The Gamma Hurdle Distribution | Towards Data Science

What result is?

Here is a general scenario: A/B test was held, where a random sample of units (for example, customers) was selected for the campaign, and they received treatment A. Another sample was chosen to receive treatment B. “A” may be communication or proposal and ” B “can be without communication or lack of proposal. “A” may contain 10%, and “B” can be 20%. Two groups, two different methods of treatment, where A and B are two separate methods of treatment, but without loss of community up to more than 2 procedures and continuous treatment methods.

Thus, the campaign passes and the results are available. With our stand -in -system, we can track which of these units have taken interest of interest (for example, made a purchase), and which are not. In addition, for those who did, we register the intensity of this action. A common script is that we can track the amount of purchase for those who have purchased. This is often called the average amount of order or income for the buyer’s metric. Or a hundred different names that mean the same thing – for those who have acquired, how much they spend, on average?

For some cases of use, the marketer is interested in the former metric-purchase machine. For example, in our acquisition campaign, we managed more (potentially for the first time) buyers with treatment A or B? Sometimes we are interested in attracting income to one buyer, so we focus on the latter.

However, more often we are interested in income in an economically effective way, and what we really care about the income that a campaign has been produced generalThe field of treatment a or b cited more income? We do not always have a balanced sample size (possibly from the cost of cost or avoiding risks), and therefore we divide the measured income by the number of candidates that were considered in each group (call these N_a and N_b calculations). We want to compare this measure between two groups, so the standard contrast is simple:

This is only the average income for treatment. Minus the average income for treatment B, where it means that this is perceived throughout the set of target units, regardless of whether they were responsible or not. Its interpretation is also simple – what is the average income for increasing the unit that comes out of treatment A compared to treatment B?

Of course, this last measure takes into account both previous ones: the answer speed multiplied by the average income to the respondent.

Uncertainty?

How much the buyer spends very varies, and a couple of large purchases in one treatment group or another can significantly distort the average value. Similarly, the change in the sample can be significant. So, we want to understand how confident in this comparison of the means and quantitatively appreciated the “significance” of the observed difference.

So, you throw data into T-criteria and look at the P-value. But wait! Unfortunately for a marketer, the vast majority of time, the purchase level is relatively low (sometimes very low), and, therefore, there are many zero revenue values ​​- often the vast majority. The assumptions of the T-test can be greatly disturbed. Very large sizes of the sample can come to the rescue, but there is a more fundamental way of analyzing these data that are useful in several ways that will be explained.

An example of a set of data

Let’s start with a data set sample to make things practical. One of my favorite direct marketing data sets is from KDD Cup 98.

url="https://kdd.ics.uci.edu/databases/kddcup98/epsilon_mirror/cup98lrn.zip"
filename="cup98LRN.txt"

r = requests.get(url)
z = zipfile.ZipFile(io.BytesIO(r.content))
z.extractall()


pdf_data = pd.read_csv(filename, sep=',')
pdf_data = pdf_data.query('TARGET_D >=0')
pdf_data['TREATMENT'] =  np.where(pdf_data.RFA_2F >1,'A','B')
pdf_data['TREATED'] =  np.where(pdf_data.RFA_2F >1,1,0)
pdf_data['GT_0'] = np.where(pdf_data.TARGET_D >0,1,0)
pdf_data = pdf_data[['TREATMENT', 'TREATED', 'GT_0', 'TARGET_D']]

In the above fragment of the code, we load the ZIP file (specifically the educational set of data), extracting it and reading it in the Pandas data frame. The nature of this set of the campaign data from a non-profit organization, which was looking for donations through direct mailings. There are no treatment options in this set of data, so we pretend instead and segment data set based on the frequency of past donations. We call this indicator CARE (as categorical and creating Processed Like a binary indicator for “A”). Consider this the results of a randomized control study in which part of the sample population was considered with a proposal, but the rest are not. We track each person and accumulate the number of their donations.

So, if we consider this set of data, we will see that there are about 95,000 people who, as a rule, are distributed equally in two processing:

Treatment A has a greater level of response, but in general, the frequency of answers in the data set is only about 5%. So, we have 95% zeros.

For those who sacrificed, treatment A, in the same way, is associated with the lower average amount of donations.

Combining everyone who was aimed, the treatment of a, in the visible one, is associated with a higher average amount of donations – a higher frequency of answers outweighs the lower amount of donations for respondents, but not many.

