Age gaps and Birth order: Reanalysis

This post fol­lows on from my pre­vi­ous post de­tailing some ar­eas where I was un­able to re­pro­duce Scott’s anal­y­sis of how the age gap be­tween siblings mod­ifies the SSC Birth or­der effect. I sug­gest you read that post first but here’s the sum­mary:

I at­tempted to re­pro­duce Scott’s anal­y­sis of Birth or­der effect vs Age gap. I found that:
There ap­peared to be an er­ror in graphs 2 & 3 where peo­ple with one sibling were counted when they shouldn’t have been (graph 2) or were counted twice (graph 3)
Com­par­ing old­est chil­dren to youngest chil­dren causes a bias in the re­sults which can be pre­vented by com­par­ing old­est chil­dren to 2nd old­est children
I was un­able to re­pro­duce Scott’s re­sult on peo­ple re­port­ing 0 year age gap – I get a non-sig­nifi­cant 58% older siblings com­pared to Scott’s 70%. I was un­able to dis­cover the cause of the differ­ence.

Summary

I re­analysed how sibling age gap mod­ifies the SSC birth or­der effect. I found that:

The birth or­der effect is rel­a­tively steady for the first 4-8 years of age gap at about 70% re­spon­dents be­ing the first­born vs sec­ond­born. For larger age gaps the effect re­duces. There is in­suffi­cient ev­i­dence to con­clude how long this re­duc­tion takes or whether the effect is com­pletely re­moved at very large age gaps.

2 other trends were noted in the data but ev­i­dence for them was not strong:

  • The re­duc­tion may not be the same (or might dis­ap­pear) for larger fam­i­lies

  • Birth or­der effect may be lower at 1 year age gap vs 2-7 year age gap

Con­sid­er­ing com­pet­ing the­o­ries on the cause of the Birth or­der effect, two the­o­ries fit the data well:

  • In­tra-fam­ily dy­nam­ics

  • De­creased parental investment

And three the­o­ries fit the data poorly:

  • Changed parental strate­gies

  • Ma­ter­nal an­ti­bod­ies

  • Ma­ter­nal vi­tamin deficiencies

Introduction

The origi­nal rea­son for me look­ing at this data was to analyse whether the data sup­port a sud­den drop be­tween years 7 and 8 or whether there is an al­ter­na­tive ex­pla­na­tion which fits the data.

I will note here that I’m not a trained statis­ti­cian and am us­ing this as prac­tice of Bayesian model com­par­i­son, in­spired by john­swent­worth’s re­cent model com­par­i­son se­quence. I’d say I’m 80% con­fi­dent in my broad con­clu­sions, less so in the speci­fics—I’d be fairly con­fi­dent there are a cou­ple of er­rors lurk­ing in here some­where.

Anal­y­sis: All fam­ily sizes combined

Is there a sud­den drop af­ter 7 years?

Get­ting back to the data, here’s the re­sult that I’m go­ing to fo­cus on, com­par­ing 1st to 2nd chil­dren in all fam­ily sizes:

Eye­bal­ling the graph makes the sud­den drop af­ter 7 years look like the most nat­u­ral ex­pla­na­tion. How­ever, we had no rea­son, a pri­ori, to think that a 7 year age gap would have any spe­cial sig­nifi­cance – a drop could have hap­pened af­ter 1 or 10 years for all we knew.

If we model a sud­den drop af­ter 6 or 8 years the model starts to match the data sig­nifi­cantly less well, any fur­ther away from 7 than that and the model performs re­ally poorly. Although a gen­eral “sud­den drop” model has a high max­i­mum like­li­hood at 7 years, the over­all model like­li­hood is lower due to the lower like­li­hoods for other drop years.

Gen­eral slope model

Imag­ine a model which is similar to a sud­den drop model but the drop is ramped down over a num­ber of years. The model is defined by 4 pa­ram­e­ters – per­centage old­est sibling be­fore the ramp (), per­centage old­est sibling af­ter the ramp (), at what age gap the ramp starts () and over how many years the ramp oc­curs ().

The sud­den drop model is nested within this model—where .

A gen­tler slope doesn’t match the data as closely as a sud­den drop but is less harshly pe­nal­ised over a range of ramp start lo­ca­tions. The graph be­low shows what some years ramps might look like.

