# INTERPRET ODDS RATIOS : What coefficients tells about monthly churn (people leaving)?

I am wondering how I can interpret results from GLM. To begin with I have 9 % monthly churn (people leaving).

I am taking as an example. The coefficient for profileid_videos says that holding all variables constant the odds of churning (churn = 1) over the odds of not churning (churn = 0) is exp(-1.99) = 0.1366954.

In terms of percent change, we can say that the odds for not churning are 86.3 % higher than the odds for churning for a one-unit increase in profileid_videos.

Now I want to know how much it has affected 9 % churn? How can I quantify my monthly churn? For example. My churn = 9% (=success). The odds are (0.09 / 0.91) = 0.1: 1 (it is possible I am finding it hard to put into words).

Or how I can interpret/relate the GLM results which are more understandable to the business manager?

enter code here##
##
## Call:
## NULL
##
## Deviance Residuals:
##     Min       1Q   Median       3Q      Max
## -3.1533  -0.5835  -0.2776  -0.0091   6.6285
##
## Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)          1.3156243  0.0430323  30.573  < 2e-16 ***
billcycle_number    -0.1739868  0.0027824 -62.531  < 2e-16 ***
avg_nr_streams       0.0052668  0.0008555   6.156 7.44e-10 ***
avg_nr_dist_titles   0.0832834  0.0036814  22.623  < 2e-16 ***
nr_streams_vod0     -0.0115866  0.0009162 -12.647  < 2e-16 ***
nr_titles_vod0      -0.0859846  0.0041068 -20.937  < 2e-16 ***
profileid_videos    -1.9924776  0.0266360 -74.804  < 2e-16 ***
avg_compl_rate      -0.0047955  0.0005568  -8.613  < 2e-16 ***
---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 77340  on 70267  degrees of freedom
## Residual deviance: 51920  on 70260  degrees of freedom
## AIC: 51936
##
## Number of Fisher Scoring iterations: 6


I would appreciate your explanation/exploration how these variables affect churn and how much it is reduced/increased. I need some number to explain to non-statistician to make a decision.

• Sorry I don't see this -.35 coefficient in your output. Also, why don't you use the option to get odds ratio directly instead of having to calculate them? – Federico Tedeschi Apr 20 '18 at 12:06
• I do not think your third pararagraph is correct. That would be OR=1.35 not 0.35. – mdewey Apr 20 '18 at 12:16
• @ Federico Tedesch thanks for your comments. I can get odd ratios as well. but my question is how to translate them into more simple and conclusive statements. – James Taylor Apr 20 '18 at 12:56

Let's take a quick step back. A logistic regression takes the following form:

$$\text{log}\frac{\pi}{1-\pi} = \beta_0 + \beta_1 x_1 + \beta_2x_2...$$

where $\pi = P(Y=1 | X)$, where $Y=1$ is churn and $Y=0$ is not churn.

Next, we know that the odds in favor of $Y=1$ can be written as:

$$\text{odds}(Y=1|X) = \frac{P(Y=1|X)}{P(Y=0|X)} = e^{\beta_0 + \beta_1 x_1 + \beta_2x_2+...}$$

In other words, a unit increase in $x_i$ increases the odds of churn - multiplicatively - by $e^{\beta_{i}}$, holding all other covariates equal.

First, let's get the odds in favor of churning:

$$\text{odds}(Y=1|X) = \frac{P(Y=1|X)}{P(Y=0|X)} = \frac{0.09}{1 - 0.09} = 0.0989$$

For your example, profileid_videos = -1.9924. So you correctly conclude that the odds in favor of churning "increase" by a factor of $e^{-1.9924} =$ 0.136 for every 1 unit increase in $x_{video}$. That is, the odds in favor of churning get multiplied by 0.136 to yield 0.0134. This is your new odds of churn.

You can say the odds of churn decrease by $\frac{0.0989 - 0.0134}{0.0989} = 0.86$. Which is equivalent to your conclusion, saying the odds of not churning increase by 86%.

To convert this back to probability, we just apply the relationship above:

$$\text{odds}(Y=1|X) = \frac{P(Y=1|X)}{1 - P(Y=1|X)} = 0.0134$$

And solve for $P(Y=1|X) = 0.0132$. This is the new probability of churn, given a 1 unit increase in $x_{video}$. There are a number of ways to interpret this new figure. One way, could be that the probability of churn decreased by $\frac{0.09 - 0.0132}{0.09}=0.853 = 85.3\%$

• thanks very much indeed @ilanman for such a detailed explanation. So, this variable takes the probability of 0.0134. Does this mean it has a strong effect on decreasing churn? Or we can say it has a small effect on non-churner. So, this variable could be major contributor to explain non-churner. So the business manager can give priority to this variable. Say if we increase profileid_videos we might bring churning significantly? – James Taylor Apr 20 '18 at 17:23
• The probability of churn is 0.0132. The odds in favor of churn is 0.0134. In terms of the strength of the effect, that's up to you and the business. What does strong or weak mean in this case? This variable is probably a good candidate for being a major contributor to decreasing churn. Before making too many strong conclusions, I might investigate what this variable represents and see if it makes intuitive or business sense. Feel free to accept the answer if it answers your question. – ilanman Apr 20 '18 at 17:35