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Today: Dummy variables.
Dummy variables in a multiple regression, regression wrap up.
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Looking back in regression, weve looked at how an interval
data response y changes as an interval data explanatory
variable x. Changes.
Example: Number of books read (y) as a function of television
watched (x).
Y = a + bX + e
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Last time, we expanded this idea to consider more than one
explanatory / independent variable at the same time, where all
the variables were interval data.
This is called multiple regression.
Example: Wins as a function of goals for and goals against.
Y = a + b1X1 + b2X2 + e
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This time, were going to drop the requirement for the
independent variables to be interval data. (Typo fixed)
Were going to look at nominal data as independent data.
Recall: Nominal means name. Its data in categorieswith
no natural order.
Example: Type of Fruit --- Kumquat, Coconut, Tomato,
Dragonfruit.
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How do you put a type of fruit into a formula like this:
= a + bXWith a dummy variable.
Dummy in this case just means a simple numbervariable (0
or 1) that we use in the place of nominal, and sometimes
ordinal, data.
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Weve already used dummy variables.
Bearded dragon gender: 0 = Male, 1 = Female
Bearded dragon colour: 0 = Green, 1 = Fancy
Other possibilities:
0 = Non-Smoker, 1 = Smoker
0 = Domestic Student, 1 = International Student
0 = Eastern, 1 = Western
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Nominal data can have more than two categories, but we cant
do this:
Favourite colour:
0 = Blue, 1 = Green, 2 = Red
This would imply an order, and that having a favourite colour
of green is somehow the middle ground between favouring
blue and favouring red.*
*If we cared about wavelength of favourite perhaps, but usually not
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Ordinal data can made into a 0,1,2, scale, as long as we
assume the differences between each category and the next
one are about the same.
0 = Against, 1 = Neutral, 2 = For
Or
-1 = Against, 0 = Neutral, 1 = For
Then were treating the ordinal data like interval data.
Handling more than two categories is a for-interest topic, at
the end of the lecture if time permits.
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Its all just words until we get up and do something about it.
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Dummy variables in regression:
Consider the NHL data set. Lets see the difference in
defensive skill between the Eastern and Western conferences,
and by how much.
Dependent variable: Goals against. (More goals against means
weaker defence)
Independent variable: Conference. (East or West)
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In our data set, we have conference listed in two different
ways. ConfName: E or W. Conf: 0 or 1.
0 = Eastern Conference, 1 = Western Conference.
ConfName is for when we need conference as nominal.
Conf is our dummy variable for when we need interval data.
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We can do a regression by using Conf as our independent.
(SPSS wont even let you put Confname in)
(Done under AnalyzeRegressionLinear)
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We get this model summary.
The conference alone explains .122 of the variance in goals
against.
Theres a lot to goals against that isnt explained simply by
whether you are in the Eastern or Western Conference.
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We get these coefficients.
The prediction formula is:
(Goals against) = 232.86717.333(Conference)
The intercept is the response (Goals against) when the
explanatory variable x = 0.
Here, x=0 means Eastern Conference.
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The intercept is the average Goals Against of teams in the
Eastern Conference.
The slope is the amount that (Goals Against) changes when
(Conference) increases by 1.
Changing x=0 to x=1 means switching for the Eastern to the
Western Conference.
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So the slope b is the difference in mean goals against between
the conferences.
Here, Western Conference teams let in 17.333 fewer goals.
Plugging in x=0 or 1
232.86717.333(0) = 232.867goals against if East
232.86717.333(1) = 215.534goals against if West
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Since theres only one independent variable, and its nominal,
so we COULD do this with a two-tailed independent t-test.
AnalyzeCompare MeansIndependent-Sample T Test
ConfName would be the grouping variable.
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We would get the same results:
A difference of 17.333 and a 2-tailed p-value of 0.059.
So why do we bother with regression and dummy variables at
all?
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Greenland has the fastest moving glaciers in the world.
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Multiple regression using a dummy variable.
Lets go back to predicting wins.
Before, we modelled wins using goals for (GF) and goals
against (GA). Now we can consider conference alongside
everything else.
Your conference (East or West) is part of what determines the
teams you play against. Teams that play against weak
opponents tend to win more.
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Will conference explain anything about wins that Goals For and
Goals Against cant?
In an SPSS multiple regression, we just include the dummy
variable in the list of independents like everything else.
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First, the model summary.
Considering goals for, goals against AND conference.
82.9% of the variance in the number of wins can be explained
by these three things together.
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Going back to last day, considering only Goals For and Goals
Against, we also got an R square of 0.829.
In other words, adding conference into our model told us
nothing moreabout wins than goals werent already
covering.
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The R square of the model is the same with or without
conference.
That means just as much variance is explained by considering
only goals for/against as by considering both goals for/against
and the conference of the team.
