
fine right but what happens when the
null hypothesis is not true
take this case where the null hypothesis
is not true now if you take the total
distribution of Y right because the null
hypothesis is not
when you you get some data points from
distribution a some data points from
distribution be some data points from
distribution C it's going to fall over a
much larger region right now if you try
to estimate the total variance let's
take a look at how MSE would estimate
the total wailings MSE would take the
variance of take the deviation of each
data point with respect to its mean not
with respect to some grand mean all
right and even that mean is essentially
a sample mean that you're going to
calculate but the idea is that your
estimate is going to a variance of cure
is going to be some value your estimate
of variance here is going to be another
value and similarly out here is going to
be some value and all you're doing with
MSE is you're taking the average of
these three numbers of one two and three
now what's the average of these three
arrows it's an arrow with a magnitude
that's much smaller than the real
variance right so when the null
hypothesis is not true the MSE where
method of calculating variance becomes
an underestimate of the total variance
now let's look at what happens on the
MSB side right now the null hypothesis
is not true and so despite the fact that
there's one source of variation which is
that the sample means are not falling
exactly on top of true means right but
in addition to that the sample means are
further separated from each other
because they are trying to go after a
totally different line the true means
themselves are different right so the
standard deviation and the distribution
essentially of the sample means is going
to be a much wider distribution because
the means are not equal right and now
when you multiply that by n you wind up
making a huge over estimate of the total
distribution of mine because you're
essentially out here you're trying to
look at different you're trying to get
sample means of distributions which have
truly different means and you're looking
at the standard deviation of the sample
means of distributions with truly
different means so it's going to be an
overestimate of the total total variance
of the distribution of Y okay so I hope
that gives you some intuition on why the
F test for the ANOVA works the way it
does but the important thing for you to
remember is that the F test the F is a
ratio of two variances okay and the core
idea is you are using the F test which
is used for asserting whether two
variances are equal you're using that in
an indirect way to make a statement
about the means of many distributions
and you're doing that by saying that if
the null hypothesis is true method a of
calculating variance should be equal to
method B of calculating variance but if
the null hypothesis which is mu a equals
mu B equals mu C is not true then these
are not accurate methods of calculating
variance great the big question is what
do you do okay so now we know the
mechanics of it we have some intuition
for the ANOVA but the idea is okay so
you did this you calculated an F
statistic you then went and plug that F
statistic and found out the probability
and let's say your p-value was really
small so you want to reject the null
hypothesis okay so what do you do after
rejecting the null hypothesis so what is
the statement you're making when you
reject a null hypothesis you're saying
I'm rejecting the null hypothesis that
mu equals mu B equals mu C equals mu D
okay great
but can you tell me which of these Meuse
are different from each others so one
great way to do this is call the Tukey
test it's not the only way to do it but
it's a fairly common way
of doing it so that's why I've called it
method one it's the two key test but
there are other methods as well the idea
here is to decide on an alpha value
which we do with many hypothesis tests
even before we start we say if that
p-value is less than a certain value I'm
going to reject it so let's say you
start with an alpha value and then you
calculate something called the critical
to key distance based on this alpha
value so there's a distribution called
the two key distribution and that's
again something that you can get usually
from software or you can get that from
the back of textbooks but you'll
essentially plug in the values of alpha
which you just decided one step above
and you know the value of I and I out
here is a number of treatments so it's
the equivalent of what we had as e but I
didn't want to confuse you between alpha
and ACE have called it I out your n is
the number of replicates so the small n
remains the same the same concept and
the large end remains the same the only
thing that has changed is I've called
what we used to call a have called that
I and the reason I've done that is so
that I avoid confusion with alpha right
out here okay so we do that and we use
that mean squared error formula out here
and we calculate something called a
critical distance and then you do a very
simple thing which is you do a complete
enumeration of all pairs so for instance
this is the mathematical way of showing
it but essentially all I'm saying is
let's calculate Y bar a dot right and
similarly calculate Y Bar B dot y bar C
dot and look at the difference between
each pair of Y bar eye dots where you
know the two eyes are not so the
different combinations you can come up
with in our particular example you'll
have a comma B a comma C a comedy B
comma C become a DC comedy so there are
six possible combinations and because
and you're essentially just taking the
absolute value so B minus a is the same
as a minus B so you just calculate the
six
differences and you see which of these
six it's six in this particular case
because we had four treatments right we
had four different behind a B C and D so
we had four possibilities four sets of
data so that's six treatments because
you get the concept here is combination
so you have four C two combinations but
how many other treatments you have you
similarly have the concept of whether
you like to use the word I or whether
you want to use a right C two
combinations you'll calculate that many
combinations of distances and you'll see
which of these distances are greater
than the two key distance and then those
are the ones that are not equal to each
other okay so you'll calculate a
critical distance and to figure out but
we see you just rejected the null
hypothesis a mu a equals mu B equals mu
C equals mu D is wrong so you've said
that you've said that that's wrong but
which of these are different from each
other for that you calculate a critical
distance and see which of these
combinations have a mean difference
that's greater than the critical
distance and those are the different
ones okay so I hope that gives you an
idea of what the two key tests are does
