Multiple regression is the appropriate statistical tool when your response

variable is quantitative.

If your response variable is categorical with two levels.

We need to use another multivariate tool, logistic regression.

Let's use the nesarc data for an example.

We will use the same stats model formula library which we imported as SMF.

The function that we will use is the logitt function.

Our response variable, nicotinedep is binary.

Coded one equals yes or zero equals no to nicotine dependents.

And so we should use a logistic regression.

We also have an explanatory variable called SOCPDLIFE that

indicated the presence or absence of social phobia.

Which is an anxiety disorder marked by a strong fear of being judged by others and

of being embarrassed.

First we create an object called lreg1 which

will include the results of our logistic regression model.

Then we type an equal sign and then the function to run the logic regression.

smf.logit.

In parentheses we type our formula and the name of the dataset, sub1, and

.fit followed by open and closed parentheses.

The same way we would for linear multiple regression analysis.

Then we ask Python to print the fit statistics for

the logistic regression model.

Let's take a look at the output here.

Similar to the multiple regression output.

You can see the number of observations,

with the complete data that we used in the model.

Here we see the name of our response variable.

Nicotinedep, also is similar to the multi regression output.

We see a table with the prime assessments and the P value.

Notice also that our regression is significant at a P value of less

than 0.0001.

Using the prime assessments we could generate the linear equation.

Nicotinedep is a function of 0.38 plus 1.23

times SOCPDLIFE But let's really think about the equation some more.

>> In a regression module, our response variable was quantitative.

And so it could theoretically take on any value.

In a logistic regression,

our response variable only takes on the values zero and one.

Therefore, if I try to use this equation as a best fit line,

I would run into some problems.

Instead of talking in decimals it may be more helpful for

us to talk about how the probability of being nicotine dependent

changes based on the presence or absence of social phobia.

For example, are those with social phobia more or

less likely to be nicotine dependent than those without social phobia?

Instead of true expected values, we want probabilities.

>> Described visually, we'll no longer find the best fit line,

shown in red, very helpful to us.

As our outcome variable cannot take on any value.

Instead, we're saying that there is somewhere along our X axis.

Where our outcome variable moves from being more likely to be a zero

to be more likely to be a one.

Our goal will be to quantify the probability

of getting a one versus a zero for given value on our X axis.

>> In order to better answer our research question,

we will choose odds ratios as opposed to coefficients.

The odds ratio is the probability of an even occurring in one group

compared to the probability of an event occurring in another group.

Odds ratios are always given in the form of odds and are not linear.

Odds ratios are often a confusing topic for students when they're first introduced

to it so it will be important to go through it conceptually and

better understand exactly what an odds ratio is and what it means.

>> An odds ratio can range from zero to positive infinity.

And is centered around the value one.

If we ran our model and got an odds ratio of one.

It would mean that there's an equal probability of nicotine dependence among

those with and without social phobia.

Those with social phobia are equally as likely to be nicotine dependent

as those without.

It's also likely then that our model would be statistically non-significant.

If an odds ratio is greater than one.

It means that the probability of becoming nicotine dependent increases among those

with social phobia compared to those without.

In contrast, if the odds ratio is below one.

It means that the probability of becoming nicotine dependent is lower among those

with social phobia than among those without.

So how do we calculate the odds ratio?

It is possible to do this by hand.

The odds ratio is the natural exponentiation of our parameter estimate.

Thus, all that we need to do is calculate the natural log

to the power of our parameter estimate.

However, we could also let Python do this for

us by adding the following Python code.

First we ask Python to print the title odds ratios.

Then in the second line of code, we ask Python to print the odds ratios

which are computer using the NumPy.exp, or exponentiate, function.

In parenthesis we add the object that contains the parameter estimates,

P-A-R-A-M-S, from our lreg1 model.

Here are the results.

Because both my explanatory and

response variables in the model are binary, coded zero and one.

I can interpret the odds ratio in the following way.

Young adult daily smokers in my sample with social phobia are 3.4 times more

likely to have nicotine dependence than young adult smokers without sociaphobia.

We can also get a confidence interval for our odds ratio.

Remember that our data set is just a sample of a population.

We do not have every young adult daily smoker in the US in our sample.

Even thought the odds ratio for our sample is 3.4, the true population

odds ratio might be slightly different due to random variation in sampling.

The code to print the confidence intervals for the odds ratio is here.

In the first line of code, we create an object called params.

P-A-R-A-M-S.

That includes the perimeter estimates from our lreg1 logistic progression model.

In the second line of code we create an object called cnof that uses a stats

model conf_int () command to return the confidence levels for

the parameters estimates.

In the third and fourth line of code, we create an odds ration object

with column labels of 'Lower CI' Upper CI and OR.

Finally, print the conference intervals using the numpy.exp function to compute

the odds ratios from the parameter estimates in the conf object.

The odds ratio indicates that there's a 95% certainty that

the 2 population odds ratio fall between 1.78 and 6.61.

It's important to keep in mind that the odds ratio is

simply a statistic calculated for the sample.

>> So looking at the confidence interval, we can get a better picture of how much

this value would change for a different sample drawn from the population.

Based on our model those with social phobia

are anywhere from 1.78 to 6.61 times

more likely to have a nicotine dependence than those without social phobias.

The odds ratio is a sample statistic and

the confidence intervals are an estimate of the population parameter.

Python Lesson 3: Logistic Regression for a Binary Response Variable, pt. 1

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