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yosimitsu kodanuri
yosimitsu kodanuri
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(edit: below this line I provide additional information in order to address considerations raised in the answer of @Ben which might be relevant to future discussants too)

"[...] Since you don't specify your sampling frame in the question, it is a bit hard to tell what exactly these data represent"

The data are a collection of (quarterly) time series documenting the absolute numbers of population, labour force, employed, unemployed and inactives for (eight) categories of educational attainment, (thirteen) districts and (six) 5-year age slots from 15-65+.

They are contained in three tables with eight, thirteen and six rows respectively and five columns each (per quarter).

In all cases, total population is decomposed into disjoint subsets of sums of individuals (eg persons with post-graduate qualification in total population or persons from District A in inactives, or unemployed aged 20-24 years old, respectively).

"[...]Since you have the counts inside the 2×2 contingency table, and not just their marginal totals, you already have a perfectly good basis for estimating the joint and conditional probabilities for the covariates in your data"

There are no observations for the number of people eg in District A who have finished primary education; that is, the number of people denoted by $x, y, a$ and $b$ in the simplified table I have included in the question, are not given;

the problem I'm facing is how to derive those joint frequencies, denoted in the question as $p_x, p_y, p_a$ and $p_b$.

"[...] If you want to find probabilities of unemployment conditional on these covariates then you will also need some data for employed people with these covariates."

I interpret this as in the following example:

Let UA denote "unemployed in District A" and UB denote "unemployed with a university degree"; also let U denote "Beeing unemployed", A denote "in District A" and B denote "with a university degree".

Then, $Pr(UA \cap UB)=Pr(U|A) \times Pr(A) + Pr(U|B) \times Pr(B)$,

where $Pr(A)$ equals "persons in District A / population" and $Pr(B)$ equals "persons with a university degree / population".

If my interpretation is correct, how should one calculate the conditional probabilities?

My understanding is that my data tables provide figures for the joint distribution of characteristics (eg number of people unemployed AND with a university degree) not conditional distributions.

(edit: below this line I provide additional information in order to address considerations raised in the answer of @Ben which might be relevant to future discussants too)

"[...] Since you don't specify your sampling frame in the question, it is a bit hard to tell what exactly these data represent"

The data are a collection of (quarterly) time series documenting the absolute numbers of population, labour force, employed, unemployed and inactives for (eight) categories of educational attainment, (thirteen) districts and (six) 5-year age slots from 15-65+.

They are contained in three tables with eight, thirteen and six rows respectively and five columns each (per quarter).

In all cases, total population is decomposed into disjoint subsets of sums of individuals (eg persons with post-graduate qualification in total population or persons from District A in inactives, or unemployed aged 20-24 years old, respectively).

"[...]Since you have the counts inside the 2×2 contingency table, and not just their marginal totals, you already have a perfectly good basis for estimating the joint and conditional probabilities for the covariates in your data"

There are no observations for the number of people eg in District A who have finished primary education; that is, the number of people denoted by $x, y, a$ and $b$ in the simplified table I have included in the question, are not given;

the problem I'm facing is how to derive those joint frequencies, denoted in the question as $p_x, p_y, p_a$ and $p_b$.

"[...] If you want to find probabilities of unemployment conditional on these covariates then you will also need some data for employed people with these covariates."

I interpret this as in the following example:

Let UA denote "unemployed in District A" and UB denote "unemployed with a university degree"; also let U denote "Beeing unemployed", A denote "in District A" and B denote "with a university degree".

Then, $Pr(UA \cap UB)=Pr(U|A) \times Pr(A) + Pr(U|B) \times Pr(B)$,

where $Pr(A)$ equals "persons in District A / population" and $Pr(B)$ equals "persons with a university degree / population".

If my interpretation is correct, how should one calculate the conditional probabilities?

My understanding is that my data tables provide figures for the joint distribution of characteristics (eg number of people unemployed AND with a university degree) not conditional distributions.

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yosimitsu kodanuri
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Is it possible to derive joint probabilities from marginals with assumptions about the conditionals?

I understand the title is too generic. I tried to look for similar questions and although there were a few that were seemingly about the same issue, either they provided answers in the negative or had no convincing answers or they suggested the use of copulas.

Since I have no working knowledge of copulas if they are actually the answer to my problem I am going to have to invest some time in getting acquainted with them, but before I do I would like to know if I should indeed invest the time, in the first place. Hence this question.

I have a population of individuals with a certain number of characteristics eg unemployed persons over some period of time; I know how many of them are located in a certain district (characteristic #1) also I know how many of them have achieved a certain education level eg MSc or relevant level (characteristic #2) but I don't have data on location and education for the same individual.

Given that the available info is something like the following table (for simplicity I don't include all the relevant characteristics-just 'location' (rows) and 'education' (columns)):

              | "MSc or higher"   "other edu"  |   sum 
   ___________|________________________________|_______________________________________      
   "Region A" |       x               a        |   n_A    (unemployed in region A)            
              |                                | 
   "rest regs"|       y               b        | n_U-n_A  (unemployed in other regions)
   ___________|________________________________|_______________________________________
      sum     |     n_MSc          n_U-n_MSc   |  n_U     (unemployed persons)
              | (unemployed       (unemployed  |
              |  with MSc)         with other  |
              |                    education)  |
  1. is it warranted to claim that eg $\frac{n_{MSc}}{n_{U}}$ is a measure of the risk of unemployment that a person with an education level equivalent or better than a MSc degree faces? Similarly, is eg $\frac{n_{A}}{n_{U}}$ a measure of the risk of unemployment for a person situated in Region A?
  2. If the table above is reinterpreted as representing the unemployment risk associated with the relevant cell each time (ie if we divide the rightmost column and bottom row with $n_U$ to obtain marginal prrobabilities for the corresponding rows/columns and replace $x,y,a$ and $b$ with $p_x,p_y,p_a$ and $p_b$-the respective--unknown--joint probabilities) is there a way to retrieve those joint probabilities using only what information is contained in the tables presented above?
  3. Are there plausible assumptions/restrictions that would assist or facilitate the calculations for finding the joint probabilities (eg some proposed/assumed relation between conditional frequencies) within reasonable bounds and for the purpose of having a rough estimate of what the actual figures about the joint instances of characteristics would be eg if more refined data sources (eg data sources detailing those joint frequencies) are considered?

( I apologize for the crude table layout but I was unable to use latex properly )