This course covers commonly used statistical inference methods for numerical and categorical data. You will learn how to set up and perform hypothesis tests, interpret p-values, and report the results of your analysis in a way that is interpretable for clients or the public. Using numerous data examples, you will learn to report estimates of quantities in a way that expresses the uncertainty of the quantity of interest. You will be guided through installing and using R and RStudio (free statistical software), and will use this software for lab exercises and a final project. The course introduces practical tools for performing data analysis and explores the fundamental concepts necessary to interpret and report results for both categorical and numerical data
ANOVA
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Del curso dictado por Duke University
Inferential Statistics
992 calificaciones
De la lección
Inference for Comparing Means
Welcome to Week Three of the course! This week we will introduce the t-distribution and comparing means as well as a simulation based method for creating a confidence interval: bootstrapping. If you have questions or discussions, please use this week's forum to ask/discuss with peers.
Conoce a los instructores
Mine Çetinkaya-RundelAssociate Professor of the Practice
Department of Statistical Science
In this video, we'll formally introduce anova.
We're going to start our discussion with variability partitioning
which means considering different factors that contribute to variability in
our response variable.
For example, at the end of this course you'll each take a final exam and
not everybody is going to get the same score on that exam.
That's expected.
The variability in your final exam scores will likely be due to a variety of
factors.
One of them might be whether or not the student completed
all components of the course leading up to the final exam, such as the videos,
the quizzes, the midterm, the lab, so on and so forth.
However, there will certainly be other factors as well.
Familiarity with the material beforehand,
number of hours per week put into the course, so on and so forth.
Suppose we're interested in studying how strongly completing
all components leading up to the final exam associated exam scores.
To study this, we would partition the total variability in exam scores
as variability due to this variable, and variability due to all other factors.
We're going to build up on this idea of variability partitioning and
the aesthetics we introduced earlier to work through our way of the analysis
of variance output.
Let's quickly remind ourselves of the data we're working with.
From the general social survey we had the vocabulary scores,
the numerical variable, social class, a categorical variable.
We have our summary statistics at the group level
as well as at the overall level.
Our null hypothesis is that the average vocabulary score is the same across all
social classes.
And the alternative hypothesis says that the average scores differ for
at least a pair of social classes.
Let's visualize this idea of variability partitioning.
Suppose the circle represents the total variability in vocabulary scores.
We partition the variability in to two
variability that can be attributed to differences in social class and
variability attributed to all other factors.
Variability attributed to social class is called the between group variability
since social class is the grouping variable in our analysis, and
the other portion of the variability is what we're not interested in.
And in fact it's somewhat of a nuisance factor for us.
Since, if everyone within a social certain class scored the same,
then we would have no variability attributed to other factors.
This portion of the variability is called our within group variability.
Here's a look at the anova output.
The first row is about between group variability, and
the second row is the within group variability.
We often refer to the first row as the group row, and
the second row as the error row, the third row displays the totals.
Next, we're going to go through all of the values n this table.
How they're calculated, and what they mean?
Let's start with the column of sum of squares.
The last value in this column is sum of squares total,
commonly referred to a as SST.
This value measures the total variability in the response variable.
In this case, that would be the variability of vocabulary scores.
This value is calculated very similarly to variance
except that it is not scaled by the sample size.
More specifically, this is calculated as the square
deviation from the mean of the response variable.
We have 795 observations in our dataset.
On the mean vocabulary, score is 6.14.
So to calculate SST, we take each individual score and subtract 6.14 from
it, square the difference, and finally add up all the values.
For example, the first is 6, so that's 6- 6.14 squared.
The next one is 9, that's 9- 6.14 squared.
Third one is also 6, so on and so forth and
we add up all of the values to get to the total sum of squares of 3,106.36.
This value represents the total variability in the response variable.
But what we're really interested in is how this variability is partitioned into
between and within group variabilities.
