Conditional vs Marginal Association: an example

• This is a $2\times2\times2$ table of fictional data, say religion by party at two different times. In each period there is no association between religion and party (perfect independence) but the distribution of religion and party change between the periods:

  Religion	Period 1				Period 2
  	Party A	Party B	Total		Party A	Party B	Total
  I	20	30	50		72	18	90
  II	20	30	50		8	2	10
  Total	40	60	100		80	20	100

  Period 1 OR: ${\frac{20}{30}}/{\frac{20}{30}} = 1$

  Period 2 OR: ${\frac{72}{18}}/{\frac{8}{2}} = 1$

  Religion by party marginal (observed):
  

  Religion	Party		Total
  	A	B
  I	92	48	140
  II	28	32	60
  Total	120	80	200

     

   
  (expected):
  

  Religion	Party		Total
  	A	B
  I	84	56	140
  II	36	24	60
  Total	120	80	200

  Pearson's $X^2$ is 6.35 for 1 df (p = 0.012).

  Observed OR is 2.19.

• The marginal table shows association.
• But when you control for the association between party and period and religion and period the association disappears.

• If we fit loglinear models to this data we get the following results (R is religion, P is party, T is time period):

  Model	$G^2$	df
  R+P+T	75.2215	4
  R+P+T+R*T	34.5218	3
  R+P+T+P*T	40.6997	3
  R+P+T+R*P	68.9432	3
  R+P+T+R*T+P*T	.0000	2

• The R+P+T model doesn't fit well but including either the religion by time or party by time 2-way interactions improve the fit by c50%.
• Adding the religion by party term reduces the deviance too, though not quite as much.
• The model with both the religion by time and party by time interactions fits perfectly: there is no religion by party association once you control for the associations with time.
• This is made-up data: in real sample data the fit wouldn't be perfect. Instead, adding the religion by time interaction to this model would not improve the fit significantly.