Finally, here the histogram of the number of donations is shown, combined on both processing, which illustrates the mass in zero and right skew.

The numerical resume of the two treatment groups quantitatively determines the phenomenon observed above – while treatment A, according to the visible, led to a significantly higher reaction, those that received on average less than a smaller amount on average. The network of these two measures, the one that we ultimately after is the general average donation per target unit is still higher for treatment A. How sure this conclusion is the subject of this analysis.

Gamma is a blessing

One of the ways of modeling this data and the answer to our question of research in terms of the difference between two appeals when creating an average donation per target unit is the distribution of a gamut of revenue. Like a more well -known distribution with a zero bloated Poisson (ZIP) or NB (Zinb), this is the distribution of mixtures when one part belongs to the mass at zero, and the other, in cases where the random variable is positive, the gamma is a function of the function

Here π is the likelihood that a random variable y is> 0. In other words, this is the probability of a gamma process. Similarly, (1- π) is the likelihood that a random variable is zero. From the point of view of our problem, this applies to the probability that the donation has been done, and if so, then this is a value.

Let’s start with the component parts of the use of this distribution in regression – logistics and gamma -regression.

Logistic regression

The Logit function is the function of the link here, connecting the chances of a magazine with a linear combination of our variables of the predictors, which with one variable, such as our binary treatment indicator, looks like:

Where π represents the likelihood that the result is “positive” (designated as 1) an event, such as a purchase and (1-note), is a probability that the result is “negative” (indicated as 0) event. In addition, π, which is an interesting interest, is determined by the reverse function of the logat:

The installation of this model is very simple, we need to find the values ​​of two beta versions that maximize the probability of data (result Y), which suggest that observations N IID are:

We could use any of the several libraries to quickly comply with this model, but demonstrate PYMC as a means to create simple Bayes logistics regression.

Without any usual stages of the Bayesian work process, we adjust this simple model using MCMC.

import pymc as pm
import arviz as az
from scipy.special import expit


with pm.Model() as logistic_model:

    # noninformative priors
    intercept = pm.Normal('intercept', 0, sigma=10)
    beta_treat = pm.Normal('beta_treat', 0, sigma=10)

    # linear combination of the treated variable 
    # through the inverse logit to squish the linear predictor between 0 and 1
    p =  pm.invlogit(intercept + beta_treat * pdf_data.TREATED)

    # Individual level binary variable (respond or not)
    pm.Bernoulli(name="logit", p=p, observed=pdf_data.GT_0)

    idata = pm.sample(nuts_sampler="numpyro")

az.summary(idata, var_names=['intercept', 'beta_treat'])

If we build the contrast of two average response indicators, we find that, as expected, the average increase in the response frequency for treatment A is 0.026 more than treatment B with a 94% reliable interval (0.024, 0.029).

# create a new column in the posterior which contrasts Treatment A - B
idata.posterior['TREATMENT A - TREATMENT B'] = expit(idata.posterior.intercept + idata.posterior.beta_treat) -  expit(idata.posterior.intercept)

az.plot_posterior(
    idata,
    var_names=['TREATMENT A - TREATMENT B']
)

Gamma is a register

The next component is the range of distribution with one of its parameterization of the function density, as shown above:

This distribution is determined for strictly positive random values ​​and when used in business for values ​​such as expenses, expenses for the requirements of customers and insurance amounts.

Since the average value and dispersion of the gamma are determined in terms of α and β in accordance with the formulas:

For gamma -regions, we can parameter with the help of α and β or μ and σ. If we make μ, defined as a linear combination of predictors, then we can define gamma from the point of view of α and β using μ:

The model of gamma -regions suggests (in this case, feedback is another common option), a link to a journal that is intended for the “linarization” of the relationship between the predictor and the result:

Following almost the same methodology as for the frequency of answers, we limit the data set only respondents and adjust the regression of the gamma using PYMC.

with pm.Model() as gamma_model:

    # noninformative priors
    intercept = pm.Normal('intercept', 0, sigma=10)
    beta_treat = pm.Normal('beta_treat', 0, sigma=10)

    shape = pm.HalfNormal('shape', 5)

    # linear combination of the treated variable 
    # through the exp to ensure the linear predictor is positive
    mu =  pm.Deterministic('mu',pm.math.exp(intercept + beta_treat * pdf_responders.TREATED))