Ramp timing and length

To find out which ramp lengths fit the data best I in­te­grate (nu­mer­i­cally) across the first 3 pa­ram­e­ters in this model (, , ) to find which value of the 4th pa­ram­e­ter () pre­dicts the data the best – how sud­den is the drop?

(Notes:

For this anal­y­sis I haven’t grouped the 10+ year age gaps to­gether but used the ac­tual val­ues for the age gaps.

For all calcu­la­tions in this post I as­sume a uniform prior across a rea­son­able range for each pa­ram­e­ter.)

Sur­pris­ingly, the like­li­hood is fairly flat over a large range of slope lengths – ev­ery­thing be­tween 0 and 10 years is within a Bayes fac­tor of 1.15 of each other.

To see what’s hap­pen­ing, let’s in­te­grate over the first two pa­ram­e­ters ( and ) and plot like­li­hood against ramp length () and start ().

This shows a max­i­mum value at , – the sud­den drop af­ter 7 years which is so vi­su­ally no­tice­able in the data.

How­ever, if you fol­low the line along (back of the graph), there is only a small range of val­ues which have a high like­li­hood. Look­ing in­stead along , the max­i­mum like­li­hood is lower (~33% lower), but there is a larger range of val­ues which provide a fairly high like­li­hood. The de­crease in max­i­mum like­li­hood is al­most ex­actly can­cel­led out by the in­crease in the width of the dis­tri­bu­tion.

So a sud­den drop pre­dicts the data ap­prox­i­mately as well as a more grad­ual drop.

We can also in­te­grate across to find the pos­te­rior prob­a­bil­ity of the var­i­ous val­ues.

I’m go­ing to de­scribe this as the ramp start­ing be­tween 4 and 8 years.

Per­centage old­est chil­dren be­fore and af­ter drop

I also in­te­grated over and in or­der to see how like­li­hood varied with and .

is very pre­cisely defined be­tween 0.70 and 0.71.

can take a large va­ri­ety of val­ues, be­tween ~0.49 & 0.62 (90% CI).

In re­al­ity, the Birth or­der effect might de­crease rel­a­tively fast to start with and then more slowly as old­est and sec­ond old­est chil­dren ap­proach par­ity. This is prob­a­bly the kind of thing which we would ex­pect in real life but which can’t be recre­ated with the ramp model.

I cre­ated an ex­po­nen­tial de­cay model (with a de­lay in the de­cay start­ing) to test whether this might be the case and it got a slightly higher over­all like­li­hood than the gen­eral ramp model (Bayes fac­tor 1.5). The start of the de­cline was in the re­gion 3-8 years, similar to the ramp model. The max­i­mum like­li­hood half-life was 5 years al­though this could be any­where be­tween 1.2-11 years (90% CI).

Ex­pected values

Us­ing these mod­els I calcu­lated ex­pected val­ues for Birth or­der effect vs age gap.

This looks fairly sen­si­ble to me. There is a grad­ual start to the slope, be­com­ing steeper into about year 8 and then shal­low­ing out as we get closer to par­ity be­tween older and younger siblings.

At larger age gaps the two mod­els di­verge which is due to a com­bi­na­tion of the differ­ing pri­ors im­plied by the mod­els and the spar­sity of data points in this re­gion—the like­li­hood isn’t suffi­cient to over­come the prior.

Com­par­i­son to con­stant birth effect model

I also com­pared the gen­eral ramp model to a con­stant Birth or­der effect model. The ramp model was preferred over the con­stant model by a Bayes fac­tor of ~1,000.

A con­stant model is ac­tu­ally nested within the ramp model where (and , be­come mean­ingless). This is illus­trated by the red line on the vs & graph where the low like­li­hood can be seen.

Anal­y­sis: Differ­ent sized families

I men­tioned in my pre­vi­ous post that it ap­peared that the drop was pre­sent in sib­ships of 2 but not in sib­ships of 3+.

Break­ing this down fur­ther, we can com­pare this effect for sib­ships of 2, sib­ships of 3 and sib­ships of 4+ (any fur­ther break­down causes the sam­ple sizes to get too small).

