Specifying correct interaction terms in logistic regression

Question

My issue is with correctly specifying and interpreting interactions terms in a logistic regression. I have not found any threads that has addressed this. I hope my explanation of the issue is clear.

The data has two continuous independent variables (IV) called value and meters. There is no correlation between the two IVs. The two IVs are used to predict an outcome (yes=1, no=0). Here is the sample of the data where the IV have been normalized (min/max) not standardized:

#Sample of data 
structure(list(outcome = c(1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 
0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 
0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0), 
    value = c(0.368748949810643, 0.331407447619786, 0.229177168560239, 
    0.263652461643853, 0.398364289684232, 0.298408882340016, 
    0.337650419429472, 0.331407447619786, 0.298408882340016, 
    0.53519588455026, 0.131929634081714, 0.773617950805899, 0.279143562500808, 
    0.205788643736994, 0.386821900810423, 0.49966393940569, 0.300861478408107, 
    0.199409955148836, 0.28220688407202, 0.82296715653961, 0.140104954308685, 
    0.0167060891594608, 0.016857962697274, 0.0214141688316724, 
    0.0214141688316724, 0.0520926234699549, 0.0232366512854318, 
    0.180425762922176, 0.180425762922176, 0.0252110072770044, 
    0.0252110072770044, 0.0205029276047927, 0.0192879393022865, 
    0.0211104217560459, 0.0856566753266897, 0.0496626468649424, 
    0.0505738880918221, 0.0856566753266897, 0.0256666278904442, 
    0.0176173303863404, 0.0262741220416974, 0.0156429743947678, 
    0.0882288314139103, 0.0505738880918221, 0.0294634663357762, 
    0.0235403983610583, 0.0364270296121085, 0.174047074334018, 
    0.0906071063890289, 0.093877234479817, 0.106214535913245, 
    0.093877234479817, 0.174047074334018, 0.0362977755373738, 
    0.380608011167552, 0.0391833727558261, 0.0241478925123115, 
    0.0150354802435147, 0.0241478925123115, 0.0968953171248724, 
    0.0186804451510334, 0.0968953171248724, 0.0476882908733698, 
    0.0192879393022865, 0.111171429679321, 0.111171429679321, 
    0.0203510540669795, 0.0280966044954567, 0.0490551527136893, 
    0.0159467214703944, 0.111171429679321, 0.111171429679321, 
    0.0441951995036643, 0.0253628808148177, 0.119676347796864, 
    0.0399427404448925, 0.152177284888906, 0.175333152377629, 
    0.152177284888906, 0.122862460739075, 0, 0.0301097367094498, 
    0.322340274277147, 0, 0.322340274277147, 0.714161076427934, 
    0.776144867966963, 0.182170692931095, 0.150955833882663, 
    0.640583194385203, 0.0590788062093657, 0.080948595654478, 
    0.080948595654478, 0.0590788062093657, 0.329623741388447, 
    0.0747023924929233, 0.376743314332984, 0.736160119947781, 
    0.561059624904675, 0.151996329184278, 0.151996329184278, 
    0.228398412759962, 0.124348882598524, 0.0799048690009952, 
    0.775996225781018, 0.419403621699174, 0.229736192433466, 
    0.149766696395104, 0.389675184510192, 0.803792314552716, 
    0.526849302674267, 0.31728643995502, 0.225425569041064, 0.390864321997751, 
    0.149320769837269, 0.482130624167927, 0.146050641746481, 
    0.168198327452273, 0.578004834102395, 0.330515594504117, 
    0.39175617511342, 0.144266935515142, 0.395174945390153, 0.0969987203846601, 
    0.0825804283480037, 0.493873356857575, 0.304800496335647, 
    0.493873356857575, 0.129551359106596, 0.186629958509442, 
    0.117659984231003, 0.0510295087052619, 0.186629958509442, 
    0.500116328667261, 0.0739591815631988, 0.618732793051301, 
    0.280274535654736, 0.169338994661807, 0.169338994661807, 
    0.0984851422441092, 0.08035079555883, 0.0686080628691819, 
    0.105917251541355, 0.0977419313143847, 0.0977419313143847, 
    0.105917251541355, 0.0686080628691819, 0.426092520066695, 
    0.100566132847338, 0.359500820763375, 0.38476999237401, 0.116768131115333, 
    0.368865278477904, 0.0987824266159991, 0.440659454289296, 
    0.386405056419404, 0.171914382100896, 0.137726679333566, 
    0.0871883361122959, 0.397404578179327, 0.0934313079219822, 
    0.257532281205165, 0.0647433660346142, 0.169387464939832, 
    0.111862938979151, 0.0868910517404061, 0.0868910517404061, 
    0.284882443419029, 0.121673323251516, 0.238060154846382, 
    0.151104476068608, 0.0900125376452493, 0.151104476068608, 
    0.0900125376452493, 0.0934313079219822, 0.0536113588480877, 
    0.169387464939832, 0.0934313079219822, 0.293949616761668, 
    0.519885739397934, 0.293949616761668, 1, 0.29974666201352, 
