# How can the Poisson GLM be used instead of logistic regression for the Titanic survival data?

I have a professor who made a very good point about the data titanic analysis during a lecture this week. I am still however trying to better understand. He argued that it is also possible to have a model of titanic data (GLM) using Poisson rather than the typical Binomial approach. I am curious to know what other people views are. His focus was on the Research Question: What characteristic of passengers survived more than others.

(i) The examination of how survival of the passengers relates to their age, sex and passenger requires the use rate=Survived/passenger.

I initially thought that this would be binomial rather than poisson. Each event either does or does not happen. There are no multiple events. However, the lecturer argued about the count data structure.

I would really appreciate whether someone could explain why Poisson would work as well.

Dataset:

   Age_Group     Mean_Age P_Class Gender Survived Passengers
1  00to13years      7.5   first female        1          2
2  00to13years      6.8   first   male        5          5
3  00to13years      6.0  second female       14         14
4  00to13years      2.5  second   male       11         11
5  00to13years      5.0   third female       15         37
6  00to13years      6.1   third   male       15         45
7  14to19years     17.8   first female       13         13
8  14to19years     17.8   first   male        1          4
9  14to19years     17.4  second female       10         11
10 14to19years     17.6  second   male        1         13
11 14to19years     17.1   third female       25         40
12 14to19years     17.5   third   male        5         75
13 20to24years     22.5   first female       20         20
14 20to24years     22.8   first   male        4         12
15 20to24years     22.6  second female       17         18
16 20to24years     22.4  second   male        2         25
17 20to24years     21.9   third female       25         51
18 20to24years     21.8   third   male       19        126
19 25to29years     26.7   first female       10         11
20 25to29years     27.4   first   male       10         16
21 25to29years     27.3  second female       14         18
22 25to29years     26.7  second   male        4         33
23 25to29years     26.9   third female       20         34
24 25to29years     27.1   third   male       18        102
25 30to34years     31.7   first female       17         17
26 30to34years     31.2   first   male        5         15
27 30to34years     31.7  second female       14         16
28 30to34years     31.5  second   male        4         29
29 30to34years     31.6   third female       10         19
30 30to34years     31.9   third   male       12         61
31 35to39years     37.1   first female       24         25
32 35to39years     36.9   first   male       11         27
33 35to39years     36.4  second female       10         11
34 35to39years     36.6  second   male        0         16
35 35to39years     37.2   third female        6         12
36 35to39years     36.8   third   male        4         36
37 40to49years     45.3   first female       28         28
38 40to49years     44.9   first   male       19         54
39 40to49years     44.5  second female       12         13
40 40to49years     43.6  second   male        1         19
41 40to49years     43.4   third female        3         18
42 40to49years     43.2   third   male        2         37
43 50yearsplus     56.4   first female       26         28
44 50yearsplus     58.0   first   male        7         44
45 50yearsplus     56.0  second female        4          6
46 50yearsplus     57.3  second   male        1         18
47 50yearsplus     61.8   third female        1          4
48 50yearsplus     58.3   third   male        0         13


For tabular data of the following form:

$$\begin{array}{c|ccc} & Y & \bar{Y} & & \\ \hline X & w_{11} & w_ {12} & w_{1.} \\ \bar{X} & w_{21} & w_{22} & w_{2.}\\ & w_{.1} & w_{.2} & w_{..}\\ \end{array}$$

A logistic regression model does not explicitly model the counts of the $X$ which is nice when $X$ is not representative of the population of interest (such as in some matched designs). The parameters estimated are the odds in those with $\bar{X}$ (intercept) and odds ratios (odds in those with $X$ divided by odds in those with $\bar{X}$). The odds are ratios of risks. In a logistic regression model, the outcome $Y$ is a binary 0/1 value where the frequency weights are given by the $w$s.

A loglinear model estimates risks. So, unconditional on $w_{1.}$ or $w_{2.}$ the loglinear model (which is a Poisson model), estimates the risks of having $X$ or $\bar{X}$ in this sample (treating the sample as representative of some generalized population). Similarly for $w_{.1}$ and $w_{.2}$. The estimated values represent risks of outcome versus risks of not having an outcome. Ratios of risks for opposite events are odds, the same odds derived from the logistic model.

In this Poisson model, the $w$s are treated as the count outcomes, and 0/1 indicators of the tabular position of the $Y$ and the $X$ are adjusted. If one further controls for the interaction between them, well you obtain ratios of ratios of risks, or ratios of odds, or an odds ratios.

Thus, using two different modeling techniques you estimate the exact same thing.

Example R code:

x <- sample(0:1, 100, replace=T)
y <- rbinom(100, 1, .3 + .2 *x) ## additive risk model, but still association
df <- aggregate(cbind('w'=rep(1, 100)), list('x'=x, 'y'=y), sum)
fit1 <- glm(y ~ x, family=binomial, weights=w, data=df)
fit2 <- glm(w ~ y * x, family=poisson, data=df)

exp(coef(fit1))

exp(coef(fit2))


With example output

> table(x,y)
y
x    0  1
0 30 18
1 24 28
> exp(coef(fit1))
(Intercept)           x
0.600000    1.944444
> exp(coef(fit2))
(Intercept)           y           x         y:x
30.000000    0.600000    0.800000    1.944444


The one-to-one correspondance of logistic regression parameters to those obtained in poisson regression is a consequence of that which I described above.

The odds ratio: ad/(bc) = 30*28/24/18 = 1.9444 is the measure of association between $x$ and $y$

• Thank you so much AdamO. Now it makes more sense why both approach are equivalent. Commented Dec 16, 2015 at 0:05
• In regards to this Poisson approach, I always thought that Normal Q-Q, Cook's D distance plots were extremely important in order to assess any kind of model. The lecturer however mentioned that in this case an "Absolute" Deviance Residual Plot was enough. (i) I am wondering why? (ii) I am also wondering why use "Absolute Deviance Residual Plot" rather than the normal "Standardised Deviance Residual Plot? Commented Dec 16, 2015 at 0:26

Your professor has created an artificial construct. You have the raw data available to you. Grouping the data is arbitrary and information-losing. It is only the grouping that makes you even entertain a Poisson model. $Y$ is binary. Keep it that way.