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I ran a logistic regression model on SPSS with a dependent variable of yes/no whether you chose bus or not (the other being personal vehicle) and 5 independent variables (Waiting Time, Trip Time, Total Daily Expense, Overall Mode Comfort, and Overall Mode Ease-of-use). While the Omnibus and Hosmer-Lemeshow tests shows the model to be very good, and the significance for the most variables adequate, the result coefficients of some of the variables are somewhat off. This affects the probability estimation in that the predictor variable goes against intuition in real life conditions.

For example, the Comfort variable has a coefficient of -0.102821; this translates to a low probability when the Comfort value is high. Who wouldn't choose the bus when the Comfort value is over the top? I'm thinking that the coefficient should be a positive instead of negative. I should probably also point out that the intercept is negative, I'm not sure how much this effects the model.

So what seems to be the problem with my model?

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    $\begingroup$ It might be the correlations between the predictors - what is the coefficient for comfort if you include no other predictors? $\endgroup$ – Jeremy Miles Jun 16 '15 at 4:24
  • $\begingroup$ What do you mean by the correlations between predictors? The coefficient for Comfort is -0.035395 when I don't include the other predictors (though statistically insignificant). Thanks for the response. $\endgroup$ – Umar Al Faruq Jun 16 '15 at 10:08
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    $\begingroup$ How exactly is "Comfort" coded? That is, what is the assignment of numerical values to comfort levels? $\endgroup$ – whuber Jun 16 '15 at 15:18
  • $\begingroup$ Comfort is coded from 1-100. $\endgroup$ – Umar Al Faruq Jun 16 '15 at 23:55
  • $\begingroup$ With 120 observations and 6 parameters, I would worry about finite sample bias of MLE. $\endgroup$ – Dimitriy V. Masterov Jun 17 '15 at 22:21
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If the beta parameter estimate is statistically significant, then the issue is not correlation among variables in your model.

Instead, the issue is one of omitted variable bias. This means there is a variable your model does not control for that is correlated with the comfort variable and the response variable. The impact of this omitted variable is absorbed by the comfort variable.

As a theoretical example, the buses that are the most comfortable might be located in the areas that are more wealthy. Perhaps people in wealthier areas are more likely to use their personal vehicle.

Because you did not control for the wealth of the driver (the variable was omitted) and this variable could be correlated with the buses that are more comfortable, the comfort variable could be negatively biased (potentially so much so to change signs of the parameter estimate).

Remember that when you interpret a beta parameter as holding everything else constant, you really mean that you are holding all of the variables in your model constant. Any variable not in your model that is correlated with a variable in your model is not considered to be held constant.

Omitted variable bias violates a fundamental assumption of linear models and leads to biased parameter estimates. Because of this negative impact, you should always include all variables you believe are part of the model. Unfortunately, there are no good ways to know what is omitted through the model alone. You must use your own experience and judgment to understand what might be omitted.

As a side note, if your goal is just to estimate the comfort variable, then you don't need to include every possible variable that you might have omitted in your model. Instead, you only need to include all variables that are correlated with the comfort and response variable.

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  • $\begingroup$ So you suggest I should put in more independent variables? Also, how can I know which variable is absorbed by the Comfort variable? Thanks for the response. $\endgroup$ – Umar Al Faruq Jun 16 '15 at 10:11
  • $\begingroup$ I updated the answer to directly answer these questions. $\endgroup$ – Jason Sanchez Jun 16 '15 at 15:00
  • $\begingroup$ I ran a correlation analysis for all variables and found that Comfort isn't correlated with my response variable, and when I didn't include it in my model the results were better. Should I omit Comfort instead? $\endgroup$ – Umar Al Faruq Jun 16 '15 at 23:55
  • $\begingroup$ 2 questions. First, is the comfort variable correlated with any of the other variables in your model? Second, do you believe the comfort variable should be part of the equation? $\endgroup$ – Jason Sanchez Jun 17 '15 at 1:04
  • $\begingroup$ Yes, Comfort is correlated with Total Daily Expense and Mode Safety (I forgot to mention this variable in my question) out of all the variables. Secondly, also yes, because ideally if an individual feel that a certain mode is more physically comfortable than another they will likely choose that one over the other, along with the other variables in mind. $\endgroup$ – Umar Al Faruq Jun 17 '15 at 1:59
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Excluded variables are likely biasing your beta estimates (potentially severely for the comfort variable) and that due to small sample sizes you will have difficulty precisely estimating the parameters.

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  • $\begingroup$ Is it possible that, aside from the small sample size (this I admit, due to time and manpower restrictions), the respondents filled in incorrect Comfort values, as in misinterpreting the question? I put in more passenger characteristics variables and the beta for Comfort turned positive, but the p value for all variables are either 1 or 0.999. $\endgroup$ – Umar Al Faruq Jun 17 '15 at 22:15
  • $\begingroup$ Bummer, I am using the mobile app and didn't realize I accidentally added a new answer. I will fix that when I get to a computer. Yes it is possible people incorrectly interpreted the question on the survey. If you can confirm that happened, then at least you will have the correct sign on the variable. $\endgroup$ – Jason Sanchez Jun 17 '15 at 23:15
  • $\begingroup$ In my report, can I just write the results down and explain that there is indeed a bias? Or is there a workaround? $\endgroup$ – Umar Al Faruq Jun 20 '15 at 16:06

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