As an aside we can see that this is an awfully tedious calculation to do by hand.
And hands for a no, we usually rely on software to do the calculations for us.
So the calculations we're going to present in this video are for
illustrative purposes and for introducing the concepts.
But you'll likely never have to calculate these by had.
You still need to understand what they mean so
that you can interpret your analysis though.
Next, let's talk about the sum of squares group, SSG.
This value measures the variability between groups and
can be thought of as the variability in the response variable
explained by explanatory variable in the analysis.
It's calculated as the deviation from group means from the overall mean
weighted by their sample sizes.
So more specifically for each group we calculate it's mean,
that's y bar j subtract the grand mean from it,
y bar square this value and multiply it for the sample size for that group.
We do this for each of the groups, and sum them up.
Here's a summary table that's going to help us.
The lower class group has a mean of 5.07 we subtract from that grand
mean of 6.14 square that value, multiply it by the sample size for the group of 41.
We do the same thing for for all of our groups and arrive at the sum of square's
group of 230.56, which on its own is not a meaningful number but it's
interesting how it compares to the total sum of squares we calculated earlier.
For example, this value is roughly 7.6% of SST.
Meaning that 7.6% of the variability in vocabulary scores
is explained by social class and the remainder is not
explained by the explanatory variable we're considering in this analysis.
This is a low percentage which I think would make sense because we would expect
vocabulary scores to be associated with, more with education or
how much people read.
The last value here is sum of squares, SSE and
it measures the variability within groups.
In other words, this is the unexplained variability and
it's the variability due to all the other variables.
The simplest way of calculating this is simply as
the difference between SST and SSG.
Now we need a way to get from the sum of squares measures to the mean square
values.
To do so we need to scale the sum of square values by values that incorporate
sample size as well as the number of groups, namely the degrees of freedom.
So next let's focus on that group.
Total degrees of freedom is calculated as sample size minus 1, 794.
Group degrees of freedom is calculated as number of groups minus 1, 3.
And the error degrees of freedom is simply the difference between these two 791.
Next stop is the mean squares column, which measures the average variability
between and within groups and is calculated as the sum of squares for
that component divided by degrees of freedom.
So we can calculate that by doing the divisions, and
we're going to next use these values for calculating our F score,
because you remember that our F statistic is the ratio of the average between and
within group variabilities.
In other words, it's MSG divided by MSE.
Once you have your F score you're finally ready to find your p-value and
conclude the hypothesis test.
The p-value in this context is the probability of at least as large a ratio
between the between and
within group variabilities, if in fact the means of all groups are equal.
This is just another way of saying p-value is the probability of observed or
more extreme outcome given the null hypothesis is true.
And it can be calculated as the area under the F distribution.
And the F statistic has two degrees of freedom, degrees of freedom group and
degrees of freedom error.
So the p-value shown on the anova table is the tail area under the F
distribution with three and 791 degrees of freedom, which is tiny.
Note that even though we're looking for differences,
we only consider the upward tail of the F distribution.
This is because the F statistic can never be negative.
Think about it, it's the ratio of two measures of variability
that can't ever be negative either.
Since the F statistic is always positive,
a more extreme statistic will always be more extreme in the positive direction.
Even though the anova table always reports the p-value, if you wanted to do so,
you could directly calculate it in R using the PF function.
This function takes the observe F score as one of its arguments as
well as the degrees of freedom.
And we need just note that we don't want the lower tail and
get this tiny P value which is in Indeed, less than 0.0001.
Now it's finally time to make a conclusion.
If the p-value is small, we reject the null hypothesis and
say that we have sufficient evidence for the alternative.
If the p-value is large, we fail to reject the null hypothesis and
conclude that the data do not provide convincing evidence that at least one pair
of population means are different from each other, the observed differences in
sample means are then attributable to sampling variability or chance.
In this case, we had a pretty tiny p-value.
So what's going to be our conclusion?
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