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Logistic regression predicts
the probability of the outcome

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variable being true.

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In this example, a
logistic regression model

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would predict the
probability that the patient

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is receiving poor care.

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Or if we denote the
PoorCare variable

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by y, the probability
that y = 1.

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For the remainder
of this lecture,

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we will denote poor care by
1, and good care by zero.

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So, since the outcome
is either 0 or 1,

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this means that the probability
that the outcome variable is 0

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is just 1 minus the probability
that the outcome variable is 1.

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So by predicting the
probability that y = 1,

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we also get the
probability that y = 0.

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Just like in linear
regression, we

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have a set of independent
variables, x1 through xk,

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where k is the total number of
independent variables we have.

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Then to predict the
probability that y = 1,

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we use what's called the
Logistic Response Function.

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This seems like a complicated,
nonlinear equation,

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but you can see the
familiar linear regression

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equation in this Logistic
Response Function.

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The Logistic
Response Function is

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used to produce a
number between 0 and 1.

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Let's understand this
function a little better.

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This plot shows the
logistic response function

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for different values of the
linear regression piece.

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The logistic response
function always

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takes values between 0
and 1, which makes sense,

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since it equals a probability.

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A positive coefficient
value for a variable

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increases the linear
regression piece,

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which increases the
probability that y = 1,

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or increases the
probability of poor care.

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On the other hand, a
negative coefficient value

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for a variable decreases
the linear regression piece,

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which in turn decreases
the probability that y = 1,

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or increases the
probability of good care.

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The coefficients, or
betas, are selected

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to predict a high probability
for the actual poor care cases,

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and to predict a low probability
for the actual good care cases.

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Another useful way to think
about the logistic response

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function is in terms of
Odds, like in gambling.

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The Odds are the
probability of 1

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divided by the probability of 0.

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The Odds are greater than 1 if 1
is more likely, and less than 1

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if 0 is more likely.

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The Odds are equal to 1 if the
outcomes are equally likely.

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If you substitute the
Logistic Response Function

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for the probabilities
in the Odds

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equation on the
previous slide, you

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can see that the Odds
are equal to "e" raised

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to the power of the linear
regression equation.

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By taking the log of both
sides, the log(Odds),

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or what we call the
Logit, looks exactly

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like the linear
regression equation.

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This helps us understand how the
coefficients, or betas, affect

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our prediction of
the probability.

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A positive beta value
increases the Logit,

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which in turn increases
the Odds of 1.

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A negative beta value
decreases the Logit,

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which in turn, decreases
the Odds of one.

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In the next video, we'll build a
logistic regression model in R,

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and predict the quality of care.