    0.192278361575349, 0.904869000995256, 0.399188284410666, 
    0.212493698863857, 0.399188284410666, 0.212493698863857, 
    0.162103997828532, 0.105619967169465, 0.246086832887407, 
    0.162103997828532, 0.641475047500872, 0.0909043907609187, 
    0.146050641746481, 0.211007277004408, 0.211007277004408, 
    0.146050641746481, 0.0909043907609187), meters = c(0.0484629861982434, 
    0.187243831033041, 0.444123797574237, 0.402969468841489, 
    0.249874529485571, 0.201589293182769, 0.208030112923463, 
    0.0936323713927227, 0.124539941447093, 0.489115432873275, 
    0.737470723546633, 0.0249634044332915, 0.708071936428273, 
    0.621183605186115, 0.746340443329151, 0.683683605186115, 
    0.579360100376412, 0.478199498117942, 0.381038268506901, 
    0.231937473860309, 0.309859891258887, 0.0484368465077374, 
    0.388592639063153, 0.746340443329151, 1, 0.124289000418235, 
    0.444123797574237, 0.249874529485571, 0.182172731074864, 
    0.489115432873275, 0.68129443747386, 0.27258469259724, 0.821099958176495, 
    0.873640736093685, 0.818224592220828, 0.58202634880803, 0.689774153074028, 
    0.798044751150146, 0.122229192806357, 0.188153492262652, 
    0.0503973232956922, 0.240955667084902, 0.166771225428691, 
    0.0390004182350481, 0.708176495190297, 0.62107904642409, 
    0.731231702216646, 0.682951693851945, 0.579360100376412, 
    0.474356963613551, 0.44487139272271, 0.490276035131744, 0.658981597657884, 
    0.125470514429109, 0.174430154746968, 0.309859891258887, 
    0.0578209953994145, 0.428272689251359, 0.153805938937683, 
    0.0484368465077374, 0.388566499372647, 0.106780635717273, 
    0.161020493517357, 0.458995713090757, 0.305241896695943, 
    0.253798985780008, 0.521155897114178, 0.587812107904642, 
    0.126803638644918, 0.785105604349644, 0.772976787954831, 
    0.804448975324132, 0.643585319949812, 0.0888488080301129, 
    0.0245190296946884, 0.707810539523212, 0.68271643663739, 
    0.579360100376412, 0.658981597657884, 0.00595984943538269, 
    0.225533249686324, 0.334431200334588, 0.0484107068172313, 
    0.24017147636972, 0.130280217482225, 0.020310539523212, 0.721063362609787, 
    0.437134044332915, 0.252248013383522, 0.0705248849853618, 
    0.734159347553325, 0.680625261396905, 0.580719364282727, 
    0.64219991635299, 0.400564617314931, 0.486511919698871, 0.125941028858218, 
    0.115677017984107, 0.0879339188624007, 0.253816394813885, 
    0.280740276035132, 0.0514951902969469, 0.407831451275617, 
    0.244719782517775, 0.00627352572145546, 0.240798828941865, 
    0.403021748222501, 0.641468005018821, 0.579307820995399, 
    0.0271329987452949, 0.352833542450857, 0.371131325805102, 
    0.232590966122961, 0.169907988289419, 0.306043496445002, 
    0.439721873693015, 0.5193956503555, 0.0209117524048515, 0.915594939355918, 
    0.865066917607695, 0.541980342952739, 0.574498117942284, 
    0.265213299874529, 0.371340443329151, 0.233009201171058, 
    0.0144291091593476, 0.279642409033877, 0.0457967377666249, 
    0.239125888749477, 0.579464659138436, 0.60461104140527, 0.643820577164366, 
    0.486198243412798, 0.306775407779172, 0.402969468841489, 
    0.0230552070263488, 0.232852363028022, 0.0987557507319113, 
    0.168653283145128, 0.374921580928482, 0.307350480970305, 
    0.77258469259724, 0.742680886658302, 0.72150773734839, 0.624712463404433, 
    0.719730238393977, 0.685853199498118, 0.444322459222083, 
    0.522898368883312, 0, 0.135246758678377, 0.279642409033877, 
    0.779642409033877, 0.480133835215391, 0.216698034295274, 
    0.000365955667084908, 0.132214554579674, 0.277132998745295, 
    0.403021748222501, 0.582340025094103, 0.643820577164366, 
    0.773055207026349, 0.812996654119615, 0.0432873274780427, 
    0.857120451693852, 0.874947720618988, 0.781890422417399, 
    0.563101212881639, 0.594730238393977, 0.299822250104559, 
    0.339293182768716, 0.41159556670849, 0.268768297783354, 0.325805102467587, 
    0.156263069845253, 0.240276035131744, 0.00026139690506065, 
    0.119615223755751, 0.122072354663321, 0.0486721037222919, 
    0.145075282308657, 0.133364700961941, 0.197354663320786, 
    0.338927227101631, 0.394134253450439, 0.428115851108323, 
    0.476787954830615, 0.518925135926391, 0.563937682977834, 
    0.696152237557507, 0.724121706398996, 0.769500209117524, 
    0.787954830614806, 0.00595984943538269, 0.559807611877875, 
    0.515108741112505, 0.486198243412798, 0.393245503973233, 
    0.424822250104559, 0.476787954830615)), row.names = c(NA, 
200L), class = "data.frame")

