Goal
My goal is to correctly model the effect of three independent variables (job autonomy, trait plasticity, and job complexity) on a dependent variable (job stress).
Problem
Having run a standard linear regression, the results were significant and in the expected direction, however the QQ plots were not good.
Alternative Solution Attempted
I opted to address this problem using a Poisson regression because the DV follows a Poisson distribution (see data below). I was conceptualising the modelling in terms of the ‘count’ of participants that scored 0-10 on the DV. Prima facie, this worked well, however upon researching Poisson regression further there appear to be a few problems with my approach, namely:
The DV is a Likert scale without a meaningful zero point (although this is debatable as it is theoretically possible to never experience job stress). The scale technically has 0 in it, but, strictly speaking, the anchor description for 0 is “strongly disagree”, rather than “none”.
I’m having difficulty making sense of the interpretation of the regression coefficients.
I’m not confident that it is correct to conceptualise a Likert scale as a ‘count’. This post starts to answer this point, but does not really explain why it is a problem.
Next Steps?
I am therefore a bit unsure how to proceed given my goal. I would appreciate if I could get advice on the following questions:
Am I correct in my thinking regarding points 1-3 above, or have I been misled? Is it, in fact, possible to run a Poisson regression on my data?
How am I best to proceed from here in terms of using alternative modelling techniques to help me achieve my goal (perhaps attempting to transform my data in some way and then going back to my standard linear modelling)?
Data
Sample data for R where all IVs are mean-centred (randomly selected 200 of my 8000+ participants):
structure(list(Job_Stress = c(4L, 1L, 4L, 4L, 2L, 0L, 0L, 2L,
2L, 1L, 12L, 6L, 0L, 5L, 3L, 0L, 3L, 4L, 1L, 2L, 5L, 4L, 5L,
2L, 8L, 11L, 10L, 2L, 6L, 11L, 2L, 0L, 0L, 2L, 0L, 2L, 0L, 1L,
2L, 0L, 5L, 8L, 7L, 2L, 0L, 2L, 6L, 0L, 1L, 1L, 1L, 0L, 0L, 2L,
1L, 3L, 10L, 1L, 1L, 0L, 0L, 0L, 8L, 3L, 2L, 5L, 1L, 3L, 5L,
5L, 4L, 7L, 0L, 3L, 9L, 4L, 2L, 3L, 2L, 2L, 3L, 6L, 0L, 6L, 0L,
2L, 5L, 2L, 2L, 0L, 3L, 5L, 4L, 0L, 0L, 1L, 2L, 3L, 2L, 2L, 1L,
2L, 2L, 2L, 9L, 0L, 4L, 5L, 4L, 2L, 7L, 0L, 2L, 0L, 0L, 0L, 0L,
6L, 2L, 4L, 1L, 4L, 1L, 8L, 6L, 1L, 7L, 0L, 1L, 0L, 1L, 0L, 3L,
5L, 6L, 7L, 8L, 5L, 5L, 2L, 10L, 2L, 3L, 9L, 3L, 0L, 5L, 4L,
4L, 2L, 2L, 2L, 0L, 7L, 4L, 3L, 6L, 5L, 7L, 8L, 0L, 4L, 2L, 1L,
0L, 1L, 0L, 8L, 2L, 5L, 0L, 0L, 0L, 4L, 2L, 0L, 0L, 8L, 10L,
0L, 5L, 0L, 6L, 0L, 2L, 1L, 4L, 0L, 1L, 3L, 12L, 4L, 0L, 4L,
8L, 6L, 0L, 8L, 2L, 9L), Autonomymc = c(0.50593505039194, 2.1726017170586,
-1.8273982829414, 1.33926838372527, 1.1726017170586, -0.160731616274729,
1.83926838372527, 1.6726017170586, -1.3273982829414, 1.1726017170586,
-1.16073161627473, 1.1726017170586, 2.83926838372527, -1.16073161627473,
-0.827398282941399, -3.16073161627473, 2.33926838372527, 1.6726017170586,
-1.16073161627473, 1.83926838372527, -0.327398282941399, 0.172601717058601,
-0.660731616274729, -0.327398282941399, -0.494064949608059, 1.83926838372527,
2.50593505039194, 0.672601717058601, 1.00593505039194, 0.172601717058601,
0.339268383725271, 1.6726017170586, -1.8273982829414, -0.994064949608059,
2.83926838372527, -1.16073161627473, -1.66073161627473, -1.49406494960806,
1.1726017170586, 1.6726017170586, 1.33926838372527, -0.660731616274729,
-0.994064949608059, 1.1726017170586, -1.3273982829414, 1.6726017170586,
-0.494064949608059, 0.839268383725271, -2.3273982829414, 1.83926838372527,
