WEBVTT
Kind: captions
Language: en
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One way to interpret the regression
analysis results from logistic regression is
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to do marginal prediction plots.
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This is a very useful technique
because it's a generic technique.
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Instead of having to memorize,
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how every possible different nonlinear
regression model is interpreted,
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you just need one tool.
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Another advantage is that this
tool gives you the effects
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on the original scale of the dependent variable.
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In the case of logistic regression analysis,
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you will directly see,
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what is the effect of each independent
variable on the predicted probability?
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To do plotting we need some data.
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I will use the Hosmer and Lemeshow data.
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So this is from a widely cited
regression analysis book.
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And the data are about babies born
to different kinds of mothers.
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The dependent variable is,
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whether the baby was born as low birth
weight defined as less than 2.5 kilos.
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And we'll be looking at the weight of
the mother at last menstrual period,
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the race of the mother,
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and whether the mother smoked during pregnancy,
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as our interesting independent variables.
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We are first going to fit a linear probability
model and logistic regression model to this data.
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And I'm using Stata here.
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We have the linear probability model here
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and we have the logistic regression model here.
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And the dependent variable
was the low birth weight.
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And we can see from the linear property model,
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it's easy to interpret,
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we get the predicted probability
of having a low birth weight baby,
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it increases for, it is 0.22 higher
for black women than for white women,
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that is the reference category.
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It is 15% higher for smokers than for non-smokers.
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So we can directly interpret the effects.
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Here, the odds ratios,
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we can say that the odds for a
black mother are 3.5 times greater,
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than for a white mother.
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But that doesn't really tell us anything
about the increase in probability,
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because the odds are a proportional effect,
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you have to know, it's a relative effect.
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You have to know,
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what is the original odds that
is being increased by 3.5?
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Plotting is very useful to understand,
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what do these effects look like?
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So when we compare the effects of race and smoke,
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we can't really, these are not really comparable.
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So it's difficult to say whether a 3.5
increase in odds is a larger effect than
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22% increase in probability,
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because they are expressed in a different scale.
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And we're usually interested in
the original scale of the variable.
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Also, we can't, from this model directly,
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say what is the expected difference between
black smokers and white non-smokers.
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The whites are the base category here,
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so black mothers is a 0.22 and smokers is 0.16,
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so it's about 40% difference between
black smokers and white non-smokers.
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Easy to see from this model.
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Here we say that the black
mother has 3.5 times greater odds,
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and smokers have 2.5 times greater odds.
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So we multiply these together
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and it's about eight or nine,
something like that, times higher
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odds for black smokers than white smokers.
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But that's difficult to interpret.
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So how we can do that is,
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we can apply the marginal predictions plots.
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The Stata's margin command or R's effects
command will do that for you quite easily.
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This is from Stata,
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so this is the linear predictions.
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And we can see from the linear model
that the effect of birth weight here
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is the same for all kinds of mothers.
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So we have three races here,
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and the effect of weight at the last
menstruation is the same for all mothers.
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So the mothers only differ
with respect to the base level.
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So what's the intercept,
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because we estimated the effect of race.
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For the logistic regression
model, we can see that it's
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the same base difference is here,
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but the shape of these curves is different.
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So this is, curves flatten here more,
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and these are lot steeper curves.
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So when we have a mother that doesn't weigh much,
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so these are pounds,
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then for all races,
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the likelihood of having a
low weight baby is large.
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And we can see that for all races
the likelihood gets smaller.
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But also that the likelihood of
probability actually converges here.
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So if you are a very big mother,
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then you're going to have a very big child.
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And which one of these fits the data
better is partly an empirical question.
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So one way to understand,
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which of these plots works better,
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is to plot the data over these plots and just see,
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which two sets of lines explains the data better.
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We can see here that the linear probability model
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predicts the negative probability
for some heavy white mothers.
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And this model always predicts between 0 and 1.
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So this is statistically more appealing.
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But if we don't have any mothers here,
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so if all white mothers are quite light,
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then the fact that we predict implausible values,
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when we go beyond our data,
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is not really a problem.
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So, which one of these is better,
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you can justify based on a theory,
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but you can also check empirically,
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which one fits the data better.
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The logistic regression analysis
is typically used by default,
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because it's a safer choice to apply.
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But this linear probability
model can be used as well,
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as long as you don't do negative predictions,
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or predictions that exceed 1 for
any of the cases in your sample.