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ANOVA Part III: F Statistic and P Value | Statistics Tutorial #27 | MarinStatsLectures - YouTube
Channel: MarinStatsLectures-R Programming & Statistics
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Let's build up the test statistic for
one-way analysis of variance so recall
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we were working with this example
comparing the weight loss on one of four
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diets A B C or D and we can see the
observations here as well as the summary
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statistics the mean weight loss and
standard deviation of weight loss for
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each of the four diets, we're working with
this null assuming that all means are
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equal and an alternative at least one
differs from the rest. We previously
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talked about how we can take the total
variability in weight loss or the total
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sum of squares and separate it into two
parts that which is explained by the
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diet and that which is not explained by
the diet, so let's look at how we can use
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that to build up the test statistic:
first just a quick note on notation and
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again we want to focus on the concepts
not on plugging into formulas but this
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helps us with understanding the formula
and what's written in the notation so i is
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used to index the group: Group one two
three or four k is to signify the
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number of groups J is used to represent
the observation number within a group
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Yij tells us the individual observations
in group i, observation number j, so for
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example Y1,3 is the observed
value for Group 1 person number 3
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Yi-bar is the mean for group i, Y-bar with
no subscript is the overall or grand
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mean: the mean weight loss for everyone
in the study; Si is the standard
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deviation for people in the group i, and
ni is the sample size for people in
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group i, so we saw that we can take the
total variability and separate it into two
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parts that which is explained by diet
and that we signified as the variance
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between diets or sometimes called the
mean square between and that was the sum
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of squares between divided by its
degrees of freedom
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the degrees of freedom between groups and looking at the formula and again we don't want to
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get stuck on this but this helps us see
the concepts we just learned from a
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slightly different angle you can think
of this as we're going to sum over all the
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groups so Group one two three four
what's the sample size in each group and
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how far is the group specific mean from
the overall mean squared divided by its
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degrees of freedom if we work that out
for example we'd find that the sum of
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squares between groups or the explained
sum of squares it's 97.3
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degrees of freedom 3 right
four groups minus one and that's going
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to come out to 32.4t, we also saw we can think of the unexplained
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and again this is the variability that's
not explained by diet or not explained
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by X, this is the variability going on
within a group or the mean square within
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and again this is the sum of squares
within groups divided by the degrees of
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freedom within and formulaically we can
think of summing over all observations
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how far is each individual from their
group specific mean square divided by
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three degrees of freedom.
Right again we have n observations and
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we lose K degrees of freedom by
estimating the K group means, we can also
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express this as summing over the groups
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each group sample size minus one times the variance the
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sample variance of each group divided by
its degrees of freedom and the reason
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why I write it this way is you can take
a moment yourself to note that this here
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is the exact formula for the pooled
variance that we talked about in the two
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sample t-test assuming equal variance in
the two groups we're taking a the sample
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variance of each group weighted by their
degrees of freedom. okay so you can take
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a moment yourself to work your way
through and convince yourself that this
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within group variance is the exact same
as the pooled variance in the two sample
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t-test assuming equal variance if you
work this out for example you're going
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to find the sum of squares within a
group is 297 its degrees of freedom 56
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and this comes out to be 5.3 so as noted
we want to compare these two to each
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other the mean square between groups to
the mean square within groups and
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the average sum of squares that can
be explained by diet the average sum of
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squares that cannot be explained by diet
so let's try and think our way through
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some stuff first suppose if the
alternative hypothesis is true if at
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least one mean differs if not all the
means are the same how would we expect
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statistically how we expect the mean
squares between groups to compare to the
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mean square within group if diets are
different we'd expect this one should be
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larger than this okay there should be
much more variability that's explained
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by diet than not explained by diet if we
take a ratio of these and this is going
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to be what we call our F statistic and
or our test statistic
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it's a mean squared between groups over
the mean square within groups if we
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expect the top to be larger than the
bottom we expect this test statistic to
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be larger than 1, if
on the other hand our null hypothesis is
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true if all the means are equal at the
population level what would we expect to
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see we'd expect the mean squares between, okay the variability that's explained by
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diet, to be roughly the same as the mean
square within or the variability that's
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not explained by diet when looking at an
F statistic you're taking the ratio of
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these two we'd expect that to come out
to be roughly 1 if we do this for our
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set of data our F statistic 32.4 over
the 5.3 that's going to come out to be
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6.1 okay so the larger our F statistic
gets the more evidence we have that the
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alternative is likely true or the null
is false well we don't want to get too
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caught up on looking things up in tables
it's important to note that this F
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statistic follows what's called an F
distribution it has degrees of freedom
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for the numerator and degrees of freedom
for the denominator so it has degrees of
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freedom for the numerator which are K
minus 1 right those are the degrees of
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freedom of what's in the numerator and
has degrees of freedom for the
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denominator n-k, ok so again
piece of software can do all these
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calculations for you we don't want to
focus on an F table and looking up an
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exact p-value from that table
so let's just jump to the interpretation
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if we were going to work this out
looking at a table or using a piece of
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software p value is going to tell us
like it always does what's the
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probability of our observed test
statistic or one even more extreme if the
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null is true and we'd expect it to be 1
so what's the probability of getting an
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F stat greater
or equal to 6.1 if it should
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be roughly equal to 1 if our null is
true we'd expect our test that to be
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roughly 1, what's the chance of seeing
an estimate of 6.1 or more
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you'll find that this comes out to 0.0011
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okay roughly 0.1 percent so again if our
null is true if all these diets are the
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same the chance of seeing an F stat like
this or the differences we saw or even
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larger is only going to happen about 0.1
percent of the time that gives us
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evidence to reject our null hypothesis
we have evidence to believe the
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alternative is likely true we have
evidence to believe at least one diet
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differs from the rest so now we need to
decide which diets might differ from the
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others and to do that we're going to
compare all possible pairwise means
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that's a topic we're going to get to
talking about in a moment
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