1(@8 gyiecless / Demographer: Statistician: Dem Stat Dem lst later Xf. Biata ORDER. At A CAUTIONARY TALE Good morning, Mr. Statistician, you are just the person I wanted to see. Good morning; what can I do for you? Well, I need to know how to measure association in a contingency table. I am interested in seeing whether first sons follow in their fathers' footsteps more often than second and later sons, so I have classified both fathers and sons as having either 'professional' or ‘non-professional’ occupations, and then recorded whether father and son were in the same occupation or not, not forgetting to separate first sons from the others. I see. And now you have a 2X 2 contingency table to interpret? Yes, here it is: Occupation Same Different sons 600 700 1300 sons 700 600 1400 1300 1300 2600 CS) UL 14 19685 a7 or Ce bp re ee Knap Lay SU Nw ad 7 Stat Dem Stat Dem Stat Dem Itm puzzled, because I should have thought that first sons would tend to follow their fathers' occupation more often than later sons, but they don't seem to. How should I measure the strength of the effect? Chi-squared comes to 200/13, if my ability to do sums in my head is not impaired at this early hour, which is 15.4, a very significant value for one degree of freedom. So you certainly need a measure of association. But let's skip that for a moment; why do you think you have found such an unexpected result? I don't know at all. I know about biassed samples, so I was careful about that. Do you know about stratification? Well, I've heard of it, but isn't that to do with correlation and heterogeneity? Yes, it is. What happens if you look at the contingency tables for the fathers' occupations separately? A professional one, and a non-professional one. I don't know; I am not interested in which occupation, but only if father and son had the same occupation. But we can look at the original figures: De Father's Occupation Professional ist sons - later sons Son's Occupation same different 300 LOO 400 600 300 900 900 400 1300 Father's Occupation Non-Professional ist sons later sons Stat Now they are very interesting figures. Sonts Occupation sane different 300 600 900 100 300 400 400 900 1300 Don't you see that the proportion of first sons with their fathers' occupation is greater than the proportion of later sons with their fathers' occupation, in both cases, just as you had expected? Dem Stat Dem Stat Dem Stat Dem Stat Dem Why, that's extraordinary! No, it's absurd! If heterogeneity can cause that, I'll eat my hat. Let's look at the strength of the association in both cases. The sample sizes are the same, so we can use chi-squared can't we? We can indeed, and in both cases we find a value of 9.02, very significant, and the association is in the same sense in each case. But if the values are the same, that means there's no heterogeneity. Oh, this is more than I can stand! I have two contingency tables in full agreement about the association, but when I ask them the same question together I get a silly answer. Is it so silly? Of course it is. I don't know whether I'm standing on my head or my heels. Exactly what was your question? I just wanted to know whether first sons followed in their fathers' footsteps more often than the others. I see. Well, the answer, according to your data, is no, they don't. (sarcastically) Very funny. What's true of the two halves of the population isn't true of the whole. You statisticians should go ANd. .eces * Stat D'you know, I think you're upset about the answer to your question because you really didn't mean to ask that question at all. What you really wanted to know was whether first sons followed in their fathers' footsteps more often than later sons, other things being equal. Dem What do you mean, ‘other things being equal'? Stat Let's look at the figures again. First, forget about the son's occupation. Then we have: Father's Occupation Professional Non-Professional number of lst sons 400 g00 1300 number of later sons 900 400 1300 1300 1300 2600 You see, the distribution of family sizes must be quite different according to the father's occupation, In the professional class larger families seem to be the rule, whilst amongst the non- professional class there were at least five hundred only sons. As a result of this there is a pretty high association between being a first son and having a father in a non-professional occupation. Dem Go on. Stat Dem Stat Dem Secondly, forget about the son's birth-order. Then: Father's Occupation Professional Non-Professional Sons Occupation same 900 400 1300 . different 400 900 1300 ‘1300 1300 2600 We find that sons of fathers in the professions tend to become professional people, but that sons of fathers not in the professions tend to break away and become professional also. This, of course, is due to the well-known fact that a larger and larger proportion of people is employed in the professions. Go and teach your grandmother to suck eggs. Patience, please! Now if there is a high association between being a first son and having a father in a non-professional occupation, and at the same time a high association between having a father in a non-professional occupation and a son in a different occupation, then it's not really surprising to find a high association between being a first son and having a different occupation from your father. No ... I mean yes, well, I suppose so. Stat Dem Stat Dem Stat Dem And that's just what you found. | Was it? In spite of the fact that first sons do tend to follow in their father's footsteps in both the professional and non-professional classes. Yes... 1 mean no. What time is it? So when I say ‘other things being equal' I mean that, so far as the data will permit, the effects of subsidiary correlations should be elimintated. So if I were you I would ask a slightly different question, one which is, I suspect, more meaningful from a demographic point of view. (after long pause) Ah, now I see! You mean that I should ask whether the first sons of bus conductors are bus conductors more frequently than the later sons, and whether the first sons of bus drivers are bus drivers more frequently than the later sons, but on no account must I ask whether the first sons of the employees of & bus company more often follow their father's occupation than the later sons? Stat No, that's not what I mean at all. What I want you to do is simply to ask the question to which you want an answer, and not some other question. You see, if we further subdivide the bus drivers into tall ones and short ones, we will get yet another answer, won't we? Dem I see. Ask a silly question and you'll get a silly answer. I wasn't brought up to think this clearly. Stat So your father wasn't a statistician? Tell me, are you an only Bon? Analysis of First Versus Later Sons' Tendency to Follow their Fathers! Occupation. Raw data. Eleven 2 xX 2 tables (one for each occupational class) each organized as follows: Occupations of Father and Son same different lst sons a b later sons c a Object of Analysis. To determine in which (if any) of the 2x2 tables the association should be judged significant. Measure of Association. log ab/cd, with large sample variance © Ire + oir + Ole 1 + Z- Father's Occupation _ Association 5.D. a Professional D135 2151 14, 37O*** Farm 305 .092 10.9'79%#* Managers 02T - 108 063 Clerical “157 294 £285 Sales 124 229 2292 Crafts -.035 .086 164 Operatives — -.2he -102 5.592* Service . 405 .266 2.298 Farm Labor L152 -429 125 Laborers - .069 .285 .058 Weighted Mean 08k -O43 3,826 Total Xo 3h .232*** Weighted Mean ~ 5.826 Heterogeneity % 30. YO6*** Notes. (1) The occupational class ‘household’ has been omitted because of insufficient data. (2) The < values have been calculated from the associations and their variances, but should not differ sensibly from the contingency ~ values. (3) The weighted mean association has not been calculated from the 2x 2 table formed by summing the separate tables, but by forming the weighted mean of the separate associations. The former technique would have been incorrect. (4) The calculations have not been checked. Conclusion. There is too much heterogeneity in this matter between occupational classes for an overall measure of association to be valid. However, two classes are so outstanding that there is little doubt that when the father is professional or a farmer, the first son tends to Pollow in the same occupation move frequently than later sons. The remaining classes can probably. be regarded as homogeneous and showing no such effect. The difference in this tendency between the professions and farmers is not significant. PROFESSIONAL _- SERVICE @ CLERICAL ° FARM LABOUR SALES ; MANAGERS — . Degree J omoaaiio Hus tisr rtiwderd erron, CRAFTS LABOURERS OPERATIVES