# Interpret logistic regression output with multiple categorical & continious variables

I'm trying to analyze some data obtained from a survey asking employers about their difficulty in hiring certain occupations (80 total occupations). But I'm uncertain about the interpretation of my results.

Dependent Variable: 0 = not difficult 1 = difficult.

Independent Variables:

1. occupation group (occ.group 1 to 9 different groups) grouped the occupations to cut down on the total number of options
2. area (0 = urban 1 = rural)
3. hourly wage of the occupation (hr.wage continuous, numeric).

The data is structured as such:

   empnum response    soc     occ.group  area    hr.wage
1  123450        1    1         1          1        70.20
2  543210        0    1         1          0        50.10
3  111111        0    1         1          1        71.10
4  222222        1    2         2          1        60.23
5  333333        1    2         4          0        100.57
6  4444444       0    80        9          1        60.18


Model is as follows:

logit1 <- glm(response~occ.group+area+hr.wage, family=binomial, data=health.sub)

Coefficients:
Estimate Std. Error  z value  Pr(>|z|)
(Intercept)     0.184978   0.466055    0.397   0.69144
occ.group2     -0.243524   0.528233   -0.461   0.64479
occ.group3      0.281285   0.407879    0.690   0.49043
occ.group4     -0.063578   0.510229   -0.125   0.90084
occ.group5     -0.039797   0.403032   -0.099   0.92134
occ.group6      0.419655   0.475109    0.883   0.37708
occ.group7     -0.898869   0.652530   -1.378   0.16835
occ.group8    -15.015216 394.6834     -0.038   0.96965
occ.group9     -0.370350   0.405532   -0.913   0.36111
area1           0.008863   0.129376    0.069   0.94538
hr.wage         0.010475   0.004028    2.600   0.00931   **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 1679.4  on 1259  degrees of freedom
Residual deviance: 1638.5  on 1249  degrees of freedom
AIC: 1660.5

Number of Fisher Scoring iterations: 13


I'm trying to get an idea if my interpretations are correct.

1. Because occ.group2 coeff is negative it would have lower odds of being difficult to hire than occ.group1 (reference group)

odds are exp(-0.2353) = .79
probability = .4414

2. An employer with occupation in occ.group2 and area1 = -0.243524 + 0.008863 = exp(-0.234661) = 0.79 odds of difficult. Would that be compared to to occ.group1 (reference group) in the same area1 or would it be area0. I guess I'm having a hard time interpreting the reference group with two categorical variables.

3. Employer with occupation in occ.group2 and area0? Would I add the coefficient for occ.group2 (-0.243524) and the intercept (-0.243524)? That doesn't seem right, but not sure how to deal with multiple reference groups. What about an employer with occ.group1 and area0?

4. Would the odds of an employer with occupation in group2 and area1 with a 1 dollar increase in hr.wage = -0.243524 + 0.008863 + 0.010475 = exp(-0.224186) = odds of .799 compared to employer with occupation group1 in area1 with 1 dollar hr.wage increase?

Thanks for the explanations and clarification @EdM.

I also had a question about doing deviation/effects coding of the occ.group categorical predictor, because I'm not convinced that comparisons between the occ.group is meaningful due to their major differences. I've read that effects coding makes it possible to compare each group to the overall mean response. But again, I'm not sure I'm fully understanding how to interpret the results.

I've used the contr.sum and contrasts commands in R to apply effects coding to the occ.group variable and end up with this output.

contr.sum(5)
contrasts(health.sub\$occ.group) = contr.sum(5)
Call:
glm(formula = response ~ occ.group + area + hr.wage, family = binomial,
data = health.sub)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-1.7637  -1.2172   0.8009   1.0462   1.1513
Coefficients:
#UPDATED devication/effects coded model based on @EdM 's comment.
#occ.group8 dropped due to limited responses
#Other occ.group's were reorganized to more appropriate categories 1-5
Estimate   Std. Error z value  Pr(>|z|)
(Intercept)   0.546408   0.147042   3.716   0.000202 ***
occ.group1   -0.265066   0.139886  -1.895   0.058109 .
occ.group2   -0.507674   0.110791  -4.582   0.0000046 ***
occ.group3    0.186679   0.154514   1.208   0.226981
occ.group4    0.347765   0.128618   2.704   0.006854 **
area1         0.003353   0.129615   0.026   0.979365
hr.wage       0.002139   0.003228   0.663   0.507424
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1679.4  on 1259  degrees of freedom
Residual deviance: 1635.1  on 1253  degrees of freedom
AIC: 1649.1
Number of Fisher Scoring iterations: 4


/#UPDATED values to reflect @EdM response.

How would I go about interpreting the odds for an employer with occ.group2 and area0 with an hr.wage = 50.00 given the effects coding?

I believe the odds would be calculated as 0.5464 +(-0.5077) + (0.5464 - 0.0034) + (0.0021*50.00)) = exp(0.6937) = odds of 2.001. Which is in comparison to the mean of the mean occ.group logits rather than any specific reference group.

• I don't often use sum contrasts. This page shows how to interpret them in R output. Note that you have a big problem with occ.group8 that you should solve before you go farther; as it stands that group leads to a situation far from having residual errors independent of the variable values, as required for interpreting p values properly, and this is harming your whole model. I suspect a data coding error. – EdM Sep 24 '15 at 0:35
• Thanks for the reference @EdM that was useful. If I'm understanding correctly the intercept would be the mean of the mean logits and occ.group5 coefficient would be the intercept minus all the other occ.group coefficients. I think i've updated the effects coding example to the proper interpretation, although I'm not 100% sure the area0 component is handled properly. – AKP Sep 25 '15 at 1:08
• You used sum contrasts for occ.group but you kept treatment contrasts for area; wage is expressed as difference from 0. So the intercept in your second model represents the case for the mean of occ.group values, at area0 and wage = 0. Your terms accounting for occ.group2 and wage in calculating odds are correct, but there should be no term for area0 as that is included in the intercept. If you went to sum contrasts for area, you would then find a coefficient reported for area0 but not for area1, as I understand the way the contrasts are handled. – EdM Sep 25 '15 at 14:24
• Seems relatively intuitive in hindsight. And, you know, having it plainly spelled out never hurts. I think I'll stick with the deviation/effects coding, but work on better groupings. There also seems to be some strong arguments for adding some interaction terms, so I'll try to wrap my ahead around interpreting those. Thanks again @EdM. And I ran sum contrasts on area and it gives a coefficient for area1 not area0. Which I assume is because the sum contrasts are 1, -1 for a dichotomous var. So to get area0 you'd subtract the area1 coefficient from the intercept. – AKP Sep 25 '15 at 23:37

So for your question 2 the reference is occ.group1 and area0, as it is for all comparisons given the way you have labeled the levels of the variables. occ.group1 and area0 as in your question 3 is the reference group with odds calculated from the intercept for a 0 wage, but you need to specify an hourly wage to get the odds for a non-zero wage.
Your interpretation in question 4 seems to be a bit off. If the areas and wages are the same for the two groups then the only difference to consider in the specified comparison is the coefficient for occ.group2 versus the occ.group1 reference.