Conference contributes nothing extra.
This is probably because the strength of your opponents is
already reflected in the goals for / goals against record. Its not
like goals against weak teams count for more.
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The coefficient table for Wins as a function of Goals
For/Against and Conference:
The fact that conference isnt improving the model any isreflected in its significance.
If its slope were really zero, wed still a sample like this .952 of
the time. (p-value = .952)
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The regression equation is:
(Estimated Wins) =
37.637 + 0.178(GF)0.167(GA) + 0.082(West Conf.)
Meaning being in the west meant winning 0.082 more games.
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But
(Estimated Wins) =
37.637 + 0.178(GF)0.167(GA) + 0.082(West Conf.)
is more complicated than
(Estimated Wins) =
37.950 + 0.177(GF)0.163(GA)
which is the model from last day that ignored conference.
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But knowing the conference doesnt change anything.
-The r2was .829 whether we included conference or not.
-We failed to reject the null that the effect of conference
was zero (controlling for Goals For/Against ).
In that case, we can use the simpler model that only uses goals
and not lose anything. We should always opt for a simpler
model when nothing is lost in doing so.
This is called theprinciple of parsimony.
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"Make everything as simple as possible, but not simpler."
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Nikola Tesla Albert Einstein
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Comments about r2in multiple regression.
Like with single variable regression, r2must be between 0 and
1.
0 is none of the variance is explained.
1 is all of it is explained.
If you add more and more variables into your model, you willeventually reach r
2= 1, where you have enough data to model
and predict the response perfectly.
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But each variable uses up a degree of freedom and makes the
results harder to interpret.
Just because you can include a variable doesnt mean you
should.
(Resting heart rate) = a + b1(Age) + b2(Body Mass Index) + b3(L
of Oxygen per Minute) + b4(Height) + b5(Number of Freckles) +
b6(Enjoyment of Sushi) + b
7(Kitchen Sinks Owned)
Again, this violates theprinciple of parsimony.
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More regression practice.
From dragons.sav, we have the weight of bearded dragons as a
function of their age, length, and sex.
What is the intercept?
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Weight of beardies as a function of age, length, and sex.
What is the intercept?
-551.125
What does it mean?
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Weight of beardies as a function of age, length, and sex.
What is the intercept?
-551.125
What does it mean?
A malebearded dragon with 0 years, 0 length,weighs
negative 551 grams. (not real-world useful)
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How much heavier is a bearded dragon if it ages two years anddoesnt get any longer or change sex? (On average)
The slope for age is 17.191, so a dragon would get
2 * 17.191 = 34.382 grams heavier with 2 extra years
(controlling for length and sex)
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Is there a significant difference in weight between male andfemale dragons of the same age and size?
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Is there a significant difference in weight between male andfemale dragons of the same age and size?
No. The p-value against there being no difference is .441, so
wefail to reject that null.
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What does the regression equation look like?
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What does the regression equation look like?
(Esimated Weight) =
-551.1 + 17.1(Age) +34.3(Length) + 4.9(Female)
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How much does the average bearded dragon weight if hes..
-Male
-3 Years Old
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24 cm long
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How much does the average bearded dragon weight if hes..
-Male
-3 Years Old
-24 cm long
(Esimated Weight) =
-551.1 + 17.1( 3) + 34.3( 24 ) + 4.9( 0 )
= 323.4 grams
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Is there a model that likely works just as well but is simpler?
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Yes. Its likely that a model without considering sex would
explain nearly as much of the variance.
From model summaries:
Model with Age, Length, Sex: r2
= .912
Model with Age, Length: r
2
= .912 (Not always so exact)
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For interest: Nominal data of 3+ categories.
Dummy variables HAVE to be 0 or 1. If not, youre treating
nominal categories as if they have some sort of order.
If you have 3 categories, you need 2 dummy variables.
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Each of the dummy variables is 1 only when a particular
category comes up, and 0 all the other times.
One of the categories is considered a baseline, or starting
point. All of the dummy variables will be 0 for that category.
(Here: Blue is the baseline, all the dummy variables are 0 for it)
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Since a colour cant be red and green at the same time, only
one of the dummy variables will ever be 1 for a particular case.
Doing a linear model with just these two dummy variables
would look like:
=a + b1(Red) + b2(Green)Which would be
= a for blue cases.
= a + b1 for red cases.
= a + b2 for green cases.
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=a + b1(Red) + b2(Green)
a , the intercept, the value when Red=0 and Green=0
is the average response for blue cases.
b1 is the average increase/decrease in the response when
the case is green instead of blue.
b2 is the average increase/decrease in the response when
the case is red instead of blue.
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Next time: Midterm 2 post-mortem.
Reintroduction to contingency, Odds and Odds Ratios.