-1.8273982829414, 0.839268383725271, 2.00593505039194, 1.00593505039194,
0.339268383725271, 1.83926838372527, -0.327398282941399, 1.50593505039194,
0.672601717058601, 1.83926838372527, 2.83926838372527, 2.6726017170586,
-0.160731616274729, 0.00593505039194042, 1.83926838372527, 0.839268383725271,
-0.660731616274729, -0.327398282941399, -1.3273982829414, 1.6726017170586,
0.839268383725271, 0.339268383725271, 2.00593505039194, 2.00593505039194,
-0.494064949608059, -3.16073161627473, -0.494064949608059, 0.172601717058601,
-1.8273982829414, 0.672601717058601, 1.83926838372527, 1.1726017170586,
1.83926838372527, 2.83926838372527, -1.66073161627473, 2.50593505039194,
0.339268383725271, -2.16073161627473, -0.160731616274729, -2.3273982829414,
1.50593505039194, 0.172601717058601, 1.1726017170586, 1.1726017170586,
-1.3273982829414, 1.83926838372527, 0.00593505039194042, 1.50593505039194,
1.6726017170586, -0.827398282941399, -0.494064949608059, -0.160731616274729,
-0.160731616274729, 1.50593505039194, 2.33926838372527, 0.00593505039194042,
-0.827398282941399, 0.50593505039194, 0.839268383725271, 1.83926838372527,
0.839268383725271, 0.672601717058601, -2.99406494960806, 2.83926838372527,
-0.994064949608059, 0.172601717058601, -0.994064949608059, 0.50593505039194,
-2.49406494960806, 2.00593505039194, 2.83926838372527, 1.33926838372527,
-0.827398282941399, 0.172601717058601, 0.339268383725271, -1.8273982829414,
-0.994064949608059, 2.83926838372527, 1.33926838372527, -1.66073161627473,
1.33926838372527, 2.83926838372527, 0.00593505039194042, 1.6726017170586,
0.339268383725271, 2.83926838372527, -2.3273982829414, -0.327398282941399,
0.50593505039194, -0.994064949608059, 0.839268383725271, -0.660731616274729,
2.00593505039194, -1.16073161627473, 0.172601717058601, 2.00593505039194,
0.50593505039194, 0.339268383725271, -0.660731616274729, -0.994064949608059,
1.1726017170586, -1.99406494960806, 1.50593505039194, 1.6726017170586,
0.50593505039194, 2.1726017170586, 0.672601717058601, -0.660731616274729,
1.1726017170586, 1.1726017170586, -0.327398282941399, -0.660731616274729,
-2.16073161627473, 0.839268383725271, 2.83926838372527, 1.6726017170586,
2.83926838372527, 1.50593505039194, 2.1726017170586, 0.672601717058601,
0.672601717058601, 0.50593505039194, -0.827398282941399, -0.660731616274729,
1.00593505039194, 0.672601717058601, 2.83926838372527, -0.160731616274729,
-2.16073161627473, -1.49406494960806, -0.160731616274729, -1.8273982829414,
-0.327398282941399, -0.660731616274729, -2.16073161627473, -1.16073161627473,
1.50593505039194, -1.66073161627473, 1.6726017170586, 1.50593505039194,
2.83926838372527, -1.16073161627473, 2.00593505039194, -1.16073161627473,
-1.16073161627473, -1.8273982829414, -1.66073161627473, -2.8273982829414,
0.839268383725271, -2.66073161627473), Job_Is_Complex_Difficultmc = c(1.97693169092945,
1.97693169092945, 1.97693169092945, 1.97693169092945, 1.97693169092945,
-1.02306830907055, -2.02306830907055, 1.97693169092945, -1.02306830907055,
-0.023068309070549, 2.97693169092945, 1.97693169092945, 2.97693169092945,
-1.02306830907055, 2.97693169092945, -3.02306830907055, 2.97693169092945,
1.97693169092945, 0.976931690929451, 0.976931690929451, -0.023068309070549,
-0.023068309070549, -2.02306830907055, 0.976931690929451, 2.97693169092945,
1.97693169092945, 2.97693169092945, 0.976931690929451, 1.97693169092945,
-1.02306830907055, -0.023068309070549, -2.02306830907055, -2.02306830907055,
-2.02306830907055, -3.02306830907055, 0.976931690929451, -0.023068309070549,
-3.02306830907055, -2.02306830907055, -3.02306830907055, 0.976931690929451,
1.97693169092945, 0.976931690929451, 0.976931690929451, -3.02306830907055,