(The very low value at 7 year age gap for 4+ chil­dren is only a sam­ple size of 11 so don’t take it too se­ri­ously!)

Here it ap­pears that the drop-off in birth effect for large age gaps be­tween first and sec­ond chil­dren hap­pens in sib­ships of 2 or 3 but doesn’t hap­pen in sib­ships of 4+.

Although the num­ber of sam­ples in the 4+ group with >7 year age gap is only 64, the differ­ence be­tween 2-3 and 4+ sib­ships is sig­nifi­cant at p<0.05 (two-tailed t-test).

This seems an odd phe­nomenon. Would hav­ing ex­tra siblings cause the birth or­der effect be­tween the old­est 2 siblings to re­main high for large age gaps?

See­ing some­thing weird like this in my data causes me to ask “how many things might I have spot­ted dur­ing my work on this pro­ject, if they had co­in­ci­den­tally shown a weird look­ing re­sult?” – when ad­just­ing for post-hoc mul­ti­ple hy­poth­e­sis test­ing I should ad­just not just for the tests that I did but also for the tests I didn’t do just be­cause noth­ing looked odd.

In this case the an­swer is quite a lot so p<0.05 is prob­a­bly not strict enough and my best bet would be that this data oc­curred by co­in­ci­dence.

That’s all a bit hand-wavey so I tried to calcu­late the Bayes fac­tor com­par­ing:

A gen­eral ramp model for all fam­ily sizes

vs

A gen­eral ramp model for fam­i­lies of 2 & 3 chil­dren com­bined with a shal­lower (or no) ramp for fam­i­lies of 4+ chil­dren (Only was changed be­tween the fam­ily sizes)

The lat­ter was preferred by a fac­tor of 5. If I were to in­clude other num­bers of chil­dren when the change might have hap­pened or pos­si­bil­ity that the change hap­pens grad­u­ally as fam­ily size got big­ger then this fac­tor would change but that would start get­ting way too com­pli­cated for me!

I still don’t re­ally be­lieve this an ac­tual effect but if some­one has an ex­pla­na­tion of what might cause this then I’m all ears.

Pos­si­ble lower birth or­der effect for 1 year age gap

One other thing which I no­ticed is the lower Birth or­der effect for age gaps of 1 year as com­pared to gaps of 2-7 years (0.66 vs 0.71 old­est siblings). A quick calcu­la­tion sug­gests Bayes fac­tor comes out at 2 in favour of the Birth or­der effect be­ing lower at 1 year age gap com­pared it be­ing con­stant across 1-7 year age gaps.

Note in this case that al­though the Bayes fac­tor isn’t huge, it seems like this is the kind of thing which might ac­tu­ally hap­pen (some of the po­ten­tial causes would give this a de­cent prior—see sec­tion be­low for more dis­cus­sion) so I’m much less in­clined to just write this one off.

Com­par­ing Ex­pla­na­tions for Birth or­der effect

Scott lists 5 po­ten­tial causes of the Birth or­der effect:

1. In­tra-fam­ily competition

2. De­creased parental investment

3. Changed par­ent­ing strategies

4. Ma­ter­nal antibodies

5. Ma­ter­nal vi­tamin deficiencies

I‘be re­named 1 to “In­tra-fam­ily dy­nam­ics” to in­clude non-com­pet­i­tive in­ter­ac­tions be­tween siblings. A few peo­ple have men­tioned other sibling dy­nam­ics which might cause a Birth or­der effect (e.g. here). The pre­dic­tions of age gap effect from com­pet­i­tive vs non-com­pet­i­tive causes seem similar to me so I’ll lump them to­gether.

My thoughts for what each of the 5 po­ten­tial causes would pre­dict re­gard­ing age gap are given be­low. The con­clu­sions for each po­ten­tial cause end up be­ing very similar to Scott’s (af­ter all that work!) ex­cept that there is no need to pos­tu­late any­thing es­pe­cially sig­nifi­cant about 7 years and that there may be a slight in­crease in birth or­der effect be­tween 1 and 2 years age gap.

In­tra-fam­ily dynamics

Pre­dic­tion: Birth or­der effect re­mains roughly con­stant with small age gaps, with less effect as the gap gets larger.