    # Individual level binary variable (respond or not)
    pm.Gamma(name="gamma", alpha = shape, beta = shape/mu,  observed=pdf_responders.TARGET_D)

    idata = pm.sample(nuts_sampler="numpyro")

az.summary(idata, var_names=['intercept', 'beta_treat'])

# create a new column in the posterior which contrasts Treatment A - B
idata.posterior['TREATMENT A - TREATMENT B'] = np.exp(idata.posterior.intercept + idata.posterior.beta_treat) -  np.exp(idata.posterior.intercept)

az.plot_posterior(
    idata,
    var_names=['TREATMENT A - TREATMENT B']
)

Again, as expected, we see that the average rise for treatment a has an expected value equal to the value of the sample -7.8. 94% of trusting interval (-8.3, -7.3).

The components, frequency of answers and the average amount for the respondent shown above are about as simple as we can get. But this is a direct expansion to add additional predictors to 1) evaluate the conditional average treatment effects (CATE) when we expect that the effect of treatment differs depending on the variables of preliminary processing.

Regression of the obstacle model (gamma)

At this stage, it should be quite easy to see where we progress. For the obstacle model, we have conditional probability, depending on whether the specific observation is 0 or more than zero, as shown above for the distribution of gamma reinforcements. We can install two models of components (logistics and gamma -regression) at the same time. We get free, their product, which in our example is an assessment of the amount of donations per target unit.

It would not be difficult to correspond to this model using the fraudulent function with the switch operator, depending on the value of the variable result, but PYMC is already encoding this distribution.

import pymc as pm
import arviz as az

with pm.Model() as hurdle_model:

    ## noninformative priors ##
    # logistic
    intercept_lr = pm.Normal('intercept_lr', 0, sigma=5)
    beta_treat_lr = pm.Normal('beta_treat_lr', 0, sigma=1)

    # gamma
    intercept_gr = pm.Normal('intercept_gr', 0, sigma=5)
    beta_treat_gr = pm.Normal('beta_treat_gr', 0, sigma=1)

    # alpha
    shape = pm.HalfNormal('shape', 1)

    ## mean functions of predictors ##
    p =  pm.Deterministic('p', pm.invlogit(intercept_lr + beta_treat_lr * pdf_data.TREATED))
    mu =  pm.Deterministic('mu',pm.math.exp(intercept_gr + beta_treat_gr * pdf_data.TREATED))
    
    ## likliehood ##
    # psi is pi
    pm.HurdleGamma(name="hurdlegamma", psi=p, alpha = shape, beta = shape/mu, observed=pdf_data.TARGET_D)

    idata = pm.sample(cores = 10)

If we look at the resume of tracer, we will see that the results are exactly the same for two component models.

As already noted, the average distribution of obstacles of the gamma is π * μ, so we can create a contrast:

# create a new column in the posterior which contrasts Treatment A - B
idata.posterior['TREATMENT A - TREATMENT B'] = ((expit(idata.posterior.intercept_lr + idata.posterior.beta_treat_lr))* np.exp(idata.posterior.intercept_gr + idata.posterior.beta_treat_gr)) - \
                                                    ((expit(idata.posterior.intercept_lr))* np.exp(idata.posterior.intercept_gr))

az.plot_posterior(
    idata,
    var_names=['TREATMENT A - TREATMENT B']

The average expected value of this model is 0.043 with 94% of a reliable interval (-0.0069, 0.092). We could interrogate the back part to see what share of donations to one buyer is predicted, which for the treatment of A and any other decision -making functions that made sense for our case, including the addition of a more complete p & l to the assessment (i.e. including margin and cost)

Notes: some implementations parameterize the Gamma-improving model in different ways, where the probability of zero is π, and, therefore, the average value of gamma-revenge includes (1 -π). Also, note that at the time of writing this article, according to the visible, there is a problem with the samplers of nuts in PYMC, and we had to return to the default Python implementation to launch the above code.

Brief content

With this approach, we get the same conclusion for both models separately and additional benefits from the third metric. The installation of these PYMC models allows us all the advantages of Bayesov’s analysis, including an injection of previous knowledge about the domain and the full back of the answer to questions and quantitatively determine uncertainty!

Loans:

1. All images are authors, unless otherwise indicated.
2. The data set used is taken from the KDD 98 CUP sponsored by Epsilon. https://kdd.ics.uci.edu/databases/kddcup98/kddcup98.html (CC according to 4.0)