At which hour of sleep, additional hour will have a negative impact on test score (curvilinear relationship) Let's assume that there is a curvilinear relationship between test score and number of sleep hours.  According to the simple example below, starting from 8 hours, test score goes down. I can see it from data.  But, how can I calculate it? how can I calculate that at which point, no. of sleep hour will have negative impact on text score?
Score   **Hr of sleep**
1.20    **1**
3.30    **2**
6.00    **3**
7.00    **4**
12.00   **5**
18.00   **6**
18.00   **7**
16.00   **8**
8.00    **9**
6.00    **10**

 A: If your "hours of sleep" is discrete and you have a sufficient number of observations, I would create a series of dummy variables "at least x hours of sleep" (maybe group them at the bottom "less than four hours" , more than four hours, more than five hours, more than six hours etc. Then do a regular linear regression with test score as the dependent and the dummys as explanatory variables and finally plot the betas. (Alternatively, if you use Stata like I do, do a regression then check out margins and marginsplot)

Score   Hr of sleep more4hours more5hours more6hours  etc
1.20    1                0            0            0
3.30    2                0            0            0
6.00    3                0            0            0
7.00    4                0            0            0
12.00   5                1            0            0
18.00   6                1            1            0
18.00   7                1            1            1
16.00   8                1            1            1
8.00    9                1            1            1
6.00    10               1            1            1