-1.02306830907055, 2.97693169092945, 2.97693169092945, -0.023068309070549,
1.97693169092945, -3.02306830907055, 1.97693169092945, -2.02306830907055,
0.976931690929451, -1.02306830907055, -0.023068309070549, 2.97693169092945,
1.97693169092945, -0.023068309070549, -0.023068309070549, -0.023068309070549,
-0.023068309070549, 0.976931690929451, 1.97693169092945, 1.97693169092945,
0.976931690929451, 1.97693169092945, -2.02306830907055, -1.02306830907055,
1.97693169092945, 2.97693169092945, 2.97693169092945, -2.02306830907055,
1.97693169092945, 1.97693169092945, -1.02306830907055, -2.02306830907055,
-1.02306830907055, 2.97693169092945, 1.97693169092945, 1.97693169092945,
1.97693169092945, 1.97693169092945, 0.976931690929451, 2.97693169092945,
1.97693169092945, 1.97693169092945, -0.023068309070549, 0.976931690929451,
-3.02306830907055, 1.97693169092945, 0.976931690929451, -0.023068309070549,
-3.02306830907055, 0.976931690929451, 0.976931690929451, -0.023068309070549,
0.976931690929451, 0.976931690929451, 0.976931690929451, -1.02306830907055,
-0.023068309070549, -2.02306830907055, 0.976931690929451, 1.97693169092945,
-3.02306830907055, -0.023068309070549, 1.97693169092945, -0.023068309070549,
1.97693169092945, -0.023068309070549, -3.02306830907055, -2.02306830907055,
-3.02306830907055, -3.02306830907055, -3.02306830907055, -3.02306830907055,
1.97693169092945, -3.02306830907055, 1.97693169092945, 1.97693169092945,
-0.023068309070549, 1.97693169092945, 1.97693169092945, -0.023068309070549,
-2.02306830907055, -3.02306830907055, 1.97693169092945, -0.023068309070549,
1.97693169092945, -0.023068309070549, -3.02306830907055, 0.976931690929451,
-1.02306830907055, -0.023068309070549, -0.023068309070549, -0.023068309070549,
1.97693169092945, 0.976931690929451, 0.976931690929451, 0.976931690929451,
-1.02306830907055, 1.97693169092945, 0.976931690929451, 1.97693169092945,
1.97693169092945, 1.97693169092945, -2.02306830907055, -1.02306830907055,
-2.02306830907055, 0.976931690929451, 1.97693169092945, -3.02306830907055,
2.97693169092945, 1.97693169092945, -0.023068309070549, -1.02306830907055,
-0.023068309070549, 1.97693169092945, 2.97693169092945, -0.023068309070549,
-0.023068309070549, 2.97693169092945, 1.97693169092945, -0.023068309070549,
1.97693169092945, -3.02306830907055, 0.976931690929451, -1.02306830907055,
2.97693169092945, -3.02306830907055, -1.02306830907055, 1.97693169092945,
-0.023068309070549, 0.976931690929451, 0.976931690929451, 0.976931690929451,
0.976931690929451, -2.02306830907055, -3.02306830907055, 0.976931690929451,
-3.02306830907055, 0.976931690929451, -1.02306830907055, -2.02306830907055,
-2.02306830907055, 2.97693169092945, -3.02306830907055, -0.023068309070549,
1.97693169092945, 0.976931690929451, -0.023068309070549, 1.97693169092945,
-1.02306830907055, -1.02306830907055, 0.976931690929451, -0.023068309070549,
-3.02306830907055, -0.023068309070549, 2.97693169092945), Plasticitymc = c(-1.56800111982083,
-0.984667786487492, 0.015332213512508, 0.431998880179169, 0.348665546845838,
1.34866554684584, 0.265332213512508, 1.09866554684584, -0.0680011198208312,
1.34866554684584, 0.598665546845838, 0.681998880179169, -1.40133445315416,
-0.0680011198208312, 1.09866554684584, -0.318001119820831, 0.015332213512508,
0.848665546845838, 0.181998880179169, 1.26533221351251, -0.151334453154162,
-0.568001119820832, -1.23466778648749, -0.651334453154162, 0.515332213512508,
2.09866554684584, 0.848665546845838, -0.318001119820831, -0.0680011198208312,
-0.484667786487492, 0.515332213512508, 0.265332213512508, -0.234667786487492,
0.015332213512508, 0.681998880179169, 0.181998880179169, -1.23466778648749,
