Polynomial Regression sensible in case with 6 cross-sectional points of experience?

Question

I have a dataset of 300 cases of police students, spread over 6 semesters. They all filled out a survey at the same time (cross section), thus their experience is interval scaled since semester 1 means approximately 6 months of studies and every further semester has each 6 additional months.

I have an integrity measure (7-point Likert scale) and found out, that integrity is lower for those in 2nd and 5th semester (which are practical semesters).

Edit: I calculated an aggregated value over 11 situations which are judged on a 7-point Likert scale.

I calculated a stepwise polynomial regression and came to the conclusion, that one with a 4th degree fits the data best.

Residuals:
     Min       1Q   Median       3Q      Max 
-1.77130 -0.31675 -0.02838  0.34768  1.50143 

Coefficients:
                                         Estimate Std. Error t value Pr(>|t|)    
(Intercept)                               7.94414    0.85978   9.240  < 2e-16 ***
data$data_crosssection_complete.SEM      -4.44345    1.41442  -3.142  0.00185 ** 
I(data$data_crosssection_complete.SEM^2)  2.18171    0.72026   3.029  0.00267 ** 
I(data$data_crosssection_complete.SEM^3) -0.42178    0.14460  -2.917  0.00380 ** 
I(data$data_crosssection_complete.SEM^4)  0.02807    0.01001   2.804  0.00538 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5719 on 297 degrees of freedom
Multiple R-squared:  0.03747,   Adjusted R-squared:  0.02451 
F-statistic: 2.891 on 4 and 297 DF,  p-value: 0.0226

Thing is, I am not sure, if I am allowed to do that. My doctor mother has not worked with polynomial regression, so she can't help me and suggested I should find studies which used one, preferably in the field of organisational psychology (am still searching). And she said the variance is too small, which I am not sure, what that means in my case. Is it that I only have 6 fixed points? How could I work around this?

Edit: Here are my boxplots:

Answer 1

You've used several parameters in your polynomial and the pattern you describe (low integrity in 2nd and 5th semester) is very odd and contrary to reasonable expectation. In this situation, I think you'd be much better off treating semester as a categorical variable and not continuous.

Other concerns:

1. Your model explains just ~3% of the variance ($R^2$), which is very low.
2. You are treating your integrity measure as continuous instead of ordinal. This may be reasonable but is worth thinking about.

EDIT based on the information that 2nd and 5th semesters are meaningfully different:

I would still be inclined to treat semester as categorical if you want to make a simple model that only includes semester. But if you are willing to consider building more complex models, you could could account for practical/university semesters with a factor and use a simpler continuous time trend for semester (I would starting by considering just a linear relationship).