I just ran the following on SMA.sav file in SAS. The data file can be accessed here.

Proc Logistic Data = sma descending;
Where Age=1;
class Genotype Treatment / param=ref;
Model sma_bin = Genotype Treatment Genotype * Treatment / CLodds=both firth;
oddsratio Genotype;
oddsratio Treatment;
run;


I'm confused in relation to the following Firth logistic regression outcome. There appears to be a counter-intuitive result.
One that emerges if you look at the 100% stacked columns, and then look at what the logistic regression is telling us.

1. The columns tell us that a combination of T1OE and NMN (the Genotype = 1, Treatment = 1 situation) REDUCES our risk of being entered into the higher smooth muscle actin category (orange), relative to the 0, 0 situation - WT and Veh, the last column (See Fig 1)
2. But the logistic regression (see Fig 2) tells us that the combination of T1OE and NMN (the 1 values of the binary variables), results in a higher risk of being entered into the elevated smooth muscle actin category (orange). If we exponentiate the beta estimate, we get something like 20x odds ratio of being entered into the orange. This seems to be telling us the opposite, since going by the visual trend, having both these conditions lowers our risk of being entered into the orange, relative to when we had neither of these conditions. How can I reconcile this? Can this be right or am I misunderstanding something here?

Fig 1: 100% stacked columns showing visual trends

Fig 2: SAS Output

I think your treatment is coded backward. Is nicotinamide mononucleotide the control? Because it's the active treatment in the model. The intercept should give us the odds from the referent treatment and genotype. Also I think you've excluded participants with missing data. A log odds of 1.09 has an odds of 3. No column in these data produces an odds of 3.

The right column is wild type and nicotinamide mononucleotide, the risk is 7/9, the odds are 7/2 =3.5, but if one of those participants were excluded, it would make sense.

• NMN is intended to represent the treatment. I have purposely excluded participants, yes - the entire analysis shown here is conducted on the resulting subset of the data. I got an odds ratio of 20 by taking the exponential of the beta estimate for the Genotype*Treatment term in the model, 3.0168? @AdamO – ptrcao Jun 14 '16 at 20:54
• @ptrcao OK then. The large interaction term makes sense. Individually, those with WT/Veh and T1OE/NMN have much more prevalent SMA whereas those with T1OE/NMN are much more like the referent group. I see nothing wrong with that. Odds ratios of 20 make sense because SMA is so prevalent in this sample, it's just important you don't interpret the OR as a RR (logistic regression is usually better at modeling rare events). – AdamO Jun 14 '16 at 21:01
• Can you please check if there isn't a typo in the last comment with respect to the labels? I'm not reading it correctly somehow... Thanks @AdamO – ptrcao Jun 14 '16 at 21:26
• There is still one thing I don't get, @AdamO. The odds ratio is relative to the referent categories as you said. But relative to the referent group (the group that is the referent values for both categories, i.e. the WT/Veh group), the interacting term T1OE/NMN should reduce the risk of SMA placement, not increase it by 20-fold? – ptrcao Jun 14 '16 at 22:25
• @ptrcao you are confusing the point of an interaction term. The T1OE/NMN is lower relative to WT/Veh. However, it is MUCH higher than what would be predicted based on the main effect, The T1OE/Veh and WT/NMN effects are VERY large, and adding them would predict a very very small risk of orange. Truth is T1OE/NMN is much higher than either of those main effects, so not only is the ORR very large, it is of opposite sign. – AdamO Jun 15 '16 at 15:57