okay so now we move on to the next topic
which is the chi-square test of
Independence right it's noteworthy at
this point to say that we've already
looked at one chi-square test we've
looked at the chi-square test for
standard deviation in the case of a
single sample tests and you might also
encounter another test called the
chi-square goodness-of-fit test and
that's used when you have a set of data
and you want to see how well it fits to
a particular distribution okay but out
here what we're interested in is
essentially using the chi-square test of
Independence and it's used here where
you have two categorical variables so
here's the first categorical variable
which is the smoking habit is it heavy
regular occasional or never and here is
the second categorical variable exercise
frequent some or not
and what you're trying to do is you're
creating a contingency table of how
frequently various values occur right
and you're trying to see if they are
independent of each other does smoking
habit have anything to do with exercise
or vice versa
you know again not implying causation
but you're taking two categorical
variables and seeing if these two
variables are independent of each other
so the core idea between that is to
convert this table to a set of
percentages so for instance you would
take each of these values and you would
see how frequently they occur within
each row essentially and and similarly
you would do that for each row and
finally what you would do is you would
create theoretical values for this table
in accordance to the assumption of
Independence so for instance if I
believe that these were completely
independent I would take the row wise
sums and the column Y sums and I would
for instance conclude that row 1 they
would be a forty eight point seven
percent chance of seeing a particular
value occur in a particular spot so I
would essentially say look since so this
is I'm getting that forty eight so the
numbers I'm getting are for instance up
here it's forty eight point seven okay
and the number for this category is
forty one point five and so this would
be I believe the remaining which is
about nine point seven percent forty-one
yeah but the idea is the following right
the idea is that you
get percentages based off of the sum of
each row so you know 7 plus 9 plus 12
plus 87 and you divide that by the total
sum of all values and that's how you get
the 48.7%
okay now given that there's a forty
eight point seven percent chance that
you will find yourself in Row one I
asked you the question that if you were
a heavy smoker right and there are a
total of 11 heavy smokers the number of
heavy smokers that frequently exercise
that I should expect is 11 times 48.7%
so just not the map that we did here we
computed the probability of being in
each row the way we did that is we took
the sum of each row so the sum of the
first row is 7 plus 9 plus 12 plus 87
and we divided that by the sum of all
the numbers here and by doing that we
got a number 48.7% ok so we told
ourselves the probability that if I did
not know anything that I would find
myself in the first row is 48.7% now
assume that I am a heavy smoker then
there are a total of 11 heavy smokers if
I were to randomly guess whether I was a
frequent exerciser or not then my
probability is nothing the number of
heavy free heavy smokers that frequently
exercised the expected value of that is
nothing but the 11 which I get from
adding 7 + 3 + 1 and multiplying that by
the 48 point 7
okay and a small correction from before
so you you would this I know for a fact
is forty eight point seven this
probability is also forty one point five
you would just need to compute the
remaining two
this value and that's close to about 10%
okay so I don't believe it's exactly
nine point seven percent that's why I'm
correcting myself but that's the core
idea the core idea is you calculate
percentages based on frequency of
occurrence and then you come up with
what is an expected value for each cell
okay once you come up with an expected
value for each cell and you can do that
row wise or column wise I explained it
to you row wise so I took each row
calculated a percentage and then I
calculated an expected value but
essentially what you've computed is a
set of expected values out here okay now
you compare each original value so you
compare that seven okay so I compare
this seven to what I get when I multiply
forty eight point seven by hundred forty
eight point seven percent times eleven I
believe that's approximately like close
to five point something so so what I'm
essentially doing is I'm I'm doing that
seven which I see - that expected value
which is the five point dot dot that I
see and I'm squaring that and I'm doing
that for each cell right so I'm doing
that for each row and each column and
thereby doing that for each cell and I'm
dividing it by their expected value so
this five will also come in the
denominator okay and that summation
should essentially give me a chi-square
distribution with R minus 1 times C
minus 1 degrees of freedom now note that
the chi-square distribution has only one
parameter associated with degrees of
freedom so if you just expand this you
will get this formula which is nothing
but just just linear algebra R times C
minus C minus R plus 1 so you've just
calculated a test statistic using this
formula and that tested test statistic
should be Chi square distributed with
this degrees of freedom and again you
can use that same core concept to
calculate a p-value associated with it
and they
by either reject the hypothesis and what
and just as a refresher what's the
hypothesis here the hypothesis the null
hypothesis is that there is no
relationship between this variable and
this variable that is these two
variables are independent of each other
and the alternate hypothesis is the
rejection of that right that there is
some relationship that depending on
whether you're a heavy regular or
occasional smoker that impacts whether
you're going to be a frequent exerciser
or not I mean it's not a causal
relationship but knowing this helps me
make say something about this other
variable or it can be the other way
around knowing that you're a frequent
exerciser does that help me in any way
predict or in any way make a statement
about your smoking habit right is there
some relationship or are these two
variables independent of each other the
null hypothesis being that they are
independent of each other and a very low
p-value in this chi-square test
statistic will enable you to reject that
null hypothesis or you would fail to
reject that hypothesis okay so I hope
this gives you an idea of the two tests
that we've introduced in inferential
statistics that have to do with multiple
samples that just don't have to do with
two samples that's the ANOVA or the
analysis of variance and the chi-square
test of Independence in our next lecture
we are going to start with a regression
thank you