Separately, they are considered as main effects. When modelled separately, both have p-values < 0.05:

#IV = meters 
model <- glm(outcome ~ meters, data = df, family= binomial)
summary(model)

#output
Call:
glm(formula = outcome ~ meters, family = binomial, data = df)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.94020  -0.24794  -0.02802  -0.00336   2.59098  

Coefficients:
            Estimate Std. Error z value    Pr(>|z|)    
(Intercept)   1.7219     0.5013   3.435    0.000594 ***
meters      -18.5005     3.6270  -5.101 0.000000338 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 169.084  on 199  degrees of freedom
Residual deviance:  77.371  on 198  degrees of freedom
AIC: 81.371

Number of Fisher Scoring iterations: 8

#IV: value
model <- glm(outcome ~ value, data = df, family= binomial)
summary(model)

#Output
Call:
glm(formula = outcome ~ value, family = binomial, data = df)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.4279  -0.5485  -0.4605  -0.4091   2.2572  

Coefficients:
            Estimate Std. Error z value          Pr(>|z|)    
(Intercept)  -2.5176     0.3282  -7.672 0.000000000000017 ***
value         3.0895     0.8801   3.511          0.000447 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 169.08  on 199  degrees of freedom
Residual deviance: 156.89  on 198  degrees of freedom
AIC: 160.89

Number of Fisher Scoring iterations: 4

The coefficients in both cases are correct in terms of the direction of change, where value is positive and meters is negative. Meaning, that meters represents "proximity" and value represents "exposure".

With that being said, together they actually have a conditional relationship, where the effect of value on the outcome (yes/no) is conditional to value's relationship with meters. In other words, the closer someone is in proximity (meters) to the exposure level (values), the more it will influence the outcome.

I have tried specifying this relationship:

model <- glm(outcome ~ value + meters + value*meters, data = df, family= binomial)
summary(model)

#output
Call:
glm(formula = outcome ~ value + meters + value * meters, family = binomial, 
    data = df)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.01741  -0.24157  -0.03552  -0.00525   2.43956  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)   
(Intercept)    1.1481     0.7867   1.459  0.14448   
value          1.9520     2.2231   0.878  0.37990   
meters       -14.7932     5.3112  -2.785  0.00535 **
value:meters -14.1441    18.6979  -0.756  0.44938   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 169.084  on 199  degrees of freedom
Residual deviance:  76.527  on 196  degrees of freedom
AIC: 84.527

Number of Fisher Scoring iterations: 8

However, I am not sure if my interpretation of this is correct. Value is no longer significant but meters still is and the interaction term is also not significant. My interpretation is that:

• When Value < 0, it will have less influence on the outcome but when Meters < 0, it will have influence on the outcome
• When Value > 0, it will have more influence on the outcome, but when Meters > 0, it will have less influence on the outcome

If this interpretation is correct, it is not what I am hoping to model. What I actually want to say, is:

• When Value > 0 and Meters < 0, it will have an influence on the outcome
• When Value < 0 and Meters > 0, it will have less influence on the outcome

Any help and clear explanation (in simple terms) would be immensely appreciated.

Answer 1

There seems to be some confusion.

First, just a very minor point with your R code. The way to specify the interaction is value:meters. When you uses the * it does specify the interaction, but also the main effects too, which you have already included. So you could just write that model as outcome ~ value * meters

Second, it is not clear, why you are interpreting the output of the model that includes the interaction in terms of the main effects being less than zero. In the supplied data, both variables never take a negative value (since you have normalized them to min-max).

Third, I don't think normalisation is very useful here.

Fourth, it is not clear why you went from two bivariate models to one with both main effects and their interaction. Why have you omitted the model outcome ~ value + meters model ?

From these models I would say that there is very little evidence that the association of either value or meters with the outcome varies depending on the other one. It is clear that there is a strong association of meters with the outcome. If value is a potential confounder of the meters - outcome association or you are sure that there is an association value with the outcome and value is not caused (either directly or indirectly) by meters, then I would also include value in the model.

However, as discussed in the comments, there appears to be repeated measures of participants in this experiment. As such, the models presented in the OP are not appropriate, since there will be dependencies within subjects. A model that includes random intercepts for subjects might be appropriate (ie a glmm) but more detail about how the experiment was conduction would be needed. Therefore I suggest asking a new question about how to model your data, giving full details about how the exsperiment was conducted along with your research questions.