-1.98466778648749, -0.818001119820832, 1.01533221351251, 1.01533221351251,
0.348665546845838, -0.0680011198208312, -0.0680011198208312,
-0.0680011198208312, 1.09866554684584, -0.401334453154162, -0.401334453154162,
-0.568001119820832, 0.015332213512508, -0.151334453154162, -0.484667786487492,
-0.151334453154162, 0.765332213512508, 0.0986655468458384, 0.181998880179169,
1.26533221351251, -0.901334453154162, 0.598665546845838, -0.651334453154162,
-0.484667786487492, 1.59866554684584, 1.43199888017917, -0.818001119820832,
-0.318001119820831, 0.015332213512508, -0.484667786487492, -1.06800111982083,
0.348665546845838, 0.765332213512508, -0.401334453154162, -0.151334453154162,
-0.818001119820832, 0.0986655468458384, -0.484667786487492, 0.0986655468458384,
-0.318001119820831, -0.151334453154162, 1.09866554684584, -0.234667786487492,
-0.484667786487492, -0.401334453154162, 0.848665546845838, 0.015332213512508,
0.181998880179169, 1.26533221351251, 0.0986655468458384, -0.234667786487492,
-0.484667786487492, -0.568001119820832, -0.484667786487492, -0.984667786487492,
-0.318001119820831, 1.51533221351251, 1.18199888017917, -0.151334453154162,
0.015332213512508, -0.651334453154162, 0.265332213512508, -0.318001119820831,
-0.901334453154162, -0.734667786487492, 0.181998880179169, 0.765332213512508,
0.931998880179169, -1.48466778648749, 0.765332213512508, 0.681998880179169,
-0.151334453154162, 0.681998880179169, -1.23466778648749, -0.734667786487492,
-0.318001119820831, 0.015332213512508, 0.181998880179169, 0.681998880179169,
-0.901334453154162, 0.515332213512508, -1.23466778648749, 1.26533221351251,
-0.734667786487492, 0.515332213512508, 0.515332213512508, -0.401334453154162,
0.0986655468458384, -0.401334453154162, 1.09866554684584, -0.318001119820831,
0.265332213512508, -0.734667786487492, -0.734667786487492, -1.31800111982083,
0.431998880179169, -2.48466778648749, -1.15133445315416, 0.598665546845838,
0.015332213512508, 1.18199888017917, -0.401334453154162, 0.348665546845838,
-0.234667786487492, -0.0680011198208312, 0.515332213512508, 0.015332213512508,
0.431998880179169, 2.09866554684584, 0.598665546845838, -0.984667786487492,
-0.234667786487492, 0.431998880179169, -0.484667786487492, 0.265332213512508,
0.015332213512508, 1.18199888017917, 0.598665546845838, 0.348665546845838,
0.681998880179169, 0.515332213512508, -0.984667786487492, -0.401334453154162,
0.0986655468458384, -0.401334453154162, 1.01533221351251, 0.515332213512508,
-0.901334453154162, 1.51533221351251, 0.931998880179169, 0.848665546845838,
0.598665546845838, 1.18199888017917, 0.515332213512508, -0.401334453154162,
0.598665546845838, 0.431998880179169, 0.015332213512508, -0.401334453154162,
0.681998880179169, 0.181998880179169, -0.484667786487492, -0.651334453154162,
0.0986655468458384, 0.0986655468458384, 0.0986655468458384, -0.818001119820832,
0.431998880179169, -0.901334453154162, 1.43199888017917, 0.015332213512508,
0.0986655468458384, 0.931998880179169, 0.681998880179169, 0.515332213512508,
-0.318001119820831, 0.931998880179169, 0.181998880179169, -0.401334453154162,
0.015332213512508, -0.401334453154162, -2.06800111982083, 0.598665546845838
)), row.names = c(NA, 200L), class = "data.frame")
R code for standard and Poisson regressions:
#A look at the distribution.
as.data.frame(table(df.stack$Job_Stress))
hist(df.stack$Job_Stress)
#Standard linear model.
lm.js <- lm(Job_Stress~
Autonomymc+Plasticitymc+Job_Is_Complex_Difficultmc,
data=df.stack)
summary(lm.js)
plot(lm.js)
#Poisson model.
glm.js <- glm(Job_Stress~
Autonomymc+Plasticitymc+Job_Is_Complex_Difficultmc,
data=df.stack,
family=poisson(link="log"))
summary(glm.js)
exp(glm.js$coefficients)
