# Can a regression coefficient be explained both ways in logistic regression?

In logistic regression, depending on the sign of the coefficient we can say, for example, that when predicting political interest, a coefficient of $1.226$ for following the news indicates that people are more likely to be interested in politics when they are following the news. However, can we say that based on this coefficient, the more people are political interested the more likely they are to follow the news?

In other words, does it work both ways?

• You may want to read this thread: Relationship between regressing Y on X and X on Y in logistic regression. Is there something else you are asking beyond the information that's listed there? If so, can you clarify what it is? – gung - Reinstate Monica Sep 20 '13 at 21:06
• @gung I believe this question has an interesting positive answer in the univariate logistic regression case and "more likely to be" is interpreted as a statement about frequencies in the population. We are, after all, simply talking about a loglinear analysis of a two-way table. – whuber Sep 20 '13 at 21:18
• @whuber, I'm not actually sure what's being asked here, \@jank below, seems to interpret the Q as being about mutual causation, the thread I link talks about the $\hat\beta$s, other possibilities are: would the be an association both ways or would the significance be the same? I'm hoping for more info from the OP here. – gung - Reinstate Monica Sep 20 '13 at 21:27

This is essentially a philosphical question, a variation of the chestnut of 'association is not causation'. The answer will depend on context. Ultimately it requires the 'counsels of the wise' in order how to determine how to interpret the association that is illustrated in the model.

Both variables here are binary: Outcome='interest in politics' and predictor='follow news'. This does seem to be something of a simplification. If that's the data you're given then it can't be helped. However it would seem that these would lend themselves better to ordinal scales, which could always be collapsed later to binary categories if needs be (e.g. low no. responses/ small vol. data).

To some extent it depends whether we're in 'hypothesis-generating' or 'hypothesis-testing' mode. Now if the data was collected in order to try to predict the likelihood of 'being interested in politics' then, in the strictest interpretation, the reverse interpretation should not made. An experiment needs to be designed with a hypothesis in mind before any data is collected (particularly in the frequentist perspective). If the data has just 'fallen into ones lap' then a both interpretations are legitimate. Just reverse the model with 'interest in politics' as the predictor. (You're using a model designed to be predictive but descriptive statistics like chi-square might be more relevant if causation has not been established).

To my mind, this type of question would seem better suited to a qualitative/dialogue-based approach rather than binary scales. It seems plausible that both are signs of something bigger/underlying e.g. 'societal engagement', 'will to power' or more basically 'literacy'. In this vein, principal components analysis might be of interest. It's an established approach to teasing this out quantitatively when you have other measures available.

• It is part of an Index scale in which it is designed to measure over all political involvement, I collected the data my self and it was indeed ordinal and indeed, I did collapse it to a binary form. I think I will just do the reverse and make "following the news"the independent variable. unless you have any other thoughts ? – yazan Sep 21 '13 at 11:47
• Good for you! Most of the authorities warn against collapsing variables like this (Frank Harrell in particular). I would tend to agree unless you have many levels of the variable with small no.s of observations. If you plan on publishing this then depending on the audience it might be advisable to state your motivations for collecting the data i.e. whether purely speculative or designed to check a hypothesis. – dardisco Sep 21 '13 at 21:30

Unfortunately, it works only non-directionally, as you cannot make causal claims without a proper design that manipulates one of the variables. A regression does allow you to estimate criterion values based on predictor values but it doesgive evidence for a causal account.

You also have to be caeful with the statement "the more... the more...". A regression coefficient can owe its direction due to a single extreme outlier.

Therefore, unfortunatley, you do not have the evidence to make either claim as a general or even causal claim.

You can make a descritpitve statement about your sample though after carefully analyzing the data. In that case you cannot restrict directionality to one direction, and both claims are equally valid descriptions. I would still try to avoid causal implications, even though it is very common practice to mix this up...

• When "more likely to be" is read as "more frequently," as it often is, no causal claims are being made in the question. – whuber Sep 20 '13 at 21:17
• That is correct, but many readers read "then this increases the frequency (or likelihood)". And it is very difficult to make this interpretation less frequent. – jank Sep 20 '13 at 21:29
• @gung, I am not concerned about causation at the moment, I am wondering if the β of the independent variable used in predicting the outcome can be used to understand the effect of the outcome on the independent variable as well. I have read the thread you suggested and one of the answers mentioned that it is possiable only in "circumstances where it makes sense and where both variables are coded by 0 and 1" ? – yazan Sep 20 '13 at 21:33
• Luckily I just checked back, but you need to post your comment under the Q where mine were in order for me to be notified, @yazan. (Now that I've commented here, it'll be OK.) I don't quite understand what "understand the effect of the outcome on the independent variable" means, it sounds like a causal interpretation. Eg, if $X$ causes $Y$, & $Y$ does not cause $X$, there can be no "effect" of $Y$ on $X$. Can you clarify further? – gung - Reinstate Monica Sep 20 '13 at 21:43
• This goes back to my point of unwarranted causal interpretation... – jank Sep 22 '13 at 5:11