Assess­ment: Find­ings match pre­dic­tion well. 4-8 years seems rea­son­able for lev­els of in­ter­ac­tions be­tween siblings to start de­creas­ing.

Po­ten­tially, for a small age gap, a very ad­vanced younger sibling might act more like an older sibling mean­ing that the 1 year age gap birth effect would be lower. This feels slightly forced to me (I would think any such effect would be fairly small) but am cu­ri­ous what oth­ers think.

De­creased parental investment

Pre­dic­tion: Birth or­der effect in­creases as age gap in­creases—the longer a first­born is the only child the longer they benefit from 100% of their par­ents’ at­ten­tion. If the ear­liest years are the most im­por­tant then birth or­der might not change af­ter that crit­i­cal pe­riod. Once older chil­dren are able to look af­ter them­selves, birth or­der effect might come down with larger age gaps.

Assess­ment: The in­crease in birth or­der effect be­tween 1 and 2 years would match the the­ory, if parental in­vest­ment is mostly im­por­tant in the first two years. If older chil­dren start be­ing able to look af­ter them­selves af­ter 4-8 years then this would ex­plain the drop in birth or­der effect af­ter this time.

The match be­tween the the­ory and re­sult is good, al­though there are a cou­ple of de­grees of free­dom to help match the pre­dic­tion to the data. 4-8 years seems rea­son­able for chil­dren start­ing to look af­ter them­selves bet­ter but 2 years seems on the low side for a pre­dic­tion of how long hav­ing ex­tra at­ten­tion is benefi­cial. Maybe be­tween 2-5 years the two effects roughly can­cel out?

Changed par­ent­ing strategies

Pre­dic­tion: Age gap has min­i­mal effect on Birth or­der effect.

Assess­ment: Pre­dic­tion matches data poorly. It is pos­si­ble that parental strate­gies start to re­set to­wards first­born strate­gies af­ter longer age gaps but I wouldn’t have put much of my prob­a­bil­ity mass on that op­tion. There is a 5 year gap be­tween my youngest chil­dren and I definitely didn’t re­set to­wards first­born strate­gies, I sus­pect this would have still been true even for a much larger gap.

Ma­ter­nal antibodies

Pre­dic­tion: Age gap has min­i­mal effect on Birth or­der effect. Gen­er­ally you don’t need top-ups of vac­cines so pre­sum­ably an­ti­bod­ies stick around in­definitely? Or is it your body’s abil­ity to make more? Any­way, Scott thinks this is un­likely and he’s a doc­tor so I’ll take his word for it.

Assess­ment: Pre­dic­tion matches data poorly. My biol­ogy knowl­edge is too poor to know how likely a de­crease in effec­tive­ness af­ter 4-8 years would be in this case.

Ma­ter­nal vi­tamin deficiencies

Pre­dic­tion: Very small age gaps have large effect. Birth or­der effect de­creases rapidly for age gaps <3 years – my es­ti­mate for how long it might take to re­build vi­tamin stock­piles.

Assess­ment: Pre­dic­tion matches data poorly. 4-8 years seems way too long for vi­tamin stock­piles to start to build back up.

Conclusions

The SSC 2019 sur­vey data sup­port a con­stant, high, birth or­der effect (~2.4 old­est siblings for ev­ery 1 sec­ond old­est sibling) for age gaps <4-8 years. This is fol­lowed by a de­cline to a lower birth or­der effect at an un­de­ter­mined rate. The de­cline does not nec­es­sar­ily com­pletely re­move any birth or­der effect al­though this may be the case for very large age gaps.

The data provide some ev­i­dence that:

  • The re­duc­tion may not be the same (or might dis­ap­pear) for larger fam­i­lies (4+ chil­dren)

  • Birth or­der effect may be lower at 1 year age gap vs 2-7 year age gap

How­ever the ev­i­dence for both of these points is rel­a­tively slim.

In­tra-fam­ily dy­nam­ics and de­creased parental in­vest­ment pre­dict the re­sults well.

Changed parental strate­gies, ma­ter­nal an­ti­bod­ies and ma­ter­nal vi­tamin defi­cien­cies do not pre­dict the re­sults well.