Probabilities for a single variable

A model for two categorical variables is characterised by the joint probabilities pxy. However we sometimes want to restrict attention to one of these variables on its own. The marginal probabilities for the variable X are defined and interpreted in a similar way to the marginal proportions that were defined earlier for bivariate categorical data.

The marginal probability, px, for the variable X is the proportion of (xy) pairs in the population for which the value of X is x . For example, consider a situation where we are interested in hair colour and eye color of teenagers. The number of blue-eyed teenagers is the sum of those with either (blue eyes and blonde hair) or (blue eyes and brunette hair) or ... or (blue eyes and red hair),

nblue eyes  =  nblue, blonde  +  nblue, brunette  +  ...

The same holds for the proportion with blue eyes — its marginal probability,

pblue eyes  =  pblue, blonde  +  pblue, brunette  +  ...

This is generalised with the formula

where the right of the equation denotes summing the joint probabilities over all possible values of y. There is a similar formula for the marginal probabilities of the other variable,

Support and grief state after neonatal death

It is difficult to find illustrative examples since population probabilities are unknown in most 'interesting' applications. The following example is based on a real data set which classifies 66 mothers who had suffered a neonatal death by the level of support that they were given and their grief state.

  Support
Grief state Good Adequate Poor
I 17 9 8
II 6 5 1
III 3 5 4
IV 1 2 5

We do not know the underlying population joint probabilities for mothers who had neonatal deaths in general. However, to provide an illustrative example, we will pretend that the population probabilities are equal to the proportions in this data set. For example, we will pretend that the joint probability for one such person getting adequate support and having grief state II is 5 / 66 = 0.0758.

Probabilities in general
  Support  
Grief state Good Adequate Poor Total
I 0.2576 0.1364 0.1212 0.5152
II 0.0909 0.0758 0.0152 0.1818
III 0.0455 0.0758 0.0606 0.1818
IV 0.0152 0.0303 0.0758 0.1212
Total 0.4091 0.3182 0.2727 1.0000

The two marginal totals (red and orange) of the table give the marginal probabilities for the two variables. For example,

The diagram below illustrates the summing of joint probabilities to give marginal ones with a 3-dimensional barchart of the joint probabilities.

Click the formula for the marginal probabilities of 'X' (the level of support) on the right. The bars stack to show the marginal probabilities for the level of support.

Similarly, clicking the formula for the marginal probabilities of 'Y' stacks the bars to show the overall probabilities that a women is in the various grief